Stationary perfect equilibria of an n-person noncooperative bargaining game and cooperative solution concepts

This paper characterizes the stationary (subgame) perfect equilibria of an n-person noncooperative bargaining model with characteristic functions, and provides strategic foundations of some cooperative solution concepts such as the core, the bargaining set and the kernel. The contribution of this paper is twofold. First, we show that a linear programming formulation successfully characterizes the stationary (subgame) perfect equilibria of our bargaining game. We suggest a linear programming formulation as an algorithm for the stationary (subgame) perfect equilibria of a class of n-person noncooperative games. Second, utilizing the linear programming formulation, we show that stationary (subgame) perfect equilibria of n-person noncooperative games provide strategic foundations for the bargaining set and the kernel.     

N-person Nim andn-person Moore's Games

We present one way of definingn-person perfect information games so that there is a reasonable outcome for every game. In particular, the theory of Nim and Moore's games is generalized ton-person games.     

Games with Ordered Outcomes

We present a brief review of the most important concepts and results concerning games in which the goal structure is formalized by binary relations called preference relations. The main part of the work is devoted to games with ordered outcomes, i.e., game-theoretic models in which preference relations of players are given by partial orders on the set of outcomes. We discuss both antagonistic games and n-person games with ordered outcomes. Optimal solutions in games with ordered outcomes are strategies of players, situations, or outcomes of the game. In the paper, we consider noncooperative and certain cooperative solutions. Special attention is paid to an extension of the order on the set of probabilistic measures since this question is substantial for constructing the mixed extension of the game with ordered outcomes. The review covers works published from 1953 until now.     

Solution concepts ofk-convexn-person games

For any positive integersk andn, the subclass ofk-convexn-person games is considered. In casek=n, we are dealing with convexn-person games. Three characterizations ofk-convexn-person games, formulated in terms of the core and certain adapted marginal worth vectors, are given. Further it is shown that fork-convexn-person games the intersection of the (pre)kernel with the core consists of a unique point (namely the nucleolus), but that the (pre)kernel may contain points outside the core. For certain 1-convex and 2-convexn-person games the part of the bargaining set outside the core is even disconnected with the core. The Shapley value of ank-convexn-person game can be expressed in terms of the extreme points of the core and a correction-vector whenever the game satisfies a certain symmetric condition. Finally, theτ-value of ank-convexn-person game is given.     

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.     

On the regions of linearity for the nucleolus and their computation

This paper describes a method for computing the linearity regions for the nucleolus for ann-person cooperative game. It also provides a way to compute the nucleolus for games with smalln.     

Monotonicity and dummy free property for multi-choice cooperative games

Given a coalition of ann-person cooperative game in characteristic function form, we can associate a zero-one vector whose non-zero coordinates identify the players in the given coalition. The cooperative game with this identification is just a map on such vectors. By allowing each coordinate to take finitely many values we can define multi-choice cooperative games. In such multi-choice games we can also define Shapley value axiomatically. We show that this multi-choice Shapley value is dummy free of actions, dummy free of players, non-decreasing for non-decreasing multi-choice games, and strictly increasing for strictly increasing cooperative games. Some of these properties are closely related to some properties of independent exponentially distributed random variables. An advantage of multi-choice formulation is that it allows to model strategic behavior of players within the context of cooperation.Partially funded by the NSF grant DMS-9024408     

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.     

A convex representation of totally balanced games

We analyze the least increment function, a convex function of n variables associated to an n-person cooperative game. Another convex representation of cooperative games, the indirect function, has previously been studied. At every point the least increment function is greater than or equal to the indirect function, and both functions coincide in the case of convex games, but an example shows that they do not necessarily coincide if the game is totally balanced but not convex. We prove that the least increment function of a game contains all the information of the game if and only if the game is totally balanced. We also give necessary and sufficient conditions for a function to be the least increment function of a game as well as an expression for the core of a game in terms of its least increment function.     

Essential stability of $$\alpha $$-core

Using the existence results in Kajii (J Econ Theory 56:194–205, 1992), we identify a class of n-person noncooperative games containing a dense residual subset of games whose cooperative equilibria are all essential. Moreover, we show that every game in this collection possesses an essential component of the \(\alpha \)-core by proving the connectivity of minimal essential subsets of the \(\alpha \)-core.     

On implementation via demand commitment games

A simple version of the Demand Commitment Game is shown to implement the Shapley value as the unique subgame perfect equilibrium outcome for any n-person characteristic function game. This improves upon previous models devoted to this implementation problem in terms of one or more of the following: a) the range of characteristic function games addressed, b) the simplicity of the underlying noncooperative game (it is a finite horizon game where individuals make demands and form coalitions rather than make comprehensive allocation proposals and c) the general acceptability of the noncooperative equilibrium concept. A complete characterization of an equilibrium strategy generating the Shapley value outcomes is provided. Furthermore, for 3 player games, it is shown that the Demand Commitment Game can implement the core for games which need not be convex but have cores with nonempty interiors. Received March 1995/Final version February 1997     

Non-cooperative stochastic dominance games

A non-cooperative stochastic dominance game is a non-cooperative game in which the only knowledge about the players' preferences and risk attitudes is presumed to be their preference orders on the set ofn-tuples of pure strategies. Stochastic dominance equilibria are defined in terms of mixed strategies for the players that are efficient in the stochastic dominance sense against the strategies of the other players. It is shown that the set of SD equilibria equals all Nash equilibria that can be obtained from combinations of utility functions that are consistent with the players' known preference orders. The latter part of the paper looks at antagonistic stochastic dominance games in which some combination of consistent utility functions is zero-sum over then-tuples of pure strategies.     

