- Email: [email protected]
Optimal information exchange
Information
Economics
Optimal Alison
and Policy
5 (1993)
5-29.
information
5
North-Holland
exchange*
J. Kirby
University of California, Berkeley,
CA 94720, USA
If information is costly and there exist many methods for partially sharing information, does the form of the optimal (i.e. expected firm profit maximizing) information exchange method vary with the motive for that exchange? In this paper it is shown that there are indeed differences. For competing firms (exchanging information purely for purposes of uncertainty resolution) the optimal method may involve sharing information in multiple pools within the industry. The existence of several trade associations operating independent exchanges of information in what appears to be a single industry might be interpreted as an example of multiple information sharing pools. By contrast, colluding firms exchanging information for the purposes of facilitating collusion (as well as resolving uncertainty) always find it optimal to share private information in just one pool. The results in this paper have implications for the design of an empirical test to distinguish between the motives for exchanging information. Keywords: Information exchange, information sharing, information collusion versus competition, partial information sharing.
pools,
trade
associations,
1. Introduction Conventional wisdom has it that the exchange of information between firms in the same industry is prima facie evidence that firms are colluding. Some theoretical models have supported this by suggesting that competing firms are not motivated to share their private information. Recently, however, more general theoretical models have shown that this is not necessarily the case. For example, competing firms with sufficiently decreasing returns to scale in production are able to earn greater expected profits, if, by mutually sharing information, they are able to reduce the uncertainty they face in the market. These results however raise a new problem of distinguishing between legitimate and illegitimate sharing of information. The question analyzed here is the following: if information is costly and there exist many methods for partially sharing information, does the method Correspondence to: Alison J. Kirby, Walter A. Haas School of Business, 350 Barrows Hall, University of California, Berkeley, CA 94720, USA. Tel. (510) 642-4789. * This paper has benefitted from comments from an anonymous referee, and from Svend Albaek, Sushi1 Bikhchandani, Joel Demski, Gerry Feltham, John Hoven, Jack Hughes, Brett Trueman and workshop participants at UBC, Berkeley, Pittsburgh, Stanford and the Antitrust Division of the Justice Department. 0167-6245/93/$06.00
0
1993-Elsevier
Science
Publishers
B.V. (North-Holland)
6
A. J. Kirby I Optimal information exchange
of the optimal (i.e. expected firm profit maximizing) information sharing regime vary with the motive for that exchange? In this paper it is shown that there are indeed differences. For competing firms (exchanging information purely for purposes of uncertainty resolution) the optimal method may involve sharing information in multiple pools within the industry.’ The existence of several trade associations operating independent exchanges of information in what appears to be a single industry might be interpreted as an example of multiple information sharing pools2 By contrast, colluding firms exchanging information for the purposes of facilitating collusion (as well as resolving uncertainty) always find it optimal to share private information in just one pool. This result has implications for the design of an empirical test to distinguish between the motives for exchanging information. More specifically the following set of questions is addressed under behavioral assumptions of collusion and competition in the product market. How can the performance of alternative partial information sharing regimes be compared? What is the optimal form of partial information sharing regime for firms collusively making the information regime choice? What is the optimal level of partially sharing information given the answer to the previous question? What is the optimal level of private information acquisition? Finally, do the answers to this series of questions jointly provide the basis for a discriminating test between the motives for information exchange. In this paper firms in a given industry are uncertain about a market demand parameter. Firms’ cost functions are identical and exhibit decreasing returns to scale. All firms acquire costly information privately and (potentially) partially share it in one of an infinite variety of ways with other firms in the industry. The resulting information structure is termed an information regime, and firms are modeled under the assumption that they commit e_xante to a particular symmetric information regime. The effects of alternative symmetric information regimes on equilibrium output and firm expected profit levels are examined under both competitive and collusive behavioral assumptions. Competing firms act as Cournot oligopolists, while colluding firms choose outputs to maximize the joint profits of the firms sharing their information.”
1 For example, firms in a 12-firm industry may be better off sharing their private information in four pools of three firms each than in one pool of all twelve firms, under some circumstances. ? For example, the electronics industry has several trade associations operating information exchange programs. 3 The mechanism by which the sharing of information translates into an ability to successfuiIy collude is not modeled, but is taken as exogenous.
A. J. Kirby
I Optimal information exchunge
7
Information can be partially shared in an infinite variety of ways. Examples include sharing only a subset of one’s privately acquired observations, the addition of noise to one’s private signal before sharing it, and/or the formation of multiple pools for the sharing of information. The first result in the paper provides a direct way of comparing the performance of alternative information sharing regimes, such as those mentioned above. It is shown that under the assumption of competitive product market behavior, two attributes of an information regime (namely, the accuracy of and the average pairwise correlation between particular sufficient statistics of the post-sharing information sets of all the firms) are sufficient to characterize the output and gross profit levels (excluding the cost of the information) expected under that information regime. Moving to an information regime of greater accuracy (holding the correlation constant) is always beneficial for competing firms (gross of the information acquisition cost), while increasing the correlation is always harmful to them. Solving for the optimal information regime under the assumption of competitive behavior takes place in three steps. First, for a given level of privately acquired information the optimal method of partially sharing that information is identified. This involves a search over all feasible methods of partially sharing information, including (but not limited to) sharing only a subset of private observations, and adding noise to information before sharing it. This search is made possible by knowing that it can be modelled as the search over a constrained set of accuracy-correlation pairs. The optimal method of partially sharing information is proved to be the sharing of information in multiple pools. Intuitively, the formation of many small pools (or trade associations within the industry) enables firms to capture the benefits of statistical aggregation which are already significant even when the number of participants is small, but more efficiently avoids the ill effects of increased correlation between firms’ signals, than does any other method of partially sharing information. Second, for a given accuracy of privately acquired information, the optimal degree of partial sharing (i.e. the optimal number of pools) is solved for as a function of industry parameters. The optimal number of pools is a monotonically increasing function of the production cost function parameter. It is shown that the larger the quadratic production cost parameter relative to other industry parameters, the more valuable is accurate information about demand and consequently the greater the preference for fewer information sharing pools. This is consistent with previous results in the literature to the extent that the quadratic cost parameter is key. However in contrast to the case for other methods of sharing information, adopting multiple pools results in an interior optimum rather than a boundary solution.
8
A.J.
Kirby
I Optimal information exchange
Third, the problem of selecting the amount of private information to be acquired (and subsequently pooled) is addressed. Total information cost is assumed to be a linear function of the accuracy of the private information that firms acquire. The results in this section illustrate the tradeoff between the two means of increasing the accuracy of information ultimately available for firms’ decisions on output levels: namely that of acquiring more private information versus that of sharing the private information in larger pools, so as to increase the benefits of aggregation. The analysis supports one’s intuition that as the cost of acquiring private information increases, the optimal information regime for firms incorporates the acquisition of less private information, but shares that private information more widely. As the quadratic production cost coefficient increases, the cost of making errors in production quantity decisions increases relative to the level of profits being made, and consequently more accurate information becomes more valuable. As a result there is a swing towards preferring the extra accuracy that can be generated when firms share their private information. Under the assumption of firms collusively selecting outputs, since firms decisions are affected primarily by the accuracy of their post-sharing signals, their expected profits are optimized by always sharing their private information in one pool. We also find that the equilibrium level of accuracy of the private information acquired varies depending on the motive for the exchange of the information. The paper is organized as follows. A brief literature review completes this section. The parameters of the model are outlined in the next section. The case of competing firms is examined in two sections beginning with a description of equilibrium in the competitive product market in Section 3. In Section 4 the optimal information regime under the assumption of competitive behavior is examined. Section 5 considers the case of colluding firms and describes the implications for designing a test for discriminating between market behavior based on the form of the information sharing program observed. Related work on the incentives for information sharing includes Novshek and Sonnenschein (1982), Clarke (1983), Vives (1984), Gal-Or (1985), Shapiro (1986) and Kirby (1988). These papers each compare equilibrium profits under a limited (usually discrete) set of information sharing alternatives. Information is either shared in one pool, kept completely private or partially shared through the revelation of a subset of each firm’s private observations (Novshek and Sonnenschein, 1982 and Vives, 1984). AS a general conclusion, the effect of sharing noisy market demand information is to reduce expected firm profits. However, Vives (1984) shows that there is a strict increase in expected profits from sharing information in a Cournot duopoly when the goods produced are complements. Kirby (1988) shows that the same is true for an oligopoly in which the cost functions exhibit
A.J.
Kirby
I Optimal
information
exchange
9
sufficiently decreasing returns to scale. These papers have not considered what are preferred methods of partial sharing of information4 The issue of optima1 research given costly information acquisition was addressed in Li et al. (1987). They did not however address the combined problem of costly information acquisition and sharing.
2. Model Consider an oligopolistic industry of y1 firms, where II is exogenous.” Firms have identical production cost functions, T(x,) = CX, + dx;, where c and d are known, d is nonnegative, and X, is firm i’s level of output. In Section 3, these outputs are assumed to be chosen competitively, i.e. firms compete as Cournot oligopolists. As d takes on different values the shape of the cost function varies from being convex (d > 0) to linear (d = 0) reflecting decreasing and constant returns to scale. The firms operate in a market with price a linear function of total industry output: P = Q - p C x,, b 2 0, where b is known by all firms, but (Y is an unknown parameter. All firms have identical prior distributions over the true value of the intercept of the inverse demand function. These are normally distributed with mean p and variance 0;. Without loss of generality, we define the parameter a as: CY- c. It is assumed that firms are willing and able to commit to a symmetric information regime, I, where symmetric means that the characteristics of firms’ information systems are identical, even though their individual information sets, I,, will generally differ. The information regime provides a (set of) noisy signal(s) about the unknown parameter, a, to each firm. This information set may be simply the result of the firm’s own private information acquisition activity. Alternatively, the information set may be the result of private information acquisition and the mutual, truthful and partial sharing of private signals by all firms in the industry. In the latter case the information set available for making output decisions is a superset of the individual firm’s original private information. For example, if the information sharing procedure requires firms to share a strict subset of their private observations, then Z, will include some signals which are common knowledge to all firms in the industry and some which are known only by firm i. Alternatively, if firms share their information in one of several small pools, then I, consists exclusively of signals which are common knowledge only to ’ In the context of accounting disclosures Kirby (1992) analyzes the optimal level of information acquisition and mutual disclosure when firms acquire multiple private observations and are required to publicly disclose a subset of those observations. ‘Thus this model may be considered as predicting the short term effects of information sharing in a free entry/exit industry. or alternatively the long term effects of an industry in which there are significant barriers to entry/exit.
