Tournament games and positive tournaments

Given a tournament T, the tournament game on T is as follows: Two players independently pick a node of T. If both pick the same node, the game is tied. Otherwise, the player whose node is at the tail of the arc connecting the two nodes wins. We show that the optimal mixed strategy for this game is unique and uses an odd number of nodes. A tournament is positive if the optimal strategy for its tournament game uses all of its nodes. The uniqueness of the optimal strategy then gives a new tournament decomposition: any tournament can be uniquely partitioned into positive subtournaments P₁, P₂, ?,P_k, so P_i “beats” P_j for all 1 ≤ i > j ≤ k. We count the number of n node positive tournaments and list them for n ≤ 7. © 1995 John Wiley & Sons, Inc.     

Static search games played over graphs and general metric spaces

We define a general game which forms a basis for modelling situations of static search and concealment over regions with spatial structure. The game involves two players, the searching player and the concealing player, and is played over a metric space. Each player simultaneously chooses to deploy at a point in the space; the searching player receiving a payoff of 1 if his opponent lies within a predetermined radius r of his position, the concealing player receiving a payoff of 1 otherwise. The concepts of dominance and equivalence of strategies are examined in the context of this game, before focusing on the more specific case of the game played over a graph. Methods are presented to simplify the analysis of such games, both by means of the iterated elimination of dominated strategies and through consideration of automorphisms of the graph. Lower and upper bounds on the value of the game are presented and optimal mixed strategies are calculated for games played over a particular family of graphs.     

Quitting games – An example

Quitting games are multi-player sequential games in which, at every stage, each player has the choice between continuing and quitting. The game ends as soon as at least one player chooses to quit; each player i then receives a payoff r _S ⁱ, which depends on the set S of players that did choose to quit. If the game never ends, the payoff to each player is zero.? We exhibit a four-player quitting game, where the “simplest” equilibrium is periodic with period two. We argue that this implies that all known methods to prove existence of an equilibrium payoff in multi-player stochastic games are therefore bound to fail in general, and provide some geometric intuition for this phenomenon. Received: October 2001     

Variational inequality formulation for the games with random payoffs

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case where the continuous strategy set of each player does not depend on the strategies of other players. If a random vector defining the payoffs of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium. We characterize the set of Nash equilibria using the solution set of a variational inequality (VI) problem. Next, we consider the case where the continuous strategy set of each player is defined by a shared constraint set. In this case, we show that there exists a generalized Nash equilibrium for elliptically symmetric distributed payoffs. Under certain conditions, we characterize the set of a generalized Nash equilibria using the solution set of a VI problem. As an application, the random payoff games arising from electricity market are studied under chance-constrained game framework.     

Stay-in-a-set games

There exists a Nash equilibrium (ε-Nash equilibrium) for every n-person stochastic game with a finite (countable) state space and finite action sets for the players if the payoff to each player i is one when the process of states remains in a given set of states G _i and is zero otherwise. Received: December 2000     

Playing off-line games with bounded rationality

We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payoff is the average of a one-shot payoff over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First we define the complexity of a sequence by its smallest period (a nonperiodic sequence being of infinite complexity) and study the maxmin of the game where player 1 is restricted to strategies with complexity at most n and player 2 is restricted to strategies with complexity at most m. We study the asymptotics of this value and a complete characterization in the matching pennies case. We extend the analysis of matching pennies to strategies with bounded recall.     

Internal correlation in repeated games

This paper characterizes the set of all the Nash equilibrium payoffs in two player repeated games where the signal that the players get after each stage is either trivial (does not reveal any information) or standard (the signal is the pair of actions played). It turns out that if the information is not always trivial then the set of all the Nash equilibrium payoffs coincides with the set of the correlated equilibrium payoffs. In particular, any correlated equilibrium payoff of the one shot game is also a Nash equilibrium payoff of the repeated game.For the proof we develop a scheme by which two players can generate any correlation device, using the signaling structure of the game. We present strategies with which the players internally correlate their actions without the need of an exogenous mediator.     

