Generalized Colonel Blotto Game

10/03/2017 ∙ by Aidin Ferdowsi, et al. ∙ University of Illinois at Urbana-Champaign Virginia Polytechnic Institute and State University 0

Competitive resource allocation between adversarial decision makers arises in a wide spectrum of real-world applications such as in communication systems, cyber-physical systems security, as well as financial, political, and electoral competition. As such, developing analytical tools to model and analyze competitive resource allocation is crucial for devising optimal allocation strategies and anticipating the potential outcomes of the competition. To this end, the Colonel Blotto game is one of the most popular game-theoretic frameworks for modeling and analyzing such competitive resource allocation problems. However, in many real-world competitive situations, the Colonel Blotto game does not admit solutions in deterministic strategies and, hence, one must rely on analytically complex mixed-strategies with their associated tractability, applicability, and practicality challenges. In this paper, a generalization of the Colonel Blotto game which enables the derivation of deterministic, practical, and implementable equilibrium strategies is proposed while accounting for the heterogeneity of the battlefields. In addition, the proposed generalized game enables accounting for the consumed resources in each battlefield, a feature that is not considered in the classical Blotto game. For the generalized game, the existence of a Nash equilibrium in pure-strategies is shown. Then, closed-form analytical expressions of the equilibrium strategies, are derived and the outcome of the game is characterized; based on the number of resources of each player as well as the valuation of each battlefield. The generated results provide invaluable insights on the outcome of the competition. For example, the results show that, when both players are fully rational, the more resourceful player can achieve a better total payoff at the Nash equilibrium, a result that is not mimicked in the classical Blotto game.

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I Introduction

Game-theoretic resource allocation models have become prevalent across a variety of domains such as wireless communications[1] and [2], cyber-physical security [3, 4, 5], financial markets, and political/electoral competitions[6, 7, 8]. One of the mostly adopted game-theoretic frameworks for modeling and analyzing such competitive resource allocation problems is the so-called Colonel Blotto game (CBG[9]. The CBG captures the competitive interactions between two players that seek to allocate resources across a set of battlefields. The player who allocates more resources to a certain battlefield wins it and receives a corresponding valuation. The fundamental question that each player seeks to answer in a CBG is how to allocate its resources to maximize the valuations acquired from the won battlefields. In recent years, many variants of the CBG have been studied including those with homogeneous valuations [8, 10], symmetric resources, multiple players [4], and heterogeneous resources [11, 12, 13].

This existing body of literature has primarily focused on analyzing the existence of the equilibrium in a CBG. For instance, the seminal work in [9]

has proven that an equilibrium in deterministic strategies (i.e. pure-strategies) of the Blotto game does not exist when the number of considered battlefields is greater than two. As a result, most of the literature has henceforth focused on analyzing the mixed-strategy equilibrium of the CBG, which essentially corresponds to a multi-variate probability distribution over the resources to be allocated over each of the battlefields. In this regard, the works in

[12, 9, 13] have studied uniform marginal distributions of resources on each battlefield.

However, relying on mixed strategies, as has been the case in all of the CBG literature [9, 12, 13, 10, 8, 11], presents serious challenges to the tractability, applicability, and implementability of the derived solutions in real-world environments. In fact, deriving such mixed strategies is often overly complex requiring various mathematical simplifications along the way for tractability [14, 2, 1, 15, 3, 4]. Moreover, in terms of applicability, the optimality of such mixed strategies assumes that the game is actually played for an infinitely large number of times [16]. In addition, previous CBG models assume win-or-lose settings over each battlefield. In other words, the player that allocates more resources over a battlefield wins all of its valuation and the other player receives nothing. However, given that resources are consumed over a certain battlefield, even when this battlefield is won or lost, these consumed resources must be accounted for in the game model. For instance, even when losing a battlefield, destroying a portion of the resources of the opponent can be considered a gain for the non-winner.

