Extensive Games with Perfect Information A Mini Tutorial

Transcription

1 Extensive Games withperfect InformationA Mini utorial p. 1/9 Extensive Games with Perfect Information A Mini utorial Krzysztof R. Apt (so not Krzystof and definitely not Krystof) CWI, Amsterdam, the Netherlands, University of Amsterdam

2 Extensive Games withperfect InformationA Mini utorial p. /9 Strategic Games: Definition Strategic game for n players: (possibly infinite) set S i of strategies, payoff function p i : S 1 S n R, for each player i. Basic assumptions: players choose their strategies simultaneously, each player is rational: his objective is to maximize his payoff, players have common knowledge of the game and of each others rationality.

3 Extensive Games with Perfect Information A Mini utorial p. 3/9 hree Examples Prisoner s Dilemma C D C, 0,3 D 3,0 1,1 he Battle of the Sexes F B F,1 0,0 B 0,0 1, Matching Pennies 1, 1 1, 1 1, 1 1, 1

4 Extensive Games with Perfect Information A Mini utorial p. 4/9 Nash Equilibrium Notation: G := (S 1,...,S n,p 1,...,p n ). Notation: s i,s i S i, s,s,(s i,s i ) S 1 S n. s i is a best response to s i if s i S i p i (s i,s i ) p i (s i,s i). s is a Nash equilibrium if i s i is a best response to s i : i {1,...,n} s i S i p i (s i,s i ) p i (s i,s i ). Intuition: In a Nash equilibrium no player can gain by unilaterally switching to another strategy.

5 Extensive Games with Perfect Information A Mini utorial p. 5/9 Nash Equilibrium Prisoner s Dilemma: 1 Nash equilibrium C D C, 0,3 D 3,0 1,1 he Battle of the Sexes: Nash equilibria F B F,1 0,0 B 0,0 1, Matching Pennies: no Nash equlibrium 1, 1 1, 1 1, 1 1, 1

6 Extensive Games with Perfect Information A Mini utorial p. 6/9 Example 1: Prisoner s Dilemma 1 C D C D C D (,) (0,3) (3,0) (1,1)

7 Extensive Games with Perfect Information A Mini utorial p. 7/9 Example : Battle of the Sexes 1 F B F B F B (,1) (0,0) (0,0) (1,)

8 Extensive Games with Perfect Information A Mini utorial p. 8/9 Example 3: Matching Pennies 1 (1,-1) (-1,1) (-1,1) (1,-1)

9 Extensive Games with Perfect Information A Mini utorial p. 9/9 Discussion hese are examples of two-player games with two stages. In general there may be more players and more stages. We limit ourselves to the games with finitely many stages (games with finite horizon) and such that at each stage exactly one player proceeds. Note. At each stage a player can have infinitely many choices. We assume here perfect information: each player knows the previous moves.

10 Extensive Games with Perfect Information A Mini utorial p. 10/9 Extensive Game: Definition Extensive game for n 1 players: game tree: a finite depth tree := (V,E) with a turn function D : V \Z {1,...,n}, where Z is the set of leaves of, outcome function o i : Z R, for each player i. We denote it by (,D,o 1,...,o n ). Given v V \Z we call {w (v,w) E} the set of actions available to player D(v) at node v. Sometimes we identify the actions with the labels put on the edges.

11 Extensive Games with Perfect Information A Mini utorial p. 11/9 Strategies Consider an extensive game EG := (,D,o 1,...,o n ). Let N i := {v V D(v) = i}. N i is the set of nodes at which player i takes an action. Strategy for player i: s i : N i V, such that for all v N i, (v,s i (v)) E. Joint strategy: s = (s 1,...,s n ). It assigns a unique edge to every node in V \Z. o each joint strategy s there corresponds a finite path path(s) := (v 1,...,v h ) in defined inductively: v 1 is the root of, if v k Z, then v k+1 := s i (v k ), where D(v k ) = i. When each player i selects s i we call (o 1 (z),...,o n (z)), where z is the last element of path(s), the outcome of EG.

