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An algorithmic solution to the Blotto game using multi-marginal couplings

02/15/2022
by   Vianney Perchet, et al.
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We describe an efficient algorithm to compute solutions for the general two-player Blotto game on n battlefields with heterogeneous values. While explicit constructions for such solutions have been limited to specific, largely symmetric or homogeneous, setups, this algorithmic resolution covers the most general situation to date: value-asymmetric game with asymmetric budget. The proposed algorithm rests on recent theoretical advances regarding Sinkhorn iterations for matrix and tensor scaling. An important case which had been out of reach of previous attempts is that of heterogeneous but symmetric battlefield values with asymmetric budget. In this case, the Blotto game is constant-sum so optimal solutions exist, and our algorithm samples from an -optimal solution in time O(n^2 + ^-4), independently of budgets and battlefield values. In the case of asymmetric values where optimal solutions need not exist but Nash equilibria do, our algorithm samples from an -Nash equilibrium with similar complexity but where implicit constants depend on various parameters of the game such as battlefield values.

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1 Introduction

A century ago, Emile Borel published his seminal paper on

the theory of play and integral equations with skew symmetric kernels

 [8], see also [42, Page 157]. While perhaps not as conspicuous, it predates von Neumann’s monumental work on the theory of games of strategy [41]

by several years. In this work, Borel describes what is now called the Blotto game and introduces the notions of strategy, mixed strategies and even foresees the fruitful interactions between game theory and economics that are to be observed throughout the century. As such, the Blotto game is considered to be the genesis of modern game theory 

[14, 30]. Despite its prestigious pedigree, equilibrium strategies for this game are only known in special cases.

Blotto is a resource-allocation game in which two players competes over different battlefields by simultaneously allocating resources to each battlefield. The following two additional characteristics are perhaps the most salient features of the Blotto game:

  1. Winner-takes-all: For each battlefield, the player allocating the most resources to a given battlefield wins the battlefield.

  2. Fixed budget: each player is subject to a fixed—and deterministic—budget that mixed strategies should satisfy almost surely.

Despite its apparent simplicity the Blotto game captures a variety of practical situations that extend far beyond the context of the above military terminology. These include political strategy [29, 24, 28], network security [22, 13], and various forms of practical auction markets [27, 17].

The goal of this paper is to efficiently construct a Nash equilibrium for this game or, when they exist, an optimal strategy.

Prior work. Despite its century-long existence, Nash equilibria for the Blotto game are only known under various restrictions on the main parameters of the problem: the budget of each player and the value given to each battlefield.

  • Budget. A large fraction of the literature considers the case where the players have symmetric budgets, starting with the original problem of Borel [8] and in most of the main contributions throughout the twentieth century [8, 9, 15, 16, 23, 39]. The case of symmetric budgets is well understood except in the setup where players may disagree on the value of battlefield that was recently introduced [21].

  • Battlefields. When the two players have a different budget the situation becomes more complex as the poorest will have to forfeit some battlefields. In this case, only partial results are known. To understand what “partial" means, recall that full generality of the battlefield values occurs when (i) players may assign a different value to a given battlefield—we say that the values are asymmetric—and (ii) these values may vary across battlefield—we say that the battlefields are heterogeneous. Partial results are known for symmetric values. Even under this simplifying assumption, the case of heterogeneous battlefields remains poorly understood, except in the case of two battlefields [26]. In the case of more than two battlefields, Nash equilibria are known for homogeneous battlefields [33] or under stringent assumptions on the battlefield values [36] that essentially reduce to the homogeneous case.

We refer the reader to Table 1 for a survey of recent advances. While we tackle the most general setup to date, we stress that an important case was not covered by prior literature: the case of asymmetric budget, heterogeneous and symmetric values. Indeed, in this case, the game is constant-sum and optimal strategies exist. Our results also cover the case of asymmetric values introduced very recently in [21] but this setup leads to only Nash equilibria rather than optimal strategies. Whenever possible, we conflate the two setups and simply refer to a solution to the Blotto.

