The textbook I am currently reading claims that, in an infinitely-repeated game with discount, there might be a payoff vector which is feasible and individually-rational, but it is not an equilibrium payoff vector in the repeated game. The example is the following basic game for three players:
L R T 0,2,5 0,0,0 M 0,1,0 2,0,5 B 1,1,0 1,1,0
The third player is a dummy player with only one possible action.
In this game:
- The minimax values of the players are 1,1,0.
- The payoff vector 1,1,5 is both individually-rational and feasible (e.g. by mixing TL and MR with equal frequencies).
The book claims that the only payoff vector in equilibrium is 1,1,0! Why?
Let $E$ be some equilibrium in the repeated game. The payoffs of the row and column players in $E$ must be 1, because:
- They must be at least 1 because these are the minimax values;
- They must be at most 1 because the sum of utilities of these players in every outcome is at most 2.
In the TR and ML cells, the sum of utilities of the row and column players is less than 2; hence, these cells are not played in equilibrium at all.
So in equilibrium, the only cells that may be played with positive frequency are: TL, MR, BL, BR.
Now, the authors claim that TL and MR are also not played at all in equilibrium. From this they conclude that the payoff of the dummy player is 0. I didn't understand this part. Is it true that TL and MR are never played in equilibrium? Why?