#### Sequential Games and its Extensive Form, Subgame Perfection and Commitment

Sequential Games:

The games we’ve studied to this point have been simultaneous move games, or at least no player knows another’s move before making his own. Now we turn to games where at least some of the moves are taken in turn.

1) Extensive Form:

A game’s extensive form specifies the players, when each player has a move, what each player can do when it’s his move, what each player knows when it’s his move, and each player’s payoffs for each combination of moves of all players. The key difference between strategic and extensive form representations of a game is that the latter specifies the order of play. This is essential in sequential games.

The game tree show a game’s extensive form. For instance a game tree for the Battle of the Sexes is:

The game tree begins with a decision node for one player and ends with terminal nodes, which list the payoffs associated with a series of strategies. In between are decision nodes for one player or another. Beside the branches we specify the strategies.

If Fred and Ethel play simultaneously, we add the shaded information box to indicate that Ethel does not have knowledge of Fred’s strategy before she plays. (We could have put Fred after Ethel, too.) In this case, the only extra information in the game tree is that the moves are indeed simultaneous.

Otherwise if Fred moves first we remove the information box. Here the game tree specifies the order of play, but the game matrix does not: the strategic form representation suppresses the order of play. To the extent order of play is important in determining the equilibrium, we must use the extensive form.

As in the one-shot version of this game, there are two Nash equilibria. Order of play has no effect on the Nash equilibria. However not all Nash equilibria are sensible in sequential games. To observe this let’s solve the game by backward induction. We start by solving each of Ethel’s two sub games. If Fred chose Boxing, the solution to the subsequent sub game is (Boxing, Boxing). If Fred chose Opera, the solution to the subsequent sub game would be (Opera, Opera). Since Ethel will condition her choice on Fred’s and Fred knows this, his best response is to choose Boxing. That is he knows that Ethel will follow him consequently he chooses his favourite.

2) Subgame Perfection:

This game is subgame perfect. To be subgame perfect, the players’ strategies must constitute a Nash equilibrium in every subgame – that is, forward from each decision node. Although going together to the Opera is a Nash equilibrium, it is not subgame perfect. The idea here is that subgame perfection throws out Nash equilibria that could be supported by only incredible threats. For instance Ethel could announce that her strategy will be Opera. If Fred believed her, he would also choose Opera. However Ethel’s strategy isn’t credible. If Fred chose Boxing, it wouldn’t be her best response to choose Opera. So choosing Opera in that subgame wouldn’t be a Nash equilibrium, which implies going together to the Opera is not subgame perfect.

3) Commitment:

If Ethel could commit to go to the Opera, she would win the Battle of the Sexes. To commit, she must take it her best response to go to the Opera no matter where Fred goes. That is, she must reduce her options to improve her payoffs.

Presume she posts a bond of \$3 with Lucy. If she goes to the Boxing match, she forfeits the bond. This changes Ethel’s payoffs. Now Opera emerges as a dominant strategy in the second subgame. Her danger is credible. Fred’s best response extant is to go to the Opera. Ethel wins. Adding the possibility of commitment would expand the strategy space (example how much to pay Lucy) so Fred and Ethel would be playing a different game.

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