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The value of information in a simple duopoly model
JOURNAL
OF ECONOMIC
THEORY
36, 36-54
(1985)
The Value of Information in a Simple Duopoly Model YASUHIRO SAKAI* Institute of Social Sciences, University Sakura,
Niihari,
Ibaraki
of Tsukuba,
305, Japan
Received November 3, 1983; revised November 8, 1984
This paper is concerned with the evaluation of information in a duopoly model in which the cost functions are subject to uncertainty. It explores how changes in information about the costs available to either firm affects the welfare of both firms along with the welfare of consumers. By comparing the ten possible types of information structures, it is shown that information may be detrimental, that improved information for one firm may or may not benefit the other firm and/or the consumer, and that it may be more desirable for a firm to gather information about the rivals cost rather than its own. All of these “irregular” results depend on the values of the variances of the costs and their correlation coefficient. Journal of Economic Literature Classification Numbers: 022, 026, 611. 0 198s Academic press, I~C.
1. INTRODUCTION
The purpose of this paper is to investigate the value of additional information in a duopoly model in which the cost functions of the firms are not known with certainty. More specifically, it deals with the question of how changes in information about the costs available to either firm affect the welfare of both firms along with the welfare of consumers. As is well known in two-person zero-sum games, information is always valuable in the sense that better information on the part of one player tends to increase the average payoff of that player but it tends to decrease the one of the other player.’ When we enter the world of nonzero sum games, however, the situation changes drastically. In fact, it has been pointed out by Levine and Ponssard [IS] that the value of information may be detrimental in those games. The present paper focuses attention on the role of information in a simple, duopolistic market model under cost uncer*I am grateful to Ryuzo Sato, Mamoru Kaneko, Masahiro Okuno, and Yozo Ito for helpful discussions. I would like to acknowledge my special thanks to an anonymous referee of this journal whose comments have substantially contributed to improvement of this study. Any errors which may remain are of course my sole responsibility. ’ See Ponssard [S] and Suzuki [ 111.
36 0022-0531/85 $3.00 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
VALUE
OF INFORMATION
37
tainty, whose analysis has not been given the prominence which, I believe, it deserves. 2 Hopefully, it will bring out some new insights about the valuation of information structures in an important cIass of market equilibrium models. The duopoly market model we are going to work with is described as follows. There are two firms, firm 1 and firm 2, with identical outputs operating in a market where the demand function is assumed to be linear. We suppose that firm i has a linear cost function: ci(xi) = cixi, where xi denotes the output level of firm i (i = 1,2). Each firm knows the probability distribution of both cost parameters c1 and cZ. To be more specific, we assume that the joint distribution of c1 and c2 is normal with means ~1~and p2, variances G: and o:, and correlation p. Each firm aims to maximize its expected profit; so no risk aversion is taken into consideration. From a game theoretic point of view, our duopoly model can be regarded as a game with incomplete information introduced by Harsanyi [4J Given a certain information structure on the costs, the resulting equilibrium will clearly be a Nash-Cournot equilibrium. However, tbere are many types of information structures conceivable; nameily, each firm is faced with the following alternatives: (A) To determine the supply on the basis of its own knowledge of the probabilities of the costs c1 and c2. (B) To pay a market research agency, which we assume faultless, and which for possibly different fees, can tell (B’): the realized values of c1 only; or (B”): the realized values of c2 only. (C) To pay the agency for information about the realized values of both c1 and c?. There are several questions that might naturally arise at this point. What is the maximum expected profit each firm can gain under any combination of the four alternatives ((A), (B’), (B”), (C)), not counting the research fees? How much should each firm be willing to pay, at most, for each of the three research services, if it tries to maximize its expected profit? In t paper, we will almost exclusively be concerned with the first question, leavring the second one for further research. Our problem here is therefore to inquire how those variations in the information structure influence the best 2 The role of information on the demand side in the duopoly market was first studied by Basar and Ho [l] and Ponssard [9]. In such a simpler case, information must be beneficial if no risk aversion is taken into account. Although Okada [7] recently examined the role of information on the cost side he made the stringent assumption that each firm always knew the realized value of its own cost, and failed to discuss the consumer point of view.
38
YASUHIRO SAKAI
decision functions of the firms and their expected profits. Put it differently, we are especially interested in seeing whether an uninformed firm has an incentive to acquire information whereas an informed firm has an incentive not to share the information. In our duopoly market game, firms 1 and 2 are active players with output levels as their strategies. Although consumers are mere passive players with no strategic choice and no explicit utility function, it is worth examining how the welfare level of passive players as a whole depends upon the gathering of information by active players. When we regard the average value of the consumer surplus as a good measure of the welfare level of the consumer, another question of interest is whether and to what extent improved information on the part of the producer decreases the expected consumer surplus. In other words, when one firm gains benefit in unilaterally conveying information to the other firm, or when the two firms gain benefit in exchanging information with each other, can we say that such a transmission of information is against the interest of the consumer? The paper is organized as follows. Section 2 discusses our duopoly model and alternative information structures in greater detail. In Section 3, for any given information structure, we first determine the best decision functions and expected profits of the two firms along with the corresponding expected consumer surplus. We then scrutinize the role of information in the model by comparing various types of information structures in terms of the expected profits and consumer surplus that can be achieved through their use. Concluding remarks are made in Section 4.
2. THE BASIC MODEL In this paper we are concerned with the following simple, duopolistic market model. Firms 1 and 2 produce and supply the same homogeneous commodity; the problem of product differentiation is not taken into account. We assume that there is no limit of capacity at each firm. Denote by xi the output level of firm i (i = 1,2). The (marginal) cost function of firm i is assumed to be linear; i.e., it is provided by MC, = cixi
(i = 1, 2).
(1)
We suppose that the cost parameters cl and c2 are random variables whose joint distribution @(c,, c2) is known to both firms. Concerning the probability distribution on (cl, cZ), it is convenient to assume that it is a bivariate normal distribution with mean (pl, pLz)> 0, variance (o:, G’,) > 0, and the correlation coefficient p ( - 1 < p < 1). It is
VALUE
OF INFORMATION
39
then well known that the regression function of cr on c2 and the one of c1 on ci become linear; indeed we have
and
where E stands for the expectations operator. Clearly, the linearity of these regression equations will make our calculations fairly manageable. Given the price p of the output, the market demand function is assumed to be given by
p=a-b(x,
-lx,),
(4)
where the demand parameters a and b are nonstochastic and positive. Without the loss of generality, we may choose the units of measurement so that b equals unity. Letting Z7j denote the profit of firm i, we obtain Il,=(a-cj-xl-x,)xj
(i= 1, 2).
(51
Each firm aims at maximizing its expected profit; namely, it is assume have a risk neutral (i.e., linear) utility function with respect to profit. In order to facilitate the following analysis, we assume
In the present model, since each firm is confronted with uncertainty about its own cost and/or the rival’s, it ought to determine its optimal output level on the basis of its estimate of the cost(s). In order to formulate such an estimate precisely, it is necessary to decide what kind of information structure we are dealing with. Let us focus on the case in which the information structure of the duopoly model is fully described by a statement as to “who knows what.” It is then useful to represent the information structure as a matrix, q = [rik] (i= 1,2; k= 1,2), such that if firm i is kept informed of ck.
qik=l
=o
if firm i is not so informed.
This case does not exhaust all possible information structures; for instance, the notation just introduced would fail to state “who knows what with what precision.” However, it does include an important class of information structures. In what follows, to save the space, the 2 x 2 matrix [qik]
40
YASUHIRO
SAKAI
will be written as a row vector, q= [vi, r2] = [ylllr12, r~~i~&l, where vi= [qilriZ] represents the information signals that can be received by firm i (i = 1,2). Since qik takes on either 1 or 0, there are 24 = 16 information structures which may be classified into the following ten types: (i) q = [00, 001: Neither its own cost ci nor the rival’s c2 is known to firm 1, and similarly for firm 2. In short, no firm has information about either cost. (ii) q = [lo, 011: c1 but not c2 is known to firm 1 whereas cz but not c1 is known to firm 2. In other words, each firm gathers information about its own cost, but not the rival’s, through the market research agency. (iii) r~= [01, lo]: In contrast to (ii), each firm acquires information about the rival’s cost only. (iv) q= [ll, 111: Both firms can know c1 as well as c2. (v) q = [lo, 001 or [00, 011: One firm can know its own cost, but not the rival’s, whereas the other firm is ignorant of the costs. (vi) y = [Ol, 001 or [00, lo]: Contrasted with (v), one firm can know the rival’s cost only, with the other firm being ignorant of the costs. (vii) r = [ll, 001 or [00, 111: One firm is kept informed of both its own cost and the rival’s whereas the other firm is not informed of either cost. (viii) r~= [ 11,011 or [lo, 111: One firm can know both costs whereas the other firm can know its own cost only. (ix) y = [ll, lo] or [01, 1 I]: By contrast, although one Iirm can know both costs, the other firm can know the rival’s cost only. (x) y = [lo, lo] or [01, 011: Either one of the costs is commonly known to both firms. From the viewpoint of the distribution of information between the firms, we are able to think of types (it(iv) as the cases of symmetric information, and types (v)-(ix) as those of nonsymmetric information. And type (x) can be regarded as the case of shared information, meaning that whenever information about either cost is gathered by one firm, it is also shared by the other firm. Let C= Ci x C2 be the set of the pair (ci, cz). Then we can consider an information structure v] = [qr, Y/~] to be a partition of C2. We say of two given information structures, yeand y’, that y is finer than q’ if 7 is a subpartition of q’; that is, if every set in y is contained in some set in 13 fl. 3 For a formal Chap. 23.
