Information patterns and Nash equilibria in extensive games: 1

In this paper we explore the relation between information patterns and Nash Equilibria in extensive games. By information we mean what players know about moves made by others, as well as by chance. For the most part we confine ourselves to pure strategies. But in Section 2 behavioral strategies are also examined. It turns out that they can be modeled as pure strategies of an appropriately enlarged game. Our results, applied to the enlarged game, can then be reinterpreted in terms of the behavioral strategies of the original game.     

A consistent value for games with n players and r alternatives

In Bolger [1993], an efficient value was obtained for a class of games called games with n players and r alternatives. In these games, each of the n players must choose one and only one of the r alternatives. This value can be used to determine a player’s “a priori” value in such a game. In this paper, we show that the value has a consistency property similar to the “consistency” for TU games in Hart/Mas-Colell [1989] and we present a set of axioms (including consistency) which characterizes this value.  The games considered in this paper differ from the multi-choice games considered by Hsiao and Raghavan [1993]. They consider games in which the actions of the players are ordered in the sense that, if i >j, then action i carries more “weight” than action j.  These games also differ from partition function games in that the worth of a coalition depends not only on the partitioning of the players but also on the action chosen by each subset of the partition. Received: April 1994/final version: June 1999     

The Compromise Value for Cooperative Games with Random Payoffs

This paper introduces and studies the compromise value for cooperative games with random payoffs, that is, for cooperative games where the payoff to a coalition of players is a random variable. This value is a compromise between utopia payoffs and minimal rights and its definition is based on the compromise value for NTU games and the τ-value for TU games. It is shown that the nonempty core of a cooperative game with random payoffs is bounded by the utopia payoffs and the minimal rights. Consequently, for such games the compromise value exists. Further, we show that the compromise value of a cooperative game with random payoffs coincides with the τ-value of a related TU game if the players have a certain type of preferences. Finally, the compromise value and the marginal value, which is defined as the average of the marginal vectors, coincide on the class of two-person games. This results in a characterization of the compromise value for two-person games.I thank Peter Borm, Ruud Hendrickx and two anonymous referees for their valuable comments.     

A polynomial time algorithm for computing the nucleolus for a class of disjunctive games with a permission structure

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many such allocation problems there is some hierarchical ordering of the players. In this paper we consider a class of games with a permission structure describing situations in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. The corresponding restricted game takes account of the limited cooperation possibilities by assigning to every coalition the worth of its largest feasible subset. In this paper we provide a polynomial time algorithm for computing the nucleolus of the restricted games corresponding to a class of games with a permission structure which economic applications include auction games, dual airport games, dual polluted river games and information market games.     

Inheritance of properties in communication situations

In this paper we consider cooperative games in which the possibilities for cooperation between the players are restricted because communication between the players is restricted. The bilateral communication possibilities are modeled by means of a (communication) graph. We are interested in how the communication restrictions influence the game. In particular, we investigate what conditions on the communication graph guarantee that certain appealing properties of the original game are inherited by the graph-restricted game, the game that arises once the communication restrictions are taken into account. We study inheritance of the following properties: average convexity, inclusion of the Shapley value in the core, inclusion of the Shapley values of a game and all its subgames in the corresponding cores, existence of a population monotonic allocation scheme, and the property that the extended Shapley value is a population monotonic allocation scheme. Received May 1998/Revised version January 2000     

Computing the value of information of quadratic decision problems and its non-negativity conditions

Abstract

There are many examples in the literature of non-cooperative games in which players prefer not to have additional information in order to improve their payoff. We present a general quadratic game in which, if one of the players improves his payoff upon obtaining more information, the other player’s payoff worsens in such a way that there is a net social loss due to having more information. How can we ensure this does not occur? The results of this paper are (1) the mathematical expression of the (social) value of information in a quadratic non-cooperative game, and (2) the conditions that ensure the social value of information is non-negative.     

The information revealed in infinitely-repeated games of incomplete information

Zero-sum two-person games with incomplete information on one side are considered. It is shown that the information revealed by the informed player in an infinitely-repeated game with simultaneous moves is essentially the same as the information revealed by him in a one-stage game in which he must move first.     

Take-and-guess games

This paper studies two classes of two-person zero-sum games in which the strategies of both players are of a special type. Each strategy can be split into two parts, a taking and a guessing part. In these games two types of asymmetry between the players can occur. In the first place, the number of objects available for taking does not need to be the same for both players. In the second place, the players can be guessing sequentially instead of simultaneously; the result is asymmetric information. The paper studies the value and equilibria of these games, for all possible numbers of objects available to the players, for the case with simultaneous guessing as well as for the variant with sequential guessing.      

