Читайте также:
|
The strategy of each player in the equilibrium constructed in the proof of Proposition 151.1, which concerns games in which the discount factor is close to 1, has the special feature that when any player deviates, the subsequent sequence of action profiles depends only on the identity of the deviant and not on the history that preceded the deviation. In this section we show that for any common discount factor a profile of such strategies can be found to support any subgame perfect equilibrium outcome.
We begin with two lemmas, the first of which extends the one deviation property proved for finite extensive games in Lemma 98.2 to infinitely repeated games with discounting.
Lemma 153.1. A strategy profile is a subgame perfect equilibrium of the 
-discounted infinitely repeated game of 
if and only if no player can gain by deviating in a single period after any history.
Exercise 153.2. Prove this result.
The next result shows that under our assumptions the set of subgame perfect equilibrium payoff profiles of any -discounted infinitely repeated game is closed.
Lemma 153.3. Let 
be a sequence of subgame perfect equilibrium payoff profiles of the - discounted infinitely repeated game of that converges to 
. Then is a subgame perfect equilibrium payoff profile of this repeated game.
Proof. For each value of 
let 
be a subgame perfect equilibrium of the repeated game that generates the payoff profile 
. We construct a strategy profile 
that we show is a subgame perfect equilibrium and yields the payoff profile . We define, by induction on the length of the history 
, an action profile 
of and an auxiliary infinite subsequence 
of the sequence 
that has the property that the payoff profile generated by the members of the subsequence in the subgame following the history has a limit and the action profile 
converges to . Assume we have done so for all histories of length 
or less, and consider a history 
of length 
, where is a history of length . Let be the sequence of strategy profiles that we chose for the history and let be the action profile we chose for that history. For 
select for a subsequence 
of for which the sequence 
converges, and let the action profile to which converges be 
. Obviously the limiting payoff profile of the subsequence that we have chosen is the same as that of . For 
choose for a subsequence 
of , for which the sequence of payoff profiles and the sequence 
both converge, and let the action profile to which converges be .
No player 
can gain in deviating from 
by changing his action after the history and inducing some outcome 
instead of since if this were so then for large enough he could profitably deviate from 
, where is the sequence that we chose for the history . Further, the payoff profile of is .
By this result the set of subgame perfect equilibrium payoff s of any player in the repeated game is closed; since it is bounded it has a minimum, which we denote 
. Let 
be the outcome of a subgame perfect equilibrium in which player ’s payoff is .
Proposition 154.1. Let 
be the outcome of a subgame perfect equilibrium of the -discounted infinitely repeated game of 
. Then the strategy profile in which each player uses the following machine is a subgame perfect equilibrium with the same outcome .
· Set of states: 
· Initial state: 
· Output function: In state 
play 
. In state 
play 
.
· Transition function:
o In state 
move to 
unless exactly one player, say, 
, deviated from 
, in which case move to 
.
o In state : Move to 
unless exactly one player, say 
, deviated from 
, in which case move to 
.
Proof. It is straightforward to verify, using Lemma 153.1, that this defines a subgame perfect equilibrium with the required property.
Дата добавления: 2015-11-14; просмотров: 55 | Нарушение авторских прав
| <== предыдущая страница | | | следующая страница ==> |
| Rewarding Players Who Punish: A Perfect Folk Theorem for the Discounting Criterion. | | | Finitely Repeated Games |