Extensive games (with perfect information) (also referred to as extensive-form games or dynamic games) DEFINITION An extensive game with perfect information has the following components A set N (the set of players). A set H of sequences (finite or infinite) satisfying: The empty sequence is a member of H If (a k ) k=1,...,k H (where K may be infinite) and L < K then (a k ) k=1,...,l H If an infinite sequence (a k ) k=1 H satisfies (ak ) k=1,...,l H for every positive integer L then (a k ) k=1 H (Each member of H is called a history; each component of a history is an action taken by a player.) A history (a k ) k=1,...,k H is called terminal (final) if it is infinite or if there is no a K+1 such that (a k ) k=1,...,k+1 H. The set of terminal histories is denoted Z. A function P that assigns to each nonterminal history (each member of H\Z) a member of N. (P is the player function, P (h) being the player who takes an action after the history h.) For each player i N a preference relation i on Z (the preference relation of player i). Denote a history h followed by action a by (h, a). Interpetation of this definition: After any nonterminal history h Player P (h) chooses a possible action a, i.e. such an action that (h, a) belongs to H. Game tree is a convenient method to represent extensive-form games. Think of a rooted tree in graph-theoretic sense. Actions are represented by branches (or edges) and histories correspond to paths and induce nodes. Non-terminal histories induce decision nodes and terminal histories induce end-nodes or end-points or leaves. Example: mini-ultimatum game 1
Strategies Action Strategy! Strategy is a plan of action for every contingency. DEFINITION A strategy of player i N in an extensive game with perfect information N, H, P, i is a function that assigns an admissible action to each nonterminal history h H\Z for which P (h) = i. A combination of strategies induces an outcome of the game - the terminal history that will result when they are implemented. Note that the strategy dictates what to do even at nodes that, under this strategy, will not be visited. Any extensive-form game can now be represented in a matrix form. Example: Entrant game Thus, also the concept of Nash Equilibrium easily applies to dynamic games it is the NE of the appropriate strategic (matrix-form) game. 2
Nash is not enough In extensive-form games players may want to revise their equilibrium strategy as the game unfolds. It seems naive to assume that they will not when it is in their best interest to do so it would be tantamount to believing non-credible threats or promises. DEFINITION The subgame of the extensive game with perfect information Γ = N, H, P, ( i ) that follows the history h is the extensive game Γ(h) = N, H h, P h, ( i h ), where H h is the set of sequences h of actions for which (h, h ) H, P h is defined by P h (h ) = P (h, h ) for each h H h, and i h is defined by h i h h if and only if (h, h ) i (h, h ). (examples) Every game is its own subgame (following an empty history). Denote by s i h the strategy induced by s i in subgame Γ(h) and by s h the strategy profile induced by s. DEFINITION Strategy profile s constitutes a subgame perfect equilibrium of a game if s h is a NE of every subgame Γ(h). One problem with the SPNE: shouldn t I give up my belief about rationality of the other player when he makes a dumb choice? (example) 3
Centipede game Is common knowledge of rationality so rational? (example) 4
Timing matters: Cournot vs Stackelberg n = 2, P = a Q, Q = q 1 + q 2, constant unit cost c 1 = c 2. Profit is given by: Π i = q i (P c i ) = q i (a q i q i c i ) FOC: Π i q i = (a q i q i c i ) q i = (a 2q i q i c i ) = 0 Thus for any q i i s BR is q i = a q i c i 2 If they move simultaneously (Counot), NE is ( a c 3, a c 3 ). If they move sequentially (Stackelberg), there are multiple NE. But the unique SPNE is ( a c ). Total output is different, player 1 is better off (why?), player 2 is worse off., a c 2 4 5
Some properties of SPNE Obviously, every SPNE is a NE but not conversely. Existence: every finitie extensive game with perfect information has a subgame perfect equilibrium (can be found by backward induction) (non)uniqueness: SPNE is in general not unique. However, it is unique if no player is indifferent between two end-nodes. (example: the gardening games) 6
Two fairly benign extensions Random moves: nature choses at some nodes, following a pre-defined distribution Simultaneous moves: more than one player moves simultaneously at some nodes Note: with simultaneous moves SPNE may fail to exist (example: matching pennies). 7
Forward induction Example: forward induction in Battle-of-the-Sexes 8