Topic 3 Games in Extensive Form 1

A. Perfect Information Games in Extensive Form. 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2

A. Perfect Information Games in Extensive Form In a perfect information extensive form game, each player knows exactly where they are in the game when they take a move. The only thing they don’t know is how future moves will be played. 3

A. Perfect Information Games in Extensive Form Strategies: Definition: A Pure Strategy for a player is an instruction book on how to play the game. This instruction book must be complete and tell the player what to do at every point at which it must make a move. Because we want to consider what happens when players make mistakes we must even include apparently redundant instructions. 4

A. Perfect Information Games in Extensive Form 1’s Instructions (Fold,Fold) 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) Raise Fold 5

A. Perfect Information Games in Extensive Form 1’s Instructions (Raise,Fold) 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) Raise Fold 6

A. Perfect Information Games in Extensive Form 1’s Instructions (Raise,Raise) 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) Raise Fold 7

A. Perfect Information Games in Extensive Form 1’s Instructions (Fold,Raise) 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) Raise Fold 8

A. Perfect Information Games in Extensive Form Strategies: In this game player 1 has 4 pure strategies 1.(Raise, Raise) 2.(Raise,Fold) 3.(Fold,Raise) 4.(Fold,Fold) Player 2 also has 4 pure strategies, but none of her instructions are ever redundant. 9

A. Perfect Information Games in Extensive Form 2’s Instructions (Fold,Fold) 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) Raise Fold 10

A. Perfect Information Games in Extensive Form 2’s Instructions (Raise,Fold) 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) Raise Fold 11

A. Perfect Information Games in Extensive Form 2’s Instructions (Raise,Raise) 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) Raise Fold 12

B. Imperfect Information Games in Extensive Form The figures above have the property that everyone knows everything about the past events in the game – this rules out: players moving simultaneously, players knowing something that others players do not. We need to have a way of representing this. We use information sets to do this. 13

B. Imperfect Information Games in Extensive Form An Information set is a collection of nodes with the property that 1.Each node in the set has the same player’s name. 2.Each node in the set has the same actions available. Information sets describe nodes that the player cannot distinguish between. 14

B. Imperfect Information Games in Extensive Form 2 sees 1’s action 1 Raise Fold 2 2 Raise (0,0) (-1,1) Raise (1,-1) (0,0) 15

B. Imperfect Information Games in Extensive Form 2 does not see 1’s action 1 Raise Fold 2 2 Raise (0,0) (-1,1) Raise (1,-1) (0,0) Information Set 16

B. Imperfect Information Games in Extensive Form 1.Each node in the set has the same player’s name. 2.Each node in the set has the same actions available. Both of these are essential, because otherwise the player would be able to distinguish between nodes in the same information set by seeing what actions they had and who they were. 17

B. Imperfect Information Games in Extensive Form Information Sets can be used to describe situations where players move simultaneously: 18

B. Imperfect Information Games in Extensive Form Simultaneously 1 Raise Fold 2 2 Raise (0,0) (-1,1) Raise (1,-1) (0,0) Information Set 19

C. Nash Equilibrium and Subgame Perfection At a Nash equilibrium each player’s strategy is a best response to the other players’ strategies: 20

C. Nash Equilibrium A Nash Equilibrium. 1 RaiseFold 2 2 Raise (0,0)(-1,1) Raise (1,-1) (-2,-2) 21

C. Nash Equilibrium A Nash Equilibrium. Checking that 1’s strategy is a best response If she folds she gets RaiseFold 2 2 Raise (0,0) ( -1,1) Raise (1,-1) (-2,-2) 22

C. Nash Equilibrium A Nash Equilibrium. Checking that 1’s strategy is a best response If she raises she gets RaiseFold 2 2 Raise (0,0) ( -1,1) Raise (1,-1) ( -2,-2) 23