Two noncooperative games between a coalition and its surrounding in a class of n-person games with constant sum

A cooperative game engendered by a noncooperative n-person game (the master game) in which any subset of n players may form a coalition playing an antagonistic game against the residual players (the surrounding) that has a (Nash equilibrium) solution, is considered, along with another noncooperative game in which both a coalition and its surrounding try to maximize their gains that also possesses a Nash equilibrium solution. It is shown that if the master game is the one with constant sum, the sets of Nash equilibrium strategies in both above-mentioned noncooperative games (in which a coalition plays with (against) its surrounding) coincide.     

Two Game Models for Cooperation with Implicit Noncooperation

We propose two flexible game models to represent and analyze cases that cannot be modeled by current game models. One is called sharing creditability game (SCG) and the other is called bottomline game (BLG). The new models transform cooperative games into new games that incorporate auxiliary information (noncooperative in nature) usually neglected in previous theories. The new games will be solved only by traditional noncooperative game theory. When the new solutions are applied to the original games, the solutions can reflect the auxiliary information in addition to the original objectives of the decision makers or players. Generally, the new solutions are different from the cooperative and the noncooperative solutions of the original games. Existing transferable utility (TU) games and noncooperative games will coincide with special cases of the two new game models. Using SCG and BLG, the prisoner’s dilemma can be reformulated and a richer set of decisions can be considered for the players. The two new game models have potential applications in military and socioeconomic situations.This research was partly funded by the College Engineering, Ohio State University.     

Constrainedn-person games

The usual properties of a characteristic function game were derived byvon Neumann andMorgenstern from the properties of a game in normal form. In this paper we give a linear programming principle for the calculation of the characteristic function. The principle is a direct application ofCharnes' linear programming method for the calculation of the optimal strategies and the value of a two-person zero-sum game. The linear programming principle gives another method for proving the standard properties of a characteristic function when it is derived from a game in normal form. Using an idea originated byCharnes for two person games, we develop the concept of a constrainedn-person game as a simple, practical extension of ann-person game. However the characteristic function for a constrainedn-person game may not satisfy properties, such as superadditivity, usually associated with a characteristic function.     

Common behaviour of power indices

The variations ensuing in a weighted majority game are studied when a player increases his weight in prejudice of others or decreases in favor, or trades shares outside the game (in particular when an-person game becomes an (n+1)-person one). An invariant behaviour for different game values is found for all these cases. Possible applications to politics, shareholdings and large games are pointed out.     

Generalized Knaster–Kuratowski–Mazurkiewicz Theorem Without Convex Hull

In this paper, a new notion of Knaster–Kuratowski–Mazurkiewicz mapping is introduced and a generalized Knaster–Kuratowski–Mazurkiewicz theorem is proved. As applications, some existence theorems of solutions for (vector) Ky Fan minimax inequality, Ky Fan section theorem, variational relation problems, n-person noncooperative game, and n-person noncooperative multiobjective game are obtained.     

Minimarg and maximarg operators

Two operators on the set ofn-person cooperative games are introduced, the minimarg operator and the maximarg operator. These operators can be seen as dual to each other. Some nice properties of these operators are given, and classes of games for which these operators yield convex (respectively, concave) games are considered. It is shown that, if these operators are applied iteratively on a game, in the limit one will yield a convex game and the other a concave game, and these two games will be dual to each other. Furthermore, it is proved that the convex games are precisely the fixed points of the minimarg operator and that the concave games are precisely the fixed points of the maximarg operator.     

The nucleolus and kernel for simple games or special valid inequalities for 0–1 linear integer programs

An equivalence between simplen-person cooperative games and linear integer programs in 0–1 variables is presented and in particular the nucleolus and kernel are shown to be special valid inequalities of the corresponding 0–1 program. In the special case of weighted majority games, corresponding to knapsack inequalities, we show a further class of games for which the nucleolus is a representation of the game, and develop a single test to show when payoff vectors giving identical amounts or zero to each player are in the kernel. Finally we give an algorithm for computing the nucleolus which has been used successfully on weighted majority games with over twenty players.     

On the balancedness of multiple machine sequencing games

This paper takes a game theoretical approach to sequencing situations with m parallel and identical machines. We show that in a cooperative environment cooperative m-sequencing games, which involve n players, give rise to m-machine games, which involve m players. Here, n corresponds to the number of jobs in an m-sequencing situation, and m corresponds to the number of machines in the same m-sequencing situation. We prove that an m-sequencing game is balanced if and only if the corresponding m-machine game is balanced. Furthermore, it is shown that m-sequencing games are balanced if m∈{1,2}. Finally, if m⩾3, balancedness is established for two special classes of m-sequencing games. Furthermore, we consider a special class of m-sequencing situations in a noncooperative setting and show that a transfer payments scheme exists that is both incentive compatible and budget balanced.