probabilities under the conjecture that every player follows the sanctioning behavior
specified by the social norm; the other is derived under the conjecture that one player
deviates from the sanctioning behavior, once.
Case 1. Off-equilibrium, everyone sanctions. Consider a player who currently selects
Z. Let ρ=(ρ1,ρ2,ρ3,ρ4) with ρd being the probability that he meets a cooperator given
that d =1,L, 4 players currently select Z. Clearly, the probability that he meets someone
who chooses Z is 1 - ρ. Recall that each player can be paired to three other players, with
equal probability. Therefore, we have
ρ=(1,2, 3,0)
Here, the transition matrix is:
123
4
0
2/3
1
1
010
0 1/3 0
000
000
The bold numbers in the rows (columns) indicate the number Dt (Dt+1 ) of players who
currently (next period) play Z. Each cell represents the corresponding conditional
probability Pr[ Dt+11 Dt ]. Clearly, Pr [2|1] = Pr [4|3] = 1 since if an odd number of players
plays Z today, then at least one of them is paired to a cooperator. The latter will choose Z
in t + 1 . Also, Pr [4|4] = 1, since the social norm does not specify reversion to
cooperation. To see why Pr [2|2] = 1/3 and Pr [4|2] = 2/3 recall that there are three
possible pairings. One of those involves the two players who currently choose Z. So, with
probability 1/3 the sanctioning behavior does not spread further. If that pairing is not
realized, then Z will be necessarily seen by the remaining two cooperators. So, with
probability 2/3, next period everyone will choose the sanction, Z.
Case 2. Off equilibrium, one player does not use the prescribed sanction.
Suppose, off-equilibrium, a player who observed a deviation in the past chooses to
deviate from the sanctioning rule and plays Y this period. Instead, everybody else follows
the social norm. Consider this player. Again, ρd is the probability that he meets a
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