A discrete version of the Blotto game where both budgets and allocations are required to be integral was also introduced in Borel’s original paper [8]. Explicit optimal solutions were provided in [18] for the homogeneous and symmetric version of the discrete game; see also [19] for partial solutions in the asymmetric-value case. More recently, this discrete version has seen significant computational advances [2, 6]. Conceptually, this line of work is close to the present paper in the sense that it provides an algorithm to sample from approximate solutions. Moreover, the discrete Blotto game can be seen as a discretization of the continuous version of interest here and that could be quantified using the arguments of Section 4. However carrying out this analysis, for instance based on [6, Theorem 4.2], leads to worse dependence on and compared to Theorem 12 here. More strikingly, the complexity bound of Theorem 12 does not depend on budgets or battlefield values while this dependence is polynomial in the bounds for discrete Blotto. The two lines of work also differ in more profound ways. First and foremost, the approach employed here is fundamentally different: it aims at mixing known solutions for the related Lotto game while solutions to the discrete Blotto games are more agnostic so that it is unclear what the marginals of the resulting strategy are. In particular, the present approach allows us sample from -Nash equilibria in the asymmetric-value case whereas this setup is currently out of reach for solutions to the discrete Blotto game.

Finally, note that our approach also yields new (existential) results for the discrete Blotto game. Since they are not the focus of our contribution, they are relegated to the appendix.

Our contributions. All of the above solutions for two-player games have consisted in constructing explicit solutions. Because of the budget constraints, these strategies can be decomposed in two parts: marginal distributions that indicate which (random) strategy to play on each battlefield and a coupling that correlates the marginal strategies in such a way to ensure that the budget constrained is satisfied almost surely.

The first question may be studied independently of the second by considering what is known as the (General) Lotto game [7]. In this game the budget constraint need only be enforced in expectation with respect to the randomization of the mixed strategies. While this setup lacks a defining characteristic of the Blotto game (fixed budget), it has the advantage of landing itself to more amenable computations. Indeed, unlike the Blotto game, a complete solution to the Lotto game was recently proposed in [21] where the authors describe an explicit Nash equilibrium in the most general case: asymmetric budget, asymmetric and heterogeneous values.

In light of this progress a natural question is whether the marginal solutions discovered in [21] can be coupled in such a way that the budget constraint is satisfied almost surely. We provide a positive answer to this question by appealing to an existing result from the theory of joint mixability [43]. Mixability asks the following question: Can random variables with prescribed marginal distributions , be coupled in such a way that . Joint mixability is precisely the step required to go from a Lotto solution to a Blotto one by coupling the marginals of the Lotto solution in such a way that the budget constraint is satisfied.

In this paper we exploit a simple and new connection between joint mixability and the theory of multi-marginal couplings that has recently received a regain of interest in the context of optimal transport [1, 12, 5]. In multi-marginal optimal transport, the goal is to optimize a cost over the space of couplings with given marginals. Unlike the case of two marginals that arises in traditional optimal transport, this question raises significant computational challenges and often leads to NP-hardness [3]. In the language of optimization, joint mixability merely asks if the set of constraints is nonempty. We propose an algorithmic solution to the Blotto problem by efficiently constructing a coupling that satisfies the budget constraint almost surely and can be easily sampled from. Our construction relies on three key steps: first we reduce the problem to a small number of marginals to bypass the inherent NP hardness of multi-marginal problems, second we discretize the marginals and finally, we employ a multi-marginal version of the Sinkhorn algorithm [37, 38] to construct a coupling of the discretized marginals. After a simple smoothing step, we produce a sampling with continuous marginals that are close to the ones prescribed by the Lotto solutions and from which it is straightforward to sample. Furthermore, we quantify the combined effect of discretization error and of the Sinkhorn algorithm on the value of the game, effectively leading to an approximate Nash equilibrium and even to an approximately optimal solution in the case of symmetric values.

The rest of this paper is organized as follows. In the next section, we recall the solution for the Lotto game and show that they can be turned into solutions for the Blotto game. This existential result simply appeals to existing results of joint mixability. We move from an existential to an algorithmic result in Section 3 by proceeding in three steps: first we reduce the problem to the case , then we discretize the problem and finally we apply Sinkhorn algorithm to couple the resulting marginals in a appropriate fashion. The main product of Section 3 is Algorithm 8 which shows how to sample from an approximate solution to the Blotto game. Finally, we provide a detailed complexity analysis for this algorithm in Section 4, showing in particular, that it runs in time polynomial in the parameters of the Blotto game and the approximation error . Finally, our techniques also yield new results for the discrete Blotto game largely studied by [18, 19] that are of independent interest. We postpone them to the appendix.

