We consider a Blotto Game with Incomplete Information. A pure-strategy symmetric monotonic Bayesian equilibrium is found and its properties are discussed. Key words: Blotto Game, Imperfect Information. Journal of Economic Literature Classification: C72, D72.

Alexander Matros

English

Research Papers in Economics

A Blotto Game with Imperfect Information
Tim Adamo
University of Pittsburgh
Alexander Matros∗
University of Pittsburgh
July 31, 2008
Abstract. We consider a Blotto Game with Imperfect Information.
A pure-strategy symmetric monotonic Bayesian equilibrium is found and its
properties are discussed.
Key words: Blotto Game, Imperfect Information.
Journal of Economic Literature Classiﬁcation: C72, D72.
1. Introduction
A topic of a particular interest in economics and political science is the behavior
of players facing resource distribution decisions. We consider a Blotto game with
Imperfect Information regarding player resources where players seek to maximize their
expected payoﬀs. Players make allocation decisions simultaneously and whichever
player spends the most on a particular contest wins that contest. The natural set
of questions to ask in this situation is: How should players allocate their private
budgets? Will all players compete for all prizes? Should we expect one player to
win all the prizes? We ﬁnd and characterize a pure-strategy symmetric monotonic
Bayesian equilibrium (SMBE) for this game and answer these key questions.
There are two directions in the literature which are close to the topic of this pa-
per. The ﬁrst one is diﬀerent Colonel Blotto games; see Borel (1921), Gross and
Wagner (1950), Gross (1951), Blackett (1958), Bellman (1969), Young (1978), Laslier
and Picard (2002), Roberson (2006), Kvasov (2007), and Weinstein (2006) among
others. In its most general case, the Blotto game presents two players with the prob-
lem of allocating a ﬁnite resource pool over N “fronts” or contests, which are then
played simultaneously. Each front or contest has its own valuation attached to it, and
this in turn impacts how players choose to distribute their resources. Virtually all
Blotto formulations are analyzed in a setting of perfect information, where all players
have perfect knowledge regarding their opponents’ budget constraints. While analy-
sis carried out under the assumption of perfect information has been insightful, the
∗Corresponding author: Department of Economics, University of Pittsburgh, Pittsburgh, PA
15260, USA, alm75@pitt.edu, Phone: (412) - 383-8156, Fax: (412) - 648-1793
1A Blotto Game with Imperfect Information 2
predictions of those papers are always mixed-strategy equilibria and ﬂawless infor-
mation about opponents, and in particular about opponents’ budgetary constraints,
is rarely available.1
A substantial body of literature exists on one- and multi- object auctions where
bidders are subjected to budgetary constraints during the game (Che and Gale, 1998,
2003; Benoˆ ıt and Krishna, 2001; Szentes and Rosenthal, 2003a,b). Such models corre-
spond closely to the one that we are analyzing as players are forced to allocate scarce
resources over multiple contests. As in the case of the Blotto literature, virtually all
analysis of such auctions has been carried out under conditions of perfect information.
In one of the few models with an incomplete information setting analyzed by Che
and Gale (1998), private information enters the picture through player valuations of
one object, not player budgets as in our model. Furthermore, work by Pitchik (1995)
in a scenario of incomplete information that extends to budget constraints is limited
by the imposition of a one-dimensional bidder “type” which determines both bidder’s
valuation and budget.
Although our model does not allow for dynamic investigations of multi-unit cases
such as those conducted by Benoˆ ıt and Krishna (2001), it has the advantage of being
more realistic insofar as bidders’ budgets are private information and that this private