The main contribution of this paper is therefore to develop a fundamentally novel generalization of the CBG dubbed as the Generalized Colonel Blotto Game (GCBG) which captures the rich and broad settings of the classical CBG, and, yet enables the derivation of tractable, deterministic, and practical solutions to the two-player competitive resource allocation problem while not being limited to studying win-or-lose settings over each battlefield. To this end, we first introduce the basics of the proposed GCBG and, then, we compare its features to the classical CBG. Subsequently, we prove the existence of an equilibrium to the GCBG – a pure-strategy Nash equilibrium (NE) – and provide a detailed derivation as well as closed-form analytical expressions of the pure NE strategies of the GCBG. In addition, using the derived NE strategies, we prove that when both players are fully rational, the more resourceful player is able to achieve an overall payoff that is greater than that of its opponent, and hence, win the game. In contrast with the classical CBG, in our proposed GCBG:

  • We derive deterministic equilibrium strategies which are practical, in the sense that the players can derive exact optimal strategies and do not need to randomize between possible strategies,

  • We characterize low complexity solutions even for a large number of battlefields with heterogeneous valuations,

  • We model realistic applications of competitive resource allocation problems by taking into consideration partial winning and losing over a battlefield.

Finally, we provide a numerical example which showcases the effect of the number of resources of each player on the NE strategies and outcomes of the GCBG.

Ii Proposed Generalized Colonel Blotto Game

Ii-a Classical Colonel Blotto Game

A standard CBG[9] is defined by six components: a) the players in the set , b) the strategy spaces for , c) the available resource for , d) a set of battlefields, , e) the normalized value of each battlefield, for , and f) the utility function, , for each player. The set of possible strategies for each player, , corresponds to the different possible distributions of its resources across the battlefields. Hence, is defined as

(1)

where is the number of allocated resource units by player to battlefield , and

is the vector of these allocated resources (an allocation strategy of player

) . To this end, the payoff of player , for a battlefield , is defined as:

(2)

where corresponds to the resources allocated by the opponent of to battlefield . Such resource allocation strategies are known as pure deterministic strategies since the allocation is not randomized over the battlefields. The utility function of each player, , over all battlefields in is defined as:

(3)

Each player in aims to choose a resource allocation strategy that maximizes its payoff over the battlefields given the potential resource allocation strategy of its opponent. As such, the pure-strategy NE of the CBG can be formally defined as follows:

Definition 1.

A strategy profile , such that and , is a pure-strategy Nash equilibrium of the CBG if

(4)

For the CBG, it has been proven [9] that for and , there exists no pure-strategy NE. As a result, for solving the CBG, a common methodology is to explore mixed strategies based on which each player

chooses a probability distribution (or a multi-variate probability density function) over

to maximize its potential expected payoff. However, the reliance on such mixed strategies introduces high analytical complexity and presents challenges in terms of the tractability of the derivation of the solutions as well as to the practicality and applicability of these solutions. Hence, to enable the derivation of tractable, deterministic, and practical solutions to the competitive resource allocation problem, we next propose a generalization of the CBG dubbed generalized Colonel Blotto game.

Ii-B Proposed Generalized Colonel Blotto Game

A GCBG is defined by seven components that include the players, strategy spaces, available resources, battlefields, and valuation of battlefields as is the case in the CBG. However, we define a new utility function as an approximation to the utility function of the CBG. This approximation is based on a continuous differentiable function and an approximation parameter, . When increases, the GCBG will very closely resemble the CBG. At the limit, when goes to infinity, the GCBG converges to the CBG. However, in contrast with the CBG, the differentiability of the approximate utility function enables the derivation of deterministic equilibrium strategies.

In this respect, as defined in (2), the payoff from each battlefield resembles a step function. Hence, we propose an approximation of this function, , referred to hereinafter as approximation, defined as follows:

(5)

and the utility function of each player is the summation of the approximation payoffs from each battlefield. Next, in Lemma 1, we will show that (5) precisely approximates as the approximation factor tends to infinity. Fig. 1 provides a number of plots of approximation functions for different values of .

Lemma 1.

The payoff from each battlefield can be expressed as follows:

(6)
Proof.

