Subgame-consistent cooperative solutions in randomly furcating stochastic differential games

The paradigm of randomly-furcating stochastic differential games incorporates additional stochastic elements via randomly branching payoffs in stochastic differential games. This paper considers dynamically stable cooperative solutions in randomly furcating stochastic differential games. Analytically tractable payoff distribution procedures contingent upon specific random realizations of the state and payoff structure are derived. This new approach widens the application of cooperative differential game theory to problems where the evolution of the state and future environments are not known with certainty. Important cases abound in regional economic cooperation, corporate joint ventures and environmental control. An illustration in cooperative resource extraction is presented.     

Subgame Consistent Solutions of a Cooperative Stochastic Differential Game with Nontransferable Payoffs

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that the extension of the solution policy to a later starting time and any possible state brought about by the prior optimal behavior of the players would remain optimal. Recently, mechanisms for the derivation of subgame consistent solutions in stochastic cooperative differential games with transferable payoffs have been found. In this paper, subgame consistent solutions are derived for a class of cooperative stochastic differential games with nontransferable payoffs. The previously intractable subgame consistent solution for games with nontransferable payoffs is rendered tractable.This research was supported by the Research Grant Council of Hong Kong, Grant HKBU2056/99H and by Hong Kong Baptist University, Grant FRG/02-03/II16.Communicated by G. Leitmann     

Computing payoff allocations in the approximate core of linear programming games in a privacy-preserving manner

We investigate privacy-preserving ways of allocating payoffs among players participating in a joint venture, using tools from cooperative game theory and differential privacy. In particular, we examine linear programming games, an important class of cooperative games that model a myriad of payoff sharing problems, including those from logistics and network design. We show that we can compute a payoff allocation in the approximate core of these games in a way that satisfies joint differential privacy.     

Subgame Consistent Cooperative Solutions in Stochastic Differential Games

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that: (i) the extension of the solution policy to a later starting time and to any possible state brought about by the prior optimal behavior of the players would remain optimal; (ii) all players do not have incentive to deviate from the initial plan. In this paper, we develop a mechanism for the derivation of the payoff distribution procedures of subgame consistent solutions in stochastic differential games with transferable payoffs. The payoff distribution procedure of the subgame consistent solution can be identified analytically under different optimality principles. Demonstration of the use of the technique for specific optimality principles is shown with an explicitly solvable game. For the first time, analytically tractable solutions of cooperative stochastic differential games with subgame consistency are derived.     

Useful New Equilibria for Differential Games with Side Interests of the Participants

For conflict static and dynamic problems (described by differential equations) considered either on a game set common for all participants or on partly intersecting game sets, we propose new notions of conflict equilibria which are efficient for seeking the solutions of coalition-free and cooperative games and for specifying the hierarchy of all known equilibria. Several examples are used to show that, without the proposed new notions of equilibrium, an actually fair sharing may be impossible in cooperative games.     

Subgame Consistent Cooperative Solution of Dynamic Games with Random Horizon

In cooperative dynamic games, a stringent condition—that of subgame consistency—is required for a dynamically stable cooperative solution. In particular, under a subgame-consistent cooperative solution an extension of the solution policy to a subgame starting at a later time with a state brought about by prior optimal behavior will remain optimal. This paper extends subgame-consistent solutions to dynamic (discrete-time) cooperative games with random horizon. In the analysis, new forms of the Bellman equation and the Isaacs–Bellman equation in discrete-time are derived. Subgame-consistent cooperative solutions are obtained for this class of dynamic games. Analytically tractable payoff distribution mechanisms, which lead to the realization of these solutions, are developed. This is the first time that subgame-consistent solutions for cooperative dynamic games with random horizon are presented.     

A fundamental study for partially defined cooperative games

The payoff of each coalition has been assumed to be known precisely in the conventional cooperative games. However, we may come across situations where some coalitional values remain unknown. This paper treats cooperative games whose coalitional values are not known completely. In the cooperative games it is assumed that some of coalitional values are known precisely but others remain unknown. Some complete games associated with such incomplete games are proposed. Solution concepts are studied in a special case where only values of the grand coalition and singleton coalitions are known. Through the investigations of solutions of complete games associated with the given incomplete game, we show a focal point solution suggested commonly from different viewpoints.     

Cores and related solution concepts for multi-choice games

A multi-choice game is a generalization of a cooperative game in which each player has several activity levels. Cooperative games form a subclass of the class of multi-choice games.This paper extends some solution concepts for cooperative games to multi-choice games. In particular, the notions of core, dominance core and Weber set are extended. Relations between cores and dominance cores and between cores and Weber sets are extensively studied. A class of flow games is introduced and relations with non-negative games with non-empty cores are investigated.     

Approximating the least core value and least core of cooperative games with supermodular costs

We study the approximation of the least core value and the least core of supermodular cost cooperative games. We provide a framework for approximation based on oracles that approximately determine maximally violated constraints. This framework yields a 3-approximation algorithm for computing the least core value of supermodular cost cooperative games, and a polynomial-time algorithm for computing a cost allocation in the 2-approximate least core of these games. This approximation framework extends naturally to submodular profit cooperative games. For scheduling games, a special class of supermodular cost cooperative games, we give a fully polynomial-time approximation scheme for computing the least core value. For matroid profit games, a special class of submodular profit cooperative games, we give exact polynomial-time algorithms for computing the least core value as well as a least core cost allocation.     

