Cooperation: Game-Theoretic Approaches

This book constitutes a systematic exposition of the various game theoretic approaches to the issue of cooperation.

There is a broad division of game theory into two approaches: the cooperative approach and the noncooperative approach. These approaches should not be considered as analyzing different kinds of games; rather, they are different ways of looking at the same game. As Joachim Rosenmüller has said, the game is one “ideal” of which the cooperative and the noncooperative approaches are two “shadows”.

In these notes we deal with the so called “bargaining problem” (Nash, 1950). Our approach is axiomatic. We search for solutions satisfying some desirable properties (axioms).

The model discussed in Part I is written under the assumption of a fixed number of agents. Here, the number of agents is allowed to vary. How solutions should respond to such changes are formulated as axioms, and additional characterizations of the main solutions are developed. A detailed account of these recent developments can be found in Thomson and Lensberg (1989).

Pure bargaining games discussed in the previous two lectures are a special case of n-person cooperative games. In the general setup coalitions other than the grand coalition matter as well. The primitive is the coalitional form (or, “coalitional function”, or “characteristic form”). The primitive can represent many different things, e.g., a simple voting game where we associate to a winning coalition the worth 1 and to a losing coalition the worth 0, or an economic market that generates a cooperative game. Von Neumann and Morgenstern (1944) suggested that one should look at what a coalition can guarantee (a kind of a constant-sum game between a coalition and its complement); however, that might not always be appropriate. Shapley and Shubik introduced the notion of a C-game (see Shubik (1982)): it is a game where there is no doubt on how to define the worth of a coalition. This happens, for example, in exchange economies where a coalition can reallocate its own resources, independent of what the complement does.

This paper surveys cooperative game theory when players have incomplete or asymmetric information, especially when the TU and NTU games are derived from economic models. First some results relating balanced games and markets are summarized, including theorems guaranteeing that the core is nonempty. Then the basic pure exchange economy is extended to include asymmetric information. The possibilities for such models to generate cooperative games are examined. Here the core is emphasized as a solution, and criteria are given for its nonemptiness. Finally, an alternative approach is explored based on Harsanyi’s formulation of games with incomplete information.

The topic to be reviewed in this lecture is included in what Bob Aumann described in his lecture as the bridges between cooperative and noncooperative theory. If I had all the time in the world, I would begin by presenting the basics of noncooperative game theory, but I cannot possibly do this. I will therefore remain very elementary, and I will be somewhat loose about the noncooperative concepts¹. The flavor of what I will be doing today consists in writing down or describing game procedures, understood as non-cooperative mechanisms for interaction, discussion, and the formulation of agreements about how to split things. These bargaining procedures will be set in a context which will stay very close to the frameworks presented by earlier lecturers. We will then see how the noncooperative solutions of the bargaining procedures relate to the axiomatic procedures presented earlier by others.

What is implementation theory all about? To answer this question we shall follow Moore (1991) by describing a classic problem known as “King Solomon’s Dilemma.” As it happened, two women approached Solomon with a newborn child. Each claimed to be the child’s mother. It was up to Solomon to decide which one was telling the truth. In his wisdom, Solomon had the women lay the child before him. He drew his sword and announced that he would settle the dispute by cutting the child in half. However, just before he brought down his sword the true mother begged that he spare the child’s life and give it to the impostor. Knowing that only the real mother would be willing to give up the child rather than allow it to die, Solomon gave it to her. Such is the nature of an implementation problem which we now describe in rather general terms.

This paper surveys implementation theory when players have incomplete or asymmetric information, especially in economic environments. After the basic problem is introduced, the theory of implementation is summarized. Some coalitional considerations for implementation problems are discussed. For economies with asymmetric information, cooperative games based on incentive compatibility constraints or Bayesian incentive compatible mechanisms are derived and examined.

Cooperative and non-cooperative approaches to game theory represent two polar, and simplifying, extremes. In the former, it is assumed that players can make commitments that are binding, i.e., an agreement once made is enforceable. In contrast, non-cooperative game theory assumes that agreements cannot be enforced and equilibrium agreements or strategies, therefore, must be self-enforcing.

The Theory of Social Situations(TOSS) is an integrative approach to the study of formal models in social and behavioral sciences. TOSS unifies the representation of “cooperative” and “non-cooperative” social environments, allowing for diverse coalitional interactions. It does so using the notion of a (social) “situation”. TOSS disassociates the solution concept from the representation of social environments. The unified solution concept in TOSS — “stable standard of behavior” — employs stability as the sole criterion. One of the important merits of TOSS is that by representing a social environment as a situation, it specifies the exact negotiation process and the way in which players and coalitions use the set of outcomes (actions, alternatives) available to them. Moreover, the flexibility of TOSS enables the analysis of social environments that cannot be studied within the classical paradigm of game theory. This lecture is divided into three parts: (1) Motivation for the notion of a social situation, (2) Formal definitions of a situation and of a stable standard of behavior, and (3) Some applications of TOSS to cooperation.¹

Cooperative game theory deals with feasible outcomes, whereas non cooperative game theory is concerned with strategic equilibrium. Repeated games provide a bridge between these two theories: folk-theorem-like results deal with the relation between feasible payoffs in a one shot game and equilibrium payoffs in the corresponding repeated game.

The purpose of this presentation is to introduce models of extension of games with preplay or intraplay information and communication. These extensions will allow us to define new notions of equilibria. The relevant question is to see how the outcomes change when communication between players is allowed, or when they are given some kind of preplay information.

Economists have for long expressed dissatisfaction with the complex models of strict rationality that are so pervasive in economic theory. There are several objections to such models. First, casual empiricism or even just simple introspection lead to the conclusion that even in quite simple decision problems, most economic agents are not in fact maximizers, in the sense that they do not scan the choice set and consciously pick a maximal element from it.

This chapter studies the implications of bounding the complexity of players’ strategies in long term interactions. The complexity of a strategy is measured by the size of the minimal automaton that can implement it.

A finite automaton has a finite number of states and an initial state. It prescribes the action to be taken as a function of the current state and its next state is a function of its current state and the actions of the other players. The size of an automaton is its number of states.

The results study the equilibrium payoffs per stage of the repeated games when players’ strategies are restricted to those implement able by automata of bounded size.

We present a selective survey of recent work on the Brown-Robinson learning process known as “fictitious play.” We study the continuous time version of the process and report convergence results for specific classes of games.

We present a selective survey of work on evolutionary models of dynamics in games. We focus on the continuous time replicator dynamics and report convergence results for specific classes of games.

There are three types of decision and game theory: