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The value of information in competitive bidding
322
European Journal of Operational Research 36 (1988) 322-333 North-Holland
Theory and Methodology
The value of information in competitive bidding Kalyan CHATTERJEE
a n d T e r r y P. H A R R I S O N
Division of Management Science, The Pennsylvania State University, College of Business Administration, University Park, PA 16802, USA
Abstract: We consider the situation where two players compete to obtain a valuable object, e.g. a stand of timber in a competitive, sealed-bid environment. Prior to submitting a bid, each player may sample the stand while incurring a common, non-zero cost for each observation. On one hand, a player wishes to take as few observations as possible due to the cost of collecting information. However, one also wishes to obtain as m a n y observations as possible to avoid a bid that overstates the value of the resource. Given different assumptions on the sampling process, the informational structure, and underlying distribution of value, we derive equilibrium bidding strategies. We use these bidding strategies to solve for equilibrium in an information collection problem from the forest products industry. Keywords: Bidding, value of information, decision theory, forestry
1. Introduction The motivation for this problem is provided by the multi-billion dollar timber industry (c.f.U.S. Department of Commerce, 1983; Pennsylvania Department of Environmental Resources 1984). Tracts (stands) of timber are frequently auctioned through a sealed, competitive bid. Prior to the auction, potential bidders are permitted to collect information about the tract. Currently, m a n y bidders follow some rule-of-thumb in deciding how much information to c o l l e c t - - f o r example, sampiing ten percent of the cells in the grid into which the tract is divided. Based on these sample observations, along with an expectation of what competitors may do, a bid is submitted. However it is costly to collect these samples, since every tree in the sampling area must be measured as to diameter, height, species, and quality. Therefore there are conflicting objectives in this bidding
Received April 1987; revised August 1987
scenario. On one hand, it is desirable to collect as few sample observations as possible to avoid the costs of sampling. On the other hand, one wishes to take m a n y sample observations so that the estimate of the timber's volume (and hence value) is as accurate as possible, thus avoiding winning the bid by being the bidder who must over-estimated the value of the stand. (This danger, known as the 'winner's curse', is documented in Capen, Clapp and Campbell (1971).) In determining the value of the stand, it is likely that each bidder has some firm-specific considerations that affect the value of the tract to the bidder. For example, a large pulp and paper mill that is facing a potential stockout of raw material for a billion dollar processing complex is likely to place a premium on obtaining a successful bid. Other examples may include the ability of one bidder to produce higher value products from the same material, the ability of one bidder to fell the trees and transport them for processing in a more economical fashion, etc. While these firm-specific considerations undoubtedly exist, we have sim-
0377-2217/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
plified the analysis by dealing exclusively with a 'common value' model. In doing so, we are restricting our attention to bidders of the same type - - s a y all pulp and paper mills within the same region, and with roughly the same inventory situat i o n - i n order to focus on the informational issues. In some sense, the industry has implicitly adopted a common value model, as evident from the regular publication of unit average prices of timber in a given area. The case where the signals (or estimates of value) are independent of each other is often called the 'independent values' model. While there has been prior empirical work on predicting a competitor's bid (Lederer, 1983), we are unaware of any work dealing explicitly with the information collection issue in sealed bid auctions for stands of timber; and of little research addressing this question in general. (We will discuss the exceptions shortly.) For example, in examining Lederer's empirical study, we observe that forest products firms frequently do not consider their opponents' actions when formulating their information collection strategy. Clearly, those firms that do adopt an explicitly competitive viewpoint will do better than those who do not, because they will be able to optimize against their opponents predicted action. Analysis of strategic behaviour in auctions is a natural arena for the use of game theory, and this paper follows a long tradition in using this method. (See the survey by Engelbrecht-Wiggans, Shubik and Stark (1983).) The crucial game-theoretic concept we use is that of non-cooperative equilibrium. This consists of a pair of strategies, one for each player, each of which is a best response to the other. Using this concept enables us to resolve the difficult question of what Player 1 expects Player 2 to do, since this depends on what Player 2 expects Player 1 expects Player 2 to do, and so on. If Player 1 expects Player 2 to play his or her equilibrium strategy (assume there is only one for the moment), Player 2 will expect Player 1 to use Player l's equilibrium strategy and the pair of expectations will be consistent with optimizing behaviour by both players. Earlier decision theoretic models assumed optimizing behaviour by only one player while the other players were treated as part of 'Nature'. There has, however, been a substantial literature in bidding using game theory.
323
Most of the work in this area (see Ortega-Reichert, 1968; Milgrom and Weber, 1982a, b, c; Rothkopf, 1969; Winkler and Brooks, 1980, Wilson, 1974; Weverbergh, 1979, among others) has concentrated on calculating equilibrium strategies in a fixed informational setting. The analysis has assumed asymmetric (nested) information so that one bidder knows everything the other knows and has additional information; or symmetric differential information in which each bidder receives a private signal and the signals are 'equally accurate' in some sense or they are independent of each other. Milgrom and Weber (1982c) have considered comparative statics on the information structure, and have obtained results on the preferences of a bidder over various states of his own and his opponent's information. This paper adopts a somewhat different approach in seeking to identify the information structure endogeneously in equilibrium. Lee (1982, 1984), Matthews (1984) and Hausch (1986) have also examined endogenously derivable information collection strategies. All of these authors consider, as we do, a two-stage game in which information collection occurs in the first stage and bidding takes place in the second. These authors use the technique of backward induction in solving for the equilibrium strategies in each stage. Matthews allows for a continuum of possible amounts of information but is unable to obtain a complete analysis of the problem. Lee considers the information structure where a perfectly informative experiment is available at a cost. Hausch permits experimenters to obtain one signal which could be either 'high' or 'low'. The signal's degree of reliability is chosen in the first stage by each player. Hausch then discusses questions of auction design. In Section 2, we discuss this basic two-stage model, adapting Lee's (1984) model slightly to illuminate the essential difference between observable information collection and secret possession of information. Results similar to Lee's are re-derived for completeness. In Sections 3 and 4, we examine a more realistic model in which bidders can choose to sample (at a cost) as many times as they wish from a Normal process prior to making their bid. The sample size is fixed in advance and information collection is observable. In Section 3 we calculate
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K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
the equilibrium bidding strategies based on the assumption of a N o r m a l sampling process with a Normal prior of very large variance. In Section 4 we use these results to numerically compute equilibrium strategies for information collection.
2. C o n c e p t u a l
models of information
UU:
Both bidders use U in the first stage, so both are uninformed. The common expected value is 6, and the equilibrium bid is fi, so the expected payoff is zero for both bidders.
II:
Both bidders are informed of the value v of the object. The equilibrium bid in the second stage is to bid the common value v. The expected profit from bidding is once again zero for either bidder, but now each bidder has paid c > 0 for the information. The payoffs are therefore ( - c , - c ) to bidders 1 and 2 respectively.
collection
2.1. Introduction
In this section, we present two models similar to Lee (1982, 1984) adapted to fit our context in which the first stage is the information collection stage, and the second is the bidding stage. We assume that within each stage players move simultaneously, and that the auction procedure is sealed bidding with the highest bidder winning. The difference between the two models is that, in the first, the cost of sampling is the same for the two bidders and is commonly known, with the activity of information collection (but not the outcome) observable. The Normal sampling model of Sections 3 and 4 shares these characteristics with our first model. This section, therefore, provides some intuition about the more complicated structure of the Normal model. The second model that we consider here has private information about costs of sampling and unobservable information collection. This model provides an indication of what more general results might be anticipated in such an informational context and is, therefore, a pointer to future research in this area.