Notation. For any integer , define . We use

to denote an all-ones vector or tensor. Note that the dimension of this vector will be clear from the context but may vary across occurrences. For any two vectors

, we denote their entrywise (Hadamard) product and their entrywise division whenever has only nonzero entries. For any two real numbers we denote by their maximum and by their minimum.

2 Solutions for Blotto and Lotto games

The goal of this section is to describe the Blotto game and its connection to the Lotto game for which explicit solutions are known. We first recall a solution for the Lotto game derived in [21] and show that it can be readily turned into a Blotto solution using the theory of joint mixability.

2.1 The Blotto game

The classical two-player Blotto game is formalized as follows. Two players, respectively denoted by and , are competing over battlefields denoted by . Since we focus on two-player games where both players obey the same rules, it will be convenient when describing the game to denote by either player and by the other player so that .

The datum of a Blotto game is as follows. Player has a total budget of to allocate across the battlefields. Moreover, she valuates battlefield to which may differ from . Without loss of generality, we assume that to break symmetry and that

(1)

Indeed, multiplying the value of all battlefields by the sames’ constant has no impact on the players’ strategies.

The rules of the Blotto game are as follows. A pure strategy for player is an allocation vector where is the amount allocated to battlefield . A mixed strategy for player

is a probability distribution over pure strategies. A salient feature of the Blotto game is that a player

is constrained to playing strategies that satisfy the budget constraint: . In turn, admissible mixed strategies for the Blotto games are random vectors such that

(2)

Given two pure strategies and for players and respectively, player wins battlefield if and receives a reward . Ties are broken arbitrarily as they are are irrelevant for our analysis.

The existence of Nash equilibria is a consequence of standard game theoretic arguments [32]. Unfortunately, these general results say little about the structure of equilibrium strategies. At the end of this section, we make partial progress towards this question by describing the marginals of such equilibrium strategies. However, these remain existential results in essence.

This is in stark contrast with the associated Lotto game, described in the following section, where the hard budget constraint is dropped in favor of a constraint in expectation, and whose explicit solutions have been computed.

2.2 The associated Lotto game

A Lotto game has the same data and rules as its associated Blotto game except for the almost sure budget constraint (2) which is relaxed to the following expected budget constraint:

(3)

This relaxation greatly simplifies the game. In fact, Kovenock and Roberson [21] have recently elicited an explicit characterization of a non-trivial Nash equilibrium game for the most general version of the Lotto game to date; see Table 1. In the rest of Section 2.2, we describe their solution in details since it is the basis for ours.

Continuous Asymm. Heterogeneous Asymm. More than 3 Complete
Strategy Budget Values Values Battlefields Results
[9]
[16]
[44]
[15]
[23]
[39]
[16]
[33]
[26]
[36]
[21]
This paper
Table 1: Variants of the continuous Blotto game and their solutions. The last column, “complete results” indicates whether results obtained hold with possibly strong assumptions on the different values (for instance, there always exist more than 3 battlefields with the exact same value).

Finding an optimal strategy for the Lotto game amounts to finding a stationary point for an optimization problem subject to constraints of the form (3). Because of linearity of expectation, the associated Lagrangian is decomposed as the sum of terms, one per battlefield, that are each mathematically equivalent to an “all-pay” auction whose solutions are well known.

More explicitly, Nash equilibria of the Lotto problem depend on two parameters and , that are set later on. First, given any , consider the subsets of battlefields that are at least -times more valuable to than to :

Given a scaling parameter to be defined later, the mixed strategy of player at equilibrium prescribes to allocate a (random) budget of to battlefield with distribution given by:

where denotes the Dirac point mass at .

The strategy of player is given by

Note that the strategy of and are the same except that the roles of and are switched. In that sense, plays the role of an “exchange" rate that accounts for discrepancies between budgets and valuations across the two players.

