Repeated Games
In these notes, we’ll discuss the repeated games, the games where a particular smaller game is
repeated; the small game is called the stage game. The stage game is repeated regardless of
what has been played in the previous games. For our analysis, it is important whether the
game is repeated finitely or infinitely many times, and whether the players observe what each
player has played in each previous game.
1.1 Finitely repeated games with observable actions
We will first consider the games where a stage game is repeated finitely many times, and at the
beginning of each repetition each player recalls what each player has played in each previous
play. Consider the following entry deterrence game, where an entrant (1) decides whether to
enter a market or not, and the incumbent (2) decides whether to fight or accommodate the
entrant if he enters.
1 Enter 2
X
Acc.
Fight
(0,2) (-1,-1)
(1,1)
Consider the game where this entry deterrence game is repeated twice, and all the previous
actions are observed. Assume that a player simply cares about the sum of his payoffs at the
stage games. This game is depicted in the following figure.
1
1 Enter 2
X
Acc.
Fight
1 Enter 2
X
Acc.
Fight
(1,3) (0,0)
(2,2)
1 Enter 2
X
Acc.
Fight
(-1,1) (-2,-2)
(0,0)
2 Enter 1
X
Acc.
Fight
(-1,1)
(1,3)
(0,4)
Note that after the each outcome of the first play, the entry deterrence game is played again
—where the payoff from the first play is added to each outcome. Since a player’s preferences
over the lotteries do not change when we add a number to his utility function, each of the three
games played on the second “day” is the same as the stage game (namely, the entry deterrence
game above). The stage game has a unique subgame perfect equilibrium, where the incumbent
accommodates the entrant, and anticipating this, the entrant enters the market.
1 Enter 2