When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the n-stage repeated version of the game exist if and only if both players have Ω(n) random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the n-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on “random-like” sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators do not exist, then Ω(n) random bits remain necessary for equilibria to exist.     

Computational Aspects of Uncertainty Profiles and Angel-Daemon Games

We analyze the complexity of equilibria problems for a class of strategic zero-sum games, called angel-daemon games. Those games were introduced to asses the performance of the execution of a web orchestration on a moderate faulty or under stress environment. Angel-daemon games are a natural example of zero-sum games whose representation is naturally succinct. We show that the problems of deciding the existence of a pure Nash equilibrium or of a dominant strategy for a given player are ${\Sigma}^{p}_{2}$ -complete. Furthermore, computing the value of an angel-daemon game is EXP-complete. Thus, our results match the already known classification of the corresponding problems for the generic families of succinctly represented games with exponential number of actions.     

On the complexity of constrained Nash equilibria in graphical games

A widely accepted rational behavior for non-cooperative players is based on the notion of Nash equilibrium. Although the existence of a Nash equilibrium is guaranteed in the mixed framework (i.e., when players select their actions in a randomized manner) in many real-world applications the existence of “any” equilibrium is not enough. Rather, it is often desirable to single out equilibria satisfying some additional requirements (in order, for instance, to guarantee a minimum payoff to certain players), which we call constrained Nash equilibria.In this paper, a formal framework for specifying these kinds of requirement is introduced and investigated in the context of graphical games, where a player p may directly be interested in some of the other players only, called the neighbors of p. This setting is very useful for modeling large population games, where typically each player does not directly depend on all the players, and representing her utility function extensively is either inconvenient or infeasible.Based on this framework, the complexity of deciding the existence and of computing constrained equilibria is then investigated, in the light of evidencing how the intrinsic difficulty of these tasks is affected by the requirements prescribed at the equilibrium and by the structure of players’ interactions. The analysis is carried out for the setting of mixed strategies as well as for the setting of pure strategies, i.e., when players are forced to deterministically choose the action to perform. In particular, for this latter case, restrictions on players’ interactions and on constraints are identified, that make the computation of Nash equilibria an easy problem, for which polynomial and highly-parallelizable algorithms are presented.     

Matrix games with missing,interval, and ambiguous lottery payoffs of pure strategy profiles and compound strategy profiles

In a matrix game, the interactions among players are based on the assumption that each player has accurate information about the payoffs of their interactions and the other players are rationally self‐interested. As a result, the players should definitely take Nash equilibrium strategies. However, in real‐life, when choosing their optimal strategies, sometimes the players have to face missing, imprecise (i.e., interval), ambiguous lottery payoffs of pure strategy profiles and even compound strategy profile, which means that it is hard to determine a Nash equilibrium. To address this issue, in this paper we introduce a new solution concept, called ambiguous Nash equilibrium, which extends the concept of Nash equilibrium to the one that can handle these types of ambiguous payoff. Moreover, we will reveal some properties of matrix games of this kind. In particular, we show that a Nash equilibrium is a special case of ambiguous Nash equilibrium if the players have accurate information of each player's payoff sets. Finally, we give an example to illustrate how our approach deals with real‐life game theory problems.     

Algorithms for Finding Repeated Game Equilibria

This paper describes computational techniques for finding all equilibria in infinitely repeated games with discounting and perfect monitoring. It illustrates these techniques with a three player Cournot game. This is the first infinitely repeated three player game ever solved. The paper also presents the solution for the set of equilibria in a two country tariff war. In both games the set of equilibria is large even when the players are not patient.     

Ranking games

The outcomes of many strategic situations such as parlor games or competitive economic scenarios are rankings of the participants, with higher ranks generally at least as desirable as lower ranks. Here we define ranking games as a class of n-player normal-form games with a payoff structure reflecting the players' von Neumann-Morgenstern preferences over their individual ranks. We investigate the computational complexity of a variety of common game-theoretic solution concepts in ranking games and deliver hardness results for iterated weak dominance and mixed Nash equilibrium when there are more than two players, and for pure Nash equilibrium when the number of players is unbounded but the game is described succinctly. This dashes hope that multi-player ranking games can be solved efficiently, despite their profound structural restrictions. Based on these findings, we provide matching upper and lower bounds for three comparative ratios, each of which relates two different solution concepts: the price of cautiousness, the mediation value, and the enforcement value.     

Discovering theorems in game theory: Two-person games with unique pure Nash equilibrium payoffs

In this paper we provide a logical framework for two-person finite games in strategic form, and use it to design a computer program for discovering some classes of games that have unique pure Nash equilibrium payoffs. The classes of games that we consider are those that can be expressed by a conjunction of two binary clauses, and our program re-discovered Kats and Thisse?s class of weakly unilaterally competitive two-person games, and came up with several other classes of games that have unique pure Nash equilibrium payoffs. It also came up with new classes of strict games that have unique pure Nash equilibria, where a game is strict if for both player different profiles have different payoffs.     

