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.     

An axiomatization of the Shapley value using a fairness property

In this paper we provide an axiomatization of the Shapley value for TU-games using a fairness property. This property states that if to a game we add another game in which two players are symmetric then their payoffs change by the same amount. We show that the Shapley value is characterized by this fairness property, efficiency and the null player property. These three axioms also characterize the Shapley value on the class of simple games. Revised August 2001     

Explicit optimal value for Dynkin's stopping game

Under the One-step Look Ahead rule of Dynamic Programming, an explicit game value of Dynkin's stopping problem for a Markov chain is obtained by using a potential operator. The condition on the One-step rule could be extended to the k-step and infinity-step rule. We shall also decompose the game value as the sum of two explicit functions under these rules.     

Pure equilibria in a simple dynamic model of strategic market game

We present a discrete model of two-person constant-sum dynamic strategic market game. We show that for every value of discount factor the game with discounted rewards possesses a pure stationary strategy equilibrium. Optimal strategies have some useful properties, such as Lipschitz property and symmetry. We also show value of the game to be nondecreasing both in state and discount factor. Further, for some values of discount factor, exact form of optimal strategies is found. For β less than , there is an equilibrium such that players make large bids. For β close to 1, there is an equilibrium with small bids. Similar result is obtained for the long run average reward game.     

The Coalitional Rationality of the Shapley Value

We establish a necessary and sufficient condition under which the Shapley value of ann-person game lies in the core.     

Values for strategic games in which players cooperate

In this paper we propose a new method to associate a coalitional game with each strategic game. The method is based on the lower value of finite two-player zero-sum games. We axiomatically characterize this new method, as well as the method that was described in Von Neumann and Morgenstern (1944). As an intermediate step, we provide axiomatic characterizations of the value and the lower value of matrix games and finite two-player zero-sum games, respectively.The authors acknowledge the financial support of Ministerio de Ciencia y Tecnologia, FEDER andXunta de Galicia through projects BEC2002-04102-C02-02 and PGIDIT03PXIC20701PN.We wish to thank Professor William Thomson as well as an anonymous referee for useful comments.     

The multichoice consistent value

We consider multichoice NTU games, i.e., cooperative NTU games in which players can participate in the game with several levels of activity. For these games, we define and characterize axiomatically the multichoice consistent value, which is a generalization of the consistent NTU value for NTU games and of the multichoice value for multichoice TU games. Moreover, we show that this value coincides with the consistent NTU value of a replicated NTU game and we provide a probabilistic interpretation. Received: May 1998/Final version: January 2000     

The optimistic <Emphasis Type="Italic">TU</Emphasis> game in minimum cost spanning tree problems

We associate an optimistic TU game with each minimum cost spanning tree problem. We define the worth of a coalition S as the cost of connecting agents in S to the source assuming that agents in N\S are already connected to the source, and agents in S can connect through agents in N\S. We study the Shapley value of this new game. We thank Hervé Moulin, William Thomson, and two referees for helpful comments. Financial support from the Ministerio de Ciencia y Tecnologia and FEDER through grant SEJ2005-07637-c02-01 is gratefully acknowledged.     

Efficient Winning Strategies in Random‐Turn Maker–Breaker Games

We consider random‐turn positional games, introduced by Peres, Schramm, Sheffield, and Wilson in 2007. A p‐random‐turn positional game is a two‐player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is p ). We analyze the random‐turn version of several classical Maker–Breaker games such as the game Box (introduced by Chvátal and Erd?s in 1987), the Hamilton cycle game and the k‐vertex‐connectivity game (both played on the edge set of ). For each of these games we provide each of the players with a (randomized) efficient strategy that typically ensures his win in the asymptotic order of the minimum value of p for which he typically wins the game, assuming optimal strategies of both players.     

k-Balanced games and capacities

In this paper, we present a generalization of the concept of balanced game for finite games. Balanced games are those having a nonempty core, and this core is usually considered as the solution of the game. Based on the concept of k-additivity, we define the so-called k-balanced games and the corresponding generalization of core, the k-additive core, whose elements are not directly imputations but k-additive games. We show that any game is k-balanced for a suitable choice of k, so that the corresponding k-additive core is not empty. For the games in the k-additive core, we propose a sharing procedure to get an imputation and a representative value for the expectations of the players based on the pessimistic criterion. Moreover, we look for necessary and sufficient conditions for a game to be k-balanced. For the general case, it is shown that any game is either balanced or 2-balanced. Finally, we treat the special case of capacities.     