10
A. J. Kirby I Optimal information exchange
other firms in the same pool. In any case it is assumed that all features of the symmetric information regime, I, are identical for all firms in the industry and are committed to simultaneously by all firms and prior to the receipt of any signals. For example, if the information regime is to be multiple pools, then all firms in the industry are assumed to agree in advance on the one size for all pools, and the one accuracy of information to be acquired and shared by each firm in all pools. Each information regime results in a different level of ex ante expected profits being earned as a result of different optimal actions taken in the product market. We use ex ante expected profits as the criterion by which firms select the optimal information regime, I*, where I* represents an optimal information policy incorporating both the level of acquisition of private information and the form and extent of partial information sharing. In the analysis which follows reference is made to the accuracy of an information regime with respect to the unknown market parameter, a. The accuracy of signal S, is defined as: var(a)lvar(s,). It is assumed that firm i privately acquires the signal, si, defined as: si = a + m,, where m, N(0, cf,,), and E(am,) = 0. The accuracy of this signal therefore equals a:/(~: + a:,). When m, has infinite variance, si has zero accuracy and when a;n = 0, S, is perfect information and has accuracy of one.’ Private information is acquired at a cost of $k per unit of accuracy.7 This cost is relevant for addressing questions relating to the optimal degree of information sharing and the optimal level of information acquisition. The notion of accuracy also simplifies notation for Bayesian updating about the market demand parameter a based on the noisy signal si. From De Groot (1970):
(1)
Assuming that the prior on a has a mean of zero, and denoting of signal S, as A(s;), the posterior can be expressed as: E(u 1s,) = A(s;)s,
.
the accuracy
(2)
‘The accuracy of S, is simply the precision of S, normalized to lie between zero and one. ’ This corresponds to an increasing marginal cost of reducing the variance in the signal, as does the cost function assumed in Li et al. (1987). The cost function used in Li et al. has marginal cost constant in the precision of the signal. The difference, however, is that the cost function used here allows for the purchase of perfect information at a finite cost, whereas the Li et al. specification makes this impossible.
A./. Kirby I Optimd
information exchange
11
More generally, we can consider firm i’s information set (after acquisition sharing) as being made up of h signals: I; = {s;, si, . . , sk}, where h E { 1,2, . . }. Each .s’, in I, is a noisy signal about a, with the noise terms being normally distributed and independent of a. The variance of the noise term and the correlation between .s: and ST may both vary across j. However, consistent with the assumption of a symmetric information regime the variance of the noise term in S: is the same for all firms i, for given signal j. Given the normality of this information structure the posterior distribution over the market parameter a is given by: and
E(a 14)=
(3)
where IS h and the I signals are conditionally independent for a given firm. The mean of the posterior dist~bution is therefore a linear function of the f signals sufficient with respect to n for the h signals in I,.” The coefficients of each of the signals reflect their relative accuracy. Firms in receipt of their post-sharing information sets either play a game of Cournot competition in the product market or they collude.
3. Competitive
product
market equilibrium
Assume that in the output game each firm independently selects a level of output so as to maximize its expected profits conditional on its information set I,, assuming that all other firms do the same. This is a game of incomplete information in which the information set f, is used by firm i to make Bayesian inferences about the unknown demand parameter u, and the information sets of the other firms. For the moment no restrictions are imposed on the form of the information exchange procedure, except that the information regime be symmetric. Formally, the firm’s problem is:
(4) =maxE r,(l,)
’ For example, if sl is simply positive weight in the calculation
[[
a - b i x, x; - d.x; 1I, 1 1 ,=I
a noisy version of s’, (relative of the mean of the posterior
(5)
to a), then only s’, will receive distribution on a.
12
A. J. Kirby I Optimal information exchange
The first order
condition
for this optimization
E(a ( I,) - 2(b + d)x,(Z,)
- b c
problem
is:
E(x, 1I,) = 0.
If’
It is conjectured that the optimal output strategy function of all signals in its information set I,: xi = q,, + q,s;
Assuming
for firm
i is a linear
+ 41s; + . . . + q&,
a symmetric
equilibrium
implies
(7) that for all j:
E(x, 11,)= qo+ q,E(s:11,)+ q,E(s;I Ii> + . . . + q,E(d II,>
(8)
Given normality E(si I I,) is . a linear function of a subset of the signals in I,. The exact form of this function depends on the particular method of information sharing assumed. Generally, then, E(x, ) I,) is a linear function of the signals in I,, as is also the first order condition. Solutions can be found for the coefficients q,, q2, . . , q,1, by requiring that the first order condition hold for every possible configuration of the information set I,. Since E(a) is assumed equal to zero, q. is also always zero. The case of partial sharing of private information via the formation of multiple information sharing pools is provided next as an example of the derivation of the output strategy coefficients. Example:
Sharing
in multiple
pools
All n firms in the industry are exogenously endowed with a private signal s, of accuracy, A(s,). Prior to receiving their signals all firms agree to honestly reveal their private signals to one of p possible trusted organizations (e.g. trade associations) and to it only. It is assumed that this occurs in such a way that equal numbers of firms participate in each pool. Each of these trusted organizations in return reveals the nip private signals to each of the n/p firms which contributed private signals and to them only.’ Thus firm i’s information set, I,, consists of nip conditionally independent signals including s,. The conjectured output strategy is: x,(Z,) = q(, + q;s, + Since the private signals arc assumed to be q;s,+, + . . . + Lp%+,,i,,A. independently drawn from a single distribution we assume without loss of generality, that q: = q’/(n/p) for all ; # 0, so that:
x,(Z;)= 4; + 9’ jF;;,<,o, 6 ” Given results
the objective
the integer
of generating
constraint
economically
on n/11 is ignored.
interesting
rather
than technically
exact
A.
J. Kirby I Optimal information exchange
Denoting the average of the signals q,f,+ q’s,. The mean of the posterior nip - 1 other firms in its pool) is:
in i’s pool distribution
13
by S,, we have: ~~(1;) = of a for firm i (and all
where A(i,)
=
a: a; + a;/(nlp)
(n/~)A(s,) = 1 + (nip - l)A(s,)
’
and
(11) Conditional on I, firm i knows that the output of other firms in the same pool will be x,(Z,), while that of firms in other pools is expected to be:
Since conditional on a, Sk is independent E(u ( I,). Substituting these expressions selecting output is:
- b(n - $)(
of S,, the best estimate of ik is just into (6), the first order condition for
q;, + q’E(u I 4)) = 0 .
For this to hold for all possible
values
of S,, implies
A@,) ” = 2(b + d) + b(n/p - 1) + b(n - n/p)A(i,)
qh = 0 and: (13)
Leaving the example of deriving output strategy coefficients for firms sharing information in multiple pools and returning to the general case of we assume that solutions for the q any information sharing procedure, coefficients of the output strategy in (4) can be found, and we restate the optimal output strategy as: qls; + 42s; + . . . + q/&,
x, = e
L
Q
11
(14)
14
A.J.
where
Q = C::, 2,
=
!!L$ Q
Kirby
I Optimal
information
exchange
qi and define: +
42$ Q
+ . . . +
!b Q
s; .
(15)
Thus zi is a weighted average of the signals in I,, and is a sufficient statistic for I, with respect to the problem of selecting the optimal output. Thus we have:
(16)
x, = Qz, .
Substitution
into the first order
condition
(eq. 6) gives:
E(a 11, > - 2(b + d) Qz, (I;> - b c E[ Qz, 1I,] = 0 , 1’1
(17)
and we find that at the optimum
Q= 2(b
ECUI Ii> + d)z,(Z;)
(18)
+ b ,z, E[z, 1Z,] ’
for all possible information sharing arrangements. In solving for the optimal we restrict attention to look at method of sharing information however, those methods of sharing information for which it is also true that:
’ = 2(b
EC”I z,> + d)z, + b ,;, E[z, 1z,]
This obviously includes but is not limited to those arrangements for which E(a 1I,) = E(a ) z,) and E(z, I I,) = E(zj I z;), i.e. for which zi is sufficient for I, with respect to a and z,. The sharing of information in multiple pools constitutes an example of an information sharing arrangement for which both conditions hold, while sharing only a subset of observations is an example of an information sharing arrangement which satisfies only the Confining attention to the set of partial information overall condition.“’ sharing methods for which expression (19) holds generates an interesting set of results. The first is given in Proposition 1. Proposition conforming
1. For all partial information sharing arrangements to condition (19) ex ante expected profits are:
“’ While the now condensed set of information sharing interesting possibilities. it is not clear whether interesting ments have been excluded in the process.
arrangements (or for that
in the set
clearly still includes matter, any) arrangc-
A. J. Kirby
15
I Optimal information exchange
(b + d)a:A(q) En,(lz)
= [2(b + d) + b(n - l)c(z,)]*
’
where zi is the sujjicient statistic for I, with respect to the output choice, A(z,) is the accuracy of this statistic, and C(z,) is the average pairwise correlation between z, and zi for all j # i. Proof:
By definition
J3a14 = Under
of the term
&
z, = A(z,)z,
1’0 =
we have:
(21)
.
I
the normality
E(Zj
accuracy,
assumption,
if z, and z, are conditionally
correlated:
cov(z,, z,) ‘1
var(z,)
(22)
1
I
assuming that E(z,) = E(zj) = 0. Because of the symmetry assumption, var(zi) = var(zj) which implies that E(zj 1zi) = corr(zi, zj)zi. Defining C(zi) as the average of these pairwise correlations between z, and the n - 1 other z, random variables, we can simplify the expression for Q given in eq. (19) to:
’ = [2(b
A(Zi)
(23)
+ d) + b(n - l)C(zi)]
It is also straightforward d) var(xi(Z,)). Substituting
but tedious to show xi = Qzj, the result follows
that EZI,(I,) immediately.
= (b + q
This proposition is appealing since it provides an intuitive way of evaluating the effects on expected profits of changes in the information regime. Changes which enhance accuracy (without changing the correlation) are beneficial, since they allow firms to introduce greater variance into their optimal output strategies. Changes which increase correlation (keeping the effect on accuracy neutral) are always detrimental since they inhibit a firm’s ability to exploit the accuracy of the information it has. More importantly, in terms of the objective of identifying the optimal information regime, this result also provides us with a way of efficiently comparing the performance of alternative partial information sharing alternatives. An information reprofits which can be gime which optimally generates z, has expected expressed as a function of just A(zi), C(z,) and industry parameters. The above result is conveniently summarized in a diagram (see fig. 1). Since both accuracy and correlation range between zero and one, a unit
A. J. Kirby
16
I Optimal information exchange
Accuracy (A)
nA(S,l
l+(n-l)A(S,)
Con-elation (C)
Fig.
1. lsoprofit
lines.
square is sufficient to represent all possible symmetric information systems. Since the minimum value of C(z,) occurs when firms keep their signals private, i.e. when cov(z;, fj) = af, correlation always weakly exceeds (T:/ var(z,) which is the definition of accuracy, i.e. only half the square is feasible. From eq. (20), the iso-ex ante-expected-profit (hereafter isoprofit) lines are given by the expression: A
=
XP(b + 4 + b(n - WI2 CT;@ + d) ’
(24)
where x is the particular level of profit. These lines are positively sloped and convex in the feasible region. For a given (A, C) point, the larger is the quadratic cost coefficient d, the smaller is the slope of the isoprofit line through that point, i.e., higher levels of correlation are tolerated for higher levels of d. Assuming private information of accuracy A(s,) and an industry of IZ firms, not sharing and completely sharing those private signals are depicted by points NS and S, respectively. With the equilibrium strategies of the output game under information regime (A, C) nested in the expected profit expression (20), we are now equipped to address the question of selection of the optimal information regime. 4. Optimal Identifying information
information
regime: I*
the optimal combination of private information sharing method is solved for in three steps:
accuracy and first, for an
A.
.I. Kirby I Optimal information exchunge
17
exogenous level of private information accuracy, the optimal method of partially sharing information is identified; second, given the optimal method of partially sharing information and an exogenous level of private information accuracy, the optimal degree of partially sharing information is derived; and third the optimal level of private information accuracy is determined knowing that any subsequent information sharing is carried out using the optimal method and degree of partial sharing identified in the first two steps. 4.1.
