multi-stage game

As an example, consider a two-stage game in which the stage game in Figure 1 is played in each of two periods:

File:Twostagegame.png

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 $2^3=8$ strategies.

For Player 2, there are $2^5=32$ strategies.

The extensive form of this multi-stage game is shown in Figure 2:

File:Twostagegameextensiveform.png

In this game, the only Nash Equilibrium in each stage is (B, b).

(BB, bb) will be the Nash Equilibrium for the entire game.

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:

File:Multistagestage1.png

File:Multistagestage2.png

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 $2^5=32$ strategies.

For Player 2, there are $2^5=32$ strategies.

The extensive form of this multi-stage game is shown in Figure 5:

File:Twostagechanginggame.png

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.

{{reflist}}

• {{Cite Fudenberg Tirole 1991}}
• {{Cite book|last=Watson|first=Joel|title=Strategy: an introduction to game theory|date=2013|isbn=978-0-393-91838-0|edition=Third |location=New York|oclc=842323069}}

Category:Game theory game classes

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