Consider the following trust game is repeatedly played 3 times:

• Ann first decides whether or not to trust Bob.
  • If Ann does not trust Bob, then the game ends and both players get a payoff of 0.
  • If Ann chooses to trust Bob, then Bob can either cooperate or defect.
    • If Bob cooperates, he provides some fair service and this leads to a payoff of 1 for each player.
    • If Bob defects, he gets away with Ann's money and gets 2, while Ann loses and gets -1.

Ann and Bob have the same discount factor 0<<1, across every period in which the game is played. Is my answer below the correct solution to finding out if cooperation can be sustained? For this game, I will evaluate using the concepts learned with regard to finitely repeated games. There are three possibilities:

BEST CASE:

Ann trusts Bob in the first and second periods, and Bob cooperates. Then, in the last period, she chooses the SPE and does not give any money to Bob.

In this case, Bob's payoff is 1+ 1 + 0²= 1 +

MID CASE:

Ann trusts Bob in the first and second periods. Bob cooperates initially but then defects in the second period. In the last period, Ann chooses the SPE.

In this case, Bob's payoff is 1+ 2 + 0²= 1 + 2

WORST CASE:

Ann trusts Bob in the first period, but he defects. Thereafter, Ann chooses the SPE.

In this case, Bob's payoff is 2+ 0 + 0²= 2

Since <1, the BEST CASE would never be better than either of the other outcomes. We would then need to see if some cooperation can be attained in the MID CASE, wherein Bob would not be able to have a profitable deviation by implementing the strategy under the WORST CASE.

The computation would be as follows:

1 + 2 2

2 1

or 0.5

Given this finding, some cooperation is possible during the 1^st time they play the trust game if 0.5. Otherwise (if < 0.5), Bob has no incentive to cooperate.

Thereafter, the 2^nd and 3^rd time this trust game is played, Ann should no longer trust Bob. He knows that the SPE is for Ann not to give him any money in the final round and would likely maximize his payoff in the 2^ndround leaving Ann worse off.