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The strategy profile defined in the proof of Proposition 149.1, in which players are punished for failing to mete out the punishment that they are assigned, may fail to be a subgame perfect equilibrium when the players' preferences are represented by the discounting criterion. The reason is as follows. Under the strategy profile a player who fails to participate in a punishment that was supposed to last, say, 
periods, is himself punished for, say, 
periods, where may be much larger then . Further deviations may require even longer punishments, with the result that the strategies should be designed to carry out punishments that are unboundedly long. However slight the discounting, there may thus be some punishment that results in losses that can never be recovered. Consequently, the strategy profile may not be a subgame perfect equilibrium if the players' preferences are represented by the discounting criterion.
To establish an analog to Proposition 149.1 for the case that the players' preferences are represented by the discounting criterion, we construct a new strategy. In this strategy players who punish deviants as the strategy dictates are subsequently rewarded, making it worthwhile for them to complete their assignments. As in the previous section we construct a strategy profile only for the case in which the equilibrium path consists of the repetition of a single (strictly enforceable) outcome. The result requires a restriction on the set of games that is usually called full dimensionality.
Proposition 151.1 (Perfect folk theorem for the discounting criterion) Let 
be a strictly enforceable outcome of 
. Assume that there is a collection 
of strictly enforceable outcomes of 
such that for every player 
we have 
and 
for all 
. Then there exists 
such that for all 
there is a subgame perfect equilibrium of the 
– discounted infinitely repeated game of than generates the path 
in which 
for all .
Proof. The strategy profile in which each player uses the following machine is a subgame perfect equilibrium that supports the outcome in every period. The machine has three types of states. In state 
the action profile chosen by the players is . For each 
the state 
is a state of "reconciliation" that is entered after any punishment of player 
is complete; in this state the action profile that is chosen is 
. For each player and period between 
and some number 
that we specify later, the state 
is one in which there remain periods in which player is supposed to be punished; in this state every player 
other than takes the action 
, which holds down to his minmax payoff. If any player deviates in any state there is a transition to the state 
, (that is, the other players plan to punish player for periods). If in none of the periods of punishment there is a deviation by a single punisher the state changes to 
. The set 
of states serves as a system that punishes players who misbehave during a punishment phase: if player does not punish player as he is supposed to, then instead of the state becoming , in which the outcome is , player is punished for periods, after which the state becomes , in which the outcome is 
.
To summarize, the machine of player is defined as follows, where for convenience we write 
; we specify later.
· Set of states 
.
· Initial state: .
· Output function: In choose 
. In choose if 
, and 
if 
.
· Transitions in response to an outcome 
:
o From stay in unless a single player 
deviated from , in which case move to 
.
o From :
§ If a single player 
deviated from 
then move to .
§ Otherwise move to 
if 
and to if 
We now specify the values of 
and . As before, let 
be the maximum of 
over all and . We choose and to be large enough that all possible deviations are deterred. To deter a deviation of any player in any state we take large enough that 
for all i 
and all 
and choose 
where 
is close enough to that for all we have
(This condition is sufficient since 
for ). If a player deviates from for 
then he obtains at most in the period that he deviates followed by periods of 
and 
subsequently. If he does not deviate then he obtains 
for between and periods and 
subsequently. Thus to deter a deviation it is sufficient to choose 
close enough to one that for all we have
(Such a value of exists because of our assumption that if )
Exercise 152.1 Consider the three-player symmetric infinitely repeated game in which each player's preferences are represented by the discounting criterion and the constituent game is 
where for 
we have 
and 
for all 
.
d) Find the set of enforceable payoffs of the constituent game.
e) Show that for any discount factor 
the payoff of any player in any subgame perfect equilibrium of the repeated game is at least 
.
f) Reconcile these results with Proposition 151.1.
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| Punishing the Punisher: A Perfect Folk Theorem for the Overtaking Criterion | | | The Structure of Subgame Perfect Equilibria Under the Discounting Criterion |