information is not restricted by the imposition of a bidder type which also determines
bidder valuations. In this sense, our model represents a case of the general multi-
object auction; we model a N-object auction with budget constraints, but with the
caveat that such constraints are private information. All that a bidder knows about
her opponent is the distribution from which her budget was drawn. Also our results
c a nb ec o m p l i m e n t e dw i t hr e s u l t so nt h es o-called Chopsticks auction obtained by
Szentes and Rosenthal (2003a,b). Their model speciﬁcally requires three objects in a
two-player auction, but each player wants to win a “majority”o fo b j e c t s .
2. The Model
There are K risk-neutral players. Each player k has her private budget Xk.W e
assume that each Xk is independently and identically distributed on the interval
[0,E], E<∞, according to the increasing distribution function F,w h e r eF (0) = 0,
F (E)=1 . T h e r ea r eN prizes. The value of prize i is Wi > 0f o ra l lp l a y e r s .
Each player k has to allocate her budget Xk across all N prizes. The players are
competing for all N prizes simultaneously. We assume that player k wins prize i if
xk
i > max
n
x1
i,...,x
k−1
i ,x
k+1
i ,...,xK
i
o
,w h e r ex
j
i is the budget allocation of player j for
prize i. We will use the upper index for the players and the lower index for the prizes.
1Snyder (1989) and Matros (2007) ﬁnd pure-strategy equilibria in stochastic versions of Blotto
games.A Blotto Game with Imperfect Information 3
All players submit their budget allocations simultaneously. A pure strategy of player
k is a N-dimensional vector
³
xk
1,...,xk
N
´
, such that
PN
j=1 xk
j = Xk.
Player k obtains the following payoﬀ
I
k
1W1 + ... + I
k
NWN,
where
I
k
i =
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
1, if xk
i > max
n
x1
i,...,x
k−1
i ,x
k+1
i ,...,xK
i
o
,
1/l, if xk
i =m a x
n
x1
i,...,xK
i
o
and there are l such players,
0, if xk
i < max
n
x1
i,...,x
k−1
i ,x
k+1
i ,...,xK
i
o
.
Our main assumptions are:
1. Each player k has her private budget Xk - the amount she has to allocate across
N prizes. Players cannot use their private budgets outside of the game.
2. A player wins a prize if she spends more than her opponents for this prize. If
several players spend the same amount, then they have an equal opportunity
to win the prize.
3. Each Xk is independently and identically distributed on the interval [0,E],
E<∞, according to the increasing distribution function F.F u n c t i o nF admits
a continuous density f ≡ F0 and has full support.
4. A player k knows the realization xk of Xk and only that the other players’
budgets are independently and identically distributed according to F.
5. Given the prize values, players maximize their expected payoﬀs.
6. The distribution function F is common knowledge.
A strategy for a player k is a function βk =
³
βk
1,...,βk
N
´
:[ 0 ,E] → [0,E]
N which
determines her budget allocation. Of course,
β
k
1
³
x
k
´
+ ... + β
k
N
³
x
k
´
≡ x
k (1)
and
β
k
i ≥ 0 for any k =1 ,...,K and i =1 ,...,N.A Blotto Game with Imperfect Information 4
2.1. Pure-Strategy Symmetric Monotonic Bayesian Equilibrium. We are
looking for a pure-strategy symmetric monotonic Bayesian equilibrium (SMBE) - an
equilibrium where all players follow the same strategy,
³
βk
1, βk
2,..., βk
N
´
=( β1, β2,..., βN)
for any k =1 ,...,K,a n di fx>y ,t h e nβi (x) > βi (y)f o rany prize i =1 ,...,N.2
Suppose that all players but player 1 follow the symmetric, increasing and diﬀer-
entiable strategy β∗ = β =( β1, β2,..., βN),
β1 (R)=α1R, β2 (R)=α2R,..., βN (R)=αNR, (2)
where αi ∈ [0,1], and α1 + α2 + ... + αN =1 .
Suppose that player 1 receives a budget, X1 = x, to allocate. We want to deter-
mine the optimal allocation (x1
1,...,x1
N), s.t. x1
1 + ... + x1
N = x. Then, player 1 solves
the following maximization problem
max
x1
1,...,x1
N
W1 Pr
n
x
1
1 > α1X
2,...,x
1
1 > α1X
K
o
+...+WN Pr
n
x
1
N > αNX
2,...,x
1
N > αNX
K
o
,
(3)
s.t.
N X
j=1
x
1
j = x. (4)
Denote G ≡ FK−1 and
T(x
1
1,...,x
1
N−1) ≡ W1G
