Multi-stage game
Multi-stage game
In game-theory, a multi-stage game is a sequence of several simultaneous-games played one after the other. This is a generalization of a repeated-game: a repeated game is a special case of a multi-stage game, in which the stage games are identical.
Multi-Stage Game with Different Information Sets As an example, consider a two-stage game in which the stage game in Figure 1 is played in each of two periods:
The payoff to each player is the simple sum of the payoffs of both games.
Players cannot observe the action of the other player within a round; however, at the beginning of Round 2, Player 2 finds out about Player 1's action in Round 1, while Player 1 does not find out about Player 2's action in Round 1.
For Player 1, there are <math display="inline">2^3=8</math> strategies.
For Player 2, there are <math display="inline">2^5=32</math> strategies.
The extensive form of this multi-stage game is shown in Figure 2:
In this game, the only Nash Equilibrium in each stage is (B, b).
(BB, bb) will be the Nash Equilibrium for the entire game.
Multi-Stage Game with Changing Payoffs In this example, consider a two-stage game in which the stage game in Figure 3 is played in the first period and the game in Figure 4 is played in the second:
The payoff to each player is the simple sum of the payoffs of both games.
Players cannot observe the action of the other player within a round; however, at the beginning of Round 2, both players find out about the other's action in Round 1.
For Player 1, there are <math display="inline">2^5=32</math> strategies.
For Player 2, there are <math display="inline">2^5=32</math> strategies.
The extensive form of this multi-stage game is shown in Figure 5:
Each of the two stages has two Nash Equilibria: which are (A, a), (B, b), (X, x), and (Y, y).
If the complete contingent strategy of Player 1 matches Player 2 (i.e. AXXXX, axxxx), it will be a Nash Equilibrium. There are 32 such combinations in this multi-stage game. Additionally, all of these equilibria are subgame-perfect.