Game Theory. Wolfgang Frimmel. Subgame Perfect Nash Equilibrium

Transcription

1 Game Theory Wolfgang Frimmel Subgame Perfect Nash Equilibrium /

2 Dynamic games of perfect information We now start analyzing dynamic games Strategic games suppress the sequential structure of decision-making The representation of extensive games explicitly allows us to study such situations, where players are free to decide whenever it is their turn to move We start with games of perfect information and finite games (no games with continuous strategy spaces) Solution concept: Backward induction We then generalize our model also for continuous strategy spaces /

3 A simple description of a dynamic game A dynamic game Think of a game of players moving sequentially: Player chooses an action a from the feasible set A Player observes a and then chooses an action a from the feasible set A 3 Payoffs are u (a, a ) and u (a, a ) This type of game can easily be extended by adding more players or by allowing players to move more than once, i.e. in a bargaining situation. Graphically, this game can be represented by a game tree 3 /

4 Nash equilibria in extensive form games A Nash equilibrium is a strategy profile s from which no player has an incentive to deviate, given the other players strategies. Nash equilibrium of extensive games The strategy profile s in an extensive game with perfect information is a Nash equilibrium if, for every player i and strategy s i, the terminal history T (s ) generated by s is at least as good as terminal history T (s i, s i ): u i (T (s )) u i (T (s i, s i )) for every strategy s i of player i 4 /

5 Example Consider again our Battle of Sexes example with Marge and Homer, where Marge decides first, and Homer chooses afterwards. Marge theater TV Homer theater TV theater TV 5 5 We can transform the game tree easily into a normal-form game: theater,theater theater,tv TV,theater TV,TV theater 5, 5,,, TV,,5,,5 5 /

6 Credibility of strategies (TV,(TV,TV)) is not a credible strategy - Marge should foresee that if she chooses theater, Homer will find it optimal to choose theater as well (theater,(theater,theater)) is also not credible - choosing theater at the second node is not a best response! Nash equilibrium basically treats strategies as choices made once and for all before the play begins it ignores the sequential structure of an extensive game It may turn out (as we have seen) that a Nash equilibrium may not be credible, so the definition of a Nash equilibrium would not suffice to rule out non-credible strategies We need to introduce a different solution concept that fulfills requirement Subgame perfect Nash equilibrium 6 /

7 Subgames A subgame of an extensive-form game is a subset of the entire game with the following properties: It must contain an information set containing a single decision node All moves and information sets from that node on must remain in the subgame 3 It does not cut any information sets 4 The game as a whole is a subgame of itself as well 5 Payoffs are defined over the remaining terminal histories 7 /

8 Subgames Consider the following extensive-form game: The game consists of two subgames 8 /

9 Subgame perfect Nash equilibrium Principle of sequential rationality: equilibrium strategies should specify optimal behavior at every point (subgame) in the game tree Subgame perfect Nash equilibrium (SPNE) A Nash equilibrium is subgame perfect if and only if it is a Nash equilibrium in every subgame of the game A SPNE is a strategy profile s with the property that in no subgame any player i can do better by choosing a strategy different from s i, given that every other player j adheres to s j Subgame perfect Nash equilibrium (SPNE) is a Nash equilibrium if it does not involve non-credible strategies Any SPNE is a Nash equilibrium, but not every Nash equilibrium is subgame perfect 9 /

10 Formal definition of a SPNE The strategy profile s = (si, s i ) in an extensive game with perfect information is a subgame perfect equilibrium if, for every player i, every history h after which it is player i s turn to move, and every strategy s i of player i the terminal history T h (s ) generated by s after history h is at least as good according to player i s preferences as terminal history T h (s i, s i ) generated by strategy profile (s i, s i ), in which player i chooses s i while every other player chooses s i. For every player i and every history h u i (T h (s )) u i (T h (s i, s i )) for every strategy s i of player i The important difference to the definition of a Nash equilibrium is that each player s strategy is required to be optimal for every history after which it is the player s turn to move, not only at the start of the game. /

11 Finding SPNE In finite games of perfect information, there is an easy solution concept available to find the outcome of such a game Backward Induction An extensive game is a game of perfect information if each information set contains only one decision node. Otherwise it is a game of imperfect information This implies that whenever it is a player s turn to move, she can observe all other player s previous moves The concept of backward induction assumes that it is common knowledge that every player will act rationally at each future node where she moves /

12 Backward induction Solve first for optimal choice at the final decision nodes (terminal histories T ), i.e. compare payoffs There is no further strategic interaction among players at these terminal histories single-person decision problem 3 Now proceed to the next-to-last decision node and determine the optimal actions by correctly anticipating the actions that will follow at the final decision nodes 4 Given the anticipation, choices at these nodes involve again a simple single-person decision problem - solve by comparing payoffs 5 Move backward through the entire game tree until the initial decision node by applying the same procedure 6 If no player has same payoffs at any two terminal nodes, the equilibrium is unique; otherwise, all equilibria are found by repeating the procedure for each optimal behavior /

13 Example - Battle of the sexes Let s solve our game with Marge and Homer by backward induction: Marge theater TV Homer theater TV theater TV 5 5 We find one subgame perfect Nash equilibrium Backward induction leads to a Nash equilibrium, but not every Nash equilibrium is reached by backward induction Note: SPNE also requires optimality at nodes which are never reached 3 /

14 This game has a unique equilibrium at (s, s ) = (rl, RRL) 4 / Example l r L R L R L R l r

15 Example: Entry deterrence game Consider the following extensive game: Invest Not invest Enter Out Enter Out How many subgames? How many Nash equilibria can we find? How many Subgame perfect Nash equilibria are in this game? 5 /

16 Example: Ultimatum game 3 Y N Y N Y N Y N 3 3 How many strategies does player have? How many strategies does player have? 3 How many pure strategy Nash equilibria are there? 4 How many Subgame Perfect Nash equilibria are there? 5 What if player has to divide Euros instead of 3? 6 /

17 General solution technique Subgame perfect Nash equilibrium is a more generally applicable concept, i.e. imperfect information or infinite moves. General solution technique: Pick a subgame that does not contain any other subgame Compute a Nash equilibrium of this game 3 Assign the payoff vector associated with this equilibrium to the starting node and eliminate the subgame 4 Iterate this procedure until there remains no subgame to eliminate 5 If there are multiple equilibria in a subgame, choose any equilibrium and repeat the procedure for every equilibrium in that subgame 6 By varying the Nash equilibrium for the subgames, one can compute all SPNE 7 /

18 Imperfect Information Consider this game tree: One cannot apply backward induction in a game of imperfect information How to solve: Use general technique 8 /

19 Imperfect Information (cont.) Look at the final subgame and find the Nash equilibrium: Eliminate the subgame and replace the eliminated node with the equilibrium payoff of this subgame (3, ) The game tree reduces to: 9 /

20 Imperfect Information (cont.) Player now decided between strategies E and X by comparing payoffs. SPNE: Player chooses E and then they play (T, R) simultaneously /

21 Example Find all Subgame Perfect Nash Equilibrium of the following game: X E 3 4 L R T B T B 5 5 /