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ECO290E: Game Theory

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 ).

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ECO290E: Game Theory

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  1. ECO290E: Game Theory Lecture 9 Subgame Perfect Equilibrium

  2. 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.

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

  4. 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.

  5. 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.

  6. 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.

  7. 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!

  8. 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.

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