A Blotto Game with Multi-Dimensional Incomplete Information

Economics Letters 113 (2011) 273–275

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Economics Letters

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A Blotto game with multi-dimensional incomplete information

Dan Kovenock a,∗, Brian Roberson b,1 a Economic Science Institute, Argyros School of Business and Economics, Chapman University, One University Drive, Orange, CA 92866, USA b Purdue University, Department of Economics, Krannert School of Management, 403 W. State Street, West Lafayette, IN 47907, USA article info a b s t r a c t

Article history: We examine a multi-dimensional incomplete information Colonel Blotto game in which each player’s n- Received 10 November 2010 tuple of battlefield valuations is drawn from a common n-variate joint distribution function that is uniform Received in revised form on the non-negative orthant of the surface of a sphere. 19 July 2011 © 2011 Elsevier B.V. All rights reserved. Accepted 18 August 2011 Available online 25 August 2011

JEL classification: C72 D72 D74

Keywords: Colonel Blotto game Conflict Multi-dimensional incomplete information Multi-dimensional action space

1. Introduction class of games with multi-dimensional incomplete information in which each player has a multi-dimensional action space. In the case in which the joint distribution of battlefield valuations is a uniform The classic formulation of the Colonel Blotto game is a two- distribution over the nonnegative points on the surface of a sphere, player constant-sum game with complete information (Borel, we show that there exists an intuitive symmetric pure-strategy 1921). There are a finite number (n) of battlefields. Each player Bayesian equilibrium. simultaneously allocates his fixed amount of a resource across the Most closely related to our focus is Adamo and Matros (2009) n battlefields. Within each battlefield, the player that allocates the who introduce incomplete information in the Colonel Blotto game higher level of the resource wins the battlefield, and each player’s framework. In that model, k players compete across a set of n payoff is equal to the number of battlefields won.2 A distinguishing battlefields with valuations that are common knowledge. These feature of the Colonel Blotto game is that, except for trivial cases, an valuations may vary across battlefields, but each battlefield’s equilibrium is a pair of non-degenerate multi-dimensional mixed valuation is the same for all players. The incomplete information strategies. is in the form of uncertainty regarding the players’ respective This note investigates a multi-dimensional incomplete infor- budgets of the resource, which are assumed to be independent mation version of the Colonel Blotto game with two risk-neutral draws from a common distribution that is absolutely continuous players. The incomplete information is multi-dimensional in that on the closed interval between 0 and the sum of the individual each player’s n-tuple of battlefield valuations is drawn from a com- battlefield valuations. Under the added assumption that the mon n-variate joint distribution. Our game is therefore within the distribution of the highest order statistic of the other k − 1 of the players’ budgets is concave, there exists a monotonic symmetric pure-strategy Bayesian equilibrium of that model in which each ∗ Corresponding author. Tel.: +1 714 628 7226. player allocates to each battlefield a share of his budget equal E-mail addresses: [email protected] (D. Kovenock), [email protected] to that battlefield’s value divided by the sum of all battlefield (B. Roberson). valuations. That is, the one-dimensional nature of the incomplete 1 Tel.: +1 765 494 4531. information allows for each element in the n-tuple of a player’s 2 For further details on this class of games, see Roberson (2006) and Kovenock allocation of force across battlefields to be conditioned on a single and Roberson (2010). variable, his budget.