2.2. A model with observable information collection and common cost
This is a two-stage game. In stage 1, each bidder decides whether to pay c and learn the value v of 5 (strategy I) or to forgo the option and bid as an uninformed player (strategy U). (Note that the unknown value is being treated as a r a n d o m variable 5 of which v is a particular realization.) The act of becoming informed is observable. In the second stage, bidder i announces a bid b~ for the object. The object is sold to the highest bidder at the price given by his bid. There are four possible second-stage bidding games that could arise as a result of different first-stage strategies. These are:
U I and I U : I n the asymmetric case, when an informed player is bidding against an uninformed player, the expected profit of the uninformed player is 0. (See Engelbrecht-Wiggans, Milgrom and Weber, 1983.) The expected profit of the informed player is f0°°(1 - F ( v ) ) F ( v )
d v - c.
(See Milgrom and Weber, 1982c, Theorem 2), where F ( - ) is the probability distribution of 5. Let fo°°(1 - F ( v ) ) F ( v ) do = v o. We can therefore set up the matrix of payoffs shown in Table 1. If the cost c of being fully informed is greater than v0, neither bidder will choose to be informed. Otherwise there are two asymmetric pure strategy equilibria and a symmetric mixed strategy equilibrium with each bidder choosing to be informed with probability ( 1 - (C/Vo)). This gives each an expected payoff of 0. Note that the bidders are the worst off when both are informed. This outcome (I, I) will occur with probability ( 1 - ( C / V o ) ) 2, which may be reasonably high if (C/Vo) is relatively small. Intuition suggests that the bidders
Table 1 Matrix of payoffs
Bidder 1
Bidder 2 u
I
U (0, 0)
(0, v0 - c)
I (% - c, 0)
( - c, - c)
K. Chatterjee, T.P. Harrison / T h e value of information in competitive bidding
325
may not gain from information collection. A similar set of equilibria is derived for the N o r m a l sampling model. However under N o r m a l sampling the players would be better off with the mixed strategies than with the average of the pure strategies.
event with probability zero in equilibrium) with a bid of 0 when the value is 1, this implies that the expected profit of the uninformed bidder is zero in equilibrium. Therefore, for the uninformed bidder, a bid b realizes an expected profit of
2.3. Unobservable information collection with differential costs
- ½b + ½ ( 1 -
In this section we consider a model with differential information about the costs of being informed. We assume that the value, v, of the tract is either 0 or 1 with equal probability, and that the cost, c, of being informed is distributed uniformly from 0 to 0.25. (These assumed values could, of course, be different without affecting the analysis.) In the first stage, the game proceeds as before, with each player drawing a cost from the commonly-known distribution above. The actual value of the cost is private information and the draws are independent of each other. Based on this draw, each of bidders 1 and 2 decides whether or not to be fully informed. The act of collecting information is unobservable--this is 'secret information' in Ponssard's (1982) terminology. In the second stage, players bid given their informationcollection decisions of the first stage. (Note that the informational assumptions here are different from those of the other models in this paper.) We now consider the second-stage bidding problem for both the asymmetric and the symmetric information structures. First, we discuss the case where Player 1 is informed about the value of v and Player 2 is not. The two-point probability distribution of v is, however, c o m m o n knowledge. Let in informed bidder use a mixed strategy Gx(.) over bids when the value is 1 and bid 0 when the value is zero. Similarly, suppose the uninformed bidder uses a mixed strategy G v (.). In equilibrium, Gu(. ) will be selected to make the informed player indifferent among all of the bids in the support of G I ( ' ) . Using this we can determine that Gu(b)
1
2(1 - b ) "
At b = 0, G U (b) = ½. Therefore, in equilibrium, a mass of ½ is placed at 0 and 0 is in the support of G~ (.). Since the uninformed bidder will never win (except if the informed bidder also bids 0, an
b)G,(b)
= O,
or
G,(b) = b / ( 1 - b ) . Note that G i ( b ) = 1 when b = ½ and is 0 when b = 0. This completes the equilibrium analysis for the asymmetric case. The symmetric case proves to be more complex. Suppose now that there is a probability, r, that a player is informed about the true value of v and a probability (1 - r ) that the player is uninformed. We look again for mixed strategies. Let G u ( - ) be the mixed strategy of the uninformed player and G I ( - ) that of the informed player when v = 1. Observe that when v = 0, the informed player will bid 0. Lee (1984) shows that there cannot be mixed strategies G I ( - ) and G U (.) whose supports intersect in a nonempty interior. We can also show that the informed bidder will not use a pure strategy in a symmetric equilibrium. Suppose not, so that an informed bidder bids x > 0 with probability 1 when v = 1. If his opponent is also informed and the equilibrium is symmetric, the opponent should bid x as well. But the opponent will be better off bidding x + ~, since by doing so he or she will capture an additional r/2 probability of winning while incurring only a small decrease in the amount of winning profits. The only symmetric equilibrium therefore could be 1. But this too is not an equilibrium. If player 1 bids 1 when informed, an informed player 2 is better off bidding b < 1 and obtaining a positive expected profit of (1 - b)(1 - r)Gu(b ). For similar reasons, a pure strategy will not be used in equilibrium by the uninformed player. If an equilibrium exists, it must therefore be in mixed strategies G u ( . ) and G I ( ' ) with non-overlapping supports (except at boundary points). It can be shown that Gu(b)
r b ( 1 - r ) 1 - 2b
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
326
and GI(bll)
1 1-r r 2-r
1 1-b
1-r r
constitute such an equilibrium. The support of G u ( ' ) and G I ( ' I 1 ) are [0, ( 1 - r ) / ( 2 - r ) ] and [(1 - r ) / ( 2 - r), (2 - r ) - 1 ] respectively. We now consider the information collection game of the first stage. Recall that the cost of being fully informed is uniformly distributed between 0 and 0.25. Under what circumstances would a player choose to be informed? If Player 2 can commit himself to being informed whatever the cost, it is clear that Player l ' s best option is to be uninformed whatever the cost. If Player 1 is uninformed Player 2 clearly should be informed. The asymmetric information structure therefore remains an equilibrium. A more interesting case would arise if players adopted the decision r u l e - - o b t a i n the information if c~< c' and remain uninformed if c > c'. If such a rule were adopted by both players the probability of facing an informed player at the second stage would be r = 4c'. The expected value to an informed Player 1 would be ½((1 - 4 c ' ) / ( 2 - 4 c ' ) } . The expected value less the cutoff cost c' should equal zero. Solving for c' yields approximately 0.15. Therefore the probability that a player will be an informed bidder is approximately 0.60. If H a y e r 2 adopts this decision rule, H a y e r l ' s expected payoff from being informed is ½{ (1 - 4 c ' ) / ( 2 - 4 c ' ) }. This is equal to the cutoff value c' that makes the player indifferent between becoming informed and not; therefore so long as c < c', it is optimal to be informed. Therefore this is an equilibrium rule in the sense of being optimal against the other person's strategy. Notice that the uninformed player is not harmed by the secrecy of the information gathering. The uninformed player is, however, much more cautious than in the asymmetric case. The informed player is substantially hurt in the symmetric case, with an expected payoff of 0.15 instead of 0.25. By similar reasoning, the seller prefers r to be as high as possible. This implies that it m a y be in the best interests of the seller to subsidize information
collection, thus moving the cost distribution to the left and encouraging information collection. This is similar to results obtained in a completely different model by Reece (1978).