A note on repeated games with non-monotonie value

We show that even when the information structure is independent of the state of nature, the value of then-stage zero-sum game with incomplete information is not necessarily monotonie with respect to the length of the game. More precisely, we give an example of such ann-stage game in whichV ₁ >V ₂ <V ₃.I am very grateful to Ehud Lehrer who introduced this question to me.     

Search game in a rectangle

We consider a search game in which the searcher (player S) moves along a continuous trajectory in a rectangleQ. The velocity vectogram of player S is a rhombus-type set. In this paper, we construct the strategies of both players which make it possible to find the asymptotic value of the game in the case of small discovery radius.The author would like to thank the referee for considerable simplification of the proof of Theorem 4.1.     

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.     

Asymptotic Lipschitz Regularity for Tug-of-War Games with Varying Probabilities

We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in Ω ? ?ⁿ. The method of the proof is based on a game-theoretic idea to estimate the value of a related game defined in Ω ×Ω via couplings.

     

Optimal Stopping with Information Constraint

We present a discrete n-person model of a dynamic strategic market game. We show that for some values of the discount factor the game possesses a stationary equilibrium where all the players make high bids. Within the class of all the high-bidding strategies we distinguish between two classes of more and less aggressive ones. We show that the set of discount factors for which these more aggressive strategies form equilibria shrinks as n goes to infinity. On the other hand, the analogous set for the less aggressive strategies grows to the whole interval (0,1) as n grows to infinity. Further we analyze the properties of the value function corresponding to these high-bidding equilibria. We also give some numerical examples contradicting some other properties that seem intuitive.     

Analyzing Risky Choices: Q‐learning for Deal‐No‐Deal

In this paper, we derive an optimal strategy for the popular Deal or No Deal game show. To do this, we use Q‐learning methods, which quantify the continuation value inherent in sequential decision making in the game. We then analyze two contestants, Frank and Susanne, risky choices from the European version of the game. Given their choices and our optimal strategy, we find what their implied bounds would be on their levels of risk aversion. Previous empirical evidence in risky decision making has suggested that past outcomes affect future choices and that contestants have time‐varying risk aversion. We demonstrate that the strategies of Frank and Susanne are consistent with constant risk aversion levels except for their final risk‐seeking choice. We conclude with directions for future research. Copyright © 2013 John Wiley & Sons, Ltd.     

Coalitional values for cooperative games withr alternatives

In this paper we consider games withn players andr alternatives. In these games the worth of a coalition depends not only on that coalition, but also on the organization of the other players in the game. We propose two coalitional values that are extensions of the Owen value (1977). We give some relations with the Owen value and an axiomatic characterization of each value introduced in this work. Finally, we compare both values. This research has been supported partially by U.P.V./E.H.U. research project 035.321-HB048/97, and the DGES of MEC project PB96-0247.     

Deterministic Time-inconsistent Optimal Control Problems - an Essentially Cooperative Approach

A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N -person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.     

Agent‐based simulation of an N‐person game with parabolic payoff functions

We report computer simulation experiments based on our agent‐based simulation tool to model a new N‐person game based on John Conway's Game of Life. The individual agents may choose between two behavior options: cooperation or defection. The payoff (reward/penalty) functions are given as two parabolas: one for each option. After a certain number of iterations, the behavior of the agents stabilizes to either a constant value or oscillates around such a value. The simulation's goal is to investigate the effects of intermediate behavior on a society of agents. We have performed a systematic investigation of this game for all six possible cases of the mutual positions of parabolic payoff functions crossing each other at two points: x = 0.3 and 0.7 where x is the ratio of the cooperation choice to the total number of agents in the agent's neighborhood. The global ratios X(t) of the total number of cooperators in the entire array of agents as functions of time (iterations) and the solutions of the game X_final as functions of X₀ were observed for each case for Pavlovian, greedy, and conformist agents. The solutions have predictable tendencies only when the neighborhood is the entire array of greedy or conformist agents. In all other cases unexpected properties emerge. © 2009 Wiley Periodicals, Inc. Complexity, 2010