discussion
on the ordering
by fineness,
see Marschack
and
Radner
[6,
VALUEOFINFORMATION
41
Given any one of the information structures, we are ready to define an equilibrium of the duopoly model a la Nash and Cournot. More exactly, the output pair (x,*(q), x:(q)) is said an equilibrium output pair under information structure 9 if
and for all x,(r);
Efl,(X?Y?)>xz*(TI))3 Efl*(X,*(Y), x2(r))
and EIIi(x,*(q), x;(q)) denotes the optimal expectedpayoff (profit) offirm i (i= 1,2). In other words, with the assumption that the firms are all risk neutral, firm i seeks to pick up a policy x,(y) so as to maximize its expected profit against some optimal policy of firm j (j f- i); which results in a NasbCournot equilibrium. When the market demand function is furnished by Eq. (4) and the equilibrium price under information structure q is denoted by p*(q), the consumer surplus is calculated as the area under the demand curve over p*(q), so that CS*(r) = +(a- P*(r))(xXr)
+ xi+(r)) = IMYt?) + xT(s)12.
67)
Throughout this paper, we regard the average value of this quantity, Z*(q) = E[CS’*(q)], as a good measure of the welfare level of consumers as a whole. 3. COMPARISON OF INFORMATION
STRUCTURES
As was seen in the last section, there are ten possible types of information structures. In this section, for any given information structure, we determine the best decision functions of the two firms and measure the resulting expected profits along with the corresponding expected consumer surplus. We then discuss the role of information in the duopoly market game by comparing various types of information structures in terms of the expected profits and consumer surplus that can be achieved through their use. A. Equilibria
under Alternative
Information
Structures
Let us start our investigation with the case of null information represented by q= [OO, 001; which serves as a basis of comparison with other information structures. Since firm i acquires no information about its own cost ci and the rival’s one cj, it seeks to maximize the expected profit E~i(x,)=E[(a-Ci-X,-X,)Xi]
(i = 1, 2).
($1
42
YASUHIRO
SAKAI
An equilibrium point (x:, x2) may be obtained derivative of Eq. (8) equal to zero: (i,j=
a-pi-2x*--xj*=O
by setting the first
1,2; i#j).
(9)
Solving these equations for xl* and x,*, we obtain two decision functions, each linear in ,ui and p2: ~300,~)
= +(a - 2~~ + fi2),
(10)
x:(00,00)
= $(a - 2pu2+ pL1).
(11)
Recalling that a > 2 max(y,, p2) (see Eq. (6)), x:(00,00) and x,*(00,00) are both positive, so that the positive solution is indeed guaranteed. In what follows, the equilibrium output pair (x:(00, 00), x,*(00,00)) under the information structure [00, 001 will simply be referred to as MYO), x2*(0)). In view of Eqs. (8), (9), and (lo), the maximum expected profit of lirm 1 is provided by Q:(O) z Ell,(x~(O))
= {xf(0)j2
= $(a- 2p1 + p2)*.
(12)
Similarly, the maximum expected profit of firm 2 is Q2,*(0)=$(a-2p2+pl)2.
(13)
And the (expected) consumer surplus under [00, 001 is given by C*(O) = CS*(O) = &(2a - pl - pL2)2
(14)
because of Eqs. (7), (lo), and (11). We turn our attention to the case [ 10, 011 in which each firm gathers private information about its own cost, but not the rivals. An equilibrium point for this case is expressed by the pair of decision functions (x:(c,), xz(c?)), where firm i chooses its optimal output level to be produced as a function of the true values of ci (i= 1,2). Since firm i maximizes its expected profit Ec,[IZiI ci] conditional OIZci, the first-order condition for such a conditional maximum results in a-ci-2x*(c,)-.E[x,*(cj)Ici]
=0
(i, j= 1, 2; i#j),
(15)
from which we find (16)
GYc2)= f(u - c2)- + g-m4
I4.
(17)
VALUE
OF INFORMATION
43
If we take the expectation over c2 conditional second equation in (15), we find
Substitution
on c1 of both sides of the
of this equation into the first equation in (15) yields
Now let
If we succeed in determining the values of the constants P and Q, it is clear under the present assumptions on the demand and cost functions that this x:(cl) represents the desired, equilibrium output level of firm 1 (see Okada [ 71). Taking the expectation over c1 conditional on c2 first and the expectation over c2 conditional on c1 next, we obtain from Eqs. (3) and (19)
Inserting Eq. (20) into Eq. (18) and making use of Eq. (2) it is not har find x:(4=,
1 po2-2c1.1 (
+ 0”
Gl
(Cl -!-%I :i
If we compare Eqs. (19) and (21), we must have
p= P2-201 a,(4-P2)
and
Q=
a--P,+h
3
By virtue of Eqs. (10) and (19) we thus obtain PO2 - 20, x:(10, 01) 3 XI*(C{) = xf(0) + a,(4-p’)
In a similar way, we can also derive
(cl--l).
.
44
YASUHJRO
SAKAI
We note that the equilibrium (Nash) expected profit of firm 1 conditional on ci is written as E,,[n: 1cl] = (x~(c~)}~ by making use of Eqs. (5) and (15). Therefore, in the light of Eq. (22), the optimal expected profit of firm 1 is (PO2 - 2d2 (4-p2)2
52:(10,01)-~~[17:Ic,]=s2:(0)+ A similar calculation firm 2:
(24)
will lead us to derive the optimal expected profit of
Q,*( 10,Ol) = g g[n:
1cz] = a;(o)
+ (PO, - 2d2 (4- P2J2
(25)
And it is not hard to prove by virtue of Eqs. (7), (22), and (23) that the expected consumer surplus under [ 10,Ol) is c*(lo,ol)
= c.c2[CS*(10, Ol)] = c*(o) +
1 2(4 - p2)2
[4(1 -P2M+4
+P2((~1-a2)2+2~1(T2(1+P)}1.
(26)
Concerning the remaining cases, we can similarly derive the best decision functions of the two firms and the resulting expected profits along with the corresponding expected consumer surpluses, and omit the proofs. Tables I, II, and III summarize the results for all sixteen information structures. B. Value of Information We are now in a position to measure the value of additional information, investigating the problem of how better information available to either firm affects the welfare of both firms along with the welfare of consumers. To begin with, it should be noticed that for any information structure ye, we have E[xF(r])] =x*(O) (i= 1,2) and E[p*(q)] = p*(O) in the light of Table I and Eq. (4). Therefore, for whatever information about the cost(s) either firm can get, the average output level of each firm in the NashCournot equilibrium remains the same, and so does the average market price. There exist many sequences of comparisons of the information structures that are worthy of investigation. First of all, we want to compare the case [OO,OO] of no information (for both firms) with any other information
VALUE OF INFORMATION TABLE I Optimal Decision Functions for Sixteen Information Information structure: 1
Structures
Optimal decision fzmction Firm 1: x:(q)
Firm 2: x;(q)
structure. In view of Tables II and III, it is straightforward following: THEOREM
1. For any information
hOid.
(4
QiW
2 GW.
(b)
Q:(q) a .n:(O).
Cc) C”(q) 2 c*(o). 642/36/l-4
structure
to obtain the
q, the following
relalions
46
YASUHIRO SARA1 TABLE II Maximum Expected Profits for Sixteen Information
Information structure: ?
!30,001 Q:(o)
Structures
Maximum expected profit Firm 2: Q:(q)
Firm 1: Q:(q)
QUO)
The economic implications of this theorem are quite clear. Let us begin with the case in which both firms are totally ignorant of the costs. Then any form of improvement of such null information tends to be beneficial to all firms and consumers.4 The fact that information is thus valuable comes from each firm’s better adjustment of quantity produced on cost, and this adjustment is also in the interest of consumers. Put it differently, the social value of information compared to the null information is always positive.5 4 This result is similar to the one of Ponssard [9] who studied the value of information on the demand. An analogous point was made by Green [3] for the framework of sequential future markets. 5 In passing, we note that 8:(10,01) -Q?(O) = (p (r2 - 2u,)*/(4 - p2)*, which measures the advantage that firm 1 can gain when each firm is kept informed of its own cost. How this advantage is affected by changes in the variances of the costs and their correlation coefficient can be studied through this equation. No doubt, there are many comparative statics analyses of this sort conceivable. Interesting as they may be, they are not pursued in the paper and will be left to the reader.