Optimal method of partially sharing information
In this section we consider an infinite variety of methods for partially sharing private information (of exogenous accuracy), and show that the optimal method is to form multiple pools. We do this by comparing the expected profit performance of any feasible partial information sharing method (as captured by the two profit relevant attributes accuracy and correlation) with the performance of the multiple pools method. Thus we begin by deriving the performance of firms under the multiple pools method in order to use it as a benchmark, which we ultimately show cannot be outperformed. Firms are endowed with private information of accuracy A(s,) and are randomly assigned to one of p pools consisting of n/p firms. To evaluate the effect of sharing information on an individual firm’s expected profit, we know from above that we need only determine the accuracy and correlation of the signals resulting from the information sharing mechanism and substitute these values into the expression for expected profit (eq. 20). From the previous section, the sufficient statistic (2,) for firm i’s information set with respect to the output choice problem is the simple average of the private signals S, of accuracy A(s,) from all n/p firms in the pool. Consequently the accuracy of z,, when information of accuracy A(s,) is acquired and shared in p pools, is denoted A(z,(p, A(s,))) and is given by:
A(z,(A(s,), P))
= A@;)
MS, > = ~+(a-p)A(s,) The correlation pairwise correlation signals for all the correlated with the
(25)
of the information system is defined as the average between firm i’s aggregated signal z, and the aggregated other firms. Firm i’s aggregated signal z, is perfectly signals of the nip - 1 other firms in its pool, and is
A. J. Kirby I Optimal information exchange
18
conditionally independent of the signals of all n - nip other industry. Consequently the average correlation for firm i under is:
C(zi(A@;),P))
= &
firms in the this method
[(;-+l+(n-;)A(z;(A(s.),p))]. (26)
Since the information system is symmetric, the average correlation is the same for all firms, even though individual correlations differ between pairs of firms. When p = II, A(z,(A(s,), p)) and C(z,(A(s,), p)) both simplify to A(si), i.e. the attributes of the private information scenario (point NS on Fig. 1). Similarly, when p = 1, A(z;(A(s;), p)) and C(zi(A(si), p)) simplify to nA(s,)/[l + (n - l)A(s,)] and 1, respectively, i.e., the attributes of the simple shared information scenario (point S on Fig. 1). As p varies from n to 1, the corresponding accuracy and correlation plot out a curve in Fig. 1, joining NS and S. The slope of this function is given by dA(z,(A(s,), p))l dC(z,(A(si),
P))’
dA(z;(A(s;)>p>>/dC(z,(A(s;)> P))
dA(z,(A(s;)p))ldp = dC(z (A@ )p)),ap I I =
and is independent line.
(n - 1Ms;) 11+ (n - lMs,)l ’
of p. We refer to this straight
line as the multiple
(27)
(28) pools
Lemma 1. Partial sharing of private information via the mechanism of multiple pools is represented by points on the straight line joining the private and shared information scenarios. It is interesting to slope of the multiple can be characterized formation of multiple be proved.
note that as the initial accuracy A(s,) increases, the pools line increases. All points in the feasible region by a pair (A(s,), p), i.e. can be implemented by the pools. One of the main results of the paper can now
Proposition 2. Sharing private information in multiple pools is the optimal method of partially sharing information for jirms competing as Cournot oligopolists in their product market. Proof:
See Appendix.
A. .I. Kirby
19
I Optimal information exchange
The proof of Proposition 2 relies on showing that the multiple pools line happens to delineate the boundary of the feasible region of partial information sharing regimes, i.e., that any point above the multiple pools line is not implementable using the exogenous level of accuracy of firms’ private information. All other methods of partially sharing information for which (19) holds (e.g., firms sharing only a subset of their private observations) lie strictly inside the feasible region.” 4.2.
Optimal degree of partially sharing information: p*
Given that multiple pools is the optimal method for partially sharing information, we can now determine the optimal degree of partially sharing information, i.e., the optimal number of pools. This is necessary so that we can derive the level of accuracy (and correlation) that will actually result when private information is shared in multiple pools, and so that we can ultimately solve for the optimal level of accuracy of private information to be acquired. Substituting A(zi(A(s,), p)) and C(z,(A(s,), p)) into the expression for ex ante expected profits (eq. 20) gives profits as a function of the number of pools:
E(4(z;(A(s;),
=
P)))
(b + 4&AMp + (n - ~)A(s,)l [(b + W[p + (n - ~)A(s;)l+ WI+ (n - WW112
Below we give the solution for the optimal value of p, recognizing number of pools must lie in the interval: [ 1, n]. It is straightforward the following proposition. Proposition 3. p*(A(s;))
Optimal pooling has p*(A(s,)) =
bn[l
and results in accuracy,
+ (n - l)A(s,)] (b + 2d)(l
that the to prove
pools, where:
- (b + 2d)nA(si) - A(s,))
’
correlation and profits:
I’ Again, it is clear that despite the imposition of the condition in (19) interesting forms of partial sharing of information are included in the comparison set. Furthermore it is not clear that this condition actually removes any information sharing methods from consideration. Even if it does, it is not obvious that the expected profits under the multiple pools method will be exceeded by a form of partial sharing of information which does not conform to expression (19). since (19) simply refers to the ability to express expected profits as a function of accuracy and correlation. Thus Proposition 2 may actually apply to more than just the partial information sharing methods for which (19) holds.
A.J.
20
Kirby
I Optimal information exchange
(b + 2d) -b
A(si) [’ + (n -
l)A(si)l
2d C(zi(A(s,),
P*(N~,))))
=
/
Pooling in p* pools is feasible 2)(n - l)A(s,) s As (b/2)(n - 1).
only
if
1 ~p”(A(s,))
s n,
i.e.
(bi
Since (~~/~~) > (<)O, whenever p < (>)p”, it follows that if p* > n then p = R is the optimal, feasible number of pools and that p = 1 is optimal when p* < 1. Since the profit function is concave when p = p*, the behavior of the slope of the profit function in the feasible region also implies that if p* is feasible, that it is a global maximum. We note that under the assumptions used to calculate the optimal number of pools, there is no guarantee that the number of pools or the number of firms in each pool will be an integer. A counterexample can however be used to contradict the hypothesis that violating the integer constraint creates the phenomenon of interest.‘* This completes the second step towards solving for the optimal information regime when private information is costly to acquire. We have that the optimal level of profits for firms with private information of accuracy A (si) is generated if those firms share those private signals in one of p* pools where p” = min{max{l, p”(A(s,))}, rz}. 4.3.
Optimal acquisition of private information: A(s,)*
The final step in the derivation of the optimal information regime, f*. is to solve for the optimal accuracy of the privately acquired information. This is carried out independently for the three parameter regions identified above. The general form of the information acquisition problem is: ~(:y EIJ,(A(s,)) subject
(b + db-:A(z,(A(si)~ P)) = max Ah,) (2(b + d) + b(n - l)C(z,(A(si), p))]’
to 0 6 A(s,) =s 1 ,
-
kA(s,) , (29)
“If d = b, n =4 and A(s,) =O.l, then p* = 1.481, nip* = 2.7 and E(IT(p*(A(s,)), A(s,))) = 0.01282mlb. The traditionally analyzed arrangements of keeping information private and sharing it in one pool result in expected profits of 0.010816mib and O.O1256mib, respectively. However, we note that organizing the firms into two pools of two firms generates a final accuracy of 2111, correlation of 5 / 11 and expected profits of ~.012~4m/b, which dominates both of the extreme information sharing arrangements.
A.J.
Kirby
21
I Optimal information exchange
where A(zi(A(sj), P)) and C(zi(A(sj), P)) are the accuracy of firms’ signals after sharing private information pools, as calculated in eqs. (25) and (26). Define a set of critical values of the information .
k
(d)
(b + 4dW
=
I
[2(b
l
k,(d)=
.
k,(d)
=
.
k,(d)
=
b(b
+ d)a2,
4(b
+
(b
.
ks(d)
=
+
and correlation of accuracy A(s,) in
p
cost parameter:
+ 4 - b(n - 111
d) +
b(n
-
’
l)]’
’
24’
+ d)g:
4b(b
+ 2d)
’
(b
+ d)cr;
n[2(b
[2(b
+ d)
+ b(n
(b
+ d)a&
+ d)
+ b(n
-
l)]’
’
l)]’ .
-
when d d b(n - 1)/2 we have k,(d)< k,(d) d k,(d) and when - 1)/2 we have k,(d) d k,(d). If the quadratic cost coefficient is sufficiently large (i.e. d > b(n - 1)/2), then constraining the number of pools to equal one is the optimal feasible number of pools. In this case the solution to the unconstrained information acquisition problem is: Thus
d 3 b(n
(b A(@‘=’
=
(n
!
1)
k[2(b
+ d)&
+ d)
+ b(n
-
l)]’
- ’I ’
(30)
This level of private information accuracy is in turn feasible only if k,(d) G k s k,(d). If k B k,(d), then A(s,) = 0 is the optimal feasible solution, and similarly if k d k,(d) then A(s,) = 1 is optimal. If d s b(n - 1)12 then sharing in multiple pools may be feasible. If multiple pools are indeed feasible, then p = p*(A(si)), and the solution to the information acquisition problem is:
A(si)*
=
~
$1)
(b
+ d)a;
4bk(b
+ 2d)
-
’
1’
(31)
Substituting this value of private information accuracy for A(si) in b(n - 1)A(sj)/2 defines the value of d at which firms are indifferent between sharing information in multiple pools and keeping their information
A. J. Kirby I Optimal information exchange
22
private (i.e. p+ = n). This condition on d is equivalent to a condition on k. Thus if k L k,(d) when d c b(rz - 1)/2 then sharing private information of accuracy A(s;)* in p*(A(s,)*) p oo 1s is optimal. This value of A(sj)* is that A(s,)* s feasible (i.e. greater than zero), if k =Zk,(d).13 The constraint 1 is never binding. The intuition for this is that if information were so cheap that perfect information would be pooled then it would be yet more preferable to keep the information private. If d G b(n - 1) 12 but k s k,(d), then the optimal number of pools is ~1. The optimal accuracy, denoted A(s~)~=~ is the solution to the following cubic equation: (b + 46]2@ + 4 - b(tr - l)A(si)l [Z(b + d) + b(n - 1)A(s,)13
_ k =1:O
This equals one (i.e. perfect information is acquired) if the cost of information is sufficiently small: k d k,(d). The constraint that A(.Y~)~=~ b 0 is never binding. This solution to the problem of simultaneously selecting the number of pools and the accuracy of the private information to be acquired, is most concisely represented in a diagram (see Fig. 2). This shows how the optimal information regime varies as a function of the information cost and production cost parameters (k and d, respectively).
Fig. 2. Optimal
disclosure
method
as a function
of industry
parameters,
” It can also be shown that this is a Nash equilibrium in that there is no incentive for firms to unilaterally defect from their pool and instead ‘create’ their own one-firm pool in which they also have the choice to independently select the accuracy of the private information that they acquire, assuming that the number of pools is not affected by the defection. This is true even when the other firms are aware of the defection by the time they choose their output levels, and adjust their strategies to recognize the reduction in the accuracy of the aggregate they receive.
