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The next result is an analog of Proposition 146.2 for the overtaking criterion; it shows how strategies di
erent from those used to prove the perfect folk theorem for the limit of means criterion can support desirable outcomes when the players' preferences are represented by the overtaking criterion. For simplicity we construct a strategy profile only for the case in which the equilibrium path consists of the repetition of a single (strictly enforceable) outcome.
Proposition 149.1 (Perfect folk theorem for the overtaking criterion) For any strictly enforceable outcome 
of 
there is a subgame perfect equilibrium of the overtaking in infinitely repeated game 
, that generates the path 
, in which 
for all 
.
Proof. Let 
– be the maximum of 
over all 
and 
. Consider the strategy profile in which each player 
uses the following machine:
· Set of states: 
. (In the state 
player 
deserves to be punished for periods more)
· Initial state: 
.
· Output function: In choose 
. In choose 
, if 
and 
if 
· Transitions in response to an outcome 
:
o From stay in , unless for some player we have 
(i.e. is the only deviant from ), in which case move to , where is the smallest integer such that 
.
o From :
§ If 
or 
for at least two players 
(i.e. all punishers punish or at least two do not do so) then move to 
if 
and to if 
.
§ If 
and 
if 
(i.e. 
is the only punisher who does not punish) then move to 
, where 
is large enough that the sum of ’s payoff in state and his payoff in subsequent periods if he does not deviate is greater than his payoff in the deviation plus 
. (Such a number exists since after periods the players were supposed to go back to the equilibrium outcome and 
.)
Under this strategy profile any attempt by a player to increase his payoff by a unilateral deviation after any history, including one after which punishment is supposed to occur, is offset by the other players' subsequent punishment. Again we leave it to you to verify that the strategy profile is a subgame perfect equilibrium.
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| Игры с конечным числом повторений | | | Rewarding Players Who Punish: A Perfect Folk Theorem for the Discounting Criterion. |