The name is absent

observe Z today and tomorrow select Z as well. Simple manipulations of (1) give:

V = h(3+δ) ₊ zδ(i+δ)

¹    3 - δ    (3 - δ)(1 + δ)

V = 2 h ₊  z (1 + δ)

² 3 - δ (3 - δ)(1 - δ)

V3 = z+3(h -z) + δj-δ

V = -z-

⁴   1 - δ

Payoff from a deviation, when a player does not follow the sanctioning rule. Suppose
d players have observed a deviation in the past, and everyone follows the social norm
except one of these players. This player defects from the sanctioning rule and cooperates.
Let V^~ _d denote the expected lifetime utility at the start of a period, before pairing takes
place, to the player that has observed a deviation in the past but selects Y currently, given

~

d . Using the vector of probabilities ρ and the transition matrix A , where we denote

~

A_d its d ^throw, we have:

When d = 1, this means that no deviation was observed previously but someone chooses
to deviate today. Therefore V^~ 1 = V₁ , since it is the first period in which a deviation is
observed. For the case d ≥ 2 notice that only d - 1 players choose Z currently, the

~
remaining one choosing Y . Therefore we must use the matrix A .

In that case, we see that l + ρ_d (y - l) is the expected current utility from meeting either a
cooperator or not. Since the player cooperates, when he meets a cooperator, he earns y ,
and otherwise he earns l . Again, the continuation payoff is 0 with probability 1 - δ, and it
~~

is Ad V with probability δ. We use V and not V in the continuation payoff, since
everyone reverts to the sanctioning rule specified by the social norm, from tomorrow on.
As done for the case above, simple calculations generate:

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