Average-discounted equilibria in stochastic games

In stochastic games with finite state and action spaces, we examine existence of equilibria where player 1 uses the limiting average reward and player 2 a discounted reward for the evaluations of the respective payoff sequences. By the nature of these rewards, the far future determines player 1's reward, while player 2 is rather interested in the near future. This gives rise to a natural cooperation between the players along the course of the play. First we show the existence of stationary ε-equilibria, for all ε>0, in these games. However, besides these stationary ε-equilibria, there also exist ε-equilibria, in terms of only slightly more complex ultimately stationary strategies, which are rather in the spirit of these games because, after a large stage when the discounted game is not interesting any longer, the players cooperate to guarantee the highest feasible reward to player 1. Moreover, we analyze an interesting example demonstrating that 0-equilibria do not necessarily exist in these games, not even in terms of history dependent strategies. Finally, we examine special classes of stochastic games with specific conditions on the transition and payoff structures. Several examples are given to clarify all these issues.     

On repeated games with incomplete information played by non-Bayesian players

Unlike in the traditional theory of games of incomplete information, the players here arenot Bayesian, i.e. a player does not necessarily have any prior probability distribution as to what game is being played. The game is infinitely repeated. A player may be absolutely uninformed, i.e. he may know only how many strategies he has. However, after each play the player is informed about his payoff and, moreover, he has perfect recall. A strategy is described, that with probability unity guarantees (in the sense of the liminf of the average payoff) in any game, whatever the player could guarantee if he had complete knowledge of the game.     

Single-payoff farsighted stable sets in strategic games with dominant punishment strategies

We investigate farsighted stable sets in a class of strategic games with dominant punishment strategies. In this class of games, each player has a strategy that uniformly minimizes the other players’ payoffs for any given strategies chosen by these other players. We particularly investigate a special class of farsighted stable sets, each of which consists of strategy profiles yielding a single payoff vector. We call such a farsighted stable set as a single-payoff farsighted stable set. We propose a concept called an inclusive set that completely characterizes single-payoff farsighted stable sets in strategic games with dominant punishment strategies. We also show that the set of payoff vectors yielded by single-payoff farsighted stable sets is closely related to the strict \(\alpha \)-core in a strategic game. Furthermore, we apply the results to strategic games where each player has two strategies and strategic games associated with some market models.     

Reduction of Silverman-like games to games on bounded sets

A two person zero sum game is regarded as Silverman-like if the strategy sets are sets of real numbers bounded below, the payoff function is bounded, the maximum payoff is achieved whenever the second player's numbery exceeds the first player's numberx by “too much”, and the minimum is achieved wheneverx exceedsy by “too much”. Explicit upper bounds are obtained for pure strategies to be included in an optimal mixed strategy in such games. In particular, if the strategy sets are discrete, the games may be reduced to games on specified finite sets.     

Layers of noncooperative games

We model and analyze classes of antagonistic stochastic games of two players. The actions of the players are formalized by marked point processes recording the cumulative damage to the players at any moment of time. The processes evolve until one of the processes crosses its fixed preassigned threshold of tolerance. Once the threshold is reached or exceeded at some point of the time (exit time), the associated player is ruined. Both stochastic processes are being “observed” by a third party point stochastic process, over which the information regarding the status of both players is obtained. We succeed in these goals by arriving at closed form joint functionals of the named elements and processes. Furthermore, we also look into the game more closely by introducing an intermediate threshold (see a layer), which a losing player is to cross prior to his ruin, in order to analyze the game more scrupulously and see what makes the player lose the game.     

Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points

Equilibrium points in mixed strategies seem to be unstable, because any player can deviate without penalty from his equilibrium strategy even if he expects all other players to stick to theirs. This paper proposes a model under which most mixed-strategy equilibrium points have full stability. It is argued that for any gameΓ the players' uncertainty about the other players' exact payoffs can be modeled as a disturbed gameΓ ^*, i.e., as a game with small random fluctuations in the payoffs. Any equilibrium point inΓ, whether it is in pure or in mixed strategies, can “almost always” be obtained as a limit of a pure-strategy equilibrium point in the corresponding disturbed gameΓ ^* when all disturbances go to zero. Accordingly, mixed-strategy equilibrium points are stable — even though the players may make no deliberate effort to use their pure strategies with the probability weights prescribed by their mixed equilibrium strategies — because the random fluctuations in their payoffs willmake them use their pure strategies approximately with the prescribed probabilities.     

Optimal strategies for equal-sum dice games

In this paper, we consider a non-cooperative two-person zero-sum matrix game, called dice game. In an (n,σ) dice game, two players can independently choose a dice from a collection of hypothetical dice having n faces and with a total of σ eyes distributed over these faces. They independently roll their dice and the player showing the highest number of eyes wins (in case of a tie, none of the players wins). The problem at hand in this paper is the characterization of all optimal strategies for these games. More precisely, we determine the (n,σ) dice games for which optimal strategies exist and derive for these games the number of optimal strategies as well as their explicit form.     

Crowding games are sequentially solvable

A sequential-move version of a given normal-form game Γ is an extensive-form game of perfect information in which each player chooses his action after observing the actions of all players who precede him and the payoffs are determined according to the payoff functions in Γ. A normal-form game Γ is sequentially solvable if each of its sequential-move versions has a subgame-perfect equilibrium in pure strategies such that the players' actions on the equilibrium path constitute an equilibrium of Γ.  A crowding game is a normal-form game in which the players share a common set of actions and the payoff a particular player receives for choosing a particular action is a nonincreasing function of the total number of players choosing that action. It is shown that every crowding game is sequentially solvable. However, not every pure-strategy equilibrium of a crowding game can be obtained in the manner described above. A sufficient, but not necessary, condition for the existence of a sequential-move version of the game that yields a given equilibrium is that there is no other equilibrium that Pareto dominates it. Received July 1997/Final version May 1998     

On the NP-completeness of finding an optimal strategy in games with common payoffs

Consider a very simple class of (finite) games: after an initial move by nature, each player makes one move. Moreover, the players have common interests: at each node, all the players get the same payoff. We show that the problem of determining whether there exists a joint strategy where each player has an expected payoff of at least r is NP-complete as a function of the number of nodes in the extensive-form representation of the game. Received January 2001/Final version May 1, 2001     

Pure subgame-perfect equilibria in free transition games

We consider a class of stochastic games, where each state is identified with a player. At any moment during play, one of the players is called active. The active player can terminate the game, or he can announce any player, who then becomes the active player. There is a non-negative payoff for each player upon termination of the game, which depends only on the player who decided to terminate. We give a combinatorial proof of the existence of subgame-perfect equilibria in pure strategies for the games in our class.     

One class of simultaneous pursuit games

Let A and B be given convex closed bounded nonempty subsets in a Hilbert space H; let the first player choose points in the set A and let the second one do those in the set B. We understand the payoff function as the mean value of the distance between these points. The goal of the first player is to minimize the mean value, while that of the second player is to maximize it. We study the structure of optimal mixed strategies and calculate the game value.     

Correlated equilibrium payoffs and public signalling in absorbing games

An absorbing game is a repeated game where some action combinations are absorbing, in the sense that whenever they are played, there is a positive probability that the game terminates, and the players receive some terminal payoff at every future stage.  We prove that every multi-player absorbing game admits a correlated equilibrium payoff. In other words, for every ε>0 there exists a probability distribution p _ε over the space of pure strategy profiles that satisfies the following. With probability at least 1−ε, if a pure strategy profile is chosen according to p _ε and each player is informed of his pure strategy, no player can profit more than ε in any sufficiently long game by deviating from the recommended strategy. Received: April 2001/Revised: June 4, 2002