It remains to find the parameters and using the budget constraints. For this set of strategies, saturating the total budget constraint (2) readily yields the following two equations:

(4)
(5)

Any pair solving the above system of two equations yields a Nash equilibrium. It remains to show that such solutions may be computed efficiently. Observe that eliminating from the equations yields the following nonlinear equation in :

(6)

Any solution to this equation readily yields a unique by plugging it into either (4) or (5); both equations will yield the same solution by (6). In turn, the existence and efficient computation of solutions to (6) are ensured by the following proposition. The bounds on presented in the following proposition depend on the distance between the vectors of battlefield values and . Interestingly the natural measure of distance that emerges is the -divergence that commonly arises in information theory and statistics; see e.g. [31]. The -divergence

between two probability vectors

and is defined by

It is clear that with equality if and only if .

Proposition 1.

Equation (6) has the following properties:

  1. It always has at least one and at most solutions .

  2. Any solution satisfies

  3. Computing all solutions can be done in operations.

The proof is based on standard computations, hence postponed to Appendix B.2, with the associated Algorithm 2.

Remark 2.

In case of symmetric values, that is when , the game is constant-sum and each player has a then unique111In the case of the Lotto game, it is natural to call a strategy an equivalence class of strategies with the same marginals. optimal strategy given by a unique pair  [21]. In fact, in that case, the unique can be computed analytically as and ; this can be easily seen from Proposition 1 (point 2.), since . With these parameters, the optimal strategy of player is to choose uniformly at random on and that of player is to forfeit each battlefield with probability and, to choose uniformly at random on on battlefield if not forfeited.

2.3 From Lotto to Blotto

In the previous section, we described how to compute solutions of a Lotto game. To turn a strategy for the Lotto game into a strategy for the Blotto game, one can couple the marginal strategies of a Lotto game, effectively turning the constraint (3) on the expected budget into the almost sure budget constraint (2).

Stated otherwise, a solution to the associated Lotto game induces a solution to the original Blotto game if the random variables (and similarly ) are jointly mixable [43].

Definition 3.

A family of random variables with finite expectations is jointly mixable if there exists a coupling such that if ,

In that case, the coupling is called a joint mix.

Obviously, not all random -tuples variables are jointly mixable. Take for example and to be Bernoulli with parameter . Then whereas there is no coupling of the s such that their some equals a fractional number.

While the full characterization of jointly mixable distribution is a complex question, some conditions, either sufficient or necessary, for joint mixability have been derived. The following proposition is a simple extension of a result of [43] (see also, [45]) on the mixability of distributions with monotone densities.

Proposition 4.

For , let , be fixed parameters and let be a random variable with distribution given by the following mixture:

(7)

Then are jointly mixable if and only if

(8)

This proposition is a consequence of few computations; its proof is delayed to Section B.3.

We are now in a position to state the main result of this section: the marginal distributions of the Lotto game described above are jointly mixable into a solution to the Blotto game. To that end, we instantiate Proposition 4 to the parameters of the marginal distributions described in Section 2.2.

Theorem 5.

Let be the parameters of Nash equilibrium of the Lotto game described in Section (2.2). Then the marginal distributions can be coupled into a Nash equilibrium for the corresponding Blotto game if and only if

(9)

Inequality (9) is simply an instantiation of (8), hence details are postponed to Section B.3

Condition (9) of the previous theorem relies on the values that define the solution of the Lotto game. In light of the bounds obtained in Proposition 1, these parameters may be eliminated to produce a sufficient condition for the existence of said solution for Blotto games with symmetric values. Recall that in this case, the game is constant-sum so a solution is, in fact, an optimal strategy. This result is captured in the following corollary which is a straightforward consequence of Theorem 5.

Corollary 6.

Assume symmetric values: . Then the marginal distributions of the optimal Lotto strategy described in Section 2.2 with and can be coupled into an optimal strategy for the corresponding Blotto game if and only if

In fact a sufficient condition may be derived in the case of non-symmetric values.

Corollary 7.

Assume that battlefield are balanced in the sense that there exits such that

Then, the marginal distributions of the optimal Lotto strategy described in Section 2.2 can be coupled into an optimal strategy for the corresponding Blotto game as long as

The proof of this result is based solely on computations; it is postponed to Section B.4

Note that the result of Corollary 7 is tight in the sense that if it recovers the result of Corollary 6. It is unclear whether the dependence in is sharp in our result and it is an interesting question to address in future work.

Under rather general conditions, the above two corollaries show the existence of solutions with marginal distributions of the Lotto game derived in [21]. It remains to show that such a coupling may be realized efficiently. This is done in the next section.