A. 1. Kirby
I Optimal
information
23
exchange
The intuition for these results is that as the production cost function becomes more convex (i.e. d increases) the optimal tradeoff of accuracy for correlation changes. For a given value of k, as d increases the marginal cost at the optimal output level also increases, the stakes in the output game increase and with it ceteris paribus the value of more accurate information. Consequently, when d is relatively small, accuracy is unimportant relative to correlation, and information systems are preferred which favor low correlation rather than high accuracy; for example, not sharing private information. However, as d increases, the preference for higher accuracy and better output decisions outweighs that for low correlation. This creates a preference for the sharing of private information in increasingly large pools. This pattern of preferences over information systems is also a function of the cost of acquiring information. If information cost is low, one approaches the case of costless information and the optimal information system is always to keep information private. However as information becomes more costly, the more economical way of achieving higher accuracy is to aggregate rather than to acquire. Consequently, we see a move towards fewer pools and lower acquired accuracy. In the limit when information is infinitely costly, no information is acquired at the optimum. The same comparative statics of the optimal information regime are depicted differently in Fig. 3. For each of three possible ranges of d values, the bold line indicates how the optimal final accuracy and correlation will change as the cost of information increases. These diagrams emphasize the surprising fact that whenever the optimal information system involves some form of information sharing, the optimal level of correlation induced by the information regime is independent of the information cost. It is as if the information sharing mechanism is designed to induce a constant level of correlation.
dz
b(n-1) 2
A
A
2d b(n - 1)
C Fig. 3. Optimal
2d b(n - 1) accuracy
C
and correlation
C
A. J. Kirby I Optimal information exchange
24
The problem of distinguishing between the motives for information exchange only arises if under collusive behavior it is also optimal to share information in multiple pools as opposed to some other form of partially sharing information. Thus the following analysis assumes that colluding firms also form pools for the purposes of sharing ~nforI~atio~ and that collusion is possible only to the extent that information sharing occurs, i.e. firms within the same pool collude in that they produce to maximize the pool’s profits, and pools compete with each other. Thus at the extremes when there are n pools, there is no information sharing and no collusion, and with one pool there is complete information sharing and monopoly output choices by the industry. The questions then become, what is the optimal number of pools? fs this different from the optimum when all firms in the industry compete regardless of the degree of information sharing? For a given level of privately acquired information and a given number of pools each pool selects output to maximize its expected profits:
max E ii xP(lP) I I
(321
where x4 is the output of the ith pool and Zf is its information set consisting of the private signals of the nip hrms in that pool. In order to minimize the pool’s production costs firms produce equal shares of the pool’s optimal output. The first order condition for this problem is: (33) Again we define 5, as the average of the private signals of the nip firms in pool i, and know therefore that:
where A(il) is the accuracy of the average signal and is given by:
(35) Conjecturing
a pool’s output strategy to be linear in S,:
A. .I. Kirby
and substituting optimal strategy
back into the first order condition for the pool has 4;; = 0 and
“’ = [2b(n/p) Consequently
25
I Optimal information exchange
(eq. 33) requires
(37)
+ 2d + b( p - l)(nlp)A(s,)]
firm i’s optimal
output
strategy
that the
is:
(38) Substituting
II,
cell
this into firm i’s profit
expression:
1
=
and taking expectations profits for firm i under
ET’” = [2bn/p Differentiating monotonically
over the information ‘partial’ collusion of azA(s,)(bn/p
ex ante expected
+ d)
+ 2d + b( p - l)(nip)A(F!)]’
with respect to p, decreasing in p. Thus
set gives
(40)
’
reveals that expected we have the following
firm profit proposition:
is
Proposition 4. If firms collude to the extent that they share information, then the optimal number of pools is one, for all industry parameters. Combining this result with those under the assumption of competitive firm behavior, we have that for d s (b/2)(n - 1) the observation of information sharing in one pool is evidence of collusion. However for values of d 3 (b/2)(n - 1) the number of pools does not provide a discriminating characteristic. Consider now the issue of the level of private information acquisition, when firms collude. The amount of information acquired and shared may instead provide a distinguishing characteristic in this case. The optimal level of expected profits under collusion. one pool and costly information acquisition is: En;“”
=
a;A(F,) 4[nb + d]
where that
A(s;)
- kA(s,)
is the accuracy
.
of the private
(41) signal purchased
by firm i. Recall
26
A. J. Kirby
A(s,) = 1 + (n and therefore given by:
- l)A(s,)
that the optimal
I Optimal information exchange
accuracy
Vf?l - 4k(nb
+ d)
- l)(nb
+ d)
A:&, >= 4k(n
(42)
’ of privately
acquired
information
is
(43)
The final issue is how this compares with the optimal level of acquired accuracy when firms compete and find it optimal to share in one pool. The latter is given by eq. (30):
A(.#‘=’
Comparing
= (n ! 1) 4~4. A(si)‘=’
k < a&[2(b
(44)
and A,*,,,(s,),
the latter
exceeds
the former
when
+ d) + b(n - l)]’
16(b + d)(nb
(45)
+ d)2
Defining this critical value of the information cost as k,(d) and referring to Fig. 2, it is easy to show that k,(d) b k,(d) holds for all relevant parameter values. Thus for those industries (i.e. points in k. d space) for which competing firms find it optimal to share in one pool, the accuracy of information acquired and subsequently shared will be less than the accuracy of information acquired by colluding firms. The results of this section are summarized in the following proposition: Proposition
5.
Firms
in industries
for
which
d < (b/2)(n
- 1) will share
their private information in more than one pool if they compete and in only one pool if they subsequently collude in the product market. However, firms in industries for which d b (b/2)(n - 1) h ave an incentive to share information in one pool irrespective of the type of behavior they subsequently adopt in the product market. For a given industry, however, the optimal level of information acquired and shared is greater if its firms subsequently collude in the product market than if they compete.
6. Conclusion
The primary conclusion of the paper is that observing the form by which firms choose to share information is likely to be much more informative
A. J. Kirby
I Optimal
information
27
exchange
about their underlying behavior in the product market than is simply observing whether information sharing takes place. Competing firms are best off sharing information in several industry subpools rather than in one large pool. By contrast colluding firms always prefer to share information in one large pool. In those industries for which competing firms actually wish to share in one pool only, the level of information they wish to share is less than that of their colluding counterparts. There are several caveats which need to be made in applying these results. First, these results were generated under and are likely to be sensitive to the assumptions of market demand uncertainty and Cournot competition. Second, attention was limited to symmetric information systems. Thirdly, and perhaps most crucially, the colluding firms were modelled as if they act with disregard for the possibility that they may suffer disutility if their behavior is determined to be collusive. In other words this model does not address the possibility that colluding firms might alter their means for exchanging information if that know that the form of information exchange could be used to detect their collusive behavior. To the extent that there are meaningful penalties for collusive behavior, this possibility undoubtedly exists. Nevertheless the tendency remains for colluding firms to want to share more information in larger pools than would competing firms.
Appendix Proof of Proposition 2 Proposition 2 implies that no method of partially sharing information exists (within the set defined by eq. 19) which produces combinations of accuracy and correlation above the multiple pools line. The proof is by contradiction. Each firm i receives a private signal, s, = a + m,, where a-N(~=O,c~),m,--N(0, af,) and E(am,) = 0, for all i, and E(m,m,) = 0 for all j # i. If the firms share their private signals in p pools of nip firms each, where 1
A(z,(A(s,),P))
= $
C(z,(A@,),P>> =
0
+ Jz,(n,p) m
3
(AlI
~ri~-li+~~-~,A(z.(A(~,,,p,,]. 642)
28
A. J. Kirby
I Optimul
infknation
exchange
Now suppose that there exists an alternative means X for firms to partially share their private information with corresponding accuracy and correlation given by A(z,(A(s,), X)) and C(zi(A(s,), X)) such that mechanism X has correlation equal to and accuracy strictly greater than that under multiple pools. Therefore, adopting abbreviated notation.
C(z,(W) = C(z,(P>> 3
A(z,(X)) = A(z,( p)) + c ,
6 >
0.
Given Proposition 1 this would imply that method X would generate strictly greater expected profits than does the multiple pools method of partially sharing information. The question is whether X is feasible given the exogenous level of accuracy of the private information. Suppose that in a second round of information sharing the information set available to each firm via method X is provided publicly to all firms. Specifically each firm then has access to z:+_,(X), where
&(X) The accuracy
= f [z,(X) + Z?(X) + . . . + z,,(X)] of the resulting
information
(A3)
set available
to all firms would be:
(A4) where var(z,,,,(X))
= $
Z? var($) i ,=I
_ var(;;l(X)) since
by assumption
A(z,(W) =
C(z,(X))
+ +2-
l) cov(z,(X)z,(X))]
[2 + (n - l>C(z,( = C(z,( p)).
=A(z,(P)) + ,,,(~i,,,
p))] ,
Also by assumption:
(A3
E.
I
and consequently expressed as:
the accuracy
A(z‘w(X))=
of this new (second
~~A(z,(P)) + ~1 2 +
(n
-
l)C(z,(
p))
round)
aggregate
can be
(Ah)
A. .I. Kirby
I Optimal information e.xchange
29
This can be compared with the accuracy resulting from directly providing all the original private information to all firms, or equivalently pooling in one pool. The correlation between firms’ information sets in this case is also one, but the accuracy would be:
(A7) into expression (A6) using cxpresSubstituting for A(z,( p)) and C(z,(p)) sions (Al) and (A2) it can be shown that A(z&) > A(z,( p = 1)) for all E > 0. Since under normality the aggregate from simple sharing is sufficient aggregates producing for the original private signals, other (compound) strictly greater accuracy for this given degree of correlation (i.e. C = 1) are not possible. Thus we have the necessary contradiction to prove that mechanisms like X cannot exist, and the multiple pools line forms the boundary of what is feasible given private signals of exogenous accuracy. Optimality of multiple pools follows given the shape of the isoprotit curves relative to the feasible region. 0
References Basar, T., 1978. Decentralized multicriteria optimization of linear stochastic systems. IEEE Transactions on Automatic Control AC-23. No. 2, 233-243. Clarke. R., 1983, Collusion and the incentives for information sharing, Bell Journal of Economics 14. 3833304. De Groot. M.. 1970, Optimal Statistical Decisions (McGraw-Hill. New York). Gal-Or, E., 1986, Information transmission-Cournot and Bertrand equilibria, Review of Economic Studies 153, 85-92. Gal-Or, E., 1985, Information sharing in oligopoly, Econometrica 53. 320-344. Kirby, A., 1992, Self-imposed information acquisition and disclosure: An economic analysis. Working Paper. University of California, Berkeley. Kirby. A., 1988, Trade associations as information exchange mechanisms. Rand Journal of Economics 19. 13X-146. Li, L., R. McKclvey and T. Page, 1987. Optimal research for Cournot oligopolists. Journal of Economic Theory 42. 140-166. Novshek. W. and H. Sonnenschein, 1982, Fulfilled expectations Cournot duopoly with information acquisition and release, Bell Journal of Economics 13. 214-218. Shapiro. C., 1986, Exchange of cost information in oligopoly, Review of Economic Studies 53. 433-446. Vives. X.. 1984, Duopoly information equilibrium: Cournot and Bertrand. Journal of Economic Theory 34, 71-04. Vives. X., 1990. Trade association disclosure rules. incentives to share information. and welfare, Rand Journal of Economics 21, 409-430.