47
VALUE OF INFORMATION TABLE
III
Expected Consumer Surpluses for Sixteen Information Information structure: 4
Structures
Expected consumer surplus: C*(q)
z*(o) 1 [4(1-p*)(aT+o:)+pZ{(oi--az)Z+2a,O?(l+P)jl 2(4 - ,L+
z*(o) +------
cw 101
z*(o)+
c11>1t1 [ 10, 001 w, 011 COL001 PO, 101 Cl& 001 cm 111 [II, 011 [lo, 11J IIlL 101
c*(o)+~{(u1-u2)2+261u2(1 +p)j z*(o)+ +cTf z*(o)+ +u; c*(o)f g&J:
Z‘*(O)
+ib(P~*
+ ~I)*
[Ol, 111 [lo, lo] w, 011
E‘*(O)
+ ?&PC,
+ az)”
x*(o)
+ #PC2
+ fJ,Y
c*(o)
+ &PO1
+ %I2
-
pz 2(4-p’)’
z*(o)
+ ~p’c$
z*(o)
+ $uf
c*(o)
+ +;
[4(1 -pp’)(o:+(i;)+p*{(o,
-az)‘+za,u~jl+
p))]
~*(0)+k[5(1-pZ)~:+4{(0,-cr,)i+2~~cr,(l+p))] c*(o)+~[5(1-p~)~:+4{(0,-~*)*+25~~*(1+pj~]
The situation becomes more complicated when we consider improvements of an existing information structure that is already better than the null information structure [00, 001. A finer information structure may or may not be Pareto superior to a coarser information structure. To approach this problem, it is useful to distinguish the two classes of cases-a class of cases of simultaneous and symmetric variations in information on the part of both firms and another class comprising all the other cases sf nonsymmetric variations. We will first deal with the first class of cases. 2. (a) !L?,*(ll, ll)>Q:(lO, Qn,*(ll, ll)>Q,*(lO, 01). c*(ll, ll)
THEOREM
(b) (c)
01).
The proof of Theorem 2 is immediate from Tables 2 and 3 and is omitted. (The proofs of the following theorems will also be omitted unless otherwise stated.) The results in Theorem 2 can be interpreted as follows.
48
YASUHIRO
SAKAI
Suppose that there is a simultaneous and symmetric improvement of information available to both firms. Then such improved information increases the expected profit of each firm, agreeing with the well-known result of one-person decision problems that making the information structure liner tends to increase the expected payoff.6 On the other hand, it results in a decrease in the expected consumer surplus. Put it differently, when each firm gathers private information about its own cost, the firms gain benefit in exchanging the information with each other; which is certainly to the detriment of the third party (namely, the consumer in the present situation). As a matter of fact, this is a rather common practice in many industries in which anonymous data are redistributed to all the firms through a syndicate, which yields a welfare loss on the part of the consumer. 7 Remark 1. The quantity (sZT( 11, 11) -QT(lO, 01)) and the one (L’*( 10,Ol) - C*(ll, ll)), respectively, measure firm i’s welfare gain and consumers’ loss due to the exchange of private information between the two firms. In the special case of equal variances (i.e., rrl = 02), it can easily be shown that this welfare gain is the larger (and so is the welfare loss), the larger is the common variance. Remark 2. It is of interest to see that
Q?(lO, 01) > s2:(01,10)
and
Q?(lO, 01) 3 Q,*(Ol, lo),
where the first equality holds if and only if p = 20,/(r,, and the second equality if and only if p = 2a,/o i . Although no comparison of fineness can be made between the structures [lo, 011 and [01, lo], it follows from these inequalities that for both firms, [ 10, 011 is more valuable than [01, 10); namely, it is, on average, more profitable to both firms for each of them to know its own cost rather than the rival’s’ We now turn to the second class of cases in which information is not symmetrically distributed between the two firms. Then we will see that the relationship between changes in the information structure and those in the welfare of the producers and consumers become much more intricate than in the first class of cases of symmetric information. 6 See Marschack and Radner [6, Part One]. ’ I am indebted to the referee for this point. *When the value of p* is near unity, we obtain .Q:(ll, (i= 1,2) and C*(ll, 11) i Z*(lO, 01) -j, Z*(Ol, lo), where to.” This is as it should be. Since near perfect correlation variable almost exactly predictable from the other, it does the consumer which one of the costs is commonly known be made for similar comparisons of information structures.
11) + Q:(lO, 01) + Q:(Ol, 10) “i” means “approximately equal (positive or negative) makes each not matter much to each firm and to the firms. The same remark will
VALUE
4
OF INFORMATION
From the viewpoint of game theory, our doupoly model can be regarde as a special sort of two-person nonzero-sum games. As is well known, zerosum games and team problems are two extreme cases of nonzero-sum games. The equilibrium solution to any zero-sum games has the property that, if a change in the information available to one player increases the optimal average payoff of one of the players, it definitely decreases the optimal average payoff of the other player; i.e., the payoffs of the players must go in opposite directions. On the other hand, the equilibrium solution to any team problem has the property that, if a variation in the information on the part of one player increases the optimal average payoff of one of the players, it also increases the optimal average payoff of the other player; namely, these two payoffs must go in the same direction. It is naturally conjectured that the duopoly model has both the zero-sum property and the team property; for it is neither an antagonistic zero-sum game nor a completely cooperative game.’ In order to develop an economic background of the interpretation of the results of the following theorems, three cases should be distinguished-independent costs, positively correlated costs, and negatively correlated costs. If the costs are positively correlated, the extent of the conflict of interests between duopolists is large, and may sometimes be so large that the zero-sum property outweighs the team property. Accordingly, it may be the best to be the only one to be informed, for in this case we may have a monopoly rent on information. And it is expected that such a rent declines with the number of firms informed. If the costs are negatively correlated, however, there is much room for cooperation between the duopolists. If so, the team property may dominate the zero-sum property, The result is that a more informed tirm may have a strong incentive to convey the information to a less informed firm, presumably at the expense of the consumer. We are ready to establish a sequence of theorems as to the cases of nonsymmetrical information. THEOREM
3.
(a)
L?,*(lO, lO)$Q:(lO,
00) according as (p-~(o,/cs,))
(P - %%/~,))i3
(b)
sZ;(lO, 10) aQ,*(lO,
00), where the equality holds q and onlv $
P = 5(~1/~2).
(6)
Z:*(lO, lO)$Z:*(lO,
00)
according
as
bP- l(a,la*))
(P + 4(~1/~2))$0.
The starting point of discussion is the case in which firm 1 has private information about its own cost whereas firm 2 is ignorant of the costs. One 9 For a discussion on the zero-sum and team properties of nonzero-sum and Ho [l].
games,
see Basaf
50
YASUHIRO
SAKAI
natural question to ask is whether the informed firm (i.e., firm 1) has an incentive to unilaterally convey the information to the uninformed firm (i.e., firm 2). Another question of interest is whether the sharing of the information by the two firms is beneficial to the third party (i.e., the consumer). The results in Theorem 3 teach us that definite answers cannot be given to these two questions; they are actually dependent on the values of 01, (72, and p. Let us limit our attention to the case of equal variances (namely, CJ~= CQ). Then we can easily see that when the correlation coefficient is large enough to exceed a half, the zero-sum property outweighs the team property; so firm 1 has no incentive to share the information with firm 2 although such transmission of the information would benefit firm 2 as well as the consumer. When the correlation coefficient is negative or zero, or when it is less than a half if positive, however, the team property plays such a dominant role that firm 1 is motivated to convey the information to the rival; and the consumer would then be worse off. 4. (a) (7(a,lo,)2+4)/16(0,/0,). THEOREM
(b) (c)
Q:(ll, c*(ll,
Q:(ll,
ll)>Q;(ll, ll)~X*(ll,
11) $ Q~(ll,OO)
according
00). 00) according as p$ (5(a,/o,)’
as
PZ
- 4)/8(o,/a,).
