In the Colonel Blotto game, each of two players simultaneously allocates his fixed budget of a resource across a finite number n of battleelds. Within each battlefield, the player that allocates the higher level of the resource wins the battlefield. Each player's payoff is equal to the sum of the values of the battlefields he wins. In this paper we examine a multi-dimensional incomplete information version of the Colonel Blotto game in which each player's n-tuple of battlefield valuations is drawn from a common n-variate joint distribution function.Colonel Blotto Game, Con
ict, Multi-dimensional Incomplete Information, Multi-dimensional Action Space

Brian Roberson

Dan Kovenock

English

Research Papers in Economics

KRANNERT SCHOOL OF 
MANAGEMENT 
 
    Purdue University 
    West Lafayette, Indiana 
 
 
A Blotto Game with Multi-Dimensional Incomplete 
Information 
 
By 
 
Dan Kovenock 
Brian Roberson 
 
Paper No. 1262 
Date: November, 2010 
Institute for Research in the 
Behavioral, Economic, and 
Management Sciences A Blotto Game with Multi-Dimensional Incomplete
Information
Dan Kovenock1 Brian Roberson2
November 2010
1Dan Kovenock, University of Iowa, Department of Economics, Tippie College of Business,
W284 PBB, 121 E. Market Street, Iowa City, IA 52242 USA t: 319-335-1810, E-mail:dan-
kovenock@uiowa.edu (Correspondent)
2Brian Roberson, Purdue University, Department of Economics, Krannert School of Man-
agement, 403 W. State Street, West Lafayette, IN 47907 USA t:765-494-4531, E-mail:
brobers@purdue.eduAbstract
In the Colonel Blotto game, each of two players simultaneously allocates his xed budget
of a resource across a nite number n of battleelds. Within each battleeld, the player
that allocates the higher level of the resource wins the battleeld. Each player's payo is
equal to the sum of the values of the battleelds he wins. In this paper we examine a
multi-dimensional incomplete information version of the Colonel Blotto game in which each
player's n-tuple of battleeld valuations is drawn from a common n-variate joint distribution
function.
JEL Classication: C72, D72, D74
Keywords: Colonel Blotto Game, Conict, Multi-dimensional Incomplete Information,
Multi-dimensional Action Space1 Introduction
The classic formulation of the Colonel Blotto game is a two-player constant-sum game with
complete information (Borel 1921). There are a nite number n of battleelds. Each player
simultaneously allocates his xed amount of a resource across the n battleelds. Within each
battleeld, the player that allocates the higher level of the resource wins the battleeld, and
each player's payo is equal to the number of battleelds won.1 A distinguishing feature
of the Colonel Blotto game is that, except for trivial cases, an equilibrium is a pair of
non-degenerate multi-dimensional mixed strategies.
This note investigates a multi-dimensional incomplete information version of the Colonel
Blotto game with two risk-neutral players. The incomplete information is multi-dimensional
in that each player's n-tuple of battleeld valuations is drawn from a common n-variate
joint distribution. Our game is therefore within the 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 battleeld valuations is a uniform distribution over
the nonnegative points on the surface of a sphere, we show that there exists an intuitive
symmetric pure-strategy Bayesian equilibrium.
Most closely related to our focus is Adamo and Matros (2009) who introduce incomplete
information in the Colonel Blotto game framework. In that model, k players compete across
a set of n battleelds with valuations that are common knowledge. These valuations may
vary across battleelds, but each battleeld's valuation is the same for all players. The
incomplete information is in the form of uncertainty regarding the players' respective budgets
of the resource, which are assumed to be independent draws from a common distribution
that is absolutely continuous on the closed interval between 0 and the sum of the individual
battleeld valuations. Under the added assumption that the distribution of the highest
1For further details on this class of games, see Roberson (2006) and Kovenock and Roberson (2010).
1order 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 player allocates
to each battleeld a share of his budget equal to that battleeld's value divided by the sum of
all battleeld valuations. That is, the one-dimensional nature of the incomplete information
allows for each element in the n-tuple of a player's allocation of force across battleelds to
be conditioned on a single variable, his budget.
Also related are Che and Gale (1998) and Pitchik (2009) who examine incomplete infor-
mation auctions with budget-constrained players. In fact, our model corresponds to the case
of two budget-constrained players competing across a set of all-pay auctions with incomplete
information with the exception that for each player any unused resources are forfeited. The
case of a single all-pay auction with incomplete information is addressed by Amann and
Leininger (1996), who examine asymmetrically distributed independent private values and
Krishna and Morgan (1997), who examine symmetrically distributed aliated signals. In
this note we show that if the joint distribution of the players' valuations takes a particu-
lar parametric form, then the analysis of the all-pay auction with incomplete information
can be extended to cover our case of multiple auctions, binding budget constraints, and
use-it-or-lose-it resources.
2 Model
There are two risk-neutral players, i 2 fA;Bg, and a nite number n of battleelds, j 2
f1;:::;ng. Each player has a budget of one unit of a homogenous resource. Let B denote
the set of feasible allocations, or n-tuples of bids,
B 
(
b 2 R
n
+
  