Complete inflation and perfect recall in extensive games

This paper considers the relation between complete inflation and perfect recall of information partitions in extensive games. It is proved that an information partition with perfect recall is completely inflated. This result, combined with Dalkey's theorem, shows that in the class of games (without chance moves) with perfect recall, a game is determinate if and only if every player has perfect information. A necessary and sufficient condition is provided for information partitions whose complete inflations have perfect recall.     

The complexity of node blocking for dags

We consider the following modification of annihilation games called node blocking. Given a directed graph, each vertex can be occupied by at most one token. There are two types of tokens, each player can move only tokens of his type. The players alternate their moves and the current player i selects one token of type i and moves the token along a directed edge to an unoccupied vertex. If a player cannot make a move then he loses. We consider the problem of determining the complexity of the game: given an arbitrary configuration of tokens in a planar directed acyclic graph (dag), does the current player have a winning strategy? We prove that the problem is PSPACE-complete.     

Some results on zero-sum games with incomplete information: The dependent case

In games with incomplete information, the players' states of information may be determined either through independent chance moves or through a unique one. Generally, a unique chance move generates some dependance in the players' state of information thus giving rise to significant complications in the analysis. However, it turns out that many results obtained in the simpler independent case have their counterpart in the dependent one. This is proved in this paper for several previous results of the authors.     

On equilibrium in pure strategies in games with many players

We demonstrate that, if there are sufficiently many players, any Bayesian equilibrium of an incomplete information game can be “ε-purified” . That is, close to any Bayesian equilibrium there is an approximate Bayesian equilibrium in pure strategies. Our main contribution is obtaining this result for games with a countable set of pure strategies. In order to do so we derive a mathematical result, in the spirit of the Shapley–Folkman Theorem, permitting countable strategy sets. Our main assumption is a “large game property,” dictating that the actions of relatively small subsets of players cannot have large affects on the payoffs of other players. E. Cartwright and M. Wooders are indebted to Phillip Reny, Frank Page and two anonymous referees for helpful comments.     

The Sprague–Grundy function of the real game Euclid

The game Euclid, introduced and named by Cole and Davie, is played with a pair of nonnegative integers. The two players move alternately, each subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who reduces one of the integers to zero wins. Unfortunately, the name Euclid has also been used for a subtle variation of this game due to Grossman in which the game stops when the two entries are equal. For that game, Straffin showed that the losing positions (a,b) with a<b are precisely the same as those for Cole and Davie’s game. Nevertheless, the Sprague–Grundy functions are not the same for the two games. We give an explicit formula for the Sprague–Grundy function for the original game of Euclid and we explain how the Sprague–Grundy functions of the two games are related.     

Information patterns and Nash equilibria in extensive games — II

In Part I of this paper we introduced extensive games with a non-atomic continuum of players. It was shown that the Nash plays (outcomes) are invariant of the information patterns on the game, provided that no player's unilateral change in moves can be observed by others. This led to an enormous reduction in the Nash plays of the these games, as exemplified in the anti-folk theorem. Our concern in this sequel is to develop a finite version of these results.     

Fast Winning Strategies in Avoider-Enforcer Games

In numerous positional games the identity of the winner is easily determined. In this case one of the more interesting questions is not who wins but rather how fast can one win. These types of problems were studied earlier for Maker-Breaker games; here we initiate their study for unbiased Avoider-Enforcer games played on the edge set of the complete graph K _n on n vertices. For several games that are known to be an Enforcer’s win, we estimate quite precisely the minimum number of moves Enforcer has to play in order to win. We consider the non-planarity game, the connectivity game and the non-bipartite game.     

Theτ-value for games on matroids

In the classical model of games with transferable utility one assumes that each subgroup of players can form and cooperate to obtain its value. However, we can think that in some situations this assumption is not realistic, that is, not all coalitions are feasible. This suggests that it is necessary to raise the whole question of generalizing the concept of transferable utility game, and therefore to introduce new solution concepts. In this paper we define games on matroids and extend theτ-value as a compromise value for these games. This work has been partially supported by the Spanish Ministery of Science and Technology under grant SEC2000-1243.     

Informational aspects of a class of subjective games of incomplete information: Static case

Subjective games of incomplete information are formulated where some of the key assumptions of Bayesian games of incomplete information are relaxed. The issues arising because of the new formulation are studied in the context of a class of nonzero-sum, two-person games, where each player has a different model of the game. The static game is investigated in this note. It is shown that the properties of the static subjective game are different from those of the corresponding Bayesian game. Counterintuitive outcomes of the game can occur because of the different beliefs of the players. These outcomes may lead the players to realize the differences in their models.This work was sponsored by the Office of Naval Research under Contract No. N00014-84-C-0485.     

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.