3 An efficient algorithm to compute solutions

Deriving solutions, either optimal strategies in the constant-sum setting or Nash equilibria, remains one of the major open problems surrounding the Blotto game. Previous attempts at this task have focused on deriving an explicit coupling between marginals. This is possible in specific cases. For example, several explicit couplings between random variables are known [20, 34]. In particular, this provides a solution to some Blotto problems with sufficient symmetry. However, this explicit approach fails for more general problems, and, in particular in the important case of asymmetric budget such as the one covered in Corollary 6. In this paper, we take another route by describing the efficient Algorithm Lotto2Blotto, whose pseudo-code is postponed to the Appendix B.1, that computes an -approximate solution with time complexity which is polynomial in and .

In light of the previous section, our goal is to find an algorithm that efficiently computes a coupling between the marginal Lotto strategies described above. This task faces two major hurdles.

On the one hand, the continuous nature of the marginals described above does not lend itself to efficient algorithms which typically work with discrete quantities. Instead, we propose to simply discretize the marginals at a scale of order . In particular, this prevents us from replicating exactly the marginals of the Lotto game but we can show that the error employed in said discretization remains of the same order once propagated to the utility of a given player.

On the other hand, the mere description of a coupling between discrete marginals on atoms is an object of size , which is exponential in the number of battlefields. To overcome this limitation, we develop a careful scheme that allows us to reduce the problem to the case of marginals instead of .

Finally, we employ recent developments in computational optimal transport, to couple our 4 marginals using a variant of Sinkhorn iterations [37, 38, 11].

3.1 Reductions

The typical size of a coupling with marginals is exponential in . While this issue is, in general, hopeless to overcome, we can exploit some of the structure of the problem at hand. Indeed, a similar principle has been recently employed in multi-marginal optimal transport to devise polynomial-time algorithms under additional structure [5]. More specifically, we reduce our problem to the case where there are only four marginals which remain mixable if the original marginals are mixable.

This reduction is done in two steps. Recall that the marginals for the Lotto game described in Section 2.2

are either uniform distributions or mixtures of a uniform distribution with a Dirac point mass at zero. Our first step reduces to the case where

marginals are uniform and only one is a mixture as above. In our second step, we further reduce to the case where there are three uniform marginals and one mixture.

Throughout this section we focus on player for brevity. Reductions for player are analogous.

3.1.1 Step 1: reduction to a single mixture

The marginal distributions described in Section 2.2 consist of uniform distributions and mixtures of a uniform distribution and a Dirac point mass at 0 and our goal is to efficiently couple them into a joint mix coupling that has these marginals and satisfies the Blotto budget constraint. For clarity, we also regard uniform distributions as mixture distributions albeit with weight zero on the point mass. Otherwise, we say that a distribution is a strict mixture. The goal of this first step is to reduce this coupling problem to the case where there are uniform distributions and one single strict mixture. To that end, we show that such a coupling may be obtained as a mixture of joint mixes :

where the marginal distributions of consist of at most one strict mixture, the rest being uniform distributions. Moreover, this decomposition can be computed efficiently as the solution of a simple greedy procedure.

Lemma 8.

Let be the parameter of a solution for the Lotto game and assume that the mixability condition (9) holds. Then, there exists a family of couplings and a set of non negative weights such that

  1. The marginal distributions of are given by

    for some with at most one in for each .

  2. Each coupling is a joint mix

  3. The mixture of couplings

    (10)

    is a solution for the Blotto game.

  4. The total complexity of computing the weights scales as .

Note that the mixture of couplings in (10) is necessarily a joint mix as a mixture of joint mixes. To sample from it, Player , simply samples with probability and plays according to the strategy prescribed by it.

The geometric proof of this lemma is delayed to Section B.5, along with the pseudo-code of associated Algorithm Decomp.

Lemma 9.

Fix , and let

be an affine hyperplane. For any point

, define ; then there exist , an extreme point and another vector satisfying (in particular, belongs to some -face of where ) such that

The overall complexity of computing and is of order .

The proof of this Lemma is based on simple geometric arguments, and is postponed to Section B.6 with the associated pseudo-code of Algorithm Extremize.

3.1.2 Step 2. Reduction to four random variables

The previous step reduces the joint mixability problem of general mixtures, to a simpler one where at most one strict mixture is involved. Still, computing—in fact even describing—a coupling of variables requires generically exponential (in ) time and memory. To overcome this limitation, we reduce the number of random variables from to a constant number.