Ã
x1
1
α1
!
+···+WN−1G
Ã
x1
N−1
αN−1
!
+WNG
Ã
x − (x1
1 + ···+ x1
N−1)
αN
!
.
Since we assume that the private budgets are distributed according to the distribution
function F on the interval [0,E], the maximization problem (3) − (4) becomes
max
x1
1,...,x1
N−1
T(x
1
1,...,x
1
N−1),
after the following substitutions: x1
N = x−
³
x1
1 + ... + x1
N−1
´
and αN =1 −(α1 + ... + αN−1).
The ﬁrst order conditions are
W1
α1
G
0
Ã
x1
1
α1
!
−
WN
(1 − (α1 + ... + αN−1))
G
0
⎛
⎝
x −
³
x1
1 + ... + x1
N−1
´
1 − (α1 + ... + αN−1)
⎞
⎠ =0 , (5)
...
WN−1
αN−1
G
0
Ã
x1
N−1
αN−1
!
−
WN
(1 − (α1 + ... + αN−1))
G
0
⎛
⎝
x −
³
x1
1 + ... + x1
N−1
´
1 − (α1 + ... + αN−1)
⎞
⎠ =0 . (6)
2Our requirement is similar to Noldeke and Samuelson’s (2003) “Net Viability” condition.A Blotto Game with Imperfect Information 5
At a symmetric monotonic equilibrium
x
1
1 = β1 (x)=α1x,..., x
1
N = βN (x)=αNx,
a n dt h u sw ec a nr e w r i t e( 5 )− (6) as
W1
α1
=
W2
α2
= ... =
WN
αN
,
and
α1 + ... + αN =1 .
Therefore, it follows that
α1 =
W1
W1 + ... + WN
,...,αN =
WN
W1 + ... + WN
. (7)
The derivation of the function β∗ =( β1, β2,..., βN) is only heuristic because
we have not formally established that if other players follow strategy β∗,t h e ni ti s
indeed optimal for a player with a private budget x to follow the same strategy β∗.
The following theorem veriﬁes when this is indeed correct.
Theorem 1. Suppose that
G
0 > 0,G
00 < 0. (8)
Then there exists a SMBE which is given by the function β∗ deﬁned in (2) and (7).
Proof: I no r d e rt op r o v et h et h e o r e m ,w eh a v et ov e r i f yt h a tT(x1
1,...,x 1
N−1)i sa
concave function. Since, by the assumption (8), function G is concave, T(x1
1,...,x 1
N−1)
must be a concave function as a sum of concave functions.3 End of proof.
L e tu sc o n s i d e rt h em o s tp o p u l a rc a s ei nt h el i t e r a t u r e ,K =2 . I ft h e r ea r et w o
players, then the main condition (8) in Theorem 1 becomes F0 > 0a n dF00 < 0.
Then, there exists a SMBE which is given by the function β∗ deﬁn e di n( 2 )a n d( 7 ) .
2.2. Properties of the SMBE. Since the SMBE is described in the previous
subsection, we can discuss its properties now. These properties provide answers to
the questions we posed in the Introduction. The next three results follow from the
structure of the SMBE.
Corollary 1. All players compete for all prizes in the SMBE.
3This is a well-known result that appears in all of the standard optimization theory texts. For
example see p. 519 of Simon and Blume (1994).A Blotto Game with Imperfect Information 6
Corollary 2. Each player spends more on more valuable prizes in the SMBE.
Corollary 3. The player with the highest budget wins all prizes in the SMBE.
The expected equilibrium payoﬀ of a player with budget x can be calculated
using the maximization problem (3) − (4). Corollary 2 provides an explanation for
the result: a player either wins all prizes or none of them.
Corollary 4. The expected equilibrium payoﬀ of a player with budget x is
(W1 + ... + WN)G(x).
If N = 2, then our model corresponds to a two-object auction with imperfect
information. The next results follow directly from Theorem 1.
Corollary 5. Suppose that N =2and
G
0 > 0,G
00 < 0.
Then there exists a SMBE in a two-object auction which is given by the function β∗
deﬁned in (2) and (7).
In this instance each bidder has to divide her budget between only two objects
and therefore there is no diﬀerence between simultaneous and sequential auctions.
However, this observation holds only for exactly two objects.
2.3. Extensions. Benoˆ ıt and Krishna (2001) have investigated the diﬀerence be-
tween auctions which are run simultaneously and auctions which are run sequentially.
In its current conception, our model, by its nature as a Blotto game, is a simultaneous-
move model; it would be a natural extension to consider a sequential Blotto game.
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A Blotto Game with Incomplete Information

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