Economics
Optimal Alison
and Policy
5 (1993)
5-29.
information
5
North-Holland
exchange*
J. Kirby
University of California, Berkeley,
CA 94720, USA
If information is costly and there exist many methods for partially sharing information, does the form of the optimal (i.e. expected firm profit maximizing) information exchange method vary with the motive for that exchange? In this paper it is shown that there are indeed differences. For competing firms (exchanging information purely for purposes of uncertainty resolution) the optimal method may involve sharing information in multiple pools within the industry. The existence of several trade associations operating independent exchanges of information in what appears to be a single industry might be interpreted as an example of multiple information sharing pools. By contrast, colluding firms exchanging information for the purposes of facilitating collusion (as well as resolving uncertainty) always find it optimal to share private information in just one pool. The results in this paper have implications for the design of an empirical test to distinguish between the motives for exchanging information. Keywords: Information exchange, information sharing, information collusion versus competition, partial information sharing.
pools,
trade
associations,
1. Introduction Conventional wisdom has it that the exchange of information between firms in the same industry is prima facie evidence that firms are colluding. Some theoretical models have supported this by suggesting that competing firms are not motivated to share their private information. Recently, however, more general theoretical models have shown that this is not necessarily the case. For example, competing firms with sufficiently decreasing returns to scale in production are able to earn greater expected profits, if, by mutually sharing information, they are able to reduce the uncertainty they face in the market. These results however raise a new problem of distinguishing between legitimate and illegitimate sharing of information. The question analyzed here is the following: if information is costly and there exist many methods for partially sharing information, does the method Correspondence to: Alison J. Kirby, Walter A. Haas School of Business, 350 Barrows Hall, University of California, Berkeley, CA 94720, USA. Tel. (510) 642-4789. * This paper has benefitted from comments from an anonymous referee, and from Svend Albaek, Sushi1 Bikhchandani, Joel Demski, Gerry Feltham, John Hoven, Jack Hughes, Brett Trueman and workshop participants at UBC, Berkeley, Pittsburgh, Stanford and the Antitrust Division of the Justice Department. 0167-6245/93/$06.00
0
1993-Elsevier
Science
Publishers
B.V. (North-Holland)
6
A. J. Kirby I Optimal information exchange
of the optimal (i.e. expected firm profit maximizing) information sharing regime vary with the motive for that exchange? In this paper it is shown that there are indeed differences. For competing firms (exchanging information purely for purposes of uncertainty resolution) the optimal method may involve sharing information in multiple pools within the industry.’ The existence of several trade associations operating independent exchanges of information in what appears to be a single industry might be interpreted as an example of multiple information sharing pools2 By contrast, colluding firms exchanging information for the purposes of facilitating collusion (as well as resolving uncertainty) always find it optimal to share private information in just one pool. This result has implications for the design of an empirical test to distinguish between the motives for exchanging information. More specifically the following set of questions is addressed under behavioral assumptions of collusion and competition in the product market. How can the performance of alternative partial information sharing regimes be compared? What is the optimal form of partial information sharing regime for firms collusively making the information regime choice? What is the optimal level of partially sharing information given the answer to the previous question? What is the optimal level of private information acquisition? Finally, do the answers to this series of questions jointly provide the basis for a discriminating test between the motives for information exchange. In this paper firms in a given industry are uncertain about a market demand parameter. Firms’ cost functions are identical and exhibit decreasing returns to scale. All firms acquire costly information privately and (potentially) partially share it in one of an infinite variety of ways with other firms in the industry. The resulting information structure is termed an information regime, and firms are modeled under the assumption that they commit e_xante to a particular symmetric information regime. The effects of alternative symmetric information regimes on equilibrium output and firm expected profit levels are examined under both competitive and collusive behavioral assumptions. Competing firms act as Cournot oligopolists, while colluding firms choose outputs to maximize the joint profits of the firms sharing their information.”
1 For example, firms in a 12-firm industry may be better off sharing their private information in four pools of three firms each than in one pool of all twelve firms, under some circumstances. ? For example, the electronics industry has several trade associations operating information exchange programs. 3 The mechanism by which the sharing of information translates into an ability to successfuiIy collude is not modeled, but is taken as exogenous.
A. J. Kirby
I Optimal information exchunge
7
Information can be partially shared in an infinite variety of ways. Examples include sharing only a subset of one’s privately acquired observations, the addition of noise to one’s private signal before sharing it, and/or the formation of multiple pools for the sharing of information. The first result in the paper provides a direct way of comparing the performance of alternative information sharing regimes, such as those mentioned above. It is shown that under the assumption of competitive product market behavior, two attributes of an information regime (namely, the accuracy of and the average pairwise correlation between particular sufficient statistics of the post-sharing information sets of all the firms) are sufficient to characterize the output and gross profit levels (excluding the cost of the information) expected under that information regime. Moving to an information regime of greater accuracy (holding the correlation constant) is always beneficial for competing firms (gross of the information acquisition cost), while increasing the correlation is always harmful to them. Solving for the optimal information regime under the assumption of competitive behavior takes place in three steps. First, for a given level of privately acquired information the optimal method of partially sharing that information is identified. This involves a search over all feasible methods of partially sharing information, including (but not limited to) sharing only a subset of private observations, and adding noise to information before sharing it. This search is made possible by knowing that it can be modelled as the search over a constrained set of accuracy-correlation pairs. The optimal method of partially sharing information is proved to be the sharing of information in multiple pools. Intuitively, the formation of many small pools (or trade associations within the industry) enables firms to capture the benefits of statistical aggregation which are already significant even when the number of participants is small, but more efficiently avoids the ill effects of increased correlation between firms’ signals, than does any other method of partially sharing information. Second, for a given accuracy of privately acquired information, the optimal degree of partial sharing (i.e. the optimal number of pools) is solved for as a function of industry parameters. The optimal number of pools is a monotonically increasing function of the production cost function parameter. It is shown that the larger the quadratic production cost parameter relative to other industry parameters, the more valuable is accurate information about demand and consequently the greater the preference for fewer information sharing pools. This is consistent with previous results in the literature to the extent that the quadratic cost parameter is key. However in contrast to the case for other methods of sharing information, adopting multiple pools results in an interior optimum rather than a boundary solution.
8
A.J.
Kirby
I Optimal information exchange
Third, the problem of selecting the amount of private information to be acquired (and subsequently pooled) is addressed. Total information cost is assumed to be a linear function of the accuracy of the private information that firms acquire. The results in this section illustrate the tradeoff between the two means of increasing the accuracy of information ultimately available for firms’ decisions on output levels: namely that of acquiring more private information versus that of sharing the private information in larger pools, so as to increase the benefits of aggregation. The analysis supports one’s intuition that as the cost of acquiring private information increases, the optimal information regime for firms incorporates the acquisition of less private information, but shares that private information more widely. As the quadratic production cost coefficient increases, the cost of making errors in production quantity decisions increases relative to the level of profits being made, and consequently more accurate information becomes more valuable. As a result there is a swing towards preferring the extra accuracy that can be generated when firms share their private information. Under the assumption of firms collusively selecting outputs, since firms decisions are affected primarily by the accuracy of their post-sharing signals, their expected profits are optimized by always sharing their private information in one pool. We also find that the equilibrium level of accuracy of the private information acquired varies depending on the motive for the exchange of the information. The paper is organized as follows. A brief literature review completes this section. The parameters of the model are outlined in the next section. The case of competing firms is examined in two sections beginning with a description of equilibrium in the competitive product market in Section 3. In Section 4 the optimal information regime under the assumption of competitive behavior is examined. Section 5 considers the case of colluding firms and describes the implications for designing a test for discriminating between market behavior based on the form of the information sharing program observed. Related work on the incentives for information sharing includes Novshek and Sonnenschein (1982), Clarke (1983), Vives (1984), Gal-Or (1985), Shapiro (1986) and Kirby (1988). These papers each compare equilibrium profits under a limited (usually discrete) set of information sharing alternatives. Information is either shared in one pool, kept completely private or partially shared through the revelation of a subset of each firm’s private observations (Novshek and Sonnenschein, 1982 and Vives, 1984). AS a general conclusion, the effect of sharing noisy market demand information is to reduce expected firm profits. However, Vives (1984) shows that there is a strict increase in expected profits from sharing information in a Cournot duopoly when the goods produced are complements. Kirby (1988) shows that the same is true for an oligopoly in which the cost functions exhibit
A.J.
Kirby
I Optimal
information
exchange
9
sufficiently decreasing returns to scale. These papers have not considered what are preferred methods of partial sharing of information4 The issue of optima1 research given costly information acquisition was addressed in Li et al. (1987). They did not however address the combined problem of costly information acquisition and sharing.
2. Model Consider an oligopolistic industry of y1 firms, where II is exogenous.” Firms have identical production cost functions, T(x,) = CX, + dx;, where c and d are known, d is nonnegative, and X, is firm i’s level of output. In Section 3, these outputs are assumed to be chosen competitively, i.e. firms compete as Cournot oligopolists. As d takes on different values the shape of the cost function varies from being convex (d > 0) to linear (d = 0) reflecting decreasing and constant returns to scale. The firms operate in a market with price a linear function of total industry output: P = Q - p C x,, b 2 0, where b is known by all firms, but (Y is an unknown parameter. All firms have identical prior distributions over the true value of the intercept of the inverse demand function. These are normally distributed with mean p and variance 0;. Without loss of generality, we define the parameter a as: CY- c. It is assumed that firms are willing and able to commit to a symmetric information regime, I, where symmetric means that the characteristics of firms’ information systems are identical, even though their individual information sets, I,, will generally differ. The information regime provides a (set of) noisy signal(s) about the unknown parameter, a, to each firm. This information set may be simply the result of the firm’s own private information acquisition activity. Alternatively, the information set may be the result of private information acquisition and the mutual, truthful and partial sharing of private signals by all firms in the industry. In the latter case the information set available for making output decisions is a superset of the individual firm’s original private information. For example, if the information sharing procedure requires firms to share a strict subset of their private observations, then Z, will include some signals which are common knowledge to all firms in the industry and some which are known only by firm i. Alternatively, if firms share their information in one of several small pools, then I, consists exclusively of signals which are common knowledge only to ’ In the context of accounting disclosures Kirby (1992) analyzes the optimal level of information acquisition and mutual disclosure when firms acquire multiple private observations and are required to publicly disclose a subset of those observations. ‘Thus this model may be considered as predicting the short term effects of information sharing in a free entry/exit industry. or alternatively the long term effects of an industry in which there are significant barriers to entry/exit.