3. Equilibrium bidding strategies in a normal model In this section, we present a partial analysis of information collection in a N o r m a l sampling process. The bidding scenario is the same as before. A tract of timber of unknown value v is being put up for auction by first-price, sealed bid. There are two bidders who share a common probability distribution on v. Each bidder i can also obtain a signal s i that gives partial information about the value v. The s i are independently drawn from Normal distributions F~(.) with mean v and precisions h i for i = 1, 2, respectively. The value v is distributed Normally with mean/~0 and precision h', where h ' is much smaller than the h i. For our analysis, we shall assume that the relative magnitude of h' is equivalent to the requirement that the prior be diffuse. We recall that precision in a decision-theoretic context refers to the reciprocal of the corresponding variance. Note that the s i are conditionally independent given v, but are not unconditionally independent. This bidding scenario corresponds to the second stage of the two-stage games we have been considering so far. The first stage corresponds to a decision by each player on the ' a m o u n t of information' to collect; the Normal process assumption facilitates the interpretation of the precision h i as a measure of the amount of (costly) information player i has chosen to collect in the first stage. In the sequel, we shall assume that h i is equal to nih, where F/i is the number of observations taken by Player i and h is the process precision. We assume that each observation costs a fixed amount, c, and that this cost is the same for each player and all observations. We first calculate the bidding strategies where h 1 and h 2 are not necessarily equal. Wilson (1974) has obtained results for the condition h 1 = h 2. Allowing for inequality is important because the precisions are chosen in the first stage by bidders who are free to choose differently. In our calculation, we assume that the seller's reservation price for the stand is zero and that
K. Chatterjee, TP. Harrison / The value of information in competitive bidding bids are zero if the value is positive but the bidding strategy dictates a negative bid. (The parameters of the distribution are chosen such that bids of less than zero are extremely unlikely to occur.) Suppose Player i observes a signal s~ and that Player 2 uses a strategy (mapping signals into bids) given by B2(" ). We write b 2 = B 2 ( s 2 ) . This function is assumed strictly monotonic and twice differentiable. Given these assumptions and one we make later about a high prior variance, we can show (see Appendix 1) that the equilibrium strategies are
Then expected payoff then becomes
1 E(P11sl)
=
±],j2
~-1 + h2l
X
"~
-~
+
-- 2 V ~
ha~-h2
.
The precision, h i , is obtained by taking n 1 observations from the sampling process of precision h. Similarly, h 2 = n 2h. The net expected payoff for Player 1, including sampling costs, for a given n 2 is
[n,+n21'J2[1
tzV2
b, = s i -
327
, n2 ]
--cna,
.
In order to derive information collection strategies, we calculate the equilibrium expected payoff to Player 1 in the diffuse prior case. Let this expected payoff be given by
where c is the cost per observation. In the next section we investigate the number of observations each player will take in equilibrium.
4. Information collection strategies 4.1. I n t r o d u c t i o n
E(P "f2(g21sa,
ha, h 2 ) d s 2 .
Substituting in the equilibrium strategies, we obtain (with k = ( ( ~ r / 2 ) ( 1 / h a + l / h 2 ) ),
E(Plsa)
-A(~21s,, h,, h2) ds2 ]12
[SlFN.((Sl_Sa)~/h
hI + h2 -
- -
k +~,
where FN.(- ) and f N * ( ' ) a r e standardized Normal distribution and density functions, respectively, and the standard formula for the incomplete expectation is used (see, for example, Raiffa and Schlaifer, 1961, p. 232). There is an identical expression, with the subscripts reversed, for Player 2's net expected payoff. The quantity h-a _ h ~-1 + h-1 2 •
In the previous section we calculated equilibrium bidding strategies and expected payoffs for players who have, with common knowledge, sampled n 1 and n 2 times from a Normal process. We also obtained the expected equilibrium payoff as a function of n I and n 2. The equilibrium pair of information choices (n l, n2) in the first stage of the game appears to be difficult to resolve explicitly. In this section we have calculated the equilibria for a numerical example, and have obtained explicit results. In these calculations we have used the diffuse prior (i.e. low precision prior) approximation used in the previous section. We note that the equilibrium bidding strategies would be asymmetric and nonlinear with a non-diffuse prior and would therefore be difficult to characterize analytically. We are aware of the problems that may arise using diffuse priors in preposterior analysis, but believe that they do not arise here. The approach we have used for this numerical example is as follows. We generate a list of payoffs for Player 1 for different combinations of na and n z. The method for determining these payoffs is discussed in Section 4.3.
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K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
4.2. Numerical example and simulation approach A value, 6, is drawn from a Normal distribution with mean $200000 and standard deviation $50 000. (The prior precision is therefore 4 × 10-10, that is, {1/500002 }.) Sample observations are drawn from a Normal distribution with mean v and standard deviation $12500. (The sampling precision is therefore 6.4 × 10-9, an order of magnitude above the prior precision.) The sample sizes n a and n 2 are permitted to range from 1 to 20. Unit sampling costs of $2 and $20 are considered. (A listing of the simulation program, along with the complete set of simulation results are available from the authors.) Appendix 2 graphically summarizes the effects on expected payoff of different n 1 and n 2. Note (1) The bidder with a higher sample size always has a higher expected value than the bidder with a lower sample size. (2) The difference in the expected values becomes smaller as the amount of information collected by the players increases. (Compare the graph for one observation by bidder 1 with that of 15 observations by bidder 1.) (3) As bidder 2's sample size increases, with bidder l's remaining fixed, bidder l's expected payoff decreases sharply, while bidder 2's expected payoff does not rise sharply. This implies that the benefit from the additional information is passed on to the seller through increased competitiveness. This is consistent with our intuition from Section 2. These results lead to the conjecture that asymmetric pure equilibria exist as well as symmetric mixed ones. We shall verify these conjectures in the next section.
4. 3. Pure strategy equilibrium sample sizes The expected payoffs of Players 1 and 2 was computed for every (integer) pair (nl, n2) with n 1 taking values from 1 to 20. We highlight some of the main features of these results below. We note that (1) When the sampling cost is $2, the best response to n 2 = 1 is n I = 20. The best response to n2=20 is n 1 ----1. Given the symmetry of the expected payoffs of Bidder 1 and 2, this implies that (1, 20) and (20, 1) are equilibria in the re-
stricted strategy space of this example. (2) When the unit sampling cost is $20, the best response to n 2 = 1 is n a = 6 (not 20 as before). The best response to n 2 = 6 is n a = 1. Thus (1, 6) and (6, 1) are both equilibria. We note that the asymmetric equilibria have one player choosing to be 'informed'. However, the amount of information the 'informed' player will collect depends on the sampling cost per unit. With high sampling cost, the 'informed' player will not choose to be 'well-informed'. Thus this result extends earlier results in the literature discussed in the introduction. (3) In neither case are there any best responses apart from the ones that determine the asymmetric equilibria. This implies that the optimal strategy of information collection for Bidder 1 given Bidder 2's sample size has a 'bang-bang' feature. That is, the extreme strategies tend to be equilibria. (4) If the strategy restrictions are dropped, our conjecture is that the asymmetric equilibria would remain. As the costs decrease, the 'less informed' player in equilibrium might be better informed. As the costs increase, the asymmetric equilibria might converge to one observation each.
4.4. Mixed strategy sample sizes Since the information game appears to have many of the features of a coordination game, a symmetric mixed strategy equilibrium should exist and should be the preferred prediction of behavior. We used the nonlinear programming formulation of the bimatrix game equilibrium problem to solve for the mixed strategy. (See, for example, Parthasarathy and Raghavan, 1971, Theorems 7.4.1 and 7.4.2.) Restricting the problem to fifteen observations per player, we calculated the symmetric equi-
Table 2 A partial listing of the payoff matrix Bidder 2 nl
Bidder l
1 15
n2 1
15
7549.65, 7549.65 7738.21, 3231.68
3231.68, 7738.21 1919.83, 1919.83
329
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
librium mixed strategy for sampling costs of $2 and $20. With the cost being $2/unit, Bidder 1 would randomize between 1 and 15 with a probability of 0.877 at 1 observation and 0.123 at 15 observations. With a cost of $20/unit, the randomization is between 1 and 5 with a weight of 0.951 placed on one observation. This is summarized in Table 2. Note that in this case, unlike Table 1, the mixed strategy expected payoff is greater than the average of the asymmetric pure strategy ones. This demonstrates that there is a substantive difference between the nested information case, analyzed in earlier work, and our example in which a firm has less accurate but still proprietary information. These results are consistent with the asymmetric equilibrium calculated earlier, with a strong incentive to take just one observation. Note, however, that in view of the diffuse prior assumption, one observation leads to considerable probability revision. The analysis of the non-diffuse prior case might be richer, although more difficult to accomplish.