12 Extensive Games with Perfect Information A Mini utorial p. 1/9 Example of Strategies: Matching Pennies 1 (1,-1) (-1,1) (-1,1) (1,-1) Strategies for player 1:,. Strategies for player :,,,. hick lines correspond with the joint strategy (,).

13 Extensive Games with Perfect Information A Mini utorial p. 13/9 Strategic Forms With each extensive game EG := (,D,o 1,...,o n ) we associate a strategic game G := (S 1,...,S n,p 1,...,p n ) defined as follows: S i is the set of strategies of player i in EG, p i (s) := o i (z), where z is the last element of path(s). G is called the strategic form of EG. s is called a Nash equilibrium of EG if it is a Nash equilibrium of G.

14 Extensive Games with Perfect Information A Mini utorial p. 14/9 Example: Matching Pennies 1 (1,-1) (-1,1) (-1,1) (1,-1) Strategic form 1, 1 1, 1 1, 1 1, 1 1, 1 1, 1 1, 1 1, 1 Note. wo Nash equilibria: (, ) and (, ).

15 Extensive Games with Perfect Information A Mini utorial p. 15/9 Win or Lose Games A two-player extensive game is called a win or lose game if the only possible outcomes are (1, 1) and ( 1,1). s i is called a winning strategy of player i in a win or lose game EG if s i S i p i (s i,s i ) = 1, where (S 1,S,p 1,p ) is the strategic form of EG. heorem (Zermelo, 1913) In every win or lose game one of the players has a winning strategy.

16 Extensive Games with Perfect Information A Mini utorial p. 16/9 Proof of Zermelo s heorem heorem In every win or lose game one of the players has a winning strategy. We can assume that the players alternate their moves. We can extend all the paths in the game so that all paths in are of the same depth, say k. Let W denote the sentence player 1 wins after k stages. hen the formula φ 1 := x 1 y 1... x k y k W denotes player 1 has a winning strategy and φ := x 1 y 1... x k y k W denotes player has a winning strategy. But φ 1 φ, i.e., φ 1 φ holds.

17 Extensive Games with Perfect Information A Mini utorial p. 17/9 Example: Ultimatum Game Player 1 claims x {0,1,...,100}. Player accepts - the outcome is then (x,100 x), or rejects - the outcome is then (0,0). For each x {0, 1,..., 100} the root 1 has the subtree 1 x A (x,100 x) R (0,0)

18 Extensive Games with Perfect Information A Mini utorial p. 18/9 Nash Equilibria in the Ultimatum Game 1 x A (x,100 x) R (0,0) Note. For each x {0,1,...,100} there is a Nash equilibrium with the outcome (x,100 x). Proof. ake x {0,1,...,100}. Strategy for player 1: x. Strategy for player : if x x then A else R fi. his is a Nash equilibrium with the outcome (x,100 x ).

19 Extensive Games with Perfect Information A Mini utorial p. 19/9 Example: Ultimatum Game, ctd Illustration. Strategy for player 1: x. Strategy for player : s :=if x x then A else R fi. Consider two deviations of player 1, x 1 < x < x. s x 1 x 1,100 x x x 1,100 x x 0, Conclusion. he notion of a Nash equilibrium is not informative here.

20 Extensive Games with Perfect Information A Mini utorial p. 0/9 Subgames Consider EG := (,D,o 1,...,o n ). We define the subgame of EG rooted at node v of, EG v, as expected. Note. Each strategy s i of player i in EG uniquely determines his strategy s v i in EGv. (s 1,...,s n ) is called a subgame perfect equilibrium in EG if for each node v of (s v 1,...,sv n) is a Nash equilibrium in EG v. Informally: s is subgame perfect equilibrium in EG if it induces a Nash equilibrium in every subgame of EG.