Theorem 4 demonstrates that when the value of correlation coefficient is small enough, the transmission of information from the informed firm (i.e., firm 1) to the uninformed firm (i.e., firm 2) may be beneficial to both lirms. This is particularly so when the costs are independent or negatively correlated (namely, p d 0). In the special case of equal variances, the results can be interpreted more easily and are summarized as follows. When the value of the correlation coefficient is large enough (more exactly, p > $), firm 1 has no incentive at all to share its information with firm 2 since the zero-sum property outweighs the team property. On the other hand, when p is nonpositive or when p is small enough if positive (more precisely, p p > $) then the formation of such a syndicate would contribute to an increase in the welfare of the consumer; hence the information structure [ 11, 1 l] would be Pareto superior to [ 11, OO].i” I0 It is not
hard
to show
that
Therefore, when firm 1 acquires complete information about the costs, firm more to inform its rival of both costs rather than only one of them. that the team property dominates the zero-sum property in the present
it would benefit that This is another proof situation.
51
VALUE OF INFORMATION
Remark 3. Ponssard
[9] has discussed the role of information in a duopoly model similar to ours when the demand side is subject to uncertainty. More specifically, the demand parameter a is a random variabie in his model in contrast to our model, where the cost parameters c1 and c2 are random variables. From a mathematical, yet not economical, point of view, the Ponssard situation may be thought of as a very special case with p + 1 and or = CJ~within our general framework. In all of the cases discussed above, there has been a positive incentive for every firm to acquire more information. However, the following theorem will show that this may not necessarily be the case. THEOREM
5.
(a)
Q,*(lO, 11)>sZ~(lO,O1),
where the equality holds if
and only if p = 2(0,/o,).
(b)
Q,*(lO, ll)$L?~(lO,
(2 - P(~ll%))2z7
01)
according
as
(4(7 - p2)/(4 - p*)“f
- 4(02)*.
To get a good interpretation of the results in Theorem 5, let us focus on the special case of equal variances. The starting point of comparison is now the case of private information: each firm is kept informed of its own cost. (By contrast, the previous cases were concerned with those where one of the two firms was totally ignorant.) In this case, each firm has a strong incentive to unilaterally convey the information to its rival, regardless of the value of p; so one firm (say, firm 1) can derive an extra benefit from moving first and taking the role of a “leader” in informational transmission. In fact, when the costs are positively correlated and this correlation is fairly large (more exactly, p > (2 $8 - 6)/7, such an initiation benefit of firm 1 is so overwhelming that even the welfare of firm 2, i.e., the firm receiving the information, would be lower. This shows the distinct possibility that more information may even be harmful to the player to be informed in our duopoly game. I1 Remark 4.
It is worth noting that Q,*(ll,
ll)>Q;(lo,
11) Q$.(lO,Ol).
So when firm 1 takes the initiative of conveying private information about c1 to firm 2, the best reaction the latter firm can take is to convey back its own information about c2 to the former so that the transmission of information between the two firms may become “two-way.” Finally, let us consider the quantity (sZf( 10, q2) - Q:(Ol, q2)). This measures the advantage (if positive) for firm 1 when that firm knows c, rather than c2, with the rival remaining to have the same information ~1~. ‘I For this possibility, see Levine and Ponssard [S].
52
YASUHIROSAKAI THEOREM 6.
(b)
(a) Q:(lO, 00) > 52:(01, 00). Q:(lO, Ol)$Q:(Ol, 01) according as
Cl- w71/~,))* + $, (c) Q$(lO, ll)$Qf(Ol,
(GJQ)~ $ ((7 - p2)/36)
11) according as a1/cr2$Z~.
Let us begin our discussion with the simplest situation that firm 2 has no information at all. Then the result agrees with common sense: it is definitely more profitable to firm 1 to be informed of its own cost rather than the rival’s (see property (a)). When firm 2 acquires some information, however, the situation becomes more complex and the “common sense” result does not always hold. This is because firm 1 is now capable of taking advantage of the information on the rival unless the latter has nothing to tell about. In fact, we note that G?~(Ol,Ol)>Q~(lO,Ol) if p=O and crJa, > 5 and that Q,*(Ol, 11) > sZ:( 10, 11) if cz/ol > 2 (see properties (b) and (c)). Therefore, when the variance of the rival’s cost is large enough relative to that of its own cost, using information about the former cost may be more valuable to firm 1 than using information about the latter. We see once more the delicate role played by information in the duopoly market model.
4. CONCLUDING REMARKS The duopoly model we have discussed in this paper is, admittedly, quite elementary. The model was selected as the simplest one which would enable us to derive the following results: (i) Compared with the case of no information for the two firms, any form of improvement in information structure tends to be beneficial to all firms and consumers; therefore, the social value of information is always positive in this sense. (ii) The situation becomes much .more complex, however, when we consider improvements of an existing information structure that is already better than the null information structure. A finer information structure may or may not be Pareto superior to a coarser information structure. (iii) If there is a simultaneous and symmetric improvement in information for both firms, then the expected profit of each firm must increase whereas the expected consumer surplus must decline. Put it differently, when each firm gathers private information about its own cost, the firms gain benefit in exchanging the information with each other, which is certainly to the detriment of the consumer as the third party. (iv) When we turn to a class of cases in which information is not
VALUE
OF INFORMATION
53
symmetrically distributed between the two firms, some seemingly counterintuitive results might be obtained. First, better information on the part of a certain firm may not help that firm in achieving a greater expecte profit, so that additional information may be detrimental to the firm to be informed. Second, the effects of better information available to one firm on the welfare of the other firm, and also on the welfare of the consumer, can be ambiguous in sign: the latter two parties would be better off or worse off. Third, it may or may not be desirable for a firm to acquire isolation about its own cost rather than the rival’s, All of these possibilities depend on the values of the variances of the costs and their correlation coefficient. We need to emphasize that, as in any theoretical analysis, there is a subtle balance between the generality and complexity of the model and its tractability and lucidity. It is worth asking how robust our results are, or in what way they depend on the specificity of the present model. Throughout this paper, we have made the assumption that the cost functions of the firms and the market demand functions are all linear. It is true that assuming instead the nonlinearity of these functions could make our model more general. However, it would make our task of deriving the best decision functions, the expected profits and the expected consumer surplus that already requires fairly lengthy calculations, an even more demanding, if not impossible, one. Besides, although changing any of these linearity assumptions would change the precise conditions under which the “irregular results” mentioned above are obtained, the possibilities still remain. We have assumed that the probability distribution on (ci, c2) is a bivariate normal one. Clearly, this assumption is a less important one and can be weakened by replacing it with a family of probability distributions whose regression equations are linear. (The normal distribution represents just one member of such a family.) Furthermore, this would ensure that the domain of (cl, c2) can be restricted to a certain set in R: (e.g., the unit square), which cannot be the case with the normal distribution. There are other directions in which we can generalize our analysis on the role of information, Although our analysis has been confined to the interaction of just two firms, some of our results may easily be extended to the case of n firms, where y1is any finite integer. We have ignored the problem of risk aversion on the part of the producer and/or the consumer along with the cost of information. Important as all these considerations may well be, our conjecture is that introducing these factors into the model would enlarge rather than lessen the possibility that the “irregular results” may be obtained. As was noted by Green [3], while the valuation of information structure has been a topic of much concern in the theory of decision and games, the economic literature dealing with this problem is still fragmentary. There is
54
YASUHIRO SAKAI
considerable scope for further development of the market model presented here, and we hope this paper will serve to stimulate further research along the lines mentioned in this final section.
REFERENCES 1. T. BA~AR AND Y. Ho, Informational properties of the Nash solutions of two stochastic nonzero-sum games, J. Econ. Theory 7 (1974), 370-387. 2. T. F. BRESNAHAM, Duopoly models with consistent conjectures, Amer. Econ. Rev. 71 (1981), 934945. 3. J. GREEN, Value of information with sequential futures markets, Econometrica 49 (1981), 335-358. 4.
5. 6. 7. 8. 9. 10. 11. 12.
.I. C. HARSANYI, Games with incomplete information played by “Bayesian” players, Part I, Management Science 13 (1967), 159-182; Part II, 14 (1967), 320-334; Part III, 15 (1968), 486-501. P. LEVINE AND .I. P. PONSSARD,The values of information in some nonzero-sum games, Int. J. Game Theory 6 (1977), 221-229. J. MARSCHACK AND R. RADNER, “Economic Theory of Teams,” Yale Univ. Press, New Haven, Conn./London, 1972. A. OKADA, Informational exchange between duopolistic limos, J. Oper. Rex Sot. Japan 25 (1982), 58-76. J. P. PONSSARD,On the concept of the value of information in competitive situations, Managem. Sci. 25 (1979), 739-747. J. P. PONSSARD,The strategic role of information on demand function in an oligopolistic market, Managem. Sci. 25 (1979), 243-250. M. SHUBIK, “Market Structure and Behavior,” Harvard Univ. Press, Cambridge, Mass., 1980. M. SUZUKI, “Theory of Games,” Kyoritsu, Tokyo, 1981. D. M. G. NEWBERY AND J. E. STIGLITZ, Pareto inferior trade, Reo. Econ. Stud. 51 (1984) 1-12.