5. Conclusions In this paper we have examined the collection of information as an explicit decision that bidders make prior to choosing their bids. In a series of simple models we show analytically that the benefits of being informed could be drastically reduced if there is a reasonable chance that the other player is as well-informed. We also show that a qualitative difference exists between 'nested information' models where the less informed player never gets a positive payoff and a model (such as our own Section 4) where a firm with a smaller amount of (private) information could still obtain a positive expected payoff. Our results also provide a beginning point for considering more complex (and more realistic) bidding situations, such as the inclusion of firmspecific characteristics in valuing the timber in the forestry example. These firm-specific characteristics can sometimes overshadow issues of information collection when the cost of sampling is relatively low or the variance of the sampling distribution is small. We hope that our work will stimulate additional research in endogenous information col -
lection models in general, and the use of bidding models in forestry in particular. Especially in the forest products industry, there is a large void in the use of game theoretic models with respect to information collection strategies.
Appendix 1. Derivation of equilibrium strategies
Player l's expected payoff from a bid b 1, with a signal of sl, is given by
Bzl(bl)
EI=L~
{E(vl(s"
~2)-bl}
•f2(Y2 Isl, hi, h 2 ) d s 2 , where f2(" I sl) is the conditional density of the other player's signal. (That is, Player 1 gets the expected value of g less his bid (given s I and s 2) if he wins the bid. But he wins only if his bid is greater than Player 2's bid. Thus winning conveys some information about the unknown s2 that must be accounted for, as we do here.) This can be written as
B21(bl)[ hlSl + h2s 2 + h'#0 o¢
L
hi + h2 + h'
] - bi
•f2(Y21sl, h,, h2) ds 2, where h' a n d / t o are the prior precision and mean. Differentiating this expression with respect to b l yields
[hisl +h2B21(b1)+h'Ito ] hi+h2+ h, -bl "f2(Bz'(bl)lsl, hi, h2)B21'(bi) = F2( B;l(bl) lsl, ha, ha). This is a first-order necessary condition for optimality. A similar condition can be derived for Player 2. Let B 2 ( s 2 ) = s 2 - k 2 and Bl(sl)=s l - k 1. Then B21(b1)= b 1 + k 2. We shall see that symmetric strategies of this form solve the differential equation above. Substituting we obtain
h,c i + h2c2 + h'#o - h'bl ] hl+h2+h' jf2(bl+c2lsl' =g2(bl + czlsl, hi, h2).
hi, h2)
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
330
We assume now that h' approaches O. This results in
hi+h2
k t =k2=k, (given that the priors can be ignored). In this case, substituting results in
A(b~+c21b~+q, h~, h2)
k f2(b, + k l b l + k, h,, h2)
=F2(b, + c z l b l +C ,, h,, h2),
= F2(bl + k l b , + k, h,, h 2 ) ,
and a sirnilar equation for the other player with f l ( ' ) and FI(- ) instead of f2(') and F 2 ( " ). However the conditional distributions Fz (.) and /'1(. ) differ only through the signals s 1 and s 2. This follows because
or, since f2 ( " ) and F 2(-) are a Normal density and distribution, respectively,
k = "2¢Tg
/1/h, + 1 / h 2 .
It follows that the equilibrium strategies are
E(g21 v ) = v ,
bi:s
and ~7 is Normal with mean s 1 and variance 02 (ignoring the effect of the prior distribution). Then the distribution of (s2 Is1) has mean s 1 and variance 02 + 02 (as shown, for example, in Lindley, 1969, p. 133). Similarly, the distribution of (Sl Is2) has mean s 2 and variance 0 2 + 0 2 ( = 1 / h 2+ 1/hi). This indicates that
i --
,/~- [1/h 1 + 1/h2] .
Appendix 2. Selected simulation graphs In Figures 1-5 some selected simulation graph are shown.
8 7.5 7
6.5 O~ >~e
6
0"0
O-C
5.5
Or" U/
5 4.5 4 3.5 3
I
I
3
I
I
5
I
I
7
I
I
9
I
I
11
I
l
I
1.3
Number of observations -- bidder 2 Bidder 1 + Bidder 2
Figure 1. Bidder I - 1 observation; sampling cost = $ 2 / u n i t
I
15
I
I
17
I
1
19
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
331
0~-~ >,w
(Ic •
3
x ~.~
1
i 1
I
I
I
3
I
5
]
I
7
[
I
9
I
Number of observations Bidder 1 +
[3
I
11
I
I
I
15
17
I
13
19
bidder 2 Bidder 2
F i g u r e 2. Bidder 1 - 10 o b s e r v a t i o n s ; s a m p l i n g cost = $ 2 / u n i t
0.-, ~w 010 0 3
0
hi
I
3
J
I
5
=
J
7
z
J
9
=
I
11
Number of observations Bidder 1 +
i
i
13
i
I
15
17
=
bidder 2 Bidder 2
F i g u r e 3. B i d d e r 1 - 20 o b s e r v a t i o n s ; s a m p l i n g cost = $ 2 / u n i t
I
I
19
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
332
7.5
7
6,5 o,.-~
6
o.t,-
s o
5.5
"~'0
~b Ld
5
4.5
4. 3.5
3
i
i
i
3
5
i
J
7
J
i
i
9
i
11
i
i
i
13
i
15
i
i
17
J
i
19
Number of observations - bidder 2 Bidder 1 + Bidder 2
[]
F i g u r e 4. B i d d e r 1 - 1 o b s e r v a t i o n s ; s a m p l i n g c o s t = $ 2 0 / u n i t
0,..~ >,M o"o a.c "o
5
o w
'~0
g~
~c
+
w
3
5
[]
7
9
11
Number of observations Bidder 1 +
13
15
bidder 2 Bidder 2
F i g u r e 5. B i d d e r 1 - 20 o b s e r v a t i o n s ; s a m p l i n g c o s t = $ 2 0 / u n i t
17
19
K. Chatterjee, TP. Harrison / The value of information in competitive bidding
Acknowledgements We wish to acknowledge Gary Lilien, Kofi Nti, and Keith Ord who have carefully read and commented on an earlier version of this paper. Conversations with Donald Hausch and John Pratt have also been very beneficial. Stephen Matthews pointed out the work of Lee [1982, 1984] to us. Also, two anonymous referees provided a thorough and helpful reviewing process.
References Capen, E.C., Clapp, R.V., and Campbell, W.M. (1971), "Competitive bidding in nigh risk situations", Journal of Petroleum Technology 23, 641-653. Engelbrecht-Wiggans, P., Milgrom, P., and Weber, R.J. (1983), "Competitive bidding with proprietary information", Journal of Mathematical Economics 11, 161-169. Engelbrecht-Wiggans, R., Shubik, M., and Stark, R.M. (eds.) (1983), Auctions, Bidding and Contracting. New York University Press. Hausch, D.B. (1986), "Auctions with endogenous information acquisition", mimeo, University of Wisconsin, Madison, WI. Lederer, P.J. (1983), "Bidding for timber", ORSA/TIMS Joint National Meeting, Chicago, IL. Lee, T.K. (1982), "Resource information policy and federal resource leasing", Bell Journal of Economics 13(2), 561-568. Lee, T.K. (1984), "Incomplete information, nigh-low bidding and public information in first price auctions", Management Science 30, 1490-1496. Lindley, D.V. (1969), Introduction to Probability and Statistics from a Bayesian Viewpoint, Part 1: Probability, Cambridge University Press. Matthews, S.A. (1984), "Information acquisition in dis-
333
criminating auctions", in: M. Boyer and R.E. Kihlstrom (eds.), Bayesian Models in Economic Theory, Elsevier, Amsterdam. Milgrom, P., and Weber, R.J. (1982a), "A theory of auctions and competitive bidding", Econometrica 50, 1089-1122. Milgrom, P., and Weber, R.J. (1982b), "Bidding with proprietary information", mimeo, Northwestern University. Milgrom, P., and Weber, R.J. (1982c), "The value of information in a sealed-bid auction", Journal of Mathematical Economics 10, 105-114. Ortega-Reichert, A. (1968), "Models for competitive bidding under uncertainty", Technical Report no. 8, Department of Operations research, Stanford University. Parthasarathy, T., and Raghavan, T.E.S. (1971), Some Topics in Two-Person Games, American Elsevier, New York. Pennsylvania Department of Environmental Resources, Division of State Forest Management (1984), Forest Products Statistical Report - - Timber Management Section.
