Noncooperative Equilibria in One-Shot Games: A Synthesis

A vast number of gaming situations that economists, political scientists, and evolutionary biologists consider have the essential feature of the prisoners' dilemma game. A cardinal feature of the equilibrium in such situations is that the outcome is individually rational (in the sense that no individual by herself has an incentive to do anything different from what she is doing in equilibrium) but is collectively irrational (in the sense that coordinated behaviour could make all players better off). Examples of such situations are the exploitation of commonly-owned resources, investment in R&D by firms when there are spillovers in knowledge, choice of subsidies given by countries to domestic firms in competition with foreign rivals, the amount spent by neighboring cities to control crime, competitive advertising by firms producing differentiated products, the amounts of time students spend studying for an examination when it marked " on the curve " , the amount of money people spend on clothes when their preferences are such that they care about how well dressed they are relative to their friends, output or price choices of firms in an oligopoly – the list is virtually endless. In some of these situations, the collectively rational (i.e., Pareto efficient) outcome calls for higher levels of individual activity than the Nash equilibrium levels. In others, Pareto efficiency calls for lower levels of individual activity than in the Nash equilibrium. What essential aspect of the situation determines whether there is too much or too little of the activity, relative to the efficient outcome? This is the central question this paper seeks to answer, and to answer it we set out a simple, yet powerful and surprisingly general framework, which we successively adapt to games of higher levels of sophistication. The answer to the question turns on the substitutability or complementary of the actions of the players. Our analysis is carried out entirely in the context of two-person games in which the strategies available to the players are continuous variables. We begin our analysis with the simplest scenario: a symmetric game involving two persons, called ONE and TWO, and a single stage in which the players move simultaneously. The actions of the players are defined to be plain substitutes (respectively plain complements) if an increase in the level of one player's activity decreases (respectively increases) the other's payoff. We demonstrate that the Nash equilibrium choices of ONE and TWO are higher than or …