OF ECONOMIC
THEORY
36, 36-54
(1985)
The Value of Information in a Simple Duopoly Model YASUHIRO SAKAI* Institute of Social Sciences, University Sakura,
Niihari,
Ibaraki
of Tsukuba,
305, Japan
Received November 3, 1983; revised November 8, 1984
This paper is concerned with the evaluation of information in a duopoly model in which the cost functions are subject to uncertainty. It explores how changes in information about the costs available to either firm affects the welfare of both firms along with the welfare of consumers. By comparing the ten possible types of information structures, it is shown that information may be detrimental, that improved information for one firm may or may not benefit the other firm and/or the consumer, and that it may be more desirable for a firm to gather information about the rivals cost rather than its own. All of these “irregular” results depend on the values of the variances of the costs and their correlation coefficient. Journal of Economic Literature Classification Numbers: 022, 026, 611. 0 198s Academic press, I~C.
1. INTRODUCTION
The purpose of this paper is to investigate the value of additional information in a duopoly model in which the cost functions of the firms are not known with certainty. More specifically, it deals with the question of how changes in information about the costs available to either firm affect the welfare of both firms along with the welfare of consumers. As is well known in two-person zero-sum games, information is always valuable in the sense that better information on the part of one player tends to increase the average payoff of that player but it tends to decrease the one of the other player.’ When we enter the world of nonzero sum games, however, the situation changes drastically. In fact, it has been pointed out by Levine and Ponssard [IS] that the value of information may be detrimental in those games. The present paper focuses attention on the role of information in a simple, duopolistic market model under cost uncer*I am grateful to Ryuzo Sato, Mamoru Kaneko, Masahiro Okuno, and Yozo Ito for helpful discussions. I would like to acknowledge my special thanks to an anonymous referee of this journal whose comments have substantially contributed to improvement of this study. Any errors which may remain are of course my sole responsibility. ’ See Ponssard [S] and Suzuki [ 111.
36 0022-0531/85 $3.00 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
VALUE
OF INFORMATION
37
tainty, whose analysis has not been given the prominence which, I believe, it deserves. 2 Hopefully, it will bring out some new insights about the valuation of information structures in an important cIass of market equilibrium models. The duopoly market model we are going to work with is described as follows. There are two firms, firm 1 and firm 2, with identical outputs operating in a market where the demand function is assumed to be linear. We suppose that firm i has a linear cost function: ci(xi) = cixi, where xi denotes the output level of firm i (i = 1,2). Each firm knows the probability distribution of both cost parameters c1 and cZ. To be more specific, we assume that the joint distribution of c1 and c2 is normal with means ~1~and p2, variances G: and o:, and correlation p. Each firm aims to maximize its expected profit; so no risk aversion is taken into consideration. From a game theoretic point of view, our duopoly model can be regarded as a game with incomplete information introduced by Harsanyi [4J Given a certain information structure on the costs, the resulting equilibrium will clearly be a Nash-Cournot equilibrium. However, tbere are many types of information structures conceivable; nameily, each firm is faced with the following alternatives: (A) To determine the supply on the basis of its own knowledge of the probabilities of the costs c1 and c2. (B) To pay a market research agency, which we assume faultless, and which for possibly different fees, can tell (B’): the realized values of c1 only; or (B”): the realized values of c2 only. (C) To pay the agency for information about the realized values of both c1 and c?. There are several questions that might naturally arise at this point. What is the maximum expected profit each firm can gain under any combination of the four alternatives ((A), (B’), (B”), (C)), not counting the research fees? How much should each firm be willing to pay, at most, for each of the three research services, if it tries to maximize its expected profit? In t paper, we will almost exclusively be concerned with the first question, leavring the second one for further research. Our problem here is therefore to inquire how those variations in the information structure influence the best 2 The role of information on the demand side in the duopoly market was first studied by Basar and Ho [l] and Ponssard [9]. In such a simpler case, information must be beneficial if no risk aversion is taken into account. Although Okada [7] recently examined the role of information on the cost side he made the stringent assumption that each firm always knew the realized value of its own cost, and failed to discuss the consumer point of view.
38
YASUHIRO SAKAI
decision functions of the firms and their expected profits. Put it differently, we are especially interested in seeing whether an uninformed firm has an incentive to acquire information whereas an informed firm has an incentive not to share the information. In our duopoly market game, firms 1 and 2 are active players with output levels as their strategies. Although consumers are mere passive players with no strategic choice and no explicit utility function, it is worth examining how the welfare level of passive players as a whole depends upon the gathering of information by active players. When we regard the average value of the consumer surplus as a good measure of the welfare level of the consumer, another question of interest is whether and to what extent improved information on the part of the producer decreases the expected consumer surplus. In other words, when one firm gains benefit in unilaterally conveying information to the other firm, or when the two firms gain benefit in exchanging information with each other, can we say that such a transmission of information is against the interest of the consumer? The paper is organized as follows. Section 2 discusses our duopoly model and alternative information structures in greater detail. In Section 3, for any given information structure, we first determine the best decision functions and expected profits of the two firms along with the corresponding expected consumer surplus. We then scrutinize the role of information in the model by comparing various types of information structures in terms of the expected profits and consumer surplus that can be achieved through their use. Concluding remarks are made in Section 4.
2. THE BASIC MODEL In this paper we are concerned with the following simple, duopolistic market model. Firms 1 and 2 produce and supply the same homogeneous commodity; the problem of product differentiation is not taken into account. We assume that there is no limit of capacity at each firm. Denote by xi the output level of firm i (i = 1,2). The (marginal) cost function of firm i is assumed to be linear; i.e., it is provided by MC, = cixi
(i = 1, 2).
(1)
We suppose that the cost parameters cl and c2 are random variables whose joint distribution @(c,, c2) is known to both firms. Concerning the probability distribution on (cl, cZ), it is convenient to assume that it is a bivariate normal distribution with mean (pl, pLz)> 0, variance (o:, G’,) > 0, and the correlation coefficient p ( - 1 < p < 1). It is
VALUE
OF INFORMATION
39
then well known that the regression function of cr on c2 and the one of c1 on ci become linear; indeed we have
and
where E stands for the expectations operator. Clearly, the linearity of these regression equations will make our calculations fairly manageable. Given the price p of the output, the market demand function is assumed to be given by
p=a-b(x,
-lx,),
(4)
where the demand parameters a and b are nonstochastic and positive. Without the loss of generality, we may choose the units of measurement so that b equals unity. Letting Z7j denote the profit of firm i, we obtain Il,=(a-cj-xl-x,)xj
(i= 1, 2).
(51
Each firm aims at maximizing its expected profit; namely, it is assume have a risk neutral (i.e., linear) utility function with respect to profit. In order to facilitate the following analysis, we assume
In the present model, since each firm is confronted with uncertainty about its own cost and/or the rival’s, it ought to determine its optimal output level on the basis of its estimate of the cost(s). In order to formulate such an estimate precisely, it is necessary to decide what kind of information structure we are dealing with. Let us focus on the case in which the information structure of the duopoly model is fully described by a statement as to “who knows what.” It is then useful to represent the information structure as a matrix, q = [rik] (i= 1,2; k= 1,2), such that if firm i is kept informed of ck.
qik=l
=o
if firm i is not so informed.
This case does not exhaust all possible information structures; for instance, the notation just introduced would fail to state “who knows what with what precision.” However, it does include an important class of information structures. In what follows, to save the space, the 2 x 2 matrix [qik]
40
YASUHIRO
SAKAI
will be written as a row vector, q= [vi, r2] = [ylllr12, r~~i~&l, where vi= [qilriZ] represents the information signals that can be received by firm i (i = 1,2). Since qik takes on either 1 or 0, there are 24 = 16 information structures which may be classified into the following ten types: (i) q = [00, 001: Neither its own cost ci nor the rival’s c2 is known to firm 1, and similarly for firm 2. In short, no firm has information about either cost. (ii) q = [lo, 011: c1 but not c2 is known to firm 1 whereas cz but not c1 is known to firm 2. In other words, each firm gathers information about its own cost, but not the rival’s, through the market research agency. (iii) r~= [01, lo]: In contrast to (ii), each firm acquires information about the rival’s cost only. (iv) q= [ll, 111: Both firms can know c1 as well as c2. (v) q = [lo, 001 or [00, 011: One firm can know its own cost, but not the rival’s, whereas the other firm is ignorant of the costs. (vi) y = [Ol, 001 or [00, lo]: Contrasted with (v), one firm can know the rival’s cost only, with the other firm being ignorant of the costs. (vii) r = [ll, 001 or [00, 111: One firm is kept informed of both its own cost and the rival’s whereas the other firm is not informed of either cost. (viii) r~= [ 11,011 or [lo, 111: One firm can know both costs whereas the other firm can know its own cost only. (ix) y = [ll, lo] or [01, 1 I]: By contrast, although one Iirm can know both costs, the other firm can know the rival’s cost only. (x) y = [lo, lo] or [01, 011: Either one of the costs is commonly known to both firms. From the viewpoint of the distribution of information between the firms, we are able to think of types (it(iv) as the cases of symmetric information, and types (v)-(ix) as those of nonsymmetric information. And type (x) can be regarded as the case of shared information, meaning that whenever information about either cost is gathered by one firm, it is also shared by the other firm. Let C= Ci x C2 be the set of the pair (ci, cz). Then we can consider an information structure v] = [qr, Y/~] to be a partition of C2. We say of two given information structures, yeand y’, that y is finer than q’ if 7 is a subpartition of q’; that is, if every set in y is contained in some set in 13 fl. 3 For a formal Chap. 23.