Ponssard, J.P. (1982), Competitive Strategies, North-Holland, Amsterdam. Raiffa, H., and Schlaifer, R. (1961), Applied Statistical Decision Theory, Harvard Business School, Division of Research. Reece, D.K. (1978), "Competitive bidding for offshore petroleum leases", Bell Journal of Economics 9(2), 369 384. Rothkopf, M. (1969), "A model of rational competitive bidding", Management Science 15, 362-373. United States Department of Commerce (1983), Statistical Abstracts of the United States - 1984, 104th Edition. Weverbergh, M. (1979), "Competitive bidding with asymetric information reanalyzed", Management Science 25, 291-294. Wilson, R.B. (1967), "Competitive bidding with asymmetric information", Management Science 13, A816-A820. Wilson, R.B. (1969), "Competitive bidding with disparate information", Management Science 15, A446-A448. Wilson, R.B. (1974), "Competitive bidding", Class Notes for Business 363, Stanford University. Winkler, R.L., and Brooks, D.G. (1980, "Competitive bidding with dependent value estimates", Operations Research 28, 603 611.
European Journal of Operational Research 36 (1988) 322-333 North-Holland
Theory and Methodology
The value of information in competitive bidding Kalyan CHATTERJEE
a n d T e r r y P. H A R R I S O N
Division of Management Science, The Pennsylvania State University, College of Business Administration, University Park, PA 16802, USA
Abstract: We consider the situation where two players compete to obtain a valuable object, e.g. a stand of timber in a competitive, sealed-bid environment. Prior to submitting a bid, each player may sample the stand while incurring a common, non-zero cost for each observation. On one hand, a player wishes to take as few observations as possible due to the cost of collecting information. However, one also wishes to obtain as m a n y observations as possible to avoid a bid that overstates the value of the resource. Given different assumptions on the sampling process, the informational structure, and underlying distribution of value, we derive equilibrium bidding strategies. We use these bidding strategies to solve for equilibrium in an information collection problem from the forest products industry. Keywords: Bidding, value of information, decision theory, forestry
1. Introduction The motivation for this problem is provided by the multi-billion dollar timber industry (c.f.U.S. Department of Commerce, 1983; Pennsylvania Department of Environmental Resources 1984). Tracts (stands) of timber are frequently auctioned through a sealed, competitive bid. Prior to the auction, potential bidders are permitted to collect information about the tract. Currently, m a n y bidders follow some rule-of-thumb in deciding how much information to c o l l e c t - - f o r example, sampiing ten percent of the cells in the grid into which the tract is divided. Based on these sample observations, along with an expectation of what competitors may do, a bid is submitted. However it is costly to collect these samples, since every tree in the sampling area must be measured as to diameter, height, species, and quality. Therefore there are conflicting objectives in this bidding
Received April 1987; revised August 1987
scenario. On one hand, it is desirable to collect as few sample observations as possible to avoid the costs of sampling. On the other hand, one wishes to take m a n y sample observations so that the estimate of the timber's volume (and hence value) is as accurate as possible, thus avoiding winning the bid by being the bidder who must over-estimated the value of the stand. (This danger, known as the 'winner's curse', is documented in Capen, Clapp and Campbell (1971).) In determining the value of the stand, it is likely that each bidder has some firm-specific considerations that affect the value of the tract to the bidder. For example, a large pulp and paper mill that is facing a potential stockout of raw material for a billion dollar processing complex is likely to place a premium on obtaining a successful bid. Other examples may include the ability of one bidder to produce higher value products from the same material, the ability of one bidder to fell the trees and transport them for processing in a more economical fashion, etc. While these firm-specific considerations undoubtedly exist, we have sim-
0377-2217/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
plified the analysis by dealing exclusively with a 'common value' model. In doing so, we are restricting our attention to bidders of the same type - - s a y all pulp and paper mills within the same region, and with roughly the same inventory situat i o n - i n order to focus on the informational issues. In some sense, the industry has implicitly adopted a common value model, as evident from the regular publication of unit average prices of timber in a given area. The case where the signals (or estimates of value) are independent of each other is often called the 'independent values' model. While there has been prior empirical work on predicting a competitor's bid (Lederer, 1983), we are unaware of any work dealing explicitly with the information collection issue in sealed bid auctions for stands of timber; and of little research addressing this question in general. (We will discuss the exceptions shortly.) For example, in examining Lederer's empirical study, we observe that forest products firms frequently do not consider their opponents' actions when formulating their information collection strategy. Clearly, those firms that do adopt an explicitly competitive viewpoint will do better than those who do not, because they will be able to optimize against their opponents predicted action. Analysis of strategic behaviour in auctions is a natural arena for the use of game theory, and this paper follows a long tradition in using this method. (See the survey by Engelbrecht-Wiggans, Shubik and Stark (1983).) The crucial game-theoretic concept we use is that of non-cooperative equilibrium. This consists of a pair of strategies, one for each player, each of which is a best response to the other. Using this concept enables us to resolve the difficult question of what Player 1 expects Player 2 to do, since this depends on what Player 2 expects Player 1 expects Player 2 to do, and so on. If Player 1 expects Player 2 to play his or her equilibrium strategy (assume there is only one for the moment), Player 2 will expect Player 1 to use Player l's equilibrium strategy and the pair of expectations will be consistent with optimizing behaviour by both players. Earlier decision theoretic models assumed optimizing behaviour by only one player while the other players were treated as part of 'Nature'. There has, however, been a substantial literature in bidding using game theory.
323
Most of the work in this area (see Ortega-Reichert, 1968; Milgrom and Weber, 1982a, b, c; Rothkopf, 1969; Winkler and Brooks, 1980, Wilson, 1974; Weverbergh, 1979, among others) has concentrated on calculating equilibrium strategies in a fixed informational setting. The analysis has assumed asymmetric (nested) information so that one bidder knows everything the other knows and has additional information; or symmetric differential information in which each bidder receives a private signal and the signals are 'equally accurate' in some sense or they are independent of each other. Milgrom and Weber (1982c) have considered comparative statics on the information structure, and have obtained results on the preferences of a bidder over various states of his own and his opponent's information. This paper adopts a somewhat different approach in seeking to identify the information structure endogeneously in equilibrium. Lee (1982, 1984), Matthews (1984) and Hausch (1986) have also examined endogenously derivable information collection strategies. All of these authors consider, as we do, a two-stage game in which information collection occurs in the first stage and bidding takes place in the second. These authors use the technique of backward induction in solving for the equilibrium strategies in each stage. Matthews allows for a continuum of possible amounts of information but is unable to obtain a complete analysis of the problem. Lee considers the information structure where a perfectly informative experiment is available at a cost. Hausch permits experimenters to obtain one signal which could be either 'high' or 'low'. The signal's degree of reliability is chosen in the first stage by each player. Hausch then discusses questions of auction design. In Section 2, we discuss this basic two-stage model, adapting Lee's (1984) model slightly to illuminate the essential difference between observable information collection and secret possession of information. Results similar to Lee's are re-derived for completeness. In Sections 3 and 4, we examine a more realistic model in which bidders can choose to sample (at a cost) as many times as they wish from a Normal process prior to making their bid. The sample size is fixed in advance and information collection is observable. In Section 3 we calculate
324
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
the equilibrium bidding strategies based on the assumption of a N o r m a l sampling process with a Normal prior of very large variance. In Section 4 we use these results to numerically compute equilibrium strategies for information collection.
