International Game Theory Review

This paper deals with the design of equilibrium solutions with coupled constraints in dynamic games of greenhouse gas (GHG) emissions abatement. Self enforcing International Environmental Agreements (IEA) among different groups of countries call for Nash equilibrium solutions when the abatement strategies of the countries are defined. In this paper we study the effect of having another party, like e.g. the United Nations which would impose to all players a coupled constraint on the total emissions allowed over the 21st century, or on the concentration of carbon reached at the end of the century. We show, using different formulations of environmental game, that the normalized equilibria obtained under a coupled constraint on emissions or concentration is close to Pareto optimality. This gives a clue on the way the post Kyoto negotiations could yield an agreement which could be close to efficiency.

In this paper we provide characterizations of convex games and total clan games by using properties of their corresponding marginal games. We show that a "dualize and restrict" procedure transforms total clan games with zero worth for the clan into monotonic convex games. Furthermore, each monotonic convex game generates a total clan game with zero worth for the clan by a "dualize and extend" procedure. These procedures are also useful for relating core elements and elements of the Weber set of the corresponding games.

The aim of the paper is to study some important differences between (classical) non-cooperative and (modern) cooperative supply chain management. For this purpose, the paper develops a differential game model involving operations and marketing activities that are performed by a manufacturer and a retailer in a simple two-member supply chain. We consider a particular single brand of the manufacturer. The manufacturer decides on production volume, production process improvement and national advertising efforts, while the retailer decides her purchase volume by the manufacturer and her pricing policy toward the final consumers. A revenue sharing contract is employed in the non-cooperative setting. Among the issues addressed on the manufacturer side are the trade-off between production and process improvement activities, the path of inventory over time, and the trade-off between attracting new customers and improving the loyalty of current customers. For the retailer we study the inventory and price evolution over time.

The effects of partially cooperating firms are examined in N-firm oligopolies. The Herfindahl-Hirschmann Index is assumed to detect the violation of the antitrust regulation by the firms, and based on this assumption a piece-wise differentiable dynamic system can be developed. The firms stop cooperating if this index becomes larger than a certain threshold and they restart or continue cooperating otherwise. The equilibria of the dynamic system are first determined. Local and global asymptotic stability of the equilibria are then investigated showing the complexity of the dynamic behavior of the system.

We consider a two-sided search model in which individuals from two distinct populations would like to form a long-term relationship with a member of the other population. The individual choice is determined by the quality of the partner. Initially the quality of individuals in the population is uniform. At every stage the individuals randomly matched from their populations recognize the quality of the partner. If they accept each other they create a couple and leave the game. The partner's quality is the payoff. Unmatched players go to the next stage. At the last stage the individuals accept any partner. Each player aims to maximize her/his expected payoff. In this paper explicit formulas for Nash equilibrium strategies are derived. Also, the model with incoming individuals is analyzed.

In this paper we consider a zero-sum differential game in feedback setting with one state coordinate and finite time horizon. The problem is characterized by two parameters. This game arises as an approximation to a nonzero-sum game. Using analytical and numerical methods we solve the HJBI equation and give the description of the optimal feedback controls for both players. These feedbacks are given in terms of switching curves and singular universal lines. We also determine the change (bifurcations) of the optimal phase portrait of the game depending upon the parameters. In conclusion, we show that for some values of the parameters the solution of the considered zero-sum game leads to the exact solution of the corresponding nonzero-sum game, and for the other values of the parameters it supplies just some approximation.

We develop a model of coalition formation based on personal proximities among the players of an n-person game. Several examples are worked out in detail, showing that certain coalitions are much more stable than others, and/or much more likely to form than others. We also consider the dynamics of such coalition-formation. By a numerical example, we show that small changes in the initial conditions can lead to very different results in the coalitions formed in a given game.

An n-person differential game Γ(x, T-t) with independent motions from the initial state x and with prescribed duration T - t is considered. Suppose that y(s) is the cooperative trajectory maximizing the sum of players' payoffs. Suppose also that before starting the game players agree to divide the joint maximal payoff V(x, T - t; N) according to the imputation α, which is considered as a solution of a cooperative version of the game Γ(x, T - t). Using individual rationality of the imputation α we prove that if in the game Γ(y(s),T - s) along the cooperative trajectory y(s), the solution will be derived from the imputation α with the use of the imputation distribution procedure (IDP), for each given ε > 0 there exists ε-Nash equilibrium in Γ(x, T - t) for which the payoffs of the players in the game will be equal exactly to the components of the imputation α (cooperative outcome). This means that the imputation α is strategically supported by some specially constructed ε-Nash equilibrium in Γ(x, T - t). A similar result is true for a discrete game with perfect information.

A finite horizon zero-sum linear-quadratic differential game with a generalized cost functional, involving a Lebesgue integral with a measure that has both discrete and distributed parts, is considered. Sufficient conditions for the solvability of such a game are established in terms of the eigenvalues of an integral operator in Hilbert space. The game solution is based on solving an impulsive Riccati matrix differential equation. These results are applied for two games associated with pursuit-evasion problems. Illustrative examples are presented.

This paper presents dynamically stable solutions to a class of cooperative differential games of pollution management in collaborative abatement with uncertain future payoffs. Collaborative abatement leads to a decrease in cost due to reduction in duplicated efforts in the process of joint development and brings about an enhancement in the effectiveness of abatement activities via the sharing of knowledge from individual nation's research. It is believed by many researchers to be the key to effective pollution reduction. Uncertainties in future economic performance are prevalent in fast developing countries like China, Brazil and India. This type of uncertainties often hinders the reaching of cooperative agreements in joint pollution control initiatives. In dynamic cooperative games, a credible cooperative agreement has to be dynamically consistent. For dynamic consistency to hold, the agreed upon optimality principle must remain in effect at any instant of time throughout the game along the optimal state trajectory contingent upon the realization of specific random events. In this analysis, dynamically consistent cooperative solutions and analytically tractable payoff distribution procedures contingent upon specific random events are derived. This approach widens the application of cooperative differential game theory to environmental problems where future payoffs are not known with certainty.

The idea leading to this study originated some time ago when I talked at a gathering of high school graduates at a summer science camp. I spoke about the theme of "the evolution of cooperation" (in Nature) and about how that topic was amenable to studies involving Game Theory (which, more frequently, has been used in research in economics).

After that event I was stimulated to think of the possibility of modeling cooperation in games through actions of acceptance in which one player could simply accept the "agency" of another player or of an existing coalition of players.

The action of acceptance would have the form of being entirely cooperative, as if "altruistic", and not at all competitive, but there was also the idea that the game would be studied under circumstances of repetition and that every player would have the possibility of reacting in a non-cooperative fashion to any undesirable pattern of behavior of any another player. Thus the game studied would be analogous to the repeated games of "Prisoner's Dilemma" variety that have been studied in theoretical biology.

These studies of "PD" (or "Prisoner's Dilemma") games have revealed the paradoxical possibility of the natural evolution of cooperative behavior when the interacting organisms or species are presumed only to be endowed with self-interested motivations, thus motivations of a non-cooperative type.