ECO290E: Game Theory Lecture 9 Subgame Perfect Equilibrium

Game Tree An extensive-form game is defined by a tree that consists of nodes connected by branches. Each branch is an arrow, pointing from one node (a predecessor) to another (a successor). For nodes x, y, and z, if x is a predecessor of y and y is a predecessor of z, then it must be that x is a predecessor of z. A tree starts with the initial node and ends at terminal nodes where payoffs are specified.

Tree Rules 1.Every node is a successor of the initial node. 2.Each node except the initial node has exactly one immediate predecessor. The initial node has no predecessor. 3.Multiple branches extending from the same node have different action labels. 4.Each information set contains decision nodes for only one of the players.

Information Set An information set for a player is a collection of decision nodes satisfying that (i) the player has the move at every node in the information set, and (ii) when the play of the game reaches a node in the information set, the player with the move does not know which node in the information set has been reached.  At every decision node in an information set, each player must (i) have the same set of feasible actions, and (ii) choose the same action.

Subgame A subgame in an extensive-form game (a) begins at some decision node n with a singleton information set, (b) includes all the decision and terminal nodes following n, and (c) does not cut any information sets.  We can analyze a subgame on its own, separating it from the other part of the game.

Subgame Perfect NE A subgame perfect Nash equilibrium (SPNE) is a combination of strategies in a extensive-form which constitutes a Nash equilibrium in every subgame.  Since the entire game itself is a subgame, it is obvious that a SPNE is a NE, i.e., SPNE is stronger solution concept than NE.

Stackelberg Model The Stackelberg model is a dynamic version of the Cournot model in which a dominant firm moves first and a subordinate firm moves second. Firm 1 (a leader) chooses a quantity first Firm 2 (a follower) observes the firm 1’s quantity and then chooses a quantity  Solve the game backwards!

Remarks A leader never becomes worse off since she could have achieved Cournot profit level in the Stackelberg game simply by choosing the Cournot output. A follower does become worse off although he has more information in the Stackelberg game than in the Cournot game, i.e., the rivals output. Note that, in single-person decision making, having more information can never make the decision maker worse off.