2. C o n c e p t u a l
models of information
UU:
Both bidders use U in the first stage, so both are uninformed. The common expected value is 6, and the equilibrium bid is fi, so the expected payoff is zero for both bidders.
II:
Both bidders are informed of the value v of the object. The equilibrium bid in the second stage is to bid the common value v. The expected profit from bidding is once again zero for either bidder, but now each bidder has paid c > 0 for the information. The payoffs are therefore ( - c , - c ) to bidders 1 and 2 respectively.
collection
2.1. Introduction
In this section, we present two models similar to Lee (1982, 1984) adapted to fit our context in which the first stage is the information collection stage, and the second is the bidding stage. We assume that within each stage players move simultaneously, and that the auction procedure is sealed bidding with the highest bidder winning. The difference between the two models is that, in the first, the cost of sampling is the same for the two bidders and is commonly known, with the activity of information collection (but not the outcome) observable. The Normal sampling model of Sections 3 and 4 shares these characteristics with our first model. This section, therefore, provides some intuition about the more complicated structure of the Normal model. The second model that we consider here has private information about costs of sampling and unobservable information collection. This model provides an indication of what more general results might be anticipated in such an informational context and is, therefore, a pointer to future research in this area.
2.2. A model with observable information collection and common cost
This is a two-stage game. In stage 1, each bidder decides whether to pay c and learn the value v of 5 (strategy I) or to forgo the option and bid as an uninformed player (strategy U). (Note that the unknown value is being treated as a r a n d o m variable 5 of which v is a particular realization.) The act of becoming informed is observable. In the second stage, bidder i announces a bid b~ for the object. The object is sold to the highest bidder at the price given by his bid. There are four possible second-stage bidding games that could arise as a result of different first-stage strategies. These are:
U I and I U : I n the asymmetric case, when an informed player is bidding against an uninformed player, the expected profit of the uninformed player is 0. (See Engelbrecht-Wiggans, Milgrom and Weber, 1983.) The expected profit of the informed player is f0°°(1 - F ( v ) ) F ( v )
d v - c.
(See Milgrom and Weber, 1982c, Theorem 2), where F ( - ) is the probability distribution of 5. Let fo°°(1 - F ( v ) ) F ( v ) do = v o. We can therefore set up the matrix of payoffs shown in Table 1. If the cost c of being fully informed is greater than v0, neither bidder will choose to be informed. Otherwise there are two asymmetric pure strategy equilibria and a symmetric mixed strategy equilibrium with each bidder choosing to be informed with probability ( 1 - (C/Vo)). This gives each an expected payoff of 0. Note that the bidders are the worst off when both are informed. This outcome (I, I) will occur with probability ( 1 - ( C / V o ) ) 2, which may be reasonably high if (C/Vo) is relatively small. Intuition suggests that the bidders
Table 1 Matrix of payoffs
Bidder 1
Bidder 2 u
I
U (0, 0)
(0, v0 - c)
I (% - c, 0)
( - c, - c)
K. Chatterjee, T.P. Harrison / T h e value of information in competitive bidding
325
may not gain from information collection. A similar set of equilibria is derived for the N o r m a l sampling model. However under N o r m a l sampling the players would be better off with the mixed strategies than with the average of the pure strategies.
event with probability zero in equilibrium) with a bid of 0 when the value is 1, this implies that the expected profit of the uninformed bidder is zero in equilibrium. Therefore, for the uninformed bidder, a bid b realizes an expected profit of
2.3. Unobservable information collection with differential costs
- ½b + ½ ( 1 -
In this section we consider a model with differential information about the costs of being informed. We assume that the value, v, of the tract is either 0 or 1 with equal probability, and that the cost, c, of being informed is distributed uniformly from 0 to 0.25. (These assumed values could, of course, be different without affecting the analysis.) In the first stage, the game proceeds as before, with each player drawing a cost from the commonly-known distribution above. The actual value of the cost is private information and the draws are independent of each other. Based on this draw, each of bidders 1 and 2 decides whether or not to be fully informed. The act of collecting information is unobservable--this is 'secret information' in Ponssard's (1982) terminology. In the second stage, players bid given their informationcollection decisions of the first stage. (Note that the informational assumptions here are different from those of the other models in this paper.) We now consider the second-stage bidding problem for both the asymmetric and the symmetric information structures. First, we discuss the case where Player 1 is informed about the value of v and Player 2 is not. The two-point probability distribution of v is, however, c o m m o n knowledge. Let in informed bidder use a mixed strategy Gx(.) over bids when the value is 1 and bid 0 when the value is zero. Similarly, suppose the uninformed bidder uses a mixed strategy G v (.). In equilibrium, Gu(. ) will be selected to make the informed player indifferent among all of the bids in the support of G I ( ' ) . Using this we can determine that Gu(b)
1
2(1 - b ) "
At b = 0, G U (b) = ½. Therefore, in equilibrium, a mass of ½ is placed at 0 and 0 is in the support of G~ (.). Since the uninformed bidder will never win (except if the informed bidder also bids 0, an
b)G,(b)
= O,
or
G,(b) = b / ( 1 - b ) . Note that G i ( b ) = 1 when b = ½ and is 0 when b = 0. This completes the equilibrium analysis for the asymmetric case. The symmetric case proves to be more complex. Suppose now that there is a probability, r, that a player is informed about the true value of v and a probability (1 - r ) that the player is uninformed. We look again for mixed strategies. Let G u ( - ) be the mixed strategy of the uninformed player and G I ( - ) that of the informed player when v = 1. Observe that when v = 0, the informed player will bid 0. Lee (1984) shows that there cannot be mixed strategies G I ( - ) and G U (.) whose supports intersect in a nonempty interior. We can also show that the informed bidder will not use a pure strategy in a symmetric equilibrium. Suppose not, so that an informed bidder bids x > 0 with probability 1 when v = 1. If his opponent is also informed and the equilibrium is symmetric, the opponent should bid x as well. But the opponent will be better off bidding x + ~, since by doing so he or she will capture an additional r/2 probability of winning while incurring only a small decrease in the amount of winning profits. The only symmetric equilibrium therefore could be 1. But this too is not an equilibrium. If player 1 bids 1 when informed, an informed player 2 is better off bidding b < 1 and obtaining a positive expected profit of (1 - b)(1 - r)Gu(b ). For similar reasons, a pure strategy will not be used in equilibrium by the uninformed player. If an equilibrium exists, it must therefore be in mixed strategies G u ( . ) and G I ( ' ) with non-overlapping supports (except at boundary points). It can be shown that Gu(b)
r b ( 1 - r ) 1 - 2b
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
326
and GI(bll)
1 1-r r 2-r
1 1-b
1-r r
constitute such an equilibrium. The support of G u ( ' ) and G I ( ' I 1 ) are [0, ( 1 - r ) / ( 2 - r ) ] and [(1 - r ) / ( 2 - r), (2 - r ) - 1 ] respectively. We now consider the information collection game of the first stage. Recall that the cost of being fully informed is uniformly distributed between 0 and 0.25. Under what circumstances would a player choose to be informed? If Player 2 can commit himself to being informed whatever the cost, it is clear that Player l ' s best option is to be uninformed whatever the cost. If Player 1 is uninformed Player 2 clearly should be informed. The asymmetric information structure therefore remains an equilibrium. A more interesting case would arise if players adopted the decision r u l e - - o b t a i n the information if c~< c' and remain uninformed if c > c'. If such a rule were adopted by both players the probability of facing an informed player at the second stage would be r = 4c'. The expected value to an informed Player 1 would be ½((1 - 4 c ' ) / ( 2 - 4 c ' ) } . The expected value less the cutoff cost c' should equal zero. Solving for c' yields approximately 0.15. Therefore the probability that a player will be an informed bidder is approximately 0.60. If H a y e r 2 adopts this decision rule, H a y e r l ' s expected payoff from being informed is ½{ (1 - 4 c ' ) / ( 2 - 4 c ' ) }. This is equal to the cutoff value c' that makes the player indifferent between becoming informed and not; therefore so long as c < c', it is optimal to be informed. Therefore this is an equilibrium rule in the sense of being optimal against the other person's strategy. Notice that the uninformed player is not harmed by the secrecy of the information gathering. The uninformed player is, however, much more cautious than in the asymmetric case. The informed player is substantially hurt in the symmetric case, with an expected payoff of 0.15 instead of 0.25. By similar reasoning, the seller prefers r to be as high as possible. This implies that it m a y be in the best interests of the seller to subsidize information
collection, thus moving the cost distribution to the left and encouraging information collection. This is similar to results obtained in a completely different model by Reece (1978).
