An “economy” is composed of four players a, b, c, and d who interact for an
indefinite number of periods denoted t = 1, 2, .... Participants are randomly paired to play
the prisoners’ dilemma of Table 2. There are three ways to pair participants in an
economy, {ab, cd}, {ac, bd}, or {ad, cb}, and in each period one pairing was randomly
chosen with equal probability.9
4.1 Equilibrium in the stage game
Consider the stage game described in Table 2A, which is a prisoners’ dilemma. The
players simultaneously and independently select an action from the set {Y,Z}. We allow
for mixed-strategies. Let π ∈ [0,1] denote the probability that the representative player
selects Y, and 1 -π the probability that he selects Z. We use Π ∈ [0,1] to denote the
given selection of the opponent.
The unique Nash equilibrium is defection. In equilibrium both players choose Z, the
minmax action, and earn z, the minmax payoff. The representative player’s payoff is
simply his expected utility, denoted U. This can be rearranged as:
U=z+Π(h-z)-π[Π(h-y)+ (1-Π)(z-l)].
The player maximizes U by choosing π, so can assure himself payoff z, independent of Π.
Notice that U is linear in π, and we have assumed y < h and l < z . It follows that the
player’s best response is to set π = 0, for any Π ≥ 0.
Since 2z < l +h < 2y , total surplus in the economy is maximized when each pair
cooperates. Thus, we refer to the outcome where both players in both pairs select Y as the
(Pareto) efficient or fully cooperative outcome. If both pairs in the economy select
{Z,Z} , then we say that the outcome is inefficient. A Nash equilibrium is a fixed point in
the players’ aggregate best response, so π = Π = 0 is the unique equilibrium.
4.2 Equilibrium in the indefinitely repeated game with private monitoring
With private monitoring indefinite repetition of the stage game with randomly
selected opponents can expand the set of equilibrium outcomes. In this section we
9 Strictly speaking, we are dealing with a game with varying opponents, since players are paired randomly
at each point in time. However, action sets and payoff functions are unchanging. Thus, we refer to it as a
supergame, following the experimental literature.
11