The limit of the -approximate payoff when can be written as:

Therefore, as compared to (2),

(7)

which proves the lemma. ∎

Hence, Lemma 1 provides an approximation for the payoff from each battlefield. Using instead of provides two additional benefits as compared to the CBG: a) the continuity and differentiability of helps in deriving pure-strategy NEs, and b) captures the notion of partial-win-or-lose by accounting for resource consumption and losses which is a practical feature in a GCBG that has not been considered in the CBG. In fact, in practice, the payoff from each battlefield must consider the portion of resources destroyed/consumed following the “battle” between the two players. This means that the payoff from each battlefield has to capture the loss due to the resources destroyed in this battlefield, which also corresponds to a gain for the opponent. The approximate utility function in (5) inherently captures this aspect. Fig. 1 shows the continuous transition from losing to winning a certain battlefield when using -approximate payoff functions. Here, we note that the larger is, the closer the -approximation approximates the classical CBG.

Figure 1: -approximate of the payoff from battlefield 1 for player when .

Now, we define the -approximate utility function per player:

(8)

From (5) and (8), and given that

is an odd function, the players’ payoffs from each battlefield can be related following

which results in the following relation between the players’ utility functions:

(9)

The relation in (9) shows that the GCBG is a constant-sum game in which the pure-strategy NE is the solution of the following minimax problem[16]:

(10)

where is the value of the GCBG with the -approximate utility function for player . Next, we solve the minimax problem in (10) and derive the pure-strategy NE of the GCBG.

Iii Solution of the GCBG

To solve the minimax problem in (10), we must derive which maximizes for all possible . Then, we choose which minimizes considering the response, , to any chosen . Given the resource limitation of each player, a limit exists on the chosen allocations as expressed in (1). First, we prove that this limit is binding, i.e. each player is always better off spending all their available resources.

Proposition 1.

All the strategies of the form are dominated by the strategies i.e, player is better off using all of its available resource.

Proof.

The proof is by contradiction. Suppose is the vector that maximizes such that: Then, the payoff for player choosing strategy is:

Now, considering arbitrarily player , we define a new allocation vector for player as . Then, the payoff for player will be:

which contradicts with the assumption maximizes . ∎

From Proposition 1, we know that the limit on the total number of allocated resources is binding. Before characterizing the solution of the game, we first define the difference between the allocated resources by each player on each battlefield , , and the difference between the total available resources, , as follows:

(11)

In addition, we consider, without loss of generality, that , which implies that the available resources of an arbitrary player are less than those of its opponent, player . We also consider the case in which , which eliminates the trivial case studied in the CBG, in which, if , player can trivially win the game. Using these two inequality assumptions on and , the constraint on can be expressed as follows:

(12)

From Proposition 1 and (11), we can write:

(13)

Since the utility functions for both players are dependent on the difference of allocated resources on each battlefield, , hereinafter, we use and interchangeably, where . In what follows, we fully characterize the pure-strategy NE for the GCB game.

First, we focus on solving the maximization component of the minimax problem in (10). In this regard, Theorem 1 characterizes the local maxima of . Here, without loss of generality, we assume that the battlefields are indexed based on an increasing order of their value, i.e. with one of these inequalities being strict to avoid solving a trivial case in which all battlefields are identical.

Theorem 1.

A local maximum of with respect to , is a solution to the following equation:

(14)

where

(15)

and

Proof.

Based on Proposition 1, can be expressed as , and player ’s utility function will be:

(16)

Hence, to find the local maxima of player ’s utility function, from the first order necessary condition of optimality, we know that, if is a local maximizer and is continuously differentiable, then Therefore, for , we must have and, hence:

(17)

Thus, must be one of two solutions:

(18)

which, from (13), must meet .