10
A. J. Kirby I Optimal information exchange
other firms in the same pool. In any case it is assumed that all features of the symmetric information regime, I, are identical for all firms in the industry and are committed to simultaneously by all firms and prior to the receipt of any signals. For example, if the information regime is to be multiple pools, then all firms in the industry are assumed to agree in advance on the one size for all pools, and the one accuracy of information to be acquired and shared by each firm in all pools. Each information regime results in a different level of ex ante expected profits being earned as a result of different optimal actions taken in the product market. We use ex ante expected profits as the criterion by which firms select the optimal information regime, I*, where I* represents an optimal information policy incorporating both the level of acquisition of private information and the form and extent of partial information sharing. In the analysis which follows reference is made to the accuracy of an information regime with respect to the unknown market parameter, a. The accuracy of signal S, is defined as: var(a)lvar(s,). It is assumed that firm i privately acquires the signal, si, defined as: si = a + m,, where m, N(0, cf,,), and E(am,) = 0. The accuracy of this signal therefore equals a:/(~: + a:,). When m, has infinite variance, si has zero accuracy and when a;n = 0, S, is perfect information and has accuracy of one.’ Private information is acquired at a cost of $k per unit of accuracy.7 This cost is relevant for addressing questions relating to the optimal degree of information sharing and the optimal level of information acquisition. The notion of accuracy also simplifies notation for Bayesian updating about the market demand parameter a based on the noisy signal si. From De Groot (1970):
(1)
Assuming that the prior on a has a mean of zero, and denoting of signal S, as A(s;), the posterior can be expressed as: E(u 1s,) = A(s;)s,
.
the accuracy
(2)
‘The accuracy of S, is simply the precision of S, normalized to lie between zero and one. ’ This corresponds to an increasing marginal cost of reducing the variance in the signal, as does the cost function assumed in Li et al. (1987). The cost function used in Li et al. has marginal cost constant in the precision of the signal. The difference, however, is that the cost function used here allows for the purchase of perfect information at a finite cost, whereas the Li et al. specification makes this impossible.
A./. Kirby I Optimd
information exchange
11
More generally, we can consider firm i’s information set (after acquisition sharing) as being made up of h signals: I; = {s;, si, . . , sk}, where h E { 1,2, . . }. Each .s’, in I, is a noisy signal about a, with the noise terms being normally distributed and independent of a. The variance of the noise term and the correlation between .s: and ST may both vary across j. However, consistent with the assumption of a symmetric information regime the variance of the noise term in S: is the same for all firms i, for given signal j. Given the normality of this information structure the posterior distribution over the market parameter a is given by: and
E(a 14)=
(3)
where IS h and the I signals are conditionally independent for a given firm. The mean of the posterior dist~bution is therefore a linear function of the f signals sufficient with respect to n for the h signals in I,.” The coefficients of each of the signals reflect their relative accuracy. Firms in receipt of their post-sharing information sets either play a game of Cournot competition in the product market or they collude.
3. Competitive
product
market equilibrium
Assume that in the output game each firm independently selects a level of output so as to maximize its expected profits conditional on its information set I,, assuming that all other firms do the same. This is a game of incomplete information in which the information set f, is used by firm i to make Bayesian inferences about the unknown demand parameter u, and the information sets of the other firms. For the moment no restrictions are imposed on the form of the information exchange procedure, except that the information regime be symmetric. Formally, the firm’s problem is:
(4) =maxE r,(l,)
’ For example, if sl is simply positive weight in the calculation
[[
a - b i x, x; - d.x; 1I, 1 1 ,=I
a noisy version of s’, (relative of the mean of the posterior
(5)
to a), then only s’, will receive distribution on a.
12
A. J. Kirby I Optimal information exchange
The first order
condition
for this optimization
E(a ( I,) - 2(b + d)x,(Z,)
- b c
problem
is:
E(x, 1I,) = 0.
If’
It is conjectured that the optimal output strategy function of all signals in its information set I,: xi = q,, + q,s;
Assuming
for firm
i is a linear
+ 41s; + . . . + q&,
a symmetric
equilibrium
implies
(7) that for all j:
E(x, 11,)= qo+ q,E(s:11,)+ q,E(s;I Ii> + . . . + q,E(d II,>
(8)
Given normality E(si I I,) is . a linear function of a subset of the signals in I,. The exact form of this function depends on the particular method of information sharing assumed. Generally, then, E(x, ) I,) is a linear function of the signals in I,, as is also the first order condition. Solutions can be found for the coefficients q,, q2, . . , q,1, by requiring that the first order condition hold for every possible configuration of the information set I,. Since E(a) is assumed equal to zero, q. is also always zero. The case of partial sharing of private information via the formation of multiple information sharing pools is provided next as an example of the derivation of the output strategy coefficients. Example:
Sharing
in multiple
pools
All n firms in the industry are exogenously endowed with a private signal s, of accuracy, A(s,). Prior to receiving their signals all firms agree to honestly reveal their private signals to one of p possible trusted organizations (e.g. trade associations) and to it only. It is assumed that this occurs in such a way that equal numbers of firms participate in each pool. Each of these trusted organizations in return reveals the nip private signals to each of the n/p firms which contributed private signals and to them only.’ Thus firm i’s information set, I,, consists of nip conditionally independent signals including s,. The conjectured output strategy is: x,(Z,) = q(, + q;s, + Since the private signals arc assumed to be q;s,+, + . . . + Lp%+,,i,,A. independently drawn from a single distribution we assume without loss of generality, that q: = q’/(n/p) for all ; # 0, so that:
x,(Z;)= 4; + 9’ jF;;,<,o, 6 ” Given results
the objective
the integer
of generating
constraint
economically
on n/11 is ignored.
interesting
rather
than technically
exact
A.
J. Kirby I Optimal information exchange
Denoting the average of the signals q,f,+ q’s,. The mean of the posterior nip - 1 other firms in its pool) is:
in i’s pool distribution
13
by S,, we have: ~~(1;) = of a for firm i (and all
where A(i,)
=
a: a; + a;/(nlp)
(n/~)A(s,) = 1 + (nip - l)A(s,)
’
and
(11) Conditional on I, firm i knows that the output of other firms in the same pool will be x,(Z,), while that of firms in other pools is expected to be:
Since conditional on a, Sk is independent E(u ( I,). Substituting these expressions selecting output is:
- b(n - $)(
of S,, the best estimate of ik is just into (6), the first order condition for
q;, + q’E(u I 4)) = 0 .
For this to hold for all possible
values
of S,, implies
A@,) ” = 2(b + d) + b(n/p - 1) + b(n - n/p)A(i,)
qh = 0 and: (13)
Leaving the example of deriving output strategy coefficients for firms sharing information in multiple pools and returning to the general case of we assume that solutions for the q any information sharing procedure, coefficients of the output strategy in (4) can be found, and we restate the optimal output strategy as: qls; + 42s; + . . . + q/&,
x, = e
L
Q
11
(14)
14
A.J.
where
Q = C::, 2,
=
!!L$ Q
Kirby
I Optimal
information
exchange
qi and define: +
42$ Q
+ . . . +
!b Q
s; .
(15)
Thus zi is a weighted average of the signals in I,, and is a sufficient statistic for I, with respect to the problem of selecting the optimal output. Thus we have:
(16)
x, = Qz, .
Substitution
into the first order
condition
(eq. 6) gives:
E(a 11, > - 2(b + d) Qz, (I;> - b c E[ Qz, 1I,] = 0 , 1’1
(17)
and we find that at the optimum
Q= 2(b
ECUI Ii> + d)z,(Z;)
(18)
+ b ,z, E[z, 1Z,] ’
for all possible information sharing arrangements. In solving for the optimal we restrict attention to look at method of sharing information however, those methods of sharing information for which it is also true that:
’ = 2(b
EC”I z,> + d)z, + b ,;, E[z, 1z,]
This obviously includes but is not limited to those arrangements for which E(a 1I,) = E(a ) z,) and E(z, I I,) = E(zj I z;), i.e. for which zi is sufficient for I, with respect to a and z,. The sharing of information in multiple pools constitutes an example of an information sharing arrangement for which both conditions hold, while sharing only a subset of observations is an example of an information sharing arrangement which satisfies only the Confining attention to the set of partial information overall condition.“’ sharing methods for which expression (19) holds generates an interesting set of results. The first is given in Proposition 1. Proposition conforming
1. For all partial information sharing arrangements to condition (19) ex ante expected profits are:
“’ While the now condensed set of information sharing interesting possibilities. it is not clear whether interesting ments have been excluded in the process.
arrangements (or for that
in the set
clearly still includes matter, any) arrangc-
A. J. Kirby
15
I Optimal information exchange
(b + d)a:A(q) En,(lz)
= [2(b + d) + b(n - l)c(z,)]*
’
where zi is the sujjicient statistic for I, with respect to the output choice, A(z,) is the accuracy of this statistic, and C(z,) is the average pairwise correlation between z, and zi for all j # i. Proof:
By definition
J3a14 = Under
of the term
&
z, = A(z,)z,
1’0 =
we have:
(21)
.
I
the normality
E(Zj
accuracy,
assumption,
if z, and z, are conditionally
correlated:
cov(z,, z,) ‘1
var(z,)
(22)
1
I
assuming that E(z,) = E(zj) = 0. Because of the symmetry assumption, var(zi) = var(zj) which implies that E(zj 1zi) = corr(zi, zj)zi. Defining C(zi) as the average of these pairwise correlations between z, and the n - 1 other z, random variables, we can simplify the expression for Q given in eq. (19) to:
’ = [2(b
A(Zi)
(23)
+ d) + b(n - l)C(zi)]
It is also straightforward d) var(xi(Z,)). Substituting
but tedious to show xi = Qzj, the result follows
that EZI,(I,) immediately.
= (b + q
This proposition is appealing since it provides an intuitive way of evaluating the effects on expected profits of changes in the information regime. Changes which enhance accuracy (without changing the correlation) are beneficial, since they allow firms to introduce greater variance into their optimal output strategies. Changes which increase correlation (keeping the effect on accuracy neutral) are always detrimental since they inhibit a firm’s ability to exploit the accuracy of the information it has. More importantly, in terms of the objective of identifying the optimal information regime, this result also provides us with a way of efficiently comparing the performance of alternative partial information sharing alternatives. An information reprofits which can be gime which optimally generates z, has expected expressed as a function of just A(zi), C(z,) and industry parameters. The above result is conveniently summarized in a diagram (see fig. 1). Since both accuracy and correlation range between zero and one, a unit
A. J. Kirby
16
I Optimal information exchange
Accuracy (A)
nA(S,l
l+(n-l)A(S,)
Con-elation (C)
Fig.
1. lsoprofit
lines.
square is sufficient to represent all possible symmetric information systems. Since the minimum value of C(z,) occurs when firms keep their signals private, i.e. when cov(z;, fj) = af, correlation always weakly exceeds (T:/ var(z,) which is the definition of accuracy, i.e. only half the square is feasible. From eq. (20), the iso-ex ante-expected-profit (hereafter isoprofit) lines are given by the expression: A
=
XP(b + 4 + b(n - WI2 CT;@ + d) ’
(24)
where x is the particular level of profit. These lines are positively sloped and convex in the feasible region. For a given (A, C) point, the larger is the quadratic cost coefficient d, the smaller is the slope of the isoprofit line through that point, i.e., higher levels of correlation are tolerated for higher levels of d. Assuming private information of accuracy A(s,) and an industry of IZ firms, not sharing and completely sharing those private signals are depicted by points NS and S, respectively. With the equilibrium strategies of the output game under information regime (A, C) nested in the expected profit expression (20), we are now equipped to address the question of selection of the optimal information regime. 4. Optimal Identifying information
information
regime: I*
the optimal combination of private information sharing method is solved for in three steps:
accuracy and first, for an
A.
.I. Kirby I Optimal information exchunge
17
exogenous level of private information accuracy, the optimal method of partially sharing information is identified; second, given the optimal method of partially sharing information and an exogenous level of private information accuracy, the optimal degree of partially sharing information is derived; and third the optimal level of private information accuracy is determined knowing that any subsequent information sharing is carried out using the optimal method and degree of partial sharing identified in the first two steps. 4.1.
