Proportional resource allocation in dynamic n-player Blotto games

In this note, we introduce a general model of dynamic n-player multi-battle Blotto contests in which asymmetric resources and non-homogeneous battlefield prizes are possible. Each player’s probability of winning the prize in a battlefield is governed by a ratio-form contest success function and players’ resource allocation on that battlefield. We show that there exists a pure subgame perfect equilibrium in which players allocate their resources in proportion to the battlefield prizes for every history. We also give a sufficient condition that if there are two players and the contest success function is of Tullock type, then the subgame perfect equilibrium is unique

The Art of Concession in General Lotto Games

Success in adversarial environments often requires investment into additional resources in order to improve one’s competitive position. But, can intentionally decreasing one’s own competitiveness ever provide strategic benefits in such settings? In this paper, we focus on characterizing the role of “concessions” as a component of strategic decision making. Specifically, we investigate whether a player can gain an advantage by either conceding budgetary resources or conceding valuable prizes to an opponent. While one might na¨ıvely assume that the player cannot, our work demonstrates that – perhaps surprisingly – concessions do offer strategic benefits when made correctly. In the context of General Lotto games, we first show that neither budgetary concessions nor value concessions can be advantageous to either player in a 1-vs.-1 scenario. However, in settings where two players compete against a common adversary, we find opportunities for one of the two players to improve her payoff by conceding a prize to the adversary. We provide a set of sufficient conditions under which such concessions exist

Efficient computation of approximate equilibria in discrete Colonel Blotto games

The Colonel Blotto game is a famous game commonly used to model resource allocation problems in many domains ranging from security to advertising. Two players distribute a fixed budget of resources on multiple battlefields to maximize the aggregate value of battlefields they win, each battlefield being won by the player who allocates more resources to it. The continuous version of the game—where players can choose any fractional allocation—has been extensively studied, albeit only with partial results to date. Recently, the discrete version—where allocations can only be integers—started to gain traction and algorithms were proposed to compute the equilibrium in polynomial time; but these remain computationally impractical for large (or even moderate) numbers of battlefields. In this paper, we propose an algorithm to compute very efficiently an approximate equilibrium for the discrete Colonel Blotto game with many battlefields. We provide a theoretical bound on the approximation error as a function of the game's parameters. We also propose an efficient dynamic programming algorithm in order to compute for each game instance the actual value of the error. We perform numerical experiments that show that the proposed strategy provides a fast and good approximation to the equilibrium even for moderate numbers of battlefields

Focality and Asymmetry in Multi-battle Contests

This article examines behavior in two-person constant-sum Colonel Blotto games in which each player maximizes the expected total value of the battlefields won. A lottery contest success function is employed in each battlefield. Recent experimental research on such games provides only partial support for Nash equilibrium behavior. We hypothesize that the salience of battlefields affects strategic behavior (the salient target hypothesis). We present a controlled test of this hypothesis – against Nash predictions – when the sources of salience come from certain asymmetries in either battlefield values or labels (as in Schelling (1960)). In both cases, subjects over-allocate the resource to the salient battlefields relative to the Nash prediction. However, the effect is stronger with salient values. In the absence of salience, we replicate previous results in the literature supporting the Nash prediction

Focality and Asymmetry in Multi-battle Contests

This article examines the influence of focality in Colonel Blotto games with a lottery contest success function (CSF), where the equilibrium is unique and in pure strategies. We hypothesise that the salience of battlefields affects strategic behaviour (the salient target hypothesis) and present a controlled test of this hypothesis against Nash predictions, checking the robustness of equilibrium play. When the sources of salience come from asymmetries in battlefield values or labels (as in Schelling, 1960), subjects over-allocate the resource to the salient battlefields relative to the Nash prediction. However, the effect is stronger with salient values. In the absence of salience, we find support for the Nash prediction

Efficient Computation of Approximate Equilibria in Discrete Colonel Blotto Games

International audienceThe Colonel Blotto game is a famous game commonly used to model resource allocation problems in many domains ranging from security to advertising. Two players distribute a fixed budget of resources on multiple battlefields to maximize the aggregate value of battlefields they win, each battlefield being won by the player who allocates more resources to it. The continuous version of the game-where players can choose any fractional allocation-has been extensively studied , albeit only with partial results to date. Recently , the discrete version-where allocations can only be integers-started to gain traction and algorithms were proposed to compute the equilibrium in polynomial time; but these remain computationally impractical for large (or even moderate) numbers of battlefields. In this paper, we propose an algorithm to compute very efficiently an approximate equilibrium for the discrete Colonel Blotto game with many battlefields. We provide a theoretical bound on the approximation error as a function of the game's parameters, in particular number of battlefields and resource budgets. We also propose an efficient dynamic programming algorithm to compute the best-response to any strategy that allows computing for each game instance the actual value of the error. We perform numerical experiments that show that the proposed strategy provides a fast and good approximation to the equilibrium even for moderate numbers of battlefields

The Optimal Defense of Networks of Targets

This paper examines a game-theoretic model of attack and defense of multiple networks of targets in which there exist intra-network strategic complementarities among targets. The defender’s objective is to successfully defend all of the networks and the attacker’s objective is to successfully attack at least one network of targets. Although there are multiple equilibria, we characterize correlation structures in the allocations of forces across targets that arise in all equilibria. For example, in all equilibria the attacker utilizes a stochastic ‘guerrilla warfare’ strategy in which a single random network is attacked

Efficient computation of approximate equilibria in discrete Colonel Blotto games

The Colonel Blotto game is a famous game commonly used to model resource allocation problems in many domains ranging from security to advertising. Two players distribute a fixed budget of resources on multiple battlefields to maximize the aggregate value of battlefields they win, each battlefield being won by the player who allocates more resources to it. The continuous version of the game—where players can choose any fractional allocation—has been extensively studied, albeit only with partial results to date. Recently, the discrete version—where allocations can only be integers—started to gain traction and algorithms were proposed to compute the equilibrium in polynomial time; but these remain computationally impractical for large (or even moderate) numbers of battlefields. In this paper, we propose an algorithm to compute very efficiently an approximate equilibrium for the discrete Colonel Blotto game with many battlefields. We provide a theoretical bound on the approximation error as a function of the game's parameters. We also propose an efficient dynamic programming algorithm in order to compute for each game instance the actual value of the error. We perform numerical experiments that show that the proposed strategy provides a fast and good approximation to the equilibrium even for moderate numbers of battlefields