The name is absent

^~   2     12

V2 = ^{l +} 3⁽ y - ^l ) ^{+ δ}(~ V2 +y V₅)

V = l + 3(y - l) + δ(3V3 + 3V4) (3)

V^~4 = l+ δV4

A3 Requirements for individual optimality

In this section we check that the actions recommended by the social norm are best
responses after any history of play. To do so we consider two issues. First, we derive a
condition ensuring that choosing Z is not a best response on the equilibrium path. Second,
we check that playing Y instead of Z, after having observed a deviation, is never optimal.

Suboptimality of a deviation, in equilibrium. We must check that deviating by
choosing Z is suboptimal, relative to cooperating. That is:

y ≥ V = h(3 + δ) ₊ zδ(1 + δ

1 - δ ¹    3 - δ   (3 - δ)(i - δ)

an inequality that is rearranged as:

δ²(h - z) +δ(2h - y - z)-3(h - y) ≥ 0

Let f (δ) define the expression on the RHS of the inequality. Notice that since h > y > z
and δ∈(0,1) , then f(δ) ≥ 0 for all δ≥ δ^* where δ^*∈(0,1) is the unique value of δthat
solves f(δ)=0 . We have δ^* >0 since f (0)<0 and f ^'(δ)> 0 . Also, δ^* <1 since f ^'(δ)> 0 for
δ>0 and f (1) =2(y -z) >0 . The parameterization of our experiment implies δ^* = 0.443 .

Suboptimality of a deviation, off-equilibrium. Here, we check that if a player has
observed Z in the past, then Y today is suboptimal. That is, since we have shown that
choosing Z is never optimal, when d =1 , we must find conditions such that V_d ≥ V^~ _d
for all d≥ 2 . To do so, use (1), (2) and (3). Clearly, V₄ ≥ V^~ ₄ since z ≥l . Now consider
the inequality V ₃ ≥ V^~ ₃ . Rearranging:

V 3 = l+3(y -l )+δ 3V - V4)+δV4,

we have V ₃ ≥ V^~ ₃ , if:

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