discussion
on the ordering
by fineness,
see Marschack
and
Radner
[6,
VALUEOFINFORMATION
41
Given any one of the information structures, we are ready to define an equilibrium of the duopoly model a la Nash and Cournot. More exactly, the output pair (x,*(q), x:(q)) is said an equilibrium output pair under information structure 9 if
and for all x,(r);
Efl,(X?Y?)>xz*(TI))3 Efl*(X,*(Y), x2(r))
and EIIi(x,*(q), x;(q)) denotes the optimal expectedpayoff (profit) offirm i (i= 1,2). In other words, with the assumption that the firms are all risk neutral, firm i seeks to pick up a policy x,(y) so as to maximize its expected profit against some optimal policy of firm j (j f- i); which results in a NasbCournot equilibrium. When the market demand function is furnished by Eq. (4) and the equilibrium price under information structure q is denoted by p*(q), the consumer surplus is calculated as the area under the demand curve over p*(q), so that CS*(r) = +(a- P*(r))(xXr)
+ xi+(r)) = IMYt?) + xT(s)12.
67)
Throughout this paper, we regard the average value of this quantity, Z*(q) = E[CS’*(q)], as a good measure of the welfare level of consumers as a whole. 3. COMPARISON OF INFORMATION
STRUCTURES
As was seen in the last section, there are ten possible types of information structures. In this section, for any given information structure, we determine the best decision functions of the two firms and measure the resulting expected profits along with the corresponding expected consumer surplus. We then discuss the role of information in the duopoly market game by comparing various types of information structures in terms of the expected profits and consumer surplus that can be achieved through their use. A. Equilibria
under Alternative
Information
Structures
Let us start our investigation with the case of null information represented by q= [OO, 001; which serves as a basis of comparison with other information structures. Since firm i acquires no information about its own cost ci and the rival’s one cj, it seeks to maximize the expected profit E~i(x,)=E[(a-Ci-X,-X,)Xi]
(i = 1, 2).
($1
42
YASUHIRO
SAKAI
An equilibrium point (x:, x2) may be obtained derivative of Eq. (8) equal to zero: (i,j=
a-pi-2x*--xj*=O
by setting the first
1,2; i#j).
(9)
Solving these equations for xl* and x,*, we obtain two decision functions, each linear in ,ui and p2: ~300,~)
= +(a - 2~~ + fi2),
(10)
x:(00,00)
= $(a - 2pu2+ pL1).
(11)
Recalling that a > 2 max(y,, p2) (see Eq. (6)), x:(00,00) and x,*(00,00) are both positive, so that the positive solution is indeed guaranteed. In what follows, the equilibrium output pair (x:(00, 00), x,*(00,00)) under the information structure [00, 001 will simply be referred to as MYO), x2*(0)). In view of Eqs. (8), (9), and (lo), the maximum expected profit of lirm 1 is provided by Q:(O) z Ell,(x~(O))
= {xf(0)j2
= $(a- 2p1 + p2)*.
(12)
Similarly, the maximum expected profit of firm 2 is Q2,*(0)=$(a-2p2+pl)2.
(13)
And the (expected) consumer surplus under [00, 001 is given by C*(O) = CS*(O) = &(2a - pl - pL2)2
(14)
because of Eqs. (7), (lo), and (11). We turn our attention to the case [ 10, 011 in which each firm gathers private information about its own cost, but not the rivals. An equilibrium point for this case is expressed by the pair of decision functions (x:(c,), xz(c?)), where firm i chooses its optimal output level to be produced as a function of the true values of ci (i= 1,2). Since firm i maximizes its expected profit Ec,[IZiI ci] conditional OIZci, the first-order condition for such a conditional maximum results in a-ci-2x*(c,)-.E[x,*(cj)Ici]
=0
(i, j= 1, 2; i#j),
(15)
from which we find (16)
GYc2)= f(u - c2)- + g-m4
I4.
(17)
VALUE
OF INFORMATION
43
If we take the expectation over c2 conditional second equation in (15), we find
Substitution
on c1 of both sides of the
of this equation into the first equation in (15) yields
Now let
If we succeed in determining the values of the constants P and Q, it is clear under the present assumptions on the demand and cost functions that this x:(cl) represents the desired, equilibrium output level of firm 1 (see Okada [ 71). Taking the expectation over c1 conditional on c2 first and the expectation over c2 conditional on c1 next, we obtain from Eqs. (3) and (19)
Inserting Eq. (20) into Eq. (18) and making use of Eq. (2) it is not har find x:(4=,
1 po2-2c1.1 (
+ 0”
Gl
(Cl -!-%I :i
If we compare Eqs. (19) and (21), we must have
p= P2-201 a,(4-P2)
and
Q=
a--P,+h
3
By virtue of Eqs. (10) and (19) we thus obtain PO2 - 20, x:(10, 01) 3 XI*(C{) = xf(0) + a,(4-p’)
In a similar way, we can also derive
(cl--l).
.
44
YASUHJRO
SAKAI
We note that the equilibrium (Nash) expected profit of firm 1 conditional on ci is written as E,,[n: 1cl] = (x~(c~)}~ by making use of Eqs. (5) and (15). Therefore, in the light of Eq. (22), the optimal expected profit of firm 1 is (PO2 - 2d2 (4-p2)2
52:(10,01)-~~[17:Ic,]=s2:(0)+ A similar calculation firm 2:
(24)
will lead us to derive the optimal expected profit of
Q,*( 10,Ol) = g g[n:
1cz] = a;(o)
+ (PO, - 2d2 (4- P2J2
(25)
And it is not hard to prove by virtue of Eqs. (7), (22), and (23) that the expected consumer surplus under [ 10,Ol) is c*(lo,ol)
= c.c2[CS*(10, Ol)] = c*(o) +
1 2(4 - p2)2
[4(1 -P2M+4
+P2((~1-a2)2+2~1(T2(1+P)}1.
(26)
Concerning the remaining cases, we can similarly derive the best decision functions of the two firms and the resulting expected profits along with the corresponding expected consumer surpluses, and omit the proofs. Tables I, II, and III summarize the results for all sixteen information structures. B. Value of Information We are now in a position to measure the value of additional information, investigating the problem of how better information available to either firm affects the welfare of both firms along with the welfare of consumers. To begin with, it should be noticed that for any information structure ye, we have E[xF(r])] =x*(O) (i= 1,2) and E[p*(q)] = p*(O) in the light of Table I and Eq. (4). Therefore, for whatever information about the cost(s) either firm can get, the average output level of each firm in the NashCournot equilibrium remains the same, and so does the average market price. There exist many sequences of comparisons of the information structures that are worthy of investigation. First of all, we want to compare the case [OO,OO] of no information (for both firms) with any other information
VALUE OF INFORMATION TABLE I Optimal Decision Functions for Sixteen Information Information structure: 1
Structures
Optimal decision fzmction Firm 1: x:(q)
Firm 2: x;(q)
structure. In view of Tables II and III, it is straightforward following: THEOREM
1. For any information
hOid.
(4
QiW
2 GW.
(b)
Q:(q) a .n:(O).
Cc) C”(q) 2 c*(o). 642/36/l-4
structure
to obtain the
q, the following
relalions
46
YASUHIRO SARA1 TABLE II Maximum Expected Profits for Sixteen Information
Information structure: ?