The following Lemma states that each can be realized as the coupling of 3 new uniform random variables and a strict mixture, thus reducing the mixability question from to only random variables. A careful inspection of the proof of Lemma 10 below indicates that the reduction may lead to three marginals rather than four. In that case, two marginals are uniform and one is a strict mixture. To handle this case, some adjustments are needed; in particular—and obvisouly—with the size of the resulting coupling. However, extensions from four to three marginals are straightforward and we omit this case for clarity.

Lemma 10.

Fix , , and assume without loss of generality that the last marginal of the coupling from Lemma 8 is a strict mixture. Then may be constructed from three uniform random variables and a partition as follows. Set for all , and

where are such that

In particular, it holds that

and are jointly mixable. The support of is where

Moreover, the ’s, the sets , and the parameters of the distributions of can each be computed in constant time.

The proof of this Lemma, based on standard mixability arguments, is postponed to Section B.7, with the pseudo-code of the corresponding Algorithm Reduc.

Note that any joint mix of readily yields a joint mix of by defining , where is the unique integer such that .

3.2 Discretization

The problem of finding a solution for the Blotto game has been reduced to the construction of a coupling of (at most) four random variables, three of them being uniform over some intervals and the fourth one being a mixture between a Dirac mass at zero and some uniform distribution. Throughout this section we denote these random variables as for simplicity; in the notation of the previous section, they correspond to , and respectively.

Unfortunately, even in this simple case, finding explicit, closed-form, couplings appears to be possible only under stringent additional conditions that limit the scope of the Blotto game. To overcome this limitation, we take an algorithmic approach, describing an efficient way to find an approximate solution. To that end, we obviously need to work with discrete random variables and describe here a coupling between these discretized random variables.

Let be jointly mixed so that

(11)

Moreover, let be some (small) discretization parameter. Define the quantized random variables by

(12)

Our goal is to compute any of the joint distributions

of the vector when ranges over joint mixes.

As a first step towards this goal, note that these discretized random variables need not be jointly mixable. Indeed, in general we have but equality may fail to hold because of discretization errors. To account for these, let be defined as

(13)

and consider the augmented random vector . In light of (13), lives almost surely on a four dimensional subspace. As such, its distribution may be represented by a 4-tensor with entries given by

In particular, note that while each range in a set of integers of size . Using (13) we can read off the distribution of from this tensor.

This tensor is subject to four sets of linear constraints, one for each of the marginal constraints given in (12). They are given by

and, in light of (13), by

Note that indeed, any draw from a distribution that satisfies the above constraints yields a random vector . Defining by solving (13) yields a vector for some joint mix defined as above. In other words, is indeed the discretization of random variables drawn from a joint mix (though it need not be jointly mixable itself).


Since the random variables , for , constructed at the previous step have a support equal to where , the reduced (to 4 random variables) and discretized problem reduces to finding some tensor with entries satisfying at most

linear constraints. Although this can be done simply via linear programming (hence polynomially in

, more precisely in with Vaidya’s algorithm), a quite efficient and more popular way is to use a variant of Sinkhorn-Knopp algorithm that quickly finds approximated solutions. This is more relevant as this linear program is already some approximation of the original problem, hence there is no point of solving it exactly.

The pseudo-code of the Algorithm Discretize can be found in Section B.8

3.3 Tensor scaling using Sinkhorn iterations

In light of the previous sections, we have reduced our problem to that of finding coupling in the form of a 4-tensor with non-negative entries subject to marginal constraints. We approach this problem from a computational perspective and propose and algorithm that converges rapidly to a feasible solution. To describe this algorithm, recall that its input are four probability vectors , with that represent the probability mass functions of the discretized random variable defined in the previous section: , , .

The linear constraints take the form

(14)
(15)
(16)
(17)

Denote by the set of tensors that satisfy these constraints.