21 Extensive Games with Perfect Information A Mini utorial p. 1/9 Backward Induction Given a tree (V,E) and v V, let desc(v) := {w (v,w) E}. Fix a finite extensive game EG := ((V,E),D,o 1,...,o n ). Backward induction algorithm while V > 1 do choose v V such that all its descendants are leaves; i := D(v); choose w desc(v) such that o i (w) is maximal; s i (v) := w; for j {1,...,n} do o j (v) := o j (w) od; V := V \desc(v); E := E (V V); od Note. his process generates a set of joint strategies. Multiple joint strategies may arise due to the second choose statement.

22 Extensive Games with Perfect Information A Mini utorial p. /9 Kuhn and Selten heorems heorem (Kuhn, 1950) Every finite extensive game (with perfect information) has a Nash equilibrium. heorem (Selten, 1965) Every finite extensive game (with perfect information) has a subgame perfect equilibrium. Proof. A stronger claim holds: A joint strategy is a subgame perfect equilibrium iff it can be generated by the backward induction algorithm.

23 Extensive Games with Perfect Information A Mini utorial p. 3/9 Example: Ultimatum Game 1 x A (x,100 x) R (0,0) Player has two best responses to the strategy 100: A and R. Note. here are two subgame perfect equilibria: (100, always A), with the outcome (100,0), (99, if x 100 then A else R fi), with the outcome (99,1).

24 Extensive Games with Perfect Information A Mini utorial p. 4/9 Example: the Centipede Game (Rosenthal, 1981) 1a C a C 1b C b C 1c C c S S S S S S C (4,4) (1,1) (0,3) (,) (1,4) (3,3) (,5) General rule: Initial situation: (1,1). If a player continues he loses 1 and the opponent gains. Note. Backward induction shows that in the unique subgame perfect equilibrium each player selects at each node S. So the outcome of the game is (1,1). (1,1) is also the outcome of the game in each Nash equilibrium.

25 Extensive Games with Perfect Information A Mini utorial p. 5/9 xtensive Games with Imperfect Information Example: Matching Pennies 1 (1,-1) (-1,1) (-1,1) (1,-1) Intuition. Player does not know the action of player 1. Imperfect information: players do not need to know some of the previous moves.

26 Extensive Games with Perfect Information A Mini utorial p. 6/9 Definition Extensive game with imperfect information for n 1 players: labelled game tree: a finite depth tree := (V,E) with labelled edges and a turn function D : V \Z {1,...,n}, where Z is the set of leaves of, such that there is a partition I of the nodes from V \Z; each element of I is called an information set, if v and v are in the same information set, then D(v) = D(v ) and E(v) = E(v ), where E(v) is the set of labels on the edges starting in v (called actions available to player D(v) at node v), outcome function o i : Z R, for each player i. We denote it by (,D,I,o 1,...,o n ).

27 Extensive Games with Perfect Information A Mini utorial p. 7/9 Strategies Consider an extensive game with imperfect information EG := (,D,I,o 1,...,o n ). Let I i := {J I for all v J,D(v) = i}. I i is the set of information sets at which player i takes an action. Strategy for player i: s i : I i v V E(v), such that for all J I i and v J, s i (J) E(v). Joint strategy: s = (s 1,...,s n ). It assigns a unique edge to every node in V \Z. If two nodes lie in the same information set, then the edges with the same label are assigned to them.

28 Extensive Games with Perfect Information A Mini utorial p. 8/9 Strategies in the Matching Pennies Game 1 (1,-1) (-1,1) (-1,1) (1,-1) Strategies for player 1:,. Strategies for player :,. hick lines correspond with the joint strategy (,). So this game coincides with the strategic game 1, 1 1, 1 1, 1 1, 1

29 Extensive Games with Perfect Information A Mini utorial p. 9/9 Strategic Forms With each extensive game with imperfect information we associate a strategic game defined as before. Note Extensive games with imperfect information do not need to have a Nash equilibrium. A fortiori these games do not need to have a subgame perfect equilibrium. Caveat he notion of a subgame has to be redefined.