In this chapter we continue the study of the testable implications of models of collective behavior. We focus here on game-theoretic models. Our first results use a version of the abstract choice environments from Chapter 2, and discuss testing Nash equilibrium as the prediction of the choices made by a collection of agents. We then turn to models of bargaining and two-sided matching.

NASH EQUILIBRIUM

Let N be a finite set of agents of cardinality n. For each i ∈ N, consider some finite set. The idea is that is some “global” set of strategies that may be available to agent i. In a specific (observable) instance, player i is restricted to choosing a strategy from, a “budget” of possible strategies. Consider the nonempty product subsets of, denoted by S. A typical element of S has the form S1 ×S2 ×· · ·×Sn; where each Si is a nonempty subset of Si. These sets are called game forms.

An element of is called a strategy profile. As usual in game theory we use (si, s−i) to denote a strategy profile in which si is i's strategy and is a strategy profile for players in N \ ﹛i﹜.

A joint choice function is a mapping c : satisfying c(S) ⊆ S. Note that c(S) need not have a product structure, but it is required to be nonempty. In particular, every joint choice function is a choice function in the sense of Chapter 2. We will be interested in notions of rationalization that reflect the product structure of S.

Let be a family of preference relations, each over. Note that these preferences define a (normal-form) game for each S = S1 ×S2×· · ·×Sn ∈ S. By _i|S we mean the restriction of preferences _i to S.