Cooperative games with large cores

A payoff vector in ann-person cooperative game is said to be acceptable if no coalition can improve upon it. The core of a game consists of all acceptable vectors which are feasible for the grand coalition. The core is said to be large if for every acceptable vectory there is a vectorx in the core withx?y. This paper examines the class of games with large cores.     

Monotonic solutions of cooperative games

The principle of monotonicity for cooperative games states that if a game changes so that some player's contribution to all coalitions increases or stays the same then the player's allocation should not decrease. There is a unique symmetric and efficient solution concept that is monotonic in this most general sense — the Shapley value. Monotonicity thus provides a simple characterization of the value without resorting to the usual “additivity” and “dummy” assumptions, and lends support to the use of the value in applications where the underlying “game” is changing, e.g. in cost allocation problems.     

On balanced games with infinitely many players: Revisiting Schmeidler's result

We consider transferable utility cooperative games with infinitely many players and the core understood in the space of bounded additive set functions. We show that, if a game is bounded below, then its core is non-empty if and only if the game is balanced. This finding generalizes Schmeidler (1967) “On Balanced Games with Infinitely Many Players”, where the game is assumed to be non-negative. We also generalize Schmeidler's (1967) result to the case of restricted cooperation too.     

On Simple MIX game

We consider zero-sum game which is called Simple MIX game. Each of two players (I and II) draws a number (x andy respectively) according to a uniform distribution on [0, 1]. After observing his number each player can then choose to offer or not offer to exchange his number for the other player's number. Conditions for an exchange are the following: 1) both players must offer for a trade to occur with certainty; 2) if only one player offers, a trade occurs with probabilityp. A player's payoff is equal to 1, 0 or — 1 if the value of the number which he finally gets is greater, equal or less than the number of his opponent. In the present paper we shall investigate Simple MIX game in which both of the players can obtain additional information about the opponent's number. Besides, we consider two-stage variant of this game.     

Characterization of optimal strategies in matrix games with convexity properties

This paper gives a full characterization of matrices with rows and columns having properties closely related to the (quasi-) convexity-concavity of functions. The matrix games described by such payoff matrices well approximate continuous games on the unit square with payoff functions F (x, y) concave in x for each y, and convex in y for each x. It is shown that the optimal strategies in such matrix games have a very simple structure and a search-procedure is given. The results have a very close relationship with the known theorem of Debreu and Glicksberg about the existence of a pure Nash equilibrium in n-person games. Received: May 1997/Final version: August 1999     

The propensity to disrupt and the disruption nucleolus of a characteristic function game

Gately [1974] recently introduced the concept of an individual player's “propensity to disrupt” a payoff vector in a three-person characteristic function game. As a generalisation of this concept we propose the “disruption nucleolus” of ann-person game. The properties and computational possibilities of this concept are analogous to those of the nucleolus itself. Two numerical examples are given.     

Guaranteed control in differential games with ellipsoidal payoff

A class of antagonistic linear differential games (DGs) in a fixed time interval with ellipsoidal payoff functional is considered. This class of DGs includes problems which assume both rigid constraints on the players' controls and requirements to minimize control expenses. Other known classes of differential games, such as linear DGs with a quadratic performance index and linear DGs with ellipsoidal terminal sets and admissible sets of controls for the players, considered in Kurzhanskii's ellipsoidal technique, are limiting cases of DGs of this class. The concept of a u-strategic function, which expresses the property of u-stability for ellipsoidal functions, is introduced. An effective algorithm is presented for computing a u-strategic function, based on Kurzhanskii's ellipsoidal technique. The main result of this paper is that a guaranteed positional strategy for player u is defined by a certain explicit formula in terms of a u-strategic function. The proof of this result is based on a viability theorem for differential equations.     

Probabilistic prospects of Stackelberg leader and follower

In this paper, we consider the family of allm×n bimatrix games, whose payoff entries are the players' orderings of the outcomes, and count the fraction of games whose Stackelberg solution is the leader'sh ^th best outcome and the follower'sk ^th best outcome (h,k≤mn). We conclude that the average leader and follower enjoy symmetric prospects, and that the advantage lies not in the leadership role, but in the relative size of the player's strategy space.     

Two and three person games: A local study

Using the tools of differential geometry two-person games in normal form and their “ordinary” points, i.e. the points which are not equilibria in any sense, are studied. The concept of reversibility if defined and characterized in terms of the derivatives of the payoff functions. Reversible points appear as those points at which the behavior may become cooperative. In the second part of the paper, three-person games in normal form are considered. All the concepts defined depend only on the preference preorderings associated with the payoff functions and do not depend on the metric of the strategy spaces.     

On equilibria in finite games

We are concerned with Nash equilibrium points forn-person games. It is proved that, given any real algebraic numberα, there exists a 3-person game with rational data which has a unique equilibrium point andα is the equilibrium payoff for some player. We also present a method which allows us to reduce an arbitraryn-person game to a 3-person one, so that a number of questions about generaln-person games can be reduced to consideration of the special 3-person case. Finally, a completely mixed game, where the equilibrium set is a manifold of dimension one, is constructed.     

Values and cores of fuzzy games with infinitely many players

The problem of the existence ofvalues (FA-valued, linear, positive, symmetric and efficient operators) on symmetric spaces of “fuzzy games” (that is, ideal set functions of bounded variation) arises naturally from [8], [18], [23] and [2], [3], [4] where it is implicitely approached for technical purposes. In our present work, this problem is approached in itself for the main reason that it is essentially related with the problem of the existence of significant countable additive measures lying in the cores of the “market games”. In fact, it is shown here that there exists a continuous value on the closed subspacebv′ICA ofIBV spanned by thebv′ functions of “fuzzy probability measures” ([9]), this values is “diagonal” onpICA, the closed subspace ofbv′ICA spanned by the natural powers of the fuzzy measures and this is used to prove the main result stating that the cooperative markets contained inpICA have unique fuzzy measures in their cores which are exactly the corresponding diagonal values. This result is of interest because it is providing a tool of determiningCA measures lying in the cores of large classes of games which are not necessarily “non-atomic” and, specially, because it is opening a way toward a new approach of the “Value Equivalence Principle” for differentiable markets with a continuum of traders which are not “perfectly competitive”.     

Strategically zero-sum games: The class of games whose completely mixed equilibria cannot be improved upon

In this paper we propose a new class of games, the “strategically zero-sum games,” which are characterized by a special payoff structure. We show that for a large body of correlation schemes which includes the correlated strategies “à la Aumann”, strategically zero-sum games are exactly these games for which no completely mixed Nash equilibrium can be improved upon.     

Cooperative games with imcomplete information

A bargaining solution concept which generalizes the Nash bargaining solution and the Shapley NTU value is defined for cooperative games with incomplete information. These bargaining solutions are efficient and equitable when interpersonal comparisons are made in terms of certainvirtual utility scales. A player's virtual utility differs from his real utility by exaggerating the difference from the preferences of false types that jeopardize his true type. In any incentive-efficient mechanism, the players always maximize their total virtual utility ex post. Conditionally-transferable virtual utility is the strongest possible transferability assumption for games with incomplete information.     

Noncooperative games and nontransitive preferences

A proof exactly analogous to Nash's proof for the existence of equilibria in finite noncooperative games with von Neumann-Morgenstern utilities shows that such games have Nash equilibria when preferences satisfy the weaker conditions of ‘SSB utility theory’. An example illustrates the dual roles of mixed strategies in the SSB game context, namely to disguise a player's actual strategy choice and to resolve the potential intrapersonal dilemma of cyclic preferences among pure strategies.     

Optimal strategies for node selection games on oriented graphs: Skew matrices and symmetric games

In the node selection game Γ_D each of the two players simultaneously selects a node from the oriented graph D. If there is an arc between the selected nodes, then there is a payoff from the “dominated” player to the “dominating” player. We investigate the set of optimal strategies for the players in the node selection game Γ_D. We point out that a classical theorem from game theory relates the dimension of the polytope of optimal strategies for Γ_D to the nullity of certain skew submatrix of the payoff matrix for Γ_D. We show that if D is bipartite (with at least two nodes in each partite set), then an optimal strategy for the node selection game Γ_D is never unique. Our work also implies that if D is a tournament, then there is a unique optimal strategy for each player, a result obtained by Fisher and Ryan [Optimal strategies for a generalized “scissors, paper, and stone” game, Amer. Math. Monthly 99 (1992) 935–942] and independently by Laffond, Laslier, and Le Breton [The bipartisan set of a tournament game, Games Econom. Behav. 5 (1993) 182–201].     

Analogues of some Tauberian theorems for bounded variation

In this paper definitions for “bounded variation”, “subsequences”, “Pringsheim limit points”, and “stretchings” of a double sequence are presented. Using these definitions and the notion of regularity for four dimensional matrices, the following two questions will be answered. First, if there exists a four dimensional regular matrix A such that Ay = Σ _k,l=1,1 ^∞∞ a _m,n,k,l y _k,l is of bounded variation (BV) for every subsequence y of x, does it necessarily follow that x ∈ BV? Second, if there exists a four dimensional regular matrix A such that Ay ∈ BV for all stretchings y of x, does it necessarily follow that x ∈ BV? Also some natural implications and variations of the two Tauberian questions above will be presented.     

Mollifiers for games in normal form and the Harsanyi-Selten valuation function

The concepts of disruption and mollifiers ofCharnes/Rousseau/Seiford [1978] for games in characteristic function form are here extended to games in normal form. We show for a large class of games that theHarsanyi-Selten [1959] modification ofvon Neumann /Morgenstern's [1953] construction of a characteristic function for games in normal form, to take better account of “disruption” or “threat” possibilities, yields a constant mollifier. In general, it can be non-superadditive when the von Neumann-Morgenstern function is superadditive, and it also fails to take account of coalitional sizes. Our extended “homomollifier” concept does, and always yields a superadditive constant sum characteristic function.     

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.     

Non-cooperative monopolistic games and monopolistic market games

A class of non-cooperative games is discussed in which one player (“the monopolist”) by choosing his strategy restricts the other players to subsets of their strategy sets. Examples of such games in various fields are given. In particular it is shown that some very important economic situations fall within this class of games. A solution concept is defined and sufficient conditions for its existence are derived. The question of the advantages a player derives from being a monopolist is raised and conditions are derived for him to benefit from being a monopolist.