3. Equilibrium bidding strategies in a normal model In this section, we present a partial analysis of information collection in a N o r m a l sampling process. The bidding scenario is the same as before. A tract of timber of unknown value v is being put up for auction by first-price, sealed bid. There are two bidders who share a common probability distribution on v. Each bidder i can also obtain a signal s i that gives partial information about the value v. The s i are independently drawn from Normal distributions F~(.) with mean v and precisions h i for i = 1, 2, respectively. The value v is distributed Normally with mean/~0 and precision h', where h ' is much smaller than the h i. For our analysis, we shall assume that the relative magnitude of h' is equivalent to the requirement that the prior be diffuse. We recall that precision in a decision-theoretic context refers to the reciprocal of the corresponding variance. Note that the s i are conditionally independent given v, but are not unconditionally independent. This bidding scenario corresponds to the second stage of the two-stage games we have been considering so far. The first stage corresponds to a decision by each player on the ' a m o u n t of information' to collect; the Normal process assumption facilitates the interpretation of the precision h i as a measure of the amount of (costly) information player i has chosen to collect in the first stage. In the sequel, we shall assume that h i is equal to nih, where F/i is the number of observations taken by Player i and h is the process precision. We assume that each observation costs a fixed amount, c, and that this cost is the same for each player and all observations. We first calculate the bidding strategies where h 1 and h 2 are not necessarily equal. Wilson (1974) has obtained results for the condition h 1 = h 2. Allowing for inequality is important because the precisions are chosen in the first stage by bidders who are free to choose differently. In our calculation, we assume that the seller's reservation price for the stand is zero and that
K. Chatterjee, TP. Harrison / The value of information in competitive bidding bids are zero if the value is positive but the bidding strategy dictates a negative bid. (The parameters of the distribution are chosen such that bids of less than zero are extremely unlikely to occur.) Suppose Player i observes a signal s~ and that Player 2 uses a strategy (mapping signals into bids) given by B2(" ). We write b 2 = B 2 ( s 2 ) . This function is assumed strictly monotonic and twice differentiable. Given these assumptions and one we make later about a high prior variance, we can show (see Appendix 1) that the equilibrium strategies are
Then expected payoff then becomes
1 E(P11sl)
=
±],j2
~-1 + h2l
X
"~
-~
+
-- 2 V ~
ha~-h2
.
The precision, h i , is obtained by taking n 1 observations from the sampling process of precision h. Similarly, h 2 = n 2h. The net expected payoff for Player 1, including sampling costs, for a given n 2 is
[n,+n21'J2[1
tzV2
b, = s i -
327
, n2 ]
--cna,
.
In order to derive information collection strategies, we calculate the equilibrium expected payoff to Player 1 in the diffuse prior case. Let this expected payoff be given by
where c is the cost per observation. In the next section we investigate the number of observations each player will take in equilibrium.
4. Information collection strategies 4.1. I n t r o d u c t i o n
E(P "f2(g21sa,
ha, h 2 ) d s 2 .
Substituting in the equilibrium strategies, we obtain (with k = ( ( ~ r / 2 ) ( 1 / h a + l / h 2 ) ),
E(Plsa)
-A(~21s,, h,, h2) ds2 ]12
[SlFN.((Sl_Sa)~/h
hI + h2 -
- -
k +~,
where FN.(- ) and f N * ( ' ) a r e standardized Normal distribution and density functions, respectively, and the standard formula for the incomplete expectation is used (see, for example, Raiffa and Schlaifer, 1961, p. 232). There is an identical expression, with the subscripts reversed, for Player 2's net expected payoff. The quantity h-a _ h ~-1 + h-1 2 •
In the previous section we calculated equilibrium bidding strategies and expected payoffs for players who have, with common knowledge, sampled n 1 and n 2 times from a Normal process. We also obtained the expected equilibrium payoff as a function of n I and n 2. The equilibrium pair of information choices (n l, n2) in the first stage of the game appears to be difficult to resolve explicitly. In this section we have calculated the equilibria for a numerical example, and have obtained explicit results. In these calculations we have used the diffuse prior (i.e. low precision prior) approximation used in the previous section. We note that the equilibrium bidding strategies would be asymmetric and nonlinear with a non-diffuse prior and would therefore be difficult to characterize analytically. We are aware of the problems that may arise using diffuse priors in preposterior analysis, but believe that they do not arise here. The approach we have used for this numerical example is as follows. We generate a list of payoffs for Player 1 for different combinations of na and n z. The method for determining these payoffs is discussed in Section 4.3.
328
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
4.2. Numerical example and simulation approach A value, 6, is drawn from a Normal distribution with mean $200000 and standard deviation $50 000. (The prior precision is therefore 4 × 10-10, that is, {1/500002 }.) Sample observations are drawn from a Normal distribution with mean v and standard deviation $12500. (The sampling precision is therefore 6.4 × 10-9, an order of magnitude above the prior precision.) The sample sizes n a and n 2 are permitted to range from 1 to 20. Unit sampling costs of $2 and $20 are considered. (A listing of the simulation program, along with the complete set of simulation results are available from the authors.) Appendix 2 graphically summarizes the effects on expected payoff of different n 1 and n 2. Note (1) The bidder with a higher sample size always has a higher expected value than the bidder with a lower sample size. (2) The difference in the expected values becomes smaller as the amount of information collected by the players increases. (Compare the graph for one observation by bidder 1 with that of 15 observations by bidder 1.) (3) As bidder 2's sample size increases, with bidder l's remaining fixed, bidder l's expected payoff decreases sharply, while bidder 2's expected payoff does not rise sharply. This implies that the benefit from the additional information is passed on to the seller through increased competitiveness. This is consistent with our intuition from Section 2. These results lead to the conjecture that asymmetric pure equilibria exist as well as symmetric mixed ones. We shall verify these conjectures in the next section.
4. 3. Pure strategy equilibrium sample sizes The expected payoffs of Players 1 and 2 was computed for every (integer) pair (nl, n2) with n 1 taking values from 1 to 20. We highlight some of the main features of these results below. We note that (1) When the sampling cost is $2, the best response to n 2 = 1 is n I = 20. The best response to n2=20 is n 1 ----1. Given the symmetry of the expected payoffs of Bidder 1 and 2, this implies that (1, 20) and (20, 1) are equilibria in the re-
stricted strategy space of this example. (2) When the unit sampling cost is $20, the best response to n 2 = 1 is n a = 6 (not 20 as before). The best response to n 2 = 6 is n a = 1. Thus (1, 6) and (6, 1) are both equilibria. We note that the asymmetric equilibria have one player choosing to be 'informed'. However, the amount of information the 'informed' player will collect depends on the sampling cost per unit. With high sampling cost, the 'informed' player will not choose to be 'well-informed'. Thus this result extends earlier results in the literature discussed in the introduction. (3) In neither case are there any best responses apart from the ones that determine the asymmetric equilibria. This implies that the optimal strategy of information collection for Bidder 1 given Bidder 2's sample size has a 'bang-bang' feature. That is, the extreme strategies tend to be equilibria. (4) If the strategy restrictions are dropped, our conjecture is that the asymmetric equilibria would remain. As the costs decrease, the 'less informed' player in equilibrium might be better informed. As the costs increase, the asymmetric equilibria might converge to one observation each.