Also related are Che and Gale (1998) and Pitchik (2009) 2, 3}, the univariate distribution function of vj, Fj(z), is equal to the who examine incomplete information auctions with budget- P-volume of the set of v ∈ S such that vj ≤ z ≤ 1. constrained players. In fact, our model corresponds to the case of two budget-constrained players competing across a set of all- 3. Results pay auctions with incomplete information with the exception that for each player any unused resources are forfeited. The case of a A strategy is a function b : S → B. Denote player i’s 3-tuple of single all-pay auction with incomplete information is addressed bids as bi({vi}3 ) = (bi ({vi}3 ), bi ({vi}3 ), bi ({vi}3 )). by Amann and Leininger (1996), who examine asymmetrically j j=1 1 j j=1 2 j j=1 3 j j=1 distributed independent private values and Krishna and Morgan (1997), who examine symmetrically distributed affiliated signals. Proposition 1. If each player’s n-tuple of battlefield valuations is In this note we show that if the joint distribution of the players’ drawn from the joint distribution P, then valuations takes a particular parametric form, then the analysis of ∗ b { }3 = { 2 2 2} (1) the all-pay auction with incomplete information can be extended ( vj j=1) (v1) , (v2) , (v3) to cover our case of multiple auctions, binding budget constraints, is a symmetric pure-strategy Bayesian equilibrium. and use-it-or-lose-it resources. The intuition for our result follows from the single all-pay We briefly outline the proof of Proposition 1. Lemma 1 states a auction with incomplete information. With one auction and useful property of the joint distribution P. uniform priors, in a symmetric Bayesian equilibrium each player’s bid is proportional to the square of his valuation. If valuations are Lemma 1. The joint distributionP satisfies the property that for each drawn uniformly over the non-negative orthant of the surface of j ∈ {1,..., 3},Fj(vj) = vj for vj ∈ [0, 1], i.e. each vj is distributed a sphere, then each valuation has a uniform prior and the sum of uniformly on [0, 1]. the squared valuations is equal to a constant. As a result, there Lemma 1 follows from the properties of spherical segments. In exists a simple equilibrium bidding correspondence that exhausts particular, note that the surface area of any spherical segment is the budget. 2πrh where h is the height between the two parallel planes that cut the sphere. The unit sphere has radius r = 1, surface area 4π, and 2. Model exactly (1/8)th of the sphere is contained in the positive orthant 3 of R . Thus, the surface area of the relevant portion of the unit There are two risk-neutral players, i ∈ {A, B}, and a finite num- sphere is (π/2). Similarly, the surface area of the spherical segment ber (n) of battlefields, j ∈ {1,..., n}. Each player has a budget of formed by cutting the surface S with two parallel planes is (πh/2). one unit of a homogeneous resource. Let B denote the set of feasi- The probability that v ≤ z ≤ 1 is equal to the probability that v is ble allocations, or n-tuples of bids, 1 1 contained in the spherical segment formed by cutting the surface   n  − S with the v2, v3-plane and a parallel plane which is height z above B ≡ b ∈ n  b ≤ 1 . R+  j the v2, v3-plane. This probability is equal to the ratio of the relevant  j=1 surface area of this spherical segment and the relevant surface area For the sake of brevity, we focus on the case that the number of of the sphere (2πz)/(2π) = z for all z ∈ [0, 1]. A similar argument battlefields, n, is equal to three. However, this approach readily applies for the v2 and v3 univariate marginal distributions. 3 ∗ { }3 generalizes to any n that is an integer multiple of three. Note that b ( vj j=1) is an admissible bidding strategy for all The two players simultaneously allocate resources across the v ∈ S. Then, because player B is following the equilibrium strategy ∗ { B}3 three battlefields. In each battlefield the player that allocates b ( vj j=1), player A’s optimization problem may be written as the higher level of the resource wins. If the players allocate the same level of the resource to a given battlefield, each player wins 3 − A A ∗ { B}3 with equal probability. Each player’s payoff equals the sum of the max vj Pr(bj > bj ( vj j=1)) (2) {bA∈ } valuations of the battlefields won. B j=1 Each player’s 3-tuple of battlefield valuations, v = (v , v , v ), 1 2 3 where for any bA ∈ B and any j ∈ {1, 2, 3}, is private information and is assumed to be independently drawn from a common 3-variate distribution function. In general, our A ∗ B 3    Pr(b > b ({v } )) = F bA = bA. problem gives rise to significant technical challenges. However, for j j j j=1 j j j the following parametric specification of the joint distribution of From Eq. (2), the Lagrangian for player A’s optimization problem the players’ types, there exists an intuitive pure-strategy Bayesian may be written as equilibrium.4 3 Consider the sphere of unit radius centered at the origin in R , 3    − A A − A + and let S denote the set of points on the surface of this unit sphere max vj bj λAbj λA. (3) A 3 {b ∈B} = that lie in R+: j 1   3  A = A 2  Solving the f.o.c. for each battlefield j, bj (vj /(2λA)) . Recalling ≡ ∈ 3 − 2 = S v R+ (vj) 1 . ∑3 A = ∑3 A 2 =  that in any optimal strategy j=1 bj 1 and j=1(vj ) 1 for  j=1 A ∈ = 1 all v S, it follows that λA 2 . Lastly, note that from (2) and Let P denote the 3-variate distribution function formed by A ∈ { (3) the expected payoff is an increasing concave function of b uniformly distributing mass over S. For each battlefield j 1, ∗ { A}3 on B. This completes the proof that the strategy b ( vj j=1) is a globally optimal response for player A given that player B is using ∗ { B}3 b ( vj j=1). The case of player B follows directly. 3 Just as in the classic complete information version of the Colonel Blotto game, The combination of multi-dimensional incomplete information moving from two to three battlefields allows for much greater flexibility in the and multi-dimensional action spaces usually entails significant choice of a relevant joint distribution (Roberson, 2006). technical difficulties. However, in the context of a Blotto game 4 Although we focus on the case in which the support of the joint distribution of battlefield valuations lies on one particular surface, this approach can be extended with multi-dimensional incomplete information we have shown to other surfaces. It is also straightforward to extend the analysis to arbitrary that if the common joint distribution of battlefield valuations symmetric budgets. is uniformly distributed on the surface S, then there exists an D. Kovenock, B. Roberson / Economics Letters 113 (2011) 273–275 275

∗ { }3 intuitive symmetric equilibrium b ( vj j=1) in which for each Amann, E., Leininger, W., 1996. Asymmetric all-pay auctions with incomplete ∗ information: the two-player case. Games and Economic Behavior 14, 1–18. battlefield j the coordinate function b { ′ }3 is a function of j ( vj j′=1) Borel, E., 1921. La théorie du jeu les équations intégrales à noyau symétrique. only that battlefield’s valuation vj and is strictly increasing in that Comptes Rendus de l’Académie 173, 1304–1308; valuation. English translation by Savage, L., 1953. The theory of play and integral equations with skew symmetric kernels. Econometrica 21, 97–100. Che, Y.-K., Gale, I., 1998. Standard auctions with financially constrained bidders. Acknowledgments Review of Economic Studies 65, 1–22. Kovenock, D., Roberson, B., 2010. Conflicts with multiple battlefields. We thank an anonymous referee for helpful comments. In: Garfinkel, M.R., Skaperdas, S. (Eds.), Oxford Handbook of the Economics of Peace and Conflict. Oxford University Press, Oxford. Krishna, V., Morgan, J., 1997. An analysis of the war of attrition and the all-pay References auction. Journal of Economic Theory 72, 343–362. Pitchik, C., 2009. Budget-constrained sequential auctions with incomplete informa- Adamo, T., Matros, A., 2009. A Blotto game with incomplete information. Economic tion. Games and Economic Behavior 66, 928–949. Letters 105, 100–102. Roberson, B., 2006. The Colonel Blotto game. Economic Theory 29, 1–24.