!30,001 Q:(o)
Structures
Maximum expected profit Firm 2: Q:(q)
Firm 1: Q:(q)
QUO)
The economic implications of this theorem are quite clear. Let us begin with the case in which both firms are totally ignorant of the costs. Then any form of improvement of such null information tends to be beneficial to all firms and consumers.4 The fact that information is thus valuable comes from each firm’s better adjustment of quantity produced on cost, and this adjustment is also in the interest of consumers. Put it differently, the social value of information compared to the null information is always positive.5 4 This result is similar to the one of Ponssard [9] who studied the value of information on the demand. An analogous point was made by Green [3] for the framework of sequential future markets. 5 In passing, we note that 8:(10,01) -Q?(O) = (p (r2 - 2u,)*/(4 - p2)*, which measures the advantage that firm 1 can gain when each firm is kept informed of its own cost. How this advantage is affected by changes in the variances of the costs and their correlation coefficient can be studied through this equation. No doubt, there are many comparative statics analyses of this sort conceivable. Interesting as they may be, they are not pursued in the paper and will be left to the reader.
47
VALUE OF INFORMATION TABLE
III
Expected Consumer Surpluses for Sixteen Information Information structure: 4
Structures
Expected consumer surplus: C*(q)
z*(o) 1 [4(1-p*)(aT+o:)+pZ{(oi--az)Z+2a,O?(l+P)jl 2(4 - ,L+
z*(o) +------
cw 101
z*(o)+
c11>1t1 [ 10, 001 w, 011 COL001 PO, 101 Cl& 001 cm 111 [II, 011 [lo, 11J IIlL 101
c*(o)+~{(u1-u2)2+261u2(1 +p)j z*(o)+ +cTf z*(o)+ +u; c*(o)f g&J:
Z‘*(O)
+ib(P~*
+ ~I)*
[Ol, 111 [lo, lo] w, 011
E‘*(O)
+ ?&PC,
+ az)”
x*(o)
+ #PC2
+ fJ,Y
c*(o)
+ &PO1
+ %I2
-
pz 2(4-p’)’
z*(o)
+ ~p’c$
z*(o)
+ $uf
c*(o)
+ +;
[4(1 -pp’)(o:+(i;)+p*{(o,
-az)‘+za,u~jl+
p))]
~*(0)+k[5(1-pZ)~:+4{(0,-cr,)i+2~~cr,(l+p))] c*(o)+~[5(1-p~)~:+4{(0,-~*)*+25~~*(1+pj~]
The situation becomes more complicated when we consider improvements of an existing information structure that is already better than the null information structure [00, 001. A finer information structure may or may not be Pareto superior to a coarser information structure. To approach this problem, it is useful to distinguish the two classes of cases-a class of cases of simultaneous and symmetric variations in information on the part of both firms and another class comprising all the other cases sf nonsymmetric variations. We will first deal with the first class of cases. 2. (a) !L?,*(ll, ll)>Q:(lO, Qn,*(ll, ll)>Q,*(lO, 01). c*(ll, ll)
THEOREM
(b) (c)
01).
The proof of Theorem 2 is immediate from Tables 2 and 3 and is omitted. (The proofs of the following theorems will also be omitted unless otherwise stated.) The results in Theorem 2 can be interpreted as follows.
48
YASUHIRO
SAKAI
Suppose that there is a simultaneous and symmetric improvement of information available to both firms. Then such improved information increases the expected profit of each firm, agreeing with the well-known result of one-person decision problems that making the information structure liner tends to increase the expected payoff.6 On the other hand, it results in a decrease in the expected consumer surplus. Put it differently, when each firm gathers private information about its own cost, the firms gain benefit in exchanging the information with each other; which is certainly to the detriment of the third party (namely, the consumer in the present situation). As a matter of fact, this is a rather common practice in many industries in which anonymous data are redistributed to all the firms through a syndicate, which yields a welfare loss on the part of the consumer. 7 Remark 1. The quantity (sZT( 11, 11) -QT(lO, 01)) and the one (L’*( 10,Ol) - C*(ll, ll)), respectively, measure firm i’s welfare gain and consumers’ loss due to the exchange of private information between the two firms. In the special case of equal variances (i.e., rrl = 02), it can easily be shown that this welfare gain is the larger (and so is the welfare loss), the larger is the common variance. Remark 2. It is of interest to see that
Q?(lO, 01) > s2:(01,10)
and
Q?(lO, 01) 3 Q,*(Ol, lo),
where the first equality holds if and only if p = 20,/(r,, and the second equality if and only if p = 2a,/o i . Although no comparison of fineness can be made between the structures [lo, 011 and [01, lo], it follows from these inequalities that for both firms, [ 10, 011 is more valuable than [01, 10); namely, it is, on average, more profitable to both firms for each of them to know its own cost rather than the rival’s’ We now turn to the second class of cases in which information is not symmetrically distributed between the two firms. Then we will see that the relationship between changes in the information structure and those in the welfare of the producers and consumers become much more intricate than in the first class of cases of symmetric information. 6 See Marschack and Radner [6, Part One]. ’ I am indebted to the referee for this point. *When the value of p* is near unity, we obtain .Q:(ll, (i= 1,2) and C*(ll, 11) i Z*(lO, 01) -j, Z*(Ol, lo), where to.” This is as it should be. Since near perfect correlation variable almost exactly predictable from the other, it does the consumer which one of the costs is commonly known be made for similar comparisons of information structures.
11) + Q:(lO, 01) + Q:(Ol, 10) “i” means “approximately equal (positive or negative) makes each not matter much to each firm and to the firms. The same remark will
VALUE
4
OF INFORMATION
From the viewpoint of game theory, our doupoly model can be regarde as a special sort of two-person nonzero-sum games. As is well known, zerosum games and team problems are two extreme cases of nonzero-sum games. The equilibrium solution to any zero-sum games has the property that, if a change in the information available to one player increases the optimal average payoff of one of the players, it definitely decreases the optimal average payoff of the other player; i.e., the payoffs of the players must go in opposite directions. On the other hand, the equilibrium solution to any team problem has the property that, if a variation in the information on the part of one player increases the optimal average payoff of one of the players, it also increases the optimal average payoff of the other player; namely, these two payoffs must go in the same direction. It is naturally conjectured that the duopoly model has both the zero-sum property and the team property; for it is neither an antagonistic zero-sum game nor a completely cooperative game.’ In order to develop an economic background of the interpretation of the results of the following theorems, three cases should be distinguished-independent costs, positively correlated costs, and negatively correlated costs. If the costs are positively correlated, the extent of the conflict of interests between duopolists is large, and may sometimes be so large that the zero-sum property outweighs the team property. Accordingly, it may be the best to be the only one to be informed, for in this case we may have a monopoly rent on information. And it is expected that such a rent declines with the number of firms informed. If the costs are negatively correlated, however, there is much room for cooperation between the duopolists. If so, the team property may dominate the zero-sum property, The result is that a more informed tirm may have a strong incentive to convey the information to a less informed firm, presumably at the expense of the consumer. We are ready to establish a sequence of theorems as to the cases of nonsymmetrical information. THEOREM
3.
(a)
L?,*(lO, lO)$Q:(lO,
00) according as (p-~(o,/cs,))
(P - %%/~,))i3
(b)
sZ;(lO, 10) aQ,*(lO,
00), where the equality holds q and onlv $
P = 5(~1/~2).
(6)
Z:*(lO, lO)$Z:*(lO,
00)
according
as
bP- l(a,la*))
(P + 4(~1/~2))$0.
The starting point of discussion is the case in which firm 1 has private information about its own cost whereas firm 2 is ignorant of the costs. One 9 For a discussion on the zero-sum and team properties of nonzero-sum and Ho [l].
games,
see Basaf
50
YASUHIRO
SAKAI
natural question to ask is whether the informed firm (i.e., firm 1) has an incentive to unilaterally convey the information to the uninformed firm (i.e., firm 2). Another question of interest is whether the sharing of the information by the two firms is beneficial to the third party (i.e., the consumer). The results in Theorem 3 teach us that definite answers cannot be given to these two questions; they are actually dependent on the values of 01, (72, and p. Let us limit our attention to the case of equal variances (namely, CJ~= CQ). Then we can easily see that when the correlation coefficient is large enough to exceed a half, the zero-sum property outweighs the team property; so firm 1 has no incentive to share the information with firm 2 although such transmission of the information would benefit firm 2 as well as the consumer. When the correlation coefficient is negative or zero, or when it is less than a half if positive, however, the team property plays such a dominant role that firm 1 is motivated to convey the information to the rival; and the consumer would then be worse off. 4. (a) (7(a,lo,)2+4)/16(0,/0,). THEOREM
(b) (c)
Q:(ll, c*(ll,
Q:(ll,
ll)>Q;(ll, ll)~X*(ll,
11) $ Q~(ll,OO)
according
00). 00) according as p$ (5(a,/o,)’
as
PZ
- 4)/8(o,/a,).