To solve this problem, we propose to project the all-ones tensor onto using the Kullback-Leibler (KL) divergence. Recall that the KL divergence between two nonnegative tensors is given by

In particular, is simply the (negative) entropy of and we aim to solve the convex optimization problem

While many algorithms are available to solve this problem [10], its specific structure can be exploited efficiently. Indeed, first order optimality conditions imply that any optimal must be of the form

(18)

for some scaling vectors . This representation readily calls for an iterative tensor scaling algorithm similar to the Sinkhorn algorithm [37, 38, 11]. Tensor scaling has been investigated in more classical setups [25, 5] that slightly differ from the present setup because the fourth marginal constraint takes a special form. Nevertheless, the implementation of Algorithm Sinkhorn remains straightforward and is presented in Section B.9. Its analysis is also a straightforward extension of that for the traditional matrix case [4]. More specifically, following the exact same lines as the one of Theorem 4.3 in [25], we readily get the following result.

Proposition 11.

Define

Algorithm Sinkhorn terminates and returns a tensor such that after at most iterations. Moreover, each marginal of has positive entries that sum to one and hence is a probability vector.

What have we accomplished so far? Through several reductions and a tensor scaling algorithm, given the datum of a Blotto game, we are able to compute a joint distribution that corresponds to an approximate solution. In Section 4, we evaluate the accuracy of this approximation in terms of the value of the game by showing that the various approximations (discretization and numerical precision of the algorithm) do not blow up when propagated back into the reductions. Before that, we investigate an important operational question: how to sample a strategy from the resulting coupling .

3.4 From coupling to sampling

Finding an efficient construction of (approximate) equilibria or optimal strategies is only relevant if it can be associated to some efficient sampling method so that a player may query a sampler and receive the allocation that they should play on each battlefield. In light of the various reduction steps employed above, it is sufficient to sample a 4-tuple

from the output of Algorithm Sinkhorn. Indeed, from , we obtain the random variables that are approximately distributed from the joint mix as follows.

To ensure that the marginal distributions are continuous, let and define

To correct for potential boundary effects, define and

Then take , , , and .

We call this procedure the smoothing procedure. Finally, as mentioned before, just define , where is the unique integer such that .

Note that the random variable is superfluous and theoretical results would follow by taking . Its role is simply to ensure, for cosmetic reasons, that the random marginal distributions are continuous apart from the potential point mass at zero.

It remains to sample from the output of Algorithm Sinkhorn. This is quite straightforward in light of the factored form of . Indeed, recall that the coupling output by Algorithm Sinkhorn has the form (18).

As a consequence, we can draw from as follows:

  1. Set with probability proportional to

  2. Set with probability proportional to

  3. Set with probability proportional to

  4. Conditionally on , set with probability proportional to

The pseudo-code of Algorithm Sample can be found in Section B.10.

4 Approximation errors and computational complexity

The construction of the previous section relies on various approximations, each of them inducing some error that can be mitigated at the cost of additional computational complexity by tuning the discretization parameter of Section 3.2 and the tolerance parameter in Algorithm Sinkhorn. In this section we study the computational complexity required to reach an -approximate solution.

4.1 From approximate strategies to approximate solutions

Note that the very notion of “approximate solution” strongly depends on whether the problem is value-symmetric ( for all ) or -asymmetric ( for some ). Indeed, in the former case, the game is constant-sum and optimal strategies do exist. This is no longer true in the latter case where only Nash equilibria are considered. As a consequence, we can consider approximation of a single optimal strategies in value-symmetric games, while we will have to consider approximations of a pair of equilibrium strategies in value-asymmetric ones. In the following two sections, we consider each case separately. In the remaining, we shall focus the analysis on Player , but it is almost identical for player ; hence we do not repeat it for the sake of clarity.

4.1.1 The value-symmetric case

A value-symmetric Blotto game, where for all is constant-sum and optimal strategies exist for each player. In particular, this allows us to provide strong approximation guarantees by controlling how sub-optimal the expected utility of a player is.

To check this well-known fact on our specific instance, consider the utility of player . Set two equilibrium parameters and (see Corollary 6) defining an optimal strategy and observe that since . For , let denote the amount allocated by player to battlefield according to this optimal strategy and denote by

its cumulative distribution function (cdf). The expected utility (a.k.a. reward) of player

if player chooses allocation depends only on the sequence of marginal cdfs rather than the whole coupling. It is given by

where we used the fact that the ’s sum to 1 and the ’s sum to at most . Moreover, if employs the mixed strategy described in Corollary 6, the utility of player , denoted , changes as follows. Let be a sequence of uniform random variables such that is independent of . In particular, and