Game Theory Sequential Games, Subgame perfection, backward induction paradox

1. Game TheorySequential Games, Subgame perfection, backward induction paradox Univ. Prof.dr. M.C.W. Janssen University of Vienna Winter semester 2011-12 Week 43 (October 24, 25)

2. Three Examples of sequential games • Take-it-or-leave it • Centipede game • Ultimatum Bargaining Game (UG) • One player proposes how to divide a pie • Other player can only accept or reject proposal • Stackelberg duopoly • Leader, follower both choose quantities • What is a strategy in each of these cases? • What is, are the Nash equilibria of these games?

3. Closer look at discrete version of UG I • Consider following Nash equilibrium • P chooses 0.5 • R accepts proposal iff it is equal split • Why is this a Nash equilibrium? 0.9, 0.1 acc R rej 0, 0 0.9 0.5, 0.5 acc 0.5 R rej 0, 0 P 0.1 0.1, 0.9 acc R rej 0, 0

4. Closer look at discrete version of UG II • Red path is equilibrium path (this is what you see when equilibrium is played) • Everything else is off the equilibrium path • What is specified off the equilibrium path is important in determining whether something is an equilibrium • If for example R would accept a proposal of 0.9, P would have an incentive to deviate • Nash equilibrium does not impose any restrictions on what is played off-the-equilibrium path 0.9, 0.1 acc R rej 0, 0 0.9 0.5, 0.5 acc 0.5 R rej 0, 0 P 0.1 0.1, 0.9 acc R rej 0, 0

5. More formally • Define history of play as follows. • Let a0 = (a01 ,a02 ,…,a0n) as the action profile that is played in stage 0, i.e., the actions played by all players • History at the beginning of stage 1, h1 = a0 • History at the beginning of stage k+1, hk+1 = (a0,…,ak) • The set Hk+1 is the set of all possible histories hk+1 and Ai(hk) is the set of actions that player i can choose after history hk and Ai(Hk) is the union of this set over all possible histories • Strategy si of player i is a sequence of mappings {ski} where each ski maps Hk to the set of feasible actions

6. Multi-stage game with observable actions • All players observe history hk • Denote the game from stage k on with history hk before stage k as G(hk) • If after history hk actions ak,…,aK are chosen, then pay-offs in G(hk) are defined as the pay-offs that accrue in the whole game if history hK+1=(hk,ak,…,aK) is played • Strategies in G(hk) are simply those strategies that are consistent with the history of play being given by hk. Denote these strategies by si|hk

7. Subgame perfect equilibrium (SPE) • A strategy combination (profile) (s*i ,s*-i )of a multi-stage game with observed actions is a subgame perfect equilibrium if for all histories hk the strategy restrictions si|hk form a Nash equilibrium of G(hk) • Can also be in mixed strategies

8. Multi-stage games with perfect information • Sequential game is a game where in each stage only one player can choose an action • Also called game with perfect information • In this type of game, subgame perfection reduces to Backward Induction • BI is an algorithm according to which you can start the analysis by considering the optimal choices in the final stage K for each history hK • The one can work back to stage stage K-1 and determine optimal choices given the choices that are fixed for period K • Etc.

9. SPE in the three examples • All three examples are sequential games with perfect information • Thus, we can apply backwards induction • Usually, this implies that there is a unique SPE • Note the value of commitment (one reduces one’s flexibility by choosing first), but because of the favourable response from opponent, this restriction of the freedom to choose may be beneficial

10. Subgames, more generally • What is a subgame? • Part of the game that can be analyzed on its own • A subgame starts in a singleton information set • A subgame includes that part of the whole game that follows opon this singleton information set • A subgame cannot cut through an information set

11. How many subgames? • There are just two subgames • How many subgames are there in the two times repeated PD game? up acc R S rej down 0.9 acc 0.5 R rej P 0.1 acc R rej

12. Backward induction paradox I • Consider the Take-it-or-Leave-it game. Start at the end, player 2 should take the 4 euros • Going one step back it is also rational for player 1 to take the (then) 3 euros • Otherwise he does not get anything • But should player 2 at her first choice moment really go for the 2 euros? • Yes, according to backwards induction and SPE • But, he may also reason as follows: • If there is common knowledge of rationality and backward induction applies, I should not have to make a choice in the first place: why did player 1 not grab the 1 euro? • His behaviour is inconsistent with CKR. I should now come up with a theory why he chose not to grab the euro. • Game theory does not necessrily say anything about this. Maybe player 1 is not interested in the money? Maybe he always leaves the money? Maybe … • But if there is a reasonably chance that he always leaves the money on the table (and this probability is actually large than 1/32), then I should also not take the 2 euro now, but rationality decide to leave it and later grab 4 euro • But then player 1 can rationally choose not to grab the euro in the first round and hope that player 2 thinks.

13. Backward induction paradox II • How can backward induction be repaired? • One way is to say that players should believe that mistakes (i.e., actions off the equilibrium path) should be uncorrelated with each other. • Thus, if player 2 makes his first move, he should belief that the fact that player 1 chose not to play according to the SPE is no indication of her not playing according the equilibrium path from that moment onwards • Another way is to say that it is not CKR that is assumed, but only K1K2R1, K2R1. If 2 then observes player choosing L, he may make the most minimal assumption to justify this and he only needs to say that apparently K1K2R1 is violated.