n X
j=1
bj  1
)
:
2For the sake of brevity, we focus on the case that the number of battleelds, n, is equal to
three. However, this approach readily generalizes to any n that is an integer multiple of
three.2
The two players simultaneously allocate resources across the three battleelds. In each
battleeld the player that allocates the higher level of the resource wins. If the players
allocate the same level of the resource to a given battleeld, each player wins with equal
probability. Each player's payo equals the sum of the valuations of the battleelds won.
Each player's 3-tuple of battleeld valuations, v = (v1;v2;v3), is private information and
is assumed to be independently drawn from a common 3-variate distribution function. In
general, our problem gives rise to signicant technical challenges. However, for the following
parametric specication of the joint distribution of the players' types, there exists an intuitive
pure-strategy Bayesian equilibrium.3
Consider the sphere of unit radius centered at the origin in R3, and let S denote the set
of points on the surface of this unit sphere that lie in R3
+:
S 
(
v 2 R
3
+
  
3 X
j=1
(vj)
2 = 1
)
:
Let b P denote the 3-variate distribution function formed by uniformly distributing mass over
S. For each battleeld j 2 f1;2;3g, the univariate distribution function of vj, Fj(z), is equal
to the b P-volume of the set of v 2 S such that vj  z  1.
2Just as in the classic complete information version of the Colonel Blotto game, moving from two to three
battleelds allows for much greater exibility in the choice of a relevant joint distribution (Roberson 2006).
3Although we focus on the case in which the support of the joint distribution of battleeld valuations lies
on one particular surface, this approach can be extended to other surfaces.
33 Results
A strategy is a function b : S ! B. Denote player i's 3-tuple of bids as b
i(fvi
jg3
j=1) =
(bi
1(fvi
jg3
j=1);bi
2(fvi
jg3
j=1);bi
3(fvi
jg3
j=1)).
Proposition 1. If each player's n-tuple of battleeld valuations is drawn from the joint
distribution b P, then
b
(fvjg
3
j=1) =

