In: Economics
I want someone to provide me with examples of repeated strategic interactions, not the definition.
Answer :-
I. Two-Stage Repeated Game with Multiple Nash Equilibria:
X | Y | Z | |
A | 5, 4 | 1,1 | 2,5 |
B | 1,1 | 3,2 | 1,1 |
Example: It shows a two-stage repeated game with multiple pure strategy Nash equilibria. Since these equilibria differ in terms of payoffs for Player 2, Player 1 can propose a strategy over multiple stages of the game that incorporates the possibility for punishment or reward for Player 2.
Let us think that Player 1 might propose that they play (A, X) in the first round. If Player 2 complies in round one, Player 1 will reward them by playing the equilibrium (A, Z) in round two, yielding a total payoff over two rounds of (7, 9).
If Player 2 deviates to (A, Z) in round one instead of playing the agreed-upon (A, X), Player 1 can threaten to punish them by playing the (B, Y) equilibrium in round two. This latter situation yields payoff (5, 7), leaving both players worse off.
In this way, the threat of punishment in a future round incentivizes a collaborative, non-equilibrium strategy in the first round.
Since the final round of any finitely repeated game, by its very nature, removes the threat of future punishment, the optimal strategy in the last round will always be one of the game’s equilibria.
It is the payoff differential between equilibria in the game represented in the Example that makes a punishment/reward strategy viable.
2. Two-Stage Repeated Game with Unique Nash Equilibrium:
M | N | O | |
C | 5,4 | 1,1 | 0,5 |
D | 1,1 | 3,2 | 1,1 |
Example: It shows a two-stage repeated game with a unique Nash equilibrium. Since there is only one equilibrium here, there is no mechanism for either player to threaten punishment or promise reward in the game’s second round.
As such, the only strategy that can be supported as a subgame perfect Nash equilibrium is that of playing the game’s unique Nash equilibrium strategy (D, N) every round.
In this case, that means playing (D, N) each stage for two stages (n=2), but it would be true for any finite number of stages n.
This result means that the very presence of a known, finite time horizon sabotages cooperation in every single round of the game. Cooperation in iterated games is only possible when the number of rounds is infinite or unknown.