Every normal-form game has a Pareto-optimal nonmyopic equilibrium

It is well known that Nash equilibria may not be Pareto-optimal; worse, a unique Nash equilibrium may be Pareto-dominated, as in Prisoners’ Dilemma. By contrast, we prove a previously conjectured result: every finite normal-form game of complete information and common knowledge has at least one Pareto-optimal nonmyopic equilibrium (NME) in pure strategies, which we define and illustrate. The outcome it gives, which depends on where play starts, may or may not coincide with that given by a Nash equilibrium. We use some simple examples to illustrate properties of NMEs—for instance, that NME outcomes are usually, though not always, maximin—and seem likely to foster cooperation in many games. Other approaches for analyzing farsighted strategic behavior in games are compared with the NME analysis.

Interactive Information Design

We study the interaction between multiple information designers who try to influence the behavior of a set of agents. When each designer can choose information policies from a compact set of statistical experiments with countable support, such games always admit subgame-perfect equilibria. When designers produce public information, every equilibrium of the simple game in which the set of messages coincides with the set of states is robust in the sense that it is an equilibrium with larger and possibly infinite and uncountable message sets. The converse is true for a class of Markovian equilibria only. When designers produce information for their own corporation of agents, robust pure-strategy equilibria exist and are characterized via an auxiliary normal-form game in which the set of strategies of each designer is the set of outcomes induced by Bayes correlated equilibria in her corporation.

Dinkelbach-Type Algorithm for Computing Quantal Stackelberg Equilibrium

Stackelberg security games (SSGs) have been deployed in many real-world situations to optimally allocate scarce resource to protect targets against attackers. However, actual human attackers are not perfectly rational and there are several behavior models that attempt to predict subrational behavior. Quantal response is among the most commonly used such models and Quantal Stackelberg Equilibrium (QSE) describes the optimal strategy to commit to when facing a subrational opponent. Non-concavity makes computing QSE computationally challenging and while there exist algorithms for computing QSE for SSGs, they cannot be directly used for solving an arbitrary game in the normal form. We (1) present a transformation of the primal problem for computing QSE using a Dinkelbach's method for any general-sum normal-form game, (2) provide a gradient-based and a MILP-based algorithm, give the convergence criteria, and bound their error, and finally (3) we experimentally demonstrate that using our novel transformation, a QSE can be closely approximated several orders of magnitude faster.

Modeling competitive situations

This chapter studies how competitive situations are conventionally modeled in noncooperative game theory. It uses two sorts or forms of models: the so-called extensive form game and the normal or strategic form game. An extensive form representation of a noncooperative game is composed of the following list of items: a list of players; a game tree; an assignment of decision nodes to players or to nature; lists of actions available at each decision node and a correspondence between immediate successors of each decision node and available actions; information sets; an assignment of payoffs for each player to terminal nodes; and probability assessments over the initial nodes and over the actions at any node that is assigned to nature. There is no single way to proceed in general from a normal form game to a corresponding extensive form game. In one obvious extensive form, the players all choose complete strategies simultaneously, but often other extensive forms could be constructed from a given normal form.

Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case

The search problem of computing a Stackelberg (or leader-follower)equilibrium (also referred to as an optimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the optimistic and pessimistic versions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomial-time algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers’ game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader’s commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless $$\textsf {P}=\textsf {NP}$$P=NP. Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies—a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier—under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as a pessimistic bilevel programming problem, we show that, with two followers, the problem is -hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless $$\textsf {P}=\textsf {NP}$$P=NP. This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave quadratic function over an unbounded nonconvex feasible region defined by linear and quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-the-art methods, we propose an exact ad hoc algorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader’s strategy where such value is attained, also an $$\alpha $$α-approximate strategy where $$\alpha > 0$$α>0 is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computational experiments on a well-established testbed of game instances.

Quantifying Commitment in Nash Equilibria

To quantify a player’s commitment in a given Nash equilibrium of a finite dynamic game, we map the corresponding normal-form game to a “canonical extension,” which allows each player to adjust his or her move with a certain probability. The commitment measure relates to the average overall adjustment probabilities for which the given Nash equilibrium can be implemented as a subgame-perfect equilibrium in the canonical extension.

The Computational Complexity of Finding a Mixed Berge Equilibrium for a k-Person Noncooperative Game in Normal Form

The mixed Berge equilibrium (MBE) is an extension of the Berge equilibrium (BE) to mixed strategies. The MBE models mutually support in a [Formula: see text]-person noncooperative game in normal form. An MBE is a mixed-strategy profile for the [Formula: see text] players in which every [Formula: see text] players have mixed strategies that maximize the expected payoff for the remaining player’s equilibrium strategy. In this paper, we study the computational complexity of determining whether an MBE exists in a [Formula: see text]-person normal-form game. For a two-person game, an MBE always exists and the problem of finding an MBE is PPAD-complete. In contrast to the mixed Nash equilibrium, the MBE is not guaranteed to exist in games with three or more players. Here we prove, when [Formula: see text], that the decision problem of asking whether an MBE exists for a [Formula: see text]-person normal-form game is NP-complete. Therefore, in the worst-case scenario there does not exist a polynomial algorithm that finds an MBE unless P=NP.

A Stochastic Game-Based Approach for Multiple Beyond-Visual-Range Air Combat

This paper addresses tactical decisions in beyond-visual-range (BVR) air combat between two adversarial teams of multiple (autonomous) aircraft. A BVR combat is formalized as a two-player stochastic game consisting of a sequence of normal-form games that determines on the number of missiles to be allocated to each adversary aircraft; within this normal-form game a continuous sub-game is embedded to determine the missile shooting times. The formulation reduces the size of decision space by taking advantage of the underlying symmetry of the combat scenario, and also facilitates incorporation of the effect of cooperative missile maneuvers and transition into within-visual-range (WVR) combat. The Nash equilibrium strategies and the associate value functions of the game are computed through linear-programming-based dynamic programming procedure. Numerical case studies on combat between airplanes with heterogeneous capabilities and cooperation effects demonstrate the validity of the proposed formulation and the effectiveness of the proposed solution scheme.