On the testable implications of collective choice theories (Q1587639)

Consider the case of two agents whose set of all conceivable actions are finite sets \(A_1\), \(A_2\), respectively; the agents' preferences are not observable; what is observed is the choice made by agents when the set of feasible actions \(B_1\), \(B_2\) alter: a joint choice function \(f\), which records for every possible pair of feasible sets \(B_1\), \(B_2\) the joint actions selected from \(B_1\times B_2\). Given a particular \(f\), it is asked whether the agents are behaving cooperatively or non-cooperatively. Non-cooperative behavior to a Nash equilibrium while cooperative behavior is related to a Pareto optimum. Formally, a joint choice function \(f\) is Nash-rationalizable if there exist preferences on \(A_1\times A_2\), one for each agent such that the joint actions selected by \(f\) from \(B_1\times B_2\) for each \(B_1\), \(B_2\) are precisely the Nash equilibria of the normal form game defined by \(B_1\), \(B_2\) and the restriction of the proposed preference to \(B_1\times B_2\). Similarly when the joint actions selected by \(f\) from \(B_1\times B_2\) coincide with the Pareto ptima of the game corresponding to \(B_1\), \(B_2\) for each \(B_1\), \(B_2\), \(f\) is said to be Pareto-rationalizable. It is shown for the two agent case if any joint choice function is Nash rationalizable, it is also Pareto rationalizable. Next, the paper allows for any number of agents and non-deterministic choice. Necessary and sufficient conditions for a joint choice function to be Nash-rationalizable are identified; for cooperative behavior, every choice function is remarkable to be partially Pareto-rationalizable.