How to dynamic repeated games complete information

Question 1

Define trigger strategies such that the outcome of all infintie stages will be (Clean,Clean) and SPNE.

Answer 1

Define the strategies.

S1: Play clean
S2: Play Clean if outcome of all previous stages was (Clean,Clean), otherwise play Don’t Clean.

(Trigger strategy: In the 1st turn,
play Clean. In every subsequent
turn, if outcome from every previous
turn was (Clean,Clean), play Clean,
otherwise play Don’t Clean.)

Step a: On the equilibrium path:
1. Define the payoff for staying with the TFT, and for deviating
2. Then write up the inequality and isolate δ
to find for what values of δ Player x
wouldn’t deviate. Be aware of the geometric series. You only need to
check the player that has the highest
incentive to deviate, because their discount factor becomes the crucial level to fulfill. If the discount factor is under this level, the player would deviate.
3. conclude, and include step b.

Step b: Off the equilibrium path: Check if the trigger strategy is credible if a
player deviated from the equilibrium
path by playing ”don’t clean” in the
previous round. It is credible if the other players BR is also “don’t clean”. Then TS have SGNE both ON and OFF EP, which means that TS constitute a SPNE.

Question 2

If we have a finite game, where we have to specify a given amount of payoffs, which is a SPNE, then we have to write down the different strategies in the stages, which is needed to fulfill the wished payoff. How do we do that?

Answer 2

1. State which strategy you would use (OSD, TS, Tit for Tat), to reach the wished payoff, there could be the two players payoff in different rounds summed up.
2. Apply the strategy to amount of stages you would use.
3. Show the summed payoff in a normal form matrix, where on payoff will show the wished amount and the others the payoff if some deviate.
4. Check that the strategy constitute to SGNE on EP in the matrix, and there is one or more SGNE, which is played OFF EP.

example if G(2), meaning a game with two stages. Then you need to find a strategy in the two stages, which will end up giving the wished amount of payoff.

Question 3

Tit for tat (copycat game), SPNE?

Answer 3

1. Set up the (copycat) strategies
   S1: Play some strategy
   S2: Play the strategy your opponent played last round.

Step a: On the equilibrium path:
1. Define the payoff for staying with the TFT, and for deviating playing NTFT.
2. Then write up the inequality and isolate δ
to find for what values of δ Player x
wouldn’t deviate. Be aware of the geometric series. Here you again only have to check for the player with the highest wish to deviate, to look on the lowest discount factor, which will make a NE possible. BUT to look at a SPNE, we need to check both players discount factor.
3. Conclude of there is a SPNE, only in the knife-edge case, were both discount factors are alike,