Optimal method of partially sharing information
In this section we consider an infinite variety of methods for partially sharing private information (of exogenous accuracy), and show that the optimal method is to form multiple pools. We do this by comparing the expected profit performance of any feasible partial information sharing method (as captured by the two profit relevant attributes accuracy and correlation) with the performance of the multiple pools method. Thus we begin by deriving the performance of firms under the multiple pools method in order to use it as a benchmark, which we ultimately show cannot be outperformed. Firms are endowed with private information of accuracy A(s,) and are randomly assigned to one of p pools consisting of n/p firms. To evaluate the effect of sharing information on an individual firm’s expected profit, we know from above that we need only determine the accuracy and correlation of the signals resulting from the information sharing mechanism and substitute these values into the expression for expected profit (eq. 20). From the previous section, the sufficient statistic (2,) for firm i’s information set with respect to the output choice problem is the simple average of the private signals S, of accuracy A(s,) from all n/p firms in the pool. Consequently the accuracy of z,, when information of accuracy A(s,) is acquired and shared in p pools, is denoted A(z,(p, A(s,))) and is given by:
A(z,(A(s,), P))
= A@;)
MS, > = ~+(a-p)A(s,) The correlation pairwise correlation signals for all the correlated with the
(25)
of the information system is defined as the average between firm i’s aggregated signal z, and the aggregated other firms. Firm i’s aggregated signal z, is perfectly signals of the nip - 1 other firms in its pool, and is
A. J. Kirby I Optimal information exchange
18
conditionally independent of the signals of all n - nip other industry. Consequently the average correlation for firm i under is:
C(zi(A@;),P))
= &
firms in the this method
[(;-+l+(n-;)A(z;(A(s.),p))]. (26)
Since the information system is symmetric, the average correlation is the same for all firms, even though individual correlations differ between pairs of firms. When p = II, A(z,(A(s,), p)) and C(z,(A(s,), p)) both simplify to A(si), i.e. the attributes of the private information scenario (point NS on Fig. 1). Similarly, when p = 1, A(z;(A(s;), p)) and C(zi(A(si), p)) simplify to nA(s,)/[l + (n - l)A(s,)] and 1, respectively, i.e., the attributes of the simple shared information scenario (point S on Fig. 1). As p varies from n to 1, the corresponding accuracy and correlation plot out a curve in Fig. 1, joining NS and S. The slope of this function is given by dA(z,(A(s,), p))l dC(z,(A(si),
P))’
dA(z;(A(s;)>p>>/dC(z,(A(s;)> P))
dA(z,(A(s;)p))ldp = dC(z (A@ )p)),ap I I =
and is independent line.
(n - 1Ms;) 11+ (n - lMs,)l ’
of p. We refer to this straight
line as the multiple
(27)
(28) pools
Lemma 1. Partial sharing of private information via the mechanism of multiple pools is represented by points on the straight line joining the private and shared information scenarios. It is interesting to slope of the multiple can be characterized formation of multiple be proved.
note that as the initial accuracy A(s,) increases, the pools line increases. All points in the feasible region by a pair (A(s,), p), i.e. can be implemented by the pools. One of the main results of the paper can now
Proposition 2. Sharing private information in multiple pools is the optimal method of partially sharing information for jirms competing as Cournot oligopolists in their product market. Proof:
See Appendix.
A. .I. Kirby
19
I Optimal information exchange
The proof of Proposition 2 relies on showing that the multiple pools line happens to delineate the boundary of the feasible region of partial information sharing regimes, i.e., that any point above the multiple pools line is not implementable using the exogenous level of accuracy of firms’ private information. All other methods of partially sharing information for which (19) holds (e.g., firms sharing only a subset of their private observations) lie strictly inside the feasible region.” 4.2.
Optimal degree of partially sharing information: p*
Given that multiple pools is the optimal method for partially sharing information, we can now determine the optimal degree of partially sharing information, i.e., the optimal number of pools. This is necessary so that we can derive the level of accuracy (and correlation) that will actually result when private information is shared in multiple pools, and so that we can ultimately solve for the optimal level of accuracy of private information to be acquired. Substituting A(zi(A(s,), p)) and C(z,(A(s,), p)) into the expression for ex ante expected profits (eq. 20) gives profits as a function of the number of pools:
E(4(z;(A(s;),
=
P)))
(b + 4&AMp + (n - ~)A(s,)l [(b + W[p + (n - ~)A(s;)l+ WI+ (n - WW112
Below we give the solution for the optimal value of p, recognizing number of pools must lie in the interval: [ 1, n]. It is straightforward the following proposition. Proposition 3. p*(A(s;))
Optimal pooling has p*(A(s,)) =
bn[l
and results in accuracy,
+ (n - l)A(s,)] (b + 2d)(l
that the to prove
pools, where:
- (b + 2d)nA(si) - A(s,))
’
correlation and profits:
I’ Again, it is clear that despite the imposition of the condition in (19) interesting forms of partial sharing of information are included in the comparison set. Furthermore it is not clear that this condition actually removes any information sharing methods from consideration. Even if it does, it is not obvious that the expected profits under the multiple pools method will be exceeded by a form of partial sharing of information which does not conform to expression (19). since (19) simply refers to the ability to express expected profits as a function of accuracy and correlation. Thus Proposition 2 may actually apply to more than just the partial information sharing methods for which (19) holds.
A.J.
20
Kirby
I Optimal information exchange
(b + 2d) -b
A(si) [’ + (n -
l)A(si)l
2d C(zi(A(s,),
P*(N~,))))
=
/
Pooling in p* pools is feasible 2)(n - l)A(s,) s As (b/2)(n - 1).
only
if
1 ~p”(A(s,))
s n,
i.e.
(bi
Since (~~/~~) > (<)O, whenever p < (>)p”, it follows that if p* > n then p = R is the optimal, feasible number of pools and that p = 1 is optimal when p* < 1. Since the profit function is concave when p = p*, the behavior of the slope of the profit function in the feasible region also implies that if p* is feasible, that it is a global maximum. We note that under the assumptions used to calculate the optimal number of pools, there is no guarantee that the number of pools or the number of firms in each pool will be an integer. A counterexample can however be used to contradict the hypothesis that violating the integer constraint creates the phenomenon of interest.‘* This completes the second step towards solving for the optimal information regime when private information is costly to acquire. We have that the optimal level of profits for firms with private information of accuracy A (si) is generated if those firms share those private signals in one of p* pools where p” = min{max{l, p”(A(s,))}, rz}. 4.3.
Optimal acquisition of private information: A(s,)*
The final step in the derivation of the optimal information regime, f*. is to solve for the optimal accuracy of the privately acquired information. This is carried out independently for the three parameter regions identified above. The general form of the information acquisition problem is: ~(:y EIJ,(A(s,)) subject
(b + db-:A(z,(A(si)~ P)) = max Ah,) (2(b + d) + b(n - l)C(z,(A(si), p))]’
to 0 6 A(s,) =s 1 ,
-
kA(s,) , (29)
“If d = b, n =4 and A(s,) =O.l, then p* = 1.481, nip* = 2.7 and E(IT(p*(A(s,)), A(s,))) = 0.01282mlb. The traditionally analyzed arrangements of keeping information private and sharing it in one pool result in expected profits of 0.010816mib and O.O1256mib, respectively. However, we note that organizing the firms into two pools of two firms generates a final accuracy of 2111, correlation of 5 / 11 and expected profits of ~.012~4m/b, which dominates both of the extreme information sharing arrangements.
A.J.
Kirby
21
I Optimal information exchange
where A(zi(A(sj), P)) and C(zi(A(sj), P)) are the accuracy of firms’ signals after sharing private information pools, as calculated in eqs. (25) and (26). Define a set of critical values of the information .
k
(d)
(b + 4dW
=
I
[2(b
l
k,(d)=
.
k,(d)
=
.
k,(d)
=
b(b
+ d)a2,
4(b
+
(b
.
ks(d)
=
+
and correlation of accuracy A(s,) in
p
cost parameter:
+ 4 - b(n - 111
d) +
b(n
-
’
l)]’
’
24’
+ d)g:
4b(b
+ 2d)
’
(b
+ d)cr;
n[2(b
[2(b
+ d)
+ b(n
(b
+ d)a&
+ d)
+ b(n
-
l)]’
’
l)]’ .
-
when d d b(n - 1)/2 we have k,(d)< k,(d) d k,(d) and when - 1)/2 we have k,(d) d k,(d). If the quadratic cost coefficient is sufficiently large (i.e. d > b(n - 1)/2), then constraining the number of pools to equal one is the optimal feasible number of pools. In this case the solution to the unconstrained information acquisition problem is: Thus
d 3 b(n
(b A(@‘=’
=
(n
!
1)
k[2(b
+ d)&
+ d)
+ b(n
-
l)]’
- ’I ’
(30)
This level of private information accuracy is in turn feasible only if k,(d) G k s k,(d). If k B k,(d), then A(s,) = 0 is the optimal feasible solution, and similarly if k d k,(d) then A(s,) = 1 is optimal. If d s b(n - 1)12 then sharing in multiple pools may be feasible. If multiple pools are indeed feasible, then p = p*(A(si)), and the solution to the information acquisition problem is:
A(si)*
=
~
$1)
(b
+ d)a;
4bk(b
+ 2d)
-
’
1’
(31)
Substituting this value of private information accuracy for A(si) in b(n - 1)A(sj)/2 defines the value of d at which firms are indifferent between sharing information in multiple pools and keeping their information
A. J. Kirby I Optimal information exchange
22
private (i.e. p+ = n). This condition on d is equivalent to a condition on k. Thus if k L k,(d) when d c b(rz - 1)/2 then sharing private information of accuracy A(s;)* in p*(A(s,)*) p oo 1s is optimal. This value of A(sj)* is that A(s,)* s feasible (i.e. greater than zero), if k =Zk,(d).13 The constraint 1 is never binding. The intuition for this is that if information were so cheap that perfect information would be pooled then it would be yet more preferable to keep the information private. If d G b(n - 1) 12 but k s k,(d), then the optimal number of pools is ~1. The optimal accuracy, denoted A(s~)~=~ is the solution to the following cubic equation: (b + 46]2@ + 4 - b(tr - l)A(si)l [Z(b + d) + b(n - 1)A(s,)13
_ k =1:O
This equals one (i.e. perfect information is acquired) if the cost of information is sufficiently small: k d k,(d). The constraint that A(.Y~)~=~ b 0 is never binding. This solution to the problem of simultaneously selecting the number of pools and the accuracy of the private information to be acquired, is most concisely represented in a diagram (see Fig. 2). This shows how the optimal information regime varies as a function of the information cost and production cost parameters (k and d, respectively).
Fig. 2. Optimal
disclosure
method
as a function
of industry
parameters,
” It can also be shown that this is a Nash equilibrium in that there is no incentive for firms to unilaterally defect from their pool and instead ‘create’ their own one-firm pool in which they also have the choice to independently select the accuracy of the private information that they acquire, assuming that the number of pools is not affected by the defection. This is true even when the other firms are aware of the defection by the time they choose their output levels, and adjust their strategies to recognize the reduction in the accuracy of the aggregate they receive.