4.4. Mixed strategy sample sizes Since the information game appears to have many of the features of a coordination game, a symmetric mixed strategy equilibrium should exist and should be the preferred prediction of behavior. We used the nonlinear programming formulation of the bimatrix game equilibrium problem to solve for the mixed strategy. (See, for example, Parthasarathy and Raghavan, 1971, Theorems 7.4.1 and 7.4.2.) Restricting the problem to fifteen observations per player, we calculated the symmetric equi-
Table 2 A partial listing of the payoff matrix Bidder 2 nl
Bidder l
1 15
n2 1
15
7549.65, 7549.65 7738.21, 3231.68
3231.68, 7738.21 1919.83, 1919.83
329
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
librium mixed strategy for sampling costs of $2 and $20. With the cost being $2/unit, Bidder 1 would randomize between 1 and 15 with a probability of 0.877 at 1 observation and 0.123 at 15 observations. With a cost of $20/unit, the randomization is between 1 and 5 with a weight of 0.951 placed on one observation. This is summarized in Table 2. Note that in this case, unlike Table 1, the mixed strategy expected payoff is greater than the average of the asymmetric pure strategy ones. This demonstrates that there is a substantive difference between the nested information case, analyzed in earlier work, and our example in which a firm has less accurate but still proprietary information. These results are consistent with the asymmetric equilibrium calculated earlier, with a strong incentive to take just one observation. Note, however, that in view of the diffuse prior assumption, one observation leads to considerable probability revision. The analysis of the non-diffuse prior case might be richer, although more difficult to accomplish.
5. Conclusions In this paper we have examined the collection of information as an explicit decision that bidders make prior to choosing their bids. In a series of simple models we show analytically that the benefits of being informed could be drastically reduced if there is a reasonable chance that the other player is as well-informed. We also show that a qualitative difference exists between 'nested information' models where the less informed player never gets a positive payoff and a model (such as our own Section 4) where a firm with a smaller amount of (private) information could still obtain a positive expected payoff. Our results also provide a beginning point for considering more complex (and more realistic) bidding situations, such as the inclusion of firmspecific characteristics in valuing the timber in the forestry example. These firm-specific characteristics can sometimes overshadow issues of information collection when the cost of sampling is relatively low or the variance of the sampling distribution is small. We hope that our work will stimulate additional research in endogenous information col -
lection models in general, and the use of bidding models in forestry in particular. Especially in the forest products industry, there is a large void in the use of game theoretic models with respect to information collection strategies.
Appendix 1. Derivation of equilibrium strategies
Player l's expected payoff from a bid b 1, with a signal of sl, is given by
Bzl(bl)
EI=L~
{E(vl(s"
~2)-bl}
•f2(Y2 Isl, hi, h 2 ) d s 2 , where f2(" I sl) is the conditional density of the other player's signal. (That is, Player 1 gets the expected value of g less his bid (given s I and s 2) if he wins the bid. But he wins only if his bid is greater than Player 2's bid. Thus winning conveys some information about the unknown s2 that must be accounted for, as we do here.) This can be written as
B21(bl)[ hlSl + h2s 2 + h'#0 o¢
L
hi + h2 + h'
] - bi
•f2(Y21sl, h,, h2) ds 2, where h' a n d / t o are the prior precision and mean. Differentiating this expression with respect to b l yields
[hisl +h2B21(b1)+h'Ito ] hi+h2+ h, -bl "f2(Bz'(bl)lsl, hi, h2)B21'(bi) = F2( B;l(bl) lsl, ha, ha). This is a first-order necessary condition for optimality. A similar condition can be derived for Player 2. Let B 2 ( s 2 ) = s 2 - k 2 and Bl(sl)=s l - k 1. Then B21(b1)= b 1 + k 2. We shall see that symmetric strategies of this form solve the differential equation above. Substituting we obtain
h,c i + h2c2 + h'#o - h'bl ] hl+h2+h' jf2(bl+c2lsl' =g2(bl + czlsl, hi, h2).
hi, h2)
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
330
We assume now that h' approaches O. This results in
hi+h2
k t =k2=k, (given that the priors can be ignored). In this case, substituting results in
A(b~+c21b~+q, h~, h2)
k f2(b, + k l b l + k, h,, h2)
=F2(b, + c z l b l +C ,, h,, h2),
= F2(bl + k l b , + k, h,, h 2 ) ,
and a sirnilar equation for the other player with f l ( ' ) and FI(- ) instead of f2(') and F 2 ( " ). However the conditional distributions Fz (.) and /'1(. ) differ only through the signals s 1 and s 2. This follows because
or, since f2 ( " ) and F 2(-) are a Normal density and distribution, respectively,
k = "2¢Tg
/1/h, + 1 / h 2 .
It follows that the equilibrium strategies are
E(g21 v ) = v ,
bi:s
and ~7 is Normal with mean s 1 and variance 02 (ignoring the effect of the prior distribution). Then the distribution of (s2 Is1) has mean s 1 and variance 02 + 02 (as shown, for example, in Lindley, 1969, p. 133). Similarly, the distribution of (Sl Is2) has mean s 2 and variance 0 2 + 0 2 ( = 1 / h 2+ 1/hi). This indicates that
i --
,/~- [1/h 1 + 1/h2] .
Appendix 2. Selected simulation graphs In Figures 1-5 some selected simulation graph are shown.
8 7.5 7
6.5 O~ >~e
6
0"0
O-C
5.5
Or" U/
5 4.5 4 3.5 3
I
I
3
I
I
5
I
I
7
I
I
9
I
I
11
I
l
I
1.3
Number of observations -- bidder 2 Bidder 1 + Bidder 2
Figure 1. Bidder I - 1 observation; sampling cost = $ 2 / u n i t
I
15
I
I
17
I
1
19
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
331
0~-~ >,w
(Ic •
3
x ~.~
1
i 1
I
I
I
3
I
5
]
I
7
[
I
9
I
Number of observations Bidder 1 +
[3
I
11
I
I
I
15
17
I
13
19
bidder 2 Bidder 2
F i g u r e 2. Bidder 1 - 10 o b s e r v a t i o n s ; s a m p l i n g cost = $ 2 / u n i t
0.-, ~w 010 0 3
0
hi
I
3
J
I
5
=
J
7
z
J
9
=
I
11
Number of observations Bidder 1 +
i
i
13
i
I
15
17
=
bidder 2 Bidder 2
F i g u r e 3. B i d d e r 1 - 20 o b s e r v a t i o n s ; s a m p l i n g cost = $ 2 / u n i t
I
I
19
K. Chatterjee, T.P. Harrison / The value of information in competitive bidding
332
7.5
7
6,5 o,.-~
6
o.t,-
s o
5.5
"~'0
~b Ld
5
4.5
4. 3.5
3
i
i
i
3
5
i
J
7
J
i
i
9
i
11
i
i
i
13
i
15
i
i
17
J
i
19
Number of observations - bidder 2 Bidder 1 + Bidder 2
[]
F i g u r e 4. B i d d e r 1 - 1 o b s e r v a t i o n s ; s a m p l i n g c o s t = $ 2 0 / u n i t
0,..~ >,M o"o a.c "o
5
o w
'~0
g~
~c
+
w
3
5
[]
7
9
11
Number of observations Bidder 1 +
13
15
bidder 2 Bidder 2
F i g u r e 5. B i d d e r 1 - 20 o b s e r v a t i o n s ; s a m p l i n g c o s t = $ 2 0 / u n i t
17
19
K. Chatterjee, TP. Harrison / The value of information in competitive bidding
Acknowledgements We wish to acknowledge Gary Lilien, Kofi Nti, and Keith Ord who have carefully read and commented on an earlier version of this paper. Conversations with Donald Hausch and John Pratt have also been very beneficial. Stephen Matthews pointed out the work of Lee [1982, 1984] to us. Also, two anonymous referees provided a thorough and helpful reviewing process.
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