Subgame Consistent Solutions for Cooperative Stochastic Dynamic Games

In cooperative games over time with uncertainty, a stringent condition (subgame consistency) is required for a dynamically stable solution. In particular, a cooperative solution is subgame consistent if an extension of the solution policy to a situation with a later starting time and any feasible state brought about by prior optimal behavior would remain optimal. This paper derives an analytically tractable payoff distribution procedure leading to the realization of subgame consistent solutions in cooperative stochastic dynamic games. This is the first time that subgame consistent solutions in discrete-time dynamic games under uncertainty are provided.     

凸随机合作对策的核心

本文将凸性扩展到随机合作对策中,从而得到凸随机合作对策具有超可加性与非空的核心,且凸随机合作对策的核心满足Minkowski和与Minkowski差.     

On ordinal equivalence of the Shapley and Banzhaf values for cooperative games

In this paper I consider the ordinal equivalence of the Shapley and Banzhaf values for TU cooperative games, i.e., cooperative games for which the preorderings on the set of players induced by these two values coincide. To this end I consider several solution concepts within semivalues and introduce three subclasses of games which are called, respectively, weakly complete, semicoherent and coherent cooperative games. A characterization theorem in terms of the ordinal equivalence of some semivalues is given for each of these three classes of cooperative games. In particular, the Shapley and Banzhaf values as well as the segment of semivalues they limit are ordinally equivalent for weakly complete, semicoherent and coherent cooperative games.     

On three Shapley-like solutions for cooperative games with random payoffs

Three solution concepts for cooperative games with random payoffs are introduced. These are the marginal value, the dividend value and the selector value. Inspiration for their definitions comes from several equivalent formulations of the Shapley value for cooperative TU games. An example shows that the equivalence is not preserved since these solutions can all be different for cooperative games with random payoffs. Properties are studied and a characterization on a subclass of games is provided.2000 Mathematics Subject Classification Number: 91A12.The authors thank two anonymous referees and an associate editor for their helpful comments.This author acknowledges financial support from the Netherlands Organization for Scientific Research (NWO) through project 613-304-059.Received: October 2000     

On some families of cooperative fuzzy games

In a fuzzy cooperative game the players may choose to partially participate in a coalition. A fuzzy coalition consists of a group of participating players along with their participation level. The characteristic function of a fuzzy game specifies the worth of each such coalition. This paper introduces well-known properties of classical cooperative games to the theory of fuzzy games, and studies their interrelations. It deals with convex games, exact games, games with a large core, extendable games and games with a stable core.     

Cooperative games with stochastic payoffs

This paper introduces a new class of cooperative games arising from cooperative decision making problems in a stochastic environment. Various examples of decision making problems that fall within this new class of games are provided. For a class of games with stochastic payoffs where the preferences are of a specific type, a balancedness concept is introduced. A variant of Farkas' lemma is used to prove that the core of a game within this class is non-empty if and only if the game is balanced. Further, other types of preferences are discussed. In particular, the effects the preferences have on the core of these games are considered.     

Linear and symmetric allocation methods for partially defined cooperative games

A partially defined cooperative game is a coalition function form game in which some of the coalitional worths are not known. An application would be cost allocation of a joint project among so many players that the determination of all coalitional worths is prohibitive. This paper generalizes the concept of the Shapley value for cooperative games to the class of partially defined cooperative games. Several allocation method characterization theorems are given utilizing linearity, symmetry, formulation independence, subsidy freedom, and monotonicity properties. Whether a value exists or is unique depends crucially on the class of games under consideration. Received June 1996/Revised August 2001     

Monitoring cooperative equilibria in a stochastic differential game

This paper deals with a class of equilibria which are based on the use of memory strategies in the context of continuous-time stochastic differential games. In order to get interpretable results, we will focus the study on a stochastic differential game model of the exploitation of one species of fish by two competing fisheries. We explore the possibility of defining a so-called cooperative equilibrium, which will implement a fishing agreement. In order to obtain that equilibrium, one defines a monitoring variable and an associated retaliation scheme. Depending on the value of the monitoring variable, which provides some evidence of a deviation from the agreement, the probability increases that the mode of a game will change from a cooperative to a punitive one. Both the monitoring variable and the parameters of this jump process are design elements of the cooperative equilibrium. A cooperative equilibrium designed in this way is a solution concept for a switching diffusion game. We solve that game using the sufficient conditions for a feedback equilibrium which are given by a set of coupled HJB equations. A numerical analysis, approximating the solution of the HJB equations through an associated Markov game, enables us to show that there exist cooperative equilibria which dominate the classical feedback Nash equilibrium of the original diffusion game model.This research was supported by FNRS-Switzerland, NSERC-Canada, FCAR-Quebec.     

Values for Interior Operator Games

The aim of this paper is to study a new class of cooperative games called interior operator games. These games are additive games restricted by antimatroids. We consider several types of cooperative games as peer group games, big boss games, clan games and information market games and show that all of them are interior operator games. Next, we analyze the properties of these games and compute the Shapley, Banzhaf and Tijs values.     

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.     

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.