Next, from the second order necessary conditions of optimality, we know that, if is a local maximizer of , then the Hessian matrix, , must be negative definite. The second order derivatives, and elements of the Hessian matrix, are calculated as follows:

(19)
(20)

From (III), we have:

(21)

Plugging (21) in the expressions of the elements of the Hessian matrix yields:

(22)
(23)

Then, the Hessian matrix at is expressed as:

(24)

where

(25)

For to be a local maximum, must be negative definite and, hence, must be positive definite. Performing row operations on , we obtain an upper-triangular matrix:

(26)

where

(27)

is positive definite if and only if its pivots (i.e. its diagonal elements) are all positive since the number of positive pivots of

is equal to the number of positive eigenvalues of

 [17]. Hence, by inspecting the diagonal elements of in (26), for to be positive definite, must be positive for . Combining this result with (18), we obtain the expressions in (15) for .

Next, we investigate the sign of in (27) which must also be positive for to be positive definite. Knowing that for , we investigate four different cases regarding the signs of and and their effect on the sign of . We prove that and as defined in (15) is the only case that leads to a positive definite and, as a result, a negative definite .

Case 1. , and : If and , then the element of in (25) is negative. In addition, for a matrix to be positive definite, none of its diagonal elements can be negative [17]. As such, if , and , then is not positive definite and hence is not negative definite.

Case 2. , and : We rewrite in (27) as , where for and ,

(28)

Based on the binomial approximation, if and , let . For practical -approximations, i.e., when is a large number (), meets the conditions of the binomial approximation and is such that for all . As such, let for all , can be approximated as:

(29)

Hence, since for we obtain is not positive definite is not negative definite; for any practically large approximation index .

Case 3. , and : We express as , where

(30)

Since , the sign of is the same as the sign of . By taking the first derivative of with respect to , we can see that , since and for all and for all . Hence is decreasing in . Knowing that , the upper bound of , , is then

(31)

Hence, is negative when

(32)

Here, we note that has the same order of magnitude as . Hence, for a practical approximation index, , the condition in (32) always holds true. Thus, this implies that is not positive definite and, hence, is not negative definite for any practically large .

Case 4. , and : For this case, , hence, is positive definite and as a result is negative definite. Hence, only for , and , is positive definite.

Therefore, the solutions proposed in (15) and meeting (14) such that , are the only possible local maxima of . ∎

In Theorem 1, we identified the local maxima of as given in (15) and meeting . We also proved that these solutions must satisfy (14). Next, we study the equality constraint in (14) which must be met. To this end, we let

(33)

is a strictly convex function with a negative minimizing as stated in Proposition 2.

Proposition 2.

is a strictly convex function whose minimum occurs at a negative .

Proof.

To prove the convexity of , we compute its second derivative with respect to .

(34)

By inspecting the numerator of each term in , every numerator can be reduced to which is positive. Hence, is a summation of positive terms (at least one of which is strictly positive since at least one for some ). Hence, and is strictly convex. To obtain the sign of which minimizes , we inspect the first derivative of . Since is a minimum of , we must have:

(35)

Since the denominator of the summation term in (35) is always positive, and since for all is always positive, must be negative to solve (35). ∎

Proposition 2 is valuable for characterizing the possible solutions in to , and equivalently to (14), as stated in Corollary 1. To this end, let be the minimum value of which is unique given that was proven to be strictly convex.

Corollary 1.

has: a) two solutions if , b) one solution if , or c) no solution if .

Proof.

From Proposition 2, we know that is a strictly convex function. Therefore any line parallel to the axis will intersect in two points if , one point if , and will not intersect if . ∎

Based on Corollary 1 and Theorem 1, we can conclude that for , we have one or two possible local maxima for . However, we must check whether these maxima are feasible as they should correspond to , which is a necessary condition as proven in Theorem 1. These maxima also must satisfy the following feasibility conditions on for :

(36)

Therefore, for any solution of , we must check that , for , satisfy

(37)
Figure 2: Comparison of solutions of equations , and .

Next, we show that only one solution of satisfies (37).

Theorem 2.

For , has a unique maximum.

Proof.

From Corollary 1, we know that, for , has two solutions which we denote as . We need to prove that only one of these solutions satisfies (37). First, we prove that has two slant asymptotes and , where . To prove this statement, we find the linear approximations of as goes to and , as follows:

(38)
(39)

Moreover, to prove that we must have:

which always holds true since for ; with at least one strict inequality for an . Hence, . Using a similar approach, we can also prove that