A. 1. Kirby
I Optimal
information
23
exchange
The intuition for these results is that as the production cost function becomes more convex (i.e. d increases) the optimal tradeoff of accuracy for correlation changes. For a given value of k, as d increases the marginal cost at the optimal output level also increases, the stakes in the output game increase and with it ceteris paribus the value of more accurate information. Consequently, when d is relatively small, accuracy is unimportant relative to correlation, and information systems are preferred which favor low correlation rather than high accuracy; for example, not sharing private information. However, as d increases, the preference for higher accuracy and better output decisions outweighs that for low correlation. This creates a preference for the sharing of private information in increasingly large pools. This pattern of preferences over information systems is also a function of the cost of acquiring information. If information cost is low, one approaches the case of costless information and the optimal information system is always to keep information private. However as information becomes more costly, the more economical way of achieving higher accuracy is to aggregate rather than to acquire. Consequently, we see a move towards fewer pools and lower acquired accuracy. In the limit when information is infinitely costly, no information is acquired at the optimum. The same comparative statics of the optimal information regime are depicted differently in Fig. 3. For each of three possible ranges of d values, the bold line indicates how the optimal final accuracy and correlation will change as the cost of information increases. These diagrams emphasize the surprising fact that whenever the optimal information system involves some form of information sharing, the optimal level of correlation induced by the information regime is independent of the information cost. It is as if the information sharing mechanism is designed to induce a constant level of correlation.
dz
b(n-1) 2
A
A
2d b(n - 1)
C Fig. 3. Optimal
2d b(n - 1) accuracy
C
and correlation
C
A. J. Kirby I Optimal information exchange
24
The problem of distinguishing between the motives for information exchange only arises if under collusive behavior it is also optimal to share information in multiple pools as opposed to some other form of partially sharing information. Thus the following analysis assumes that colluding firms also form pools for the purposes of sharing ~nforI~atio~ and that collusion is possible only to the extent that information sharing occurs, i.e. firms within the same pool collude in that they produce to maximize the pool’s profits, and pools compete with each other. Thus at the extremes when there are n pools, there is no information sharing and no collusion, and with one pool there is complete information sharing and monopoly output choices by the industry. The questions then become, what is the optimal number of pools? fs this different from the optimum when all firms in the industry compete regardless of the degree of information sharing? For a given level of privately acquired information and a given number of pools each pool selects output to maximize its expected profits:
max E ii xP(lP) I I
(321
where x4 is the output of the ith pool and Zf is its information set consisting of the private signals of the nip hrms in that pool. In order to minimize the pool’s production costs firms produce equal shares of the pool’s optimal output. The first order condition for this problem is: (33) Again we define 5, as the average of the private signals of the nip firms in pool i, and know therefore that:
where A(il) is the accuracy of the average signal and is given by:
(35) Conjecturing
a pool’s output strategy to be linear in S,:
A. .I. Kirby
and substituting optimal strategy
back into the first order condition for the pool has 4;; = 0 and
“’ = [2b(n/p) Consequently
25
I Optimal information exchange
(eq. 33) requires
(37)
+ 2d + b( p - l)(nlp)A(s,)]
firm i’s optimal
output
strategy
that the
is:
(38) Substituting
II,
cell
this into firm i’s profit
expression:
1
=
and taking expectations profits for firm i under
ET’” = [2bn/p Differentiating monotonically
over the information ‘partial’ collusion of azA(s,)(bn/p
ex ante expected
+ d)
+ 2d + b( p - l)(nip)A(F!)]’
with respect to p, decreasing in p. Thus
set gives
(40)
’
reveals that expected we have the following
firm profit proposition:
is
Proposition 4. If firms collude to the extent that they share information, then the optimal number of pools is one, for all industry parameters. Combining this result with those under the assumption of competitive firm behavior, we have that for d s (b/2)(n - 1) the observation of information sharing in one pool is evidence of collusion. However for values of d 3 (b/2)(n - 1) the number of pools does not provide a discriminating characteristic. Consider now the issue of the level of private information acquisition, when firms collude. The amount of information acquired and shared may instead provide a distinguishing characteristic in this case. The optimal level of expected profits under collusion. one pool and costly information acquisition is: En;“”
=
a;A(F,) 4[nb + d]
where that
A(s;)
- kA(s,)
is the accuracy
.
of the private
(41) signal purchased
by firm i. Recall
26
A. J. Kirby
A(s,) = 1 + (n and therefore given by:
- l)A(s,)
that the optimal
I Optimal information exchange
accuracy
Vf?l - 4k(nb
+ d)
- l)(nb
+ d)
A:&, >= 4k(n
(42)
’ of privately
acquired
information
is
(43)
The final issue is how this compares with the optimal level of acquired accuracy when firms compete and find it optimal to share in one pool. The latter is given by eq. (30):
A(.#‘=’
Comparing
= (n ! 1) 4~4. A(si)‘=’
k < a&[2(b
(44)
and A,*,,,(s,),
the latter
exceeds
the former
when
+ d) + b(n - l)]’
16(b + d)(nb
(45)
+ d)2
Defining this critical value of the information cost as k,(d) and referring to Fig. 2, it is easy to show that k,(d) b k,(d) holds for all relevant parameter values. Thus for those industries (i.e. points in k. d space) for which competing firms find it optimal to share in one pool, the accuracy of information acquired and subsequently shared will be less than the accuracy of information acquired by colluding firms. The results of this section are summarized in the following proposition: Proposition
5.
Firms
in industries
for
which
d < (b/2)(n
- 1) will share
their private information in more than one pool if they compete and in only one pool if they subsequently collude in the product market. However, firms in industries for which d b (b/2)(n - 1) h ave an incentive to share information in one pool irrespective of the type of behavior they subsequently adopt in the product market. For a given industry, however, the optimal level of information acquired and shared is greater if its firms subsequently collude in the product market than if they compete.
6. Conclusion
The primary conclusion of the paper is that observing the form by which firms choose to share information is likely to be much more informative
A. J. Kirby
I Optimal
information
27
exchange
about their underlying behavior in the product market than is simply observing whether information sharing takes place. Competing firms are best off sharing information in several industry subpools rather than in one large pool. By contrast colluding firms always prefer to share information in one large pool. In those industries for which competing firms actually wish to share in one pool only, the level of information they wish to share is less than that of their colluding counterparts. There are several caveats which need to be made in applying these results. First, these results were generated under and are likely to be sensitive to the assumptions of market demand uncertainty and Cournot competition. Second, attention was limited to symmetric information systems. Thirdly, and perhaps most crucially, the colluding firms were modelled as if they act with disregard for the possibility that they may suffer disutility if their behavior is determined to be collusive. In other words this model does not address the possibility that colluding firms might alter their means for exchanging information if that know that the form of information exchange could be used to detect their collusive behavior. To the extent that there are meaningful penalties for collusive behavior, this possibility undoubtedly exists. Nevertheless the tendency remains for colluding firms to want to share more information in larger pools than would competing firms.
Appendix Proof of Proposition 2 Proposition 2 implies that no method of partially sharing information exists (within the set defined by eq. 19) which produces combinations of accuracy and correlation above the multiple pools line. The proof is by contradiction. Each firm i receives a private signal, s, = a + m,, where a-N(~=O,c~),m,--N(0, af,) and E(am,) = 0, for all i, and E(m,m,) = 0 for all j # i. If the firms share their private signals in p pools of nip firms each, where 1
A(z,(A(s,),P))
= $
C(z,(A@,),P>> =
0
+ Jz,(n,p) m
3
(AlI
~ri~-li+~~-~,A(z.(A(~,,,p,,]. 642)
28
A. J. Kirby
I Optimul
infknation
exchange
Now suppose that there exists an alternative means X for firms to partially share their private information with corresponding accuracy and correlation given by A(z,(A(s,), X)) and C(zi(A(s,), X)) such that mechanism X has correlation equal to and accuracy strictly greater than that under multiple pools. Therefore, adopting abbreviated notation.
C(z,(W) = C(z,(P>> 3
A(z,(X)) = A(z,( p)) + c ,
6 >
0.
Given Proposition 1 this would imply that method X would generate strictly greater expected profits than does the multiple pools method of partially sharing information. The question is whether X is feasible given the exogenous level of accuracy of the private information. Suppose that in a second round of information sharing the information set available to each firm via method X is provided publicly to all firms. Specifically each firm then has access to z:+_,(X), where
&(X) The accuracy
= f [z,(X) + Z?(X) + . . . + z,,(X)] of the resulting
information
(A3)
set available
to all firms would be:
(A4) where var(z,,,,(X))
= $
Z? var($) i ,=I
_ var(;;l(X)) since
by assumption
A(z,(W) =
C(z,(X))
+ +2-
l) cov(z,(X)z,(X))]
[2 + (n - l>C(z,( = C(z,( p)).
=A(z,(P)) + ,,,(~i,,,
p))] ,
Also by assumption:
(A3
E.
I
and consequently expressed as:
the accuracy
A(z‘w(X))=
of this new (second
~~A(z,(P)) + ~1 2 +
(n
-
l)C(z,(
p))
round)
aggregate
can be
(Ah)
A. .I. Kirby
I Optimal information e.xchange
29
This can be compared with the accuracy resulting from directly providing all the original private information to all firms, or equivalently pooling in one pool. The correlation between firms’ information sets in this case is also one, but the accuracy would be:
(A7) into expression (A6) using cxpresSubstituting for A(z,( p)) and C(z,(p)) sions (Al) and (A2) it can be shown that A(z&) > A(z,( p = 1)) for all E > 0. Since under normality the aggregate from simple sharing is sufficient aggregates producing for the original private signals, other (compound) strictly greater accuracy for this given degree of correlation (i.e. C = 1) are not possible. Thus we have the necessary contradiction to prove that mechanisms like X cannot exist, and the multiple pools line forms the boundary of what is feasible given private signals of exogenous accuracy. Optimality of multiple pools follows given the shape of the isoprotit curves relative to the feasible region. 0
References Basar, T., 1978. Decentralized multicriteria optimization of linear stochastic systems. IEEE Transactions on Automatic Control AC-23. No. 2, 233-243. Clarke. R., 1983, Collusion and the incentives for information sharing, Bell Journal of Economics 14. 3833304. De Groot. M.. 1970, Optimal Statistical Decisions (McGraw-Hill. New York). Gal-Or, E., 1986, Information transmission-Cournot and Bertrand equilibria, Review of Economic Studies 153, 85-92. Gal-Or, E., 1985, Information sharing in oligopoly, Econometrica 53. 320-344. Kirby, A., 1992, Self-imposed information acquisition and disclosure: An economic analysis. Working Paper. University of California, Berkeley. Kirby. A., 1988, Trade associations as information exchange mechanisms. Rand Journal of Economics 19. 13X-146. Li, L., R. McKclvey and T. Page, 1987. Optimal research for Cournot oligopolists. Journal of Economic Theory 42. 140-166. Novshek. W. and H. Sonnenschein, 1982, Fulfilled expectations Cournot duopoly with information acquisition and release, Bell Journal of Economics 13. 214-218. Shapiro. C., 1986, Exchange of cost information in oligopoly, Review of Economic Studies 53. 433-446. Vives. X.. 1984, Duopoly information equilibrium: Cournot and Bertrand. Journal of Economic Theory 34, 71-04. Vives. X., 1990. Trade association disclosure rules. incentives to share information. and welfare, Rand Journal of Economics 21, 409-430.