Theorem 4 demonstrates that when the value of correlation coefficient is small enough, the transmission of information from the informed firm (i.e., firm 1) to the uninformed firm (i.e., firm 2) may be beneficial to both lirms. This is particularly so when the costs are independent or negatively correlated (namely, p d 0). In the special case of equal variances, the results can be interpreted more easily and are summarized as follows. When the value of the correlation coefficient is large enough (more exactly, p > $), firm 1 has no incentive at all to share its information with firm 2 since the zero-sum property outweighs the team property. On the other hand, when p is nonpositive or when p is small enough if positive (more precisely, p
hard
to show
that
Therefore, when firm 1 acquires complete information about the costs, firm more to inform its rival of both costs rather than only one of them. that the team property dominates the zero-sum property in the present
it would benefit that This is another proof situation.
51
VALUE OF INFORMATION
Remark 3. Ponssard
[9] has discussed the role of information in a duopoly model similar to ours when the demand side is subject to uncertainty. More specifically, the demand parameter a is a random variabie in his model in contrast to our model, where the cost parameters c1 and c2 are random variables. From a mathematical, yet not economical, point of view, the Ponssard situation may be thought of as a very special case with p + 1 and or = CJ~within our general framework. In all of the cases discussed above, there has been a positive incentive for every firm to acquire more information. However, the following theorem will show that this may not necessarily be the case. THEOREM
5.
(a)
Q,*(lO, 11)>sZ~(lO,O1),
where the equality holds if
and only if p = 2(0,/o,).
(b)
Q,*(lO, ll)$L?~(lO,
(2 - P(~ll%))2z7
01)
according
as
(4(7 - p2)/(4 - p*)“f
- 4(02)*.
To get a good interpretation of the results in Theorem 5, let us focus on the special case of equal variances. The starting point of comparison is now the case of private information: each firm is kept informed of its own cost. (By contrast, the previous cases were concerned with those where one of the two firms was totally ignorant.) In this case, each firm has a strong incentive to unilaterally convey the information to its rival, regardless of the value of p; so one firm (say, firm 1) can derive an extra benefit from moving first and taking the role of a “leader” in informational transmission. In fact, when the costs are positively correlated and this correlation is fairly large (more exactly, p > (2 $8 - 6)/7, such an initiation benefit of firm 1 is so overwhelming that even the welfare of firm 2, i.e., the firm receiving the information, would be lower. This shows the distinct possibility that more information may even be harmful to the player to be informed in our duopoly game. I1 Remark 4.
It is worth noting that Q,*(ll,
ll)>Q;(lo,
11) Q$.(lO,Ol).
So when firm 1 takes the initiative of conveying private information about c1 to firm 2, the best reaction the latter firm can take is to convey back its own information about c2 to the former so that the transmission of information between the two firms may become “two-way.” Finally, let us consider the quantity (sZf( 10, q2) - Q:(Ol, q2)). This measures the advantage (if positive) for firm 1 when that firm knows c, rather than c2, with the rival remaining to have the same information ~1~. ‘I For this possibility, see Levine and Ponssard [S].
52
YASUHIROSAKAI THEOREM 6.
(b)
(a) Q:(lO, 00) > 52:(01, 00). Q:(lO, Ol)$Q:(Ol, 01) according as
Cl- w71/~,))* + $, (c) Q$(lO, ll)$Qf(Ol,
(GJQ)~ $ ((7 - p2)/36)
11) according as a1/cr2$Z~.
Let us begin our discussion with the simplest situation that firm 2 has no information at all. Then the result agrees with common sense: it is definitely more profitable to firm 1 to be informed of its own cost rather than the rival’s (see property (a)). When firm 2 acquires some information, however, the situation becomes more complex and the “common sense” result does not always hold. This is because firm 1 is now capable of taking advantage of the information on the rival unless the latter has nothing to tell about. In fact, we note that G?~(Ol,Ol)>Q~(lO,Ol) if p=O and crJa, > 5 and that Q,*(Ol, 11) > sZ:( 10, 11) if cz/ol > 2 (see properties (b) and (c)). Therefore, when the variance of the rival’s cost is large enough relative to that of its own cost, using information about the former cost may be more valuable to firm 1 than using information about the latter. We see once more the delicate role played by information in the duopoly market model.
4. CONCLUDING REMARKS The duopoly model we have discussed in this paper is, admittedly, quite elementary. The model was selected as the simplest one which would enable us to derive the following results: (i) Compared with the case of no information for the two firms, any form of improvement in information structure tends to be beneficial to all firms and consumers; therefore, the social value of information is always positive in this sense. (ii) The situation becomes much .more complex, however, when we consider improvements of an existing information structure that is already better than the null information structure. A finer information structure may or may not be Pareto superior to a coarser information structure. (iii) If there is a simultaneous and symmetric improvement in information for both firms, then the expected profit of each firm must increase whereas the expected consumer surplus must decline. Put it differently, when each firm gathers private information about its own cost, the firms gain benefit in exchanging the information with each other, which is certainly to the detriment of the consumer as the third party. (iv) When we turn to a class of cases in which information is not
VALUE
OF INFORMATION
53
symmetrically distributed between the two firms, some seemingly counterintuitive results might be obtained. First, better information on the part of a certain firm may not help that firm in achieving a greater expecte profit, so that additional information may be detrimental to the firm to be informed. Second, the effects of better information available to one firm on the welfare of the other firm, and also on the welfare of the consumer, can be ambiguous in sign: the latter two parties would be better off or worse off. Third, it may or may not be desirable for a firm to acquire isolation about its own cost rather than the rival’s, All of these possibilities depend on the values of the variances of the costs and their correlation coefficient. We need to emphasize that, as in any theoretical analysis, there is a subtle balance between the generality and complexity of the model and its tractability and lucidity. It is worth asking how robust our results are, or in what way they depend on the specificity of the present model. Throughout this paper, we have made the assumption that the cost functions of the firms and the market demand functions are all linear. It is true that assuming instead the nonlinearity of these functions could make our model more general. However, it would make our task of deriving the best decision functions, the expected profits and the expected consumer surplus that already requires fairly lengthy calculations, an even more demanding, if not impossible, one. Besides, although changing any of these linearity assumptions would change the precise conditions under which the “irregular results” mentioned above are obtained, the possibilities still remain. We have assumed that the probability distribution on (ci, c2) is a bivariate normal one. Clearly, this assumption is a less important one and can be weakened by replacing it with a family of probability distributions whose regression equations are linear. (The normal distribution represents just one member of such a family.) Furthermore, this would ensure that the domain of (cl, c2) can be restricted to a certain set in R: (e.g., the unit square), which cannot be the case with the normal distribution. There are other directions in which we can generalize our analysis on the role of information, Although our analysis has been confined to the interaction of just two firms, some of our results may easily be extended to the case of n firms, where y1is any finite integer. We have ignored the problem of risk aversion on the part of the producer and/or the consumer along with the cost of information. Important as all these considerations may well be, our conjecture is that introducing these factors into the model would enlarge rather than lessen the possibility that the “irregular results” may be obtained. As was noted by Green [3], while the valuation of information structure has been a topic of much concern in the theory of decision and games, the economic literature dealing with this problem is still fragmentary. There is
54
YASUHIRO SAKAI
considerable scope for further development of the market model presented here, and we hope this paper will serve to stimulate further research along the lines mentioned in this final section.
REFERENCES 1. T. BA~AR AND Y. Ho, Informational properties of the Nash solutions of two stochastic nonzero-sum games, J. Econ. Theory 7 (1974), 370-387. 2. T. F. BRESNAHAM, Duopoly models with consistent conjectures, Amer. Econ. Rev. 71 (1981), 934945. 3. J. GREEN, Value of information with sequential futures markets, Econometrica 49 (1981), 335-358. 4.
5. 6. 7. 8. 9. 10. 11. 12.
.I. C. HARSANYI, Games with incomplete information played by “Bayesian” players, Part I, Management Science 13 (1967), 159-182; Part II, 14 (1967), 320-334; Part III, 15 (1968), 486-501. P. LEVINE AND .I. P. PONSSARD,The values of information in some nonzero-sum games, Int. J. Game Theory 6 (1977), 221-229. J. MARSCHACK AND R. RADNER, “Economic Theory of Teams,” Yale Univ. Press, New Haven, Conn./London, 1972. A. OKADA, Informational exchange between duopolistic limos, J. Oper. Rex Sot. Japan 25 (1982), 58-76. J. P. PONSSARD,On the concept of the value of information in competitive situations, Managem. Sci. 25 (1979), 739-747. J. P. PONSSARD,The strategic role of information on demand function in an oligopolistic market, Managem. Sci. 25 (1979), 243-250. M. SHUBIK, “Market Structure and Behavior,” Harvard Univ. Press, Cambridge, Mass., 1980. M. SUZUKI, “Theory of Games,” Kyoritsu, Tokyo, 1981. D. M. G. NEWBERY AND J. E. STIGLITZ, Pareto inferior trade, Reo. Econ. Stud. 51 (1984) 1-12.