(v1)
2;(v2)
2;(v3)
2	
(1)
is a symmetric pure-strategy Bayesian equilibrium.
We briey outline the proof of Proposition 1. Lemma 1 states a useful property of the
joint distribution b P.
Lemma 1. The joint distribution b P satises the property that for each j 2 f1;:::;3g,
Fj(vj) = vj for vj 2 [0;1], i.e. each vj is distributed uniformly on [0;1].
Lemma 1 follows from the properties of spherical segments. In particular, note that the
surface area of any spherical segment is 2rh where h is the height between the two parallel
planes that cut the sphere. The unit sphere has radius r = 1, surface area 4, and exactly
(1=8)th of the sphere is contained in the positive orthant of R3. Thus, the surface area of
the relevant portion of the unit sphere is (=2). Similarly, the surface area of the spherical
segment formed by cutting the surface S with two parallel planes is (h=2). The probability
that v1  z  1 is equal to the probability that v1 is contained in the spherical segment
formed by cutting the surface S with the v2;v3-plane and a parallel plane which is height
z above the v2;v3-plane. This probability is equal to the ratio of the relevant surface area
of this spherical segment and the relevant surface area of the sphere (2z)=(2) = z for all
z 2 [0;1]. A similar argument applies for the v2 and v3 univariate marginal distributions.
Note that b
(fvjg3
j=1) is an admissible bidding strategy for all v 2 S. Then, because
player B is following the equilibrium strategy b
(fvB
j g3
j=1), player A's optimization problem
4may be written as
max
fbA2Bg
3 X
j=1
v
A
j Pr(b
A
j > b

j(fv
B
j g
3
j=1)) (2)
where for any b
A 2 B and any j 2 f1;2;3g,
Pr(b
A
j > b

j(fv
B
j g
3
j=1)) = Fj
q
bA
j

=
q
bA
j :
From equation (2), the Lagrangian for player A's optimization problem may be written as
max
fbA2Bg
3 X
j=1
h
v
A
j
q
bA
j   Ab
A
j
i
+ A: (3)
Solving the f.o.c. for each battleeld j, bA
j = (vA
j =(2A))2. Recalling that in any optimal
strategy
P3
j=1 bA
j = 1 and
P3
j=1(vA
j )2 = 1 for all vA 2 S, it follows that A = 1
2. Lastly, note
that from (2) and (3) the expected payo is an increasing concave function of b
A on B. This
completes the proof that the strategy b
(fvA
j g3
j=1) is a globally optimal response for player
A given that player B is using b
(fvB
j g3
j=1). The case of player B follows directly.
The combination of multi-dimensional incomplete information and multi-dimensional ac-
tion spaces usually entails signicant technical diculties. However, in the context of a
Blotto game with multi-dimensional incomplete information we have shown that if the com-
mon joint distribution of battleeld valuations is uniformly distributed on the surface S, then
there exists an intuitive symmetric equilibrium b
(fvjg3
j=1) in which for each battleeld j
the coordinate function b
j(fvj0g3
j0=1) is a function of only that battleeld's valuation vj and
is strictly increasing in that valuation.
5References
[1] Adamo, T., & Matros, A. (2009) A Blotto game with incomplete information. Economic
Letters. 105:100-102.
[2] Amann, E., & Leininger, W. (1996) Asymmetric all-pay auctions with incomplete in-
formation: the two-player case. Games and Economic Behavior. 14:1-18.
[3] Borel, E. (1921) La th eorie du jeu les  equations int egrales  a noyau sym etrique. Comptes
Rendus de l'Acad emie. 173:1304-1308; English translation by Savage, L. (1953) The
theory of play and integral equations with skew symmetric kernels. Econometrica. 21:97-
100.
[4] Che, Y.-K., & Gale, I. (1998) Standard auctions with nancially constrained bidders.
Review of Economic Studies. 65:1-22.
[5] Kovenock, D., & Roberson, B. (2010) Conicts with multiple battleelds. in Garnkel,
M.R., & Skaperdas, S., eds. Oxford Handbook of the Economics of Peace and Conict,
Oxford University Press, forthcoming.
[6] Krishna, V., & and Morgan, J. (1997) An analysis of the war of attrition and the all-pay
auction. Journal of Economic Theory. 72:343-362.
[7] Pitchik, C. (2009) Budget-constrained sequential auctions with incomplete information.
Games and Economic Behavior. 66:928-949.
[8] Roberson, B. (2006) The Colonel Blotto game. Economic Theory. 29:1-24.
6

A Blotto Game with Multi-Dimensional Incomplete Information

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A Blotto Game with Multi-Dimensional Incomplete Information

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