Establishing Nash equilibria of strategic games: a multistart Fuzzy Adaptive Simulated Annealing approach

This paper's proposal is to show some significant results obtained by the application of the optimization algorithm known as Fuzzy Adaptive Simulated Annealing (Fuzzy ASA) to the task of finding all Nash equilibria of normal form games. To that end, a special version of Fuzzy ASA, that utilizes space-filling curves to find good seeds, is applied to several well-known strategic games, showing its effectiveness in obtaining all Nash equilibria in all cases. The results are compared to previous work that also used computational intelligence techniques in order to solve the same problem but could not find all equilibria in all tests. Game theory is a very important subject, modeling interactions between generic agents, and Nash equilibrium represents a powerful concept portraying situations in which joint strategies are optimal in the sense that no player can benefit from changing her/his strategy while the other players do not change their strategies as well. So, new techniques are always welcome, mainly those that can find the whole set of solutions for a given strategic game.     

Quantum games and quaternionic strategies

It is well-known that the phenomenon of entanglement plays a fundamental role in quantum game theory. Occasionally, games constructed via maximally entangled initial states (MEIS) will have new Nash equilibria yielding to the players higher payoffs than the ones they receive in the classical version of the game. When examining these new games for Nash equilibrium payoffs, a fundamental question arises; does a suitable choice of an MEIS improve the lot of the players? In this paper, we show that the answer to this question is yes for at least the case of a variant of the well-known two player, two strategy game of Chicken. To that end, we generalize Landsburg’s quaternionic representation of the payoff function of two player, two strategy maximally entangled states to games where the initial state is chosen arbitrarily from a circle of maximally entangled initial states and for the corresponding quantized games show the existence of superior Nash equilibrium payoffs when an MEIS is appropriately chosen.     

Chain equilibria in secure strategies

In this paper we introduce a modification of the concept of Equilibrium in Secure Strategies (EinSS), which takes into account the non-uniform attitudes of players to security in non-cooperative games. In particular, we examine an asymmetric attitude of players to mutual threats in the simplest case, when all players are strictly ordered by their relation to security. Namely, we assume that the players can be reindexed so that each player i in his behavior takes into account the threats posed by players j > i but ignores the threats of players j < i provided that these threats are effectively contained by some counterthreats. A corresponding equilibrium will be called a Chain EinSS. The conceptual meaning of this equilibrium is illustrated by two continuous games that have no pure Nash equilibrium or (conventional) EinSS. The Colonel Blotto two-player game (Borel 1953; Owen 1968) for two battlefields with different price always admits a Chain EinSS with intuitive interpretation. The product competition of many players on a segment (Eaton, Lipsey 1975; Shaked 1975) with the linear distribution of consumer preferences always admits a unique Chain EinSS solution (up to a permutation of players). Finally, we compare Chain EinSS with Stackelberg equilibrium.     

Symmetries and the complexity of pure Nash equilibrium

Strategic games may exhibit symmetries in a variety of ways. A characteristic feature, enabling the compact representation of games even when the number of players is unbounded, is that players cannot, or need not, distinguish between the other players. We investigate the computational complexity of pure Nash equilibria in four classes of symmetric games obtained by considering two additional properties: identical payoff functions for all players and the ability to distinguish oneself from the other players. In contrast to other types of succinctly representable multi-player games, the pure equilibrium problem is tractable in all four classes when only a constant number of actions is available to each player. Identical payoff functions make the difference between TC⁰-completeness and membership in AC⁰, while a growing number of actions renders the equilibrium problem NP-hard for three of the classes and PLS-hard for the most restricted class for which the existence of a pure equilibrium is guaranteed. Our results also extend to larger classes of threshold symmetric games where players are unable to determine the exact number of players playing a certain action.     

Graphical Congestion Games

We consider congestion games with linear latency functions in which each player is aware only of a subset of all the other players. This is modeled by means of a social knowledge graph G in which nodes represent players and there is an edge from i to j if i knows j. Under the assumption that the payoff of each player is affected only by the strategies of the adjacent ones, we first give a complete characterization of the games possessing pure Nash equilibria. Namely, if the social graph G is undirected, the game is an exact potential game and thus isomorphic to a classical congestion game. As a consequence, it always converges and possesses Nash equilibria. On the other hand, if G is directed an equilibrium is not guaranteed to exist, but the game is always convergent and an equilibrium can be found in polynomial time if G is acyclic, even if finding the best equilibrium remains an intractable problem.     

多组对策系统非劣Nash策略的最优均衡解算法

多组对策系统中求解组与组之间的非劣Nash策略至关重要.如何针对一般问题解析求出非劣Nash策略还没有有效的方法.本文阐述了一种利用组与组之间的非劣反应集构造求解非劣Nash策略的迭代算法.为此首先引进多组对策系统组内部合作对策的最优均衡值和最优均衡解的概念,然后通过证明最优均衡解是组内部隐含某一权重向量的合作对策的非劣解,得到求解合作对策的单目标规划问题.进一步说明在组内部该问题的解不仅是非劣解而且对所有局中人都优于不合作时的Nash平衡策略.最后给出了验证该算法有效性的一个实际例子.     

