The name is absent

Appendix A

Proof of Lemma 1

In this section we develop the proof of the Proposition. We start by discussing payoffs,
given a deviation. A player deviates from desired play in two instances: In- or off-
equilibrium, if she has not observed a deviation in the past but chooses Z, currently. Off-
equilibrium, if he has observed a deviation in the past but plays y, currently. Since the
environment is stationary, by the unimprovability criterion we restrict attention to one-
time deviations. We also consider only single-player deviations. While this simplifies the
analysis, deriving off-equilibrium payoffs still requires a bit of work (which is why we
include the proof in the appendix). The problem is that players observe only the actions in
their pair; in order to calculate expected values, we must know how uncooperative
behavior spreads to the economy after a defection is observed.

A1 The diffusion of sanctions in the economy

Consider a representative period t and recall that there are 3 possible ways to pair four
players. Thus, if d = 1, L ,4 is the number of players who choose Z currently, then
d^' =d,L,4 is the number of deviators tomorrow, which depends on the current realization
of the random pair.²⁵ As noted above, the central concern of a player is the likelihood that
her/his opponent does not cooperate. Thus, we report the probabilities ρ_d that a player
who selects Z today will meet a cooperator today, given that d players choose Z today.
We also calculate Pr[ d '∣ d ] (i.e., the probability that tomorrow there are d ' individuals
who play Z, given that today there are d ).

The first set of probabilities is needed to determine the expected current utility to a player
who is aware of a deviation or that deviates, selecting Z. The second set of probabilities is
needed to calculate the continuation payoff for a player who is aware of a deviation or
that deviates, selecting Z. Indeed, they will give us transition matrices, allowing us to
calculate the various probabilities that the sanction spreads to the rest of the economy.
Notice that there will be two contingencies to consider. In one case we calculate

²⁵ Clearly, if D_t=0, then D_t+1=0 with certainty.

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