Chapter 6 Extensive Games, perfect info Detailed description of the sequential structure of strategic situations as opposed to Strategic Games Players perfectly informed of occurred events Initially decisions are not made at the same time, no randomness

Example 91.1 / Def. 89.1 Extensive game: Players to share two objects Conventional definition with trees as primitives: 1 2 (0,2) (2,0) no yes 2,0 0,0 1,1 0,2 (1,1)

Definition with players´ actions as primitives: G = (N, H, P, (i)) A Set of Players N = {1,2} A Set of possible histories (sequences, finite or infinite) H = {, (2, 0), (1, 1), (0, 2), ((2, 0), yes), ((2, 0), no), ((1, 1), yes), ((1, 0), no), ((0, 2), yes), ((0, 2), no)} Terminal histories Z = {((2, 0), yes), ((2, 0), no), ((1, 1), yes), ((1, 1), no), ((0, 2), yes), ((0, 2), no)} 1 2 (0,2) (2,0) no yes 2,0 0,0 1,1 0,2 (1,1)

A player function that assigns a player to each non terminal history P() = 1 and P(h) = 2 for every non terminal h   A preference relation for each player on Z: i: ((2, 0), yes) >1 ((1, 1), yes) >1 ((0, 2), yes) ~1 ((2, 0), no) ~1 ((1, 1) ~1 yes) ~1((1, 1), no) and ((2, 0), yes) >2 ((1, 1), yes) >2 ((0, 2), yes) ~2 ((2, 0), no) ~2 ((1, 1) ~2 yes) ~2((1, 1), no) 1 2 (0,2) (2,0) no yes 2,0 0,0 1,1 0,2 (1,1)

Def. 92.1 Strategies A strategy of player i is a function that assigns an action to each nonterminal history Even for histories that, if strategy is followed, are never reached Player 1 below has AE, AF, BE, BF The outcome O(s) of strategy profile s = (si)iN yields the terminal history when each player i follows si 1 A B 2 d C D 1 E c F a b

Def. 93.1 Nash Equilibrium Nash Equilibrium for an extensive game with perfect info is a strategy profile s* such that for every player iN we have O(s*-i, s*i) i O(s*-i, si) for every strategy si of player i (If other players follows s* you would better follow s* too... ) Alternatively it is the Nash Equilibrium of a strategic game derived from the extensive game

Equivalent strategic games 1 A B 2 d C D 1 E c F a b Extensive Game Equivalent Strategic Game Equivalent Strategic Game Reduced form

Example 95.2 1 A B 2 L R 1, 2 2, 1 0, 0 Given that player 2 chooses L it is optimal for player 1 to choose B The Nash equilibrium (B,L) lacks plausibility since P2 wouldn’t choose L after A.

Def. 97.1 Subgame (h) = (N, H|h, P|h, (i)|h) is the subgame to  = (N, H, P, (i)) that follows the history h h (h)

Def. 97.2 Subgame Perfect Equlibrium A subgame perfect equilibrium is a strategy profile s* such that for any history h the strategy profile s*|h is a Nash equlibrium of the subgame (h) OR?

Example 95.2 again (A) (B,L) is a Nash equilibrium 1 A B 2 L R 1, 2 0, 0 (A) 2, 1 (B,L) is a Nash equilibrium Is (B,L) a subgame perfect equilibrium? The strategy profile s*|h = (B,L)|A in the subgame (A) is for instance no Nash Equilibrium Player 2 wouldn’t chose L given that player 1 has chosen A

Prop. 99.2 Kuhn´s Theorem Every finite extensive game with perfect info has a subgame perfect equilibrium. E.g chess is draw once a position is repeated three times => chess is finite

Two Extensions to Extensive Games with perfect info Exogenous uncertainty The Player function P(h) has a probability that chance determines the action after the history h Definition of a subgame perfect equilibrium and Kuhn’s theorem still OK Simultaneous moves The Player function P(h) assigns a set of players that make choices after the history h

6.5.1 The Chain Store Game Multitude of Nash equilibria Every terminal history which the outcome in any period is either Out or (In,C) Intuitively unappealing for small K Unique Subgame Perfect Equilibrium Always (In, C) Not that appealing for large K k In Out CS F C 5 ,1 2, 2 0, 0

Ex. 110.1 BoS with an outside option Elimination of dominated actions yields: (B, B) Interpretation BB >1 Book >1 SS >1 BS ~1 SB Player 2 knows that if player 1 selects concert he would choose Bach otherwise he would better stay home reading the book Thus player 1 can select B knowing that player 2 also selects B 1 Book Concert 2, 2

Ex. 111.1 Burning money Elimination of dominated actions yields: (0B, BB) Interpretation: P2 thinks that if P1 spends D then he wants to go Bach otherwise he would loose compared to not spending D => P2 chooses B if P1 chooses D P1 knows this and can expect a payoff of 2 by choosing DB P2 knows that the rationality of P1 choosing 0 is that he expects to gain better than 2 (by choosing DB) Thus P1 can choose 0B and gain 3 Authors think that this example is implausible 1 D