When ignorance helps: Graphical multicast cost sharing games

In non-cooperative games played on highly decentralized networks the assumption that each player knows the strategy adopted by any other player may be too optimistic or even infeasible. In such situations, the set of players of which each player knows the chosen strategy can be modeled by means of a social knowledge graph in which nodes represent players and there is an edge from i to j if i knows j. Following the framework introduced in [7], we study the impact of social knowledge graphs on the fundamental multicast cost sharing game in which all the players want to receive the same communication from a given source in an undirected network. In the classical complete information case, such a game is known to be highly inefficient, since its price of anarchy can be as high as the total number of players ρ. We first show that, under our incomplete information setting, pure Nash equilibria always exist only if the social knowledge graph is directed acyclic (DAG). We then prove that the price of stability of any DAG is at least and provide a DAG lowering the classical price of anarchy to a value between and log²ρ. If specific instances of the game are concerned, that is if the social knowledge graph can be selected as a function of the instance, we show that the price of stability is at least , and that the same bound holds also for the price of anarchy of any social knowledge graph (not only DAGs). Moreover, we provide a nearly matching upper bound by proving that, for any fixed instance, there always exists a DAG yielding a price of anarchy less than 4. Our results open a new window on how the performances of non-cooperative systems may benefit from the lack of total knowledge among players.     

<Emphasis Type="Italic">N</Emphasis>-person credibilistic strategic game

This paper enlarges the scope of fuzzy-payoff game to n-person form from the previous two-person form. Based on credibility theory, three credibilistic approaches are introduced to model the behaviors of players in different decision situations. Accordingly, three new definitions of Nash equilibrium are proposed for n-person credibilistic strategic game. Moreover, existence theorems are proved for further research into credibilistic equilibrium strategies. Finally, two numerical examples are given to illustrate the significance of credibilistic equilibria in practical strategic games.     

Distributed Nash Equilibrium Seeking for General Networked Games With Bounded Disturbances

This paper is concerned with anti-disturbance Nash equilibrium seeking for games with partial information. First, reduced-order disturbance observer-based algorithms are proposed to achieve Nash equilibrium seeking for games with first-order and second-order players, respectively. In the developed algorithms, the observed disturbance values are included in control signals to eliminate the influence of disturbances, based on which a gradient-like optimization method is implemented for each player. Second, a signum function based distributed algorithm is proposed to attenuate disturbances for games with second-order integrator-type players. To be more specific, a signum function is involved in the proposed seeking strategy to dominate disturbances, based on which the feedback of the velocity-like states and the gradients of the functions associated with players achieves stabilization of system dynamics and optimization of players’ objective functions. Through Lyapunov stability analysis, it is proven that the players’ actions can approach a small region around the Nash equilibrium by utilizing disturbance observer-based strategies with appropriate control gains. Moreover, exponential (asymptotic) convergence can be achieved when the signum function based control strategy (with an adaptive control gain) is employed. The performance of the proposed algorithms is tested by utilizing an integrated simulation platform of virtual robot experimentation platform (V-REP) and MATLAB.      

Random Bimatrix Games Are Asymptotically Easy to Solve (A Simple Proof)

We focus on the problem of computing approximate Nash equilibria and well-supported approximate Nash equilibria in random bimatrix games, where each player’s payoffs are bounded and independent random variables, not necessarily identically distributed, but with almost common expectations. We show that the completely mixed uniform strategy profile, i.e., the combination of mixed strategies (one per player) where each player plays with equal probability each one of her available pure strategies, is with high probability a $\sqrt{\frac{\ln n}{n}}$ -Nash equilibrium and a $\sqrt{\frac{3\ln n}{n}}$ -well supported Nash equilibrium, where n is the number of pure strategies available to each player. This asserts that the completely mixed, uniform strategy profile is an almost Nash equilibrium for random bimatrix games, since it is, with high probability, an ?-well-supported Nash equilibrium where ? tends to zero as n tends to infinity.     

On the Structure of Weakly Acyclic Games

The class of weakly acyclic games, which includes potential games and dominance-solvable games, captures many practical application domains. In a weakly acyclic game, from any starting state, there is a sequence of better-response moves that leads to a pure Nash equilibrium; informally, these are games in which natural distributed dynamics, such as better-response dynamics, cannot enter inescapable oscillations. We establish a novel link between such games and the existence of pure Nash equilibria in subgames. Specifically, we show that the existence of a unique pure Nash equilibrium in every subgame implies the weak acyclicity of a game. In contrast, the possible existence of multiple pure Nash equilibria in every subgame is insufficient for weak acyclicity in general; here, we also systematically identify the special cases (in terms of the number of players and strategies) for which this is sufficient to guarantee weak acyclicity.