C. Nash Equilibrium A Nash Equilibrium. Checking that 2’s strategy is a best response If she changes her action on the right her payoff is unaltered. 1 RaiseFold 2 2 Raise (0,0) (-1, 1 ) Raise (1,-1) (-2,-2) 24

C. Nash Equilibrium A Nash Equilibrium. Checking that 2’s strategy is a best response If she changes her action on the right her payoff is unaltered. 1 RaiseFold 2 2 Raise (0,0) (-1, 1 ) Raise (1,-1) (-2,-2) 25

C. Nash Equilibrium and Subgame Perfection A Nash Equilibrium. Checking that 2’s strategy is a best response If she changes her action on the left her payoff goes down. 1 RaiseFold 2 2 Raise (0,0) (-1, 1 ) Raise (1,-1) (-2,-2) 26

C. Nash Equilibrium and Subgame Perfection A Nash Equilibrium. Checking that 2’s strategy is a best response If she changes her action on the left her payoff goes down. 1 RaiseFold 2 2 Raise (0, 0 )(-1, 1 ) Raise (1,-1) (-2,-2) 27

C. Nash Equilibrium Nash equilibrium alone in an extensive form game is un satisfactory because it builds in non-credible threats. At a Nash equilibrium each player assumes the others will stick to their equilibrium actions when they test their own action. They assume other players are committed to playing their strategy. This may not be a good assumption in a dynamic model because players may make threats that are not rational for them. 28

C. Nash Equilibrium A Non-Credible Threat. 1 RaiseFold 2 2 Raise (0,0) (-1, 1 ) Raise (1,-1) ( -2,-2) 2 is making a threat here that she would not carry out. 29

C. Nash Equilibrium A Non-Credible Threat. 1 RaiseFold 2 2 Raise (0,0) (-1, 1 ) Raise (1,- 1 ) (-2, -2 ) 2 is making a threat here that she would not carry out. 30

C. Nash Equilibrium A Non-Credible Threat. Once player 1 realizes this she is better off raising too 1 RaiseFold 2 2 Raise (0,0) ( -1,1) Raise (1, - 1) (-2,-2 ) 2 is making a threat here that she would not carry out. 31

C. Nash Equilibrium A Non-Credible Threat. Once player 1 realizes this she is better off raising too 1 RaiseFold 2 2 Raise (0,0) ( -1,1) Raise ( 1, - 1) (-2,-2 ) 2 is making a threat here that she would not carry out. 32

C. Nash Equilibrium This is another Nash equilibrium.(Check) At this Nash equilibrium no non-credible threats are made. Can we always find such a Nash equilibrium? Answer: Yes by using a process called backwards induction. This always works in games of perfect information (i.e. games without information sets). 33

D. Backwards Induction Example Start at the last move and figure out what is optimal there 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) ( -1,1) (2,-2) Raise Fold 34

D. Backwards Induction Example Start at the last move and figure out what is optimal there 1 RaiseFold Raise (0,0) (-1,1) Raise (1,-1) ( -1,1) (2,-2) Raise Fold 35

D. Backwards Induction Example Do all last moves 1 RaiseFold Raise (0, 0 ) (-1,1) Raise (1,-1) ( - 1,1) (2,-2) Raise Fold 36

D. Backwards Induction Example Do all last moves 1 RaiseFold Raise (0,0) (-1, 1 ) Raise (1,-1) ( - 1,1) (2,-2) Raise Fold 37

D. Backwards Induction Example Now do the second last move. 1 RaiseFold Raise (0,0) (-1, 1 ) Raise (1,-1) ( - 1,1) (2,-2) Raise Fold 38

D. Backwards Induction Example Now do the second last move. 1 RaiseFold Raise (0,0) (-1, 1 ) Raise (1,-1) ( - 1,1) (2,-2) Raise Fold 39

D. Backwards Induction Example: Finally do the first move 1 RaiseFold Raise (0,0) (-1, 1 ) Raise (1,-1) ( - 1,1) (2,-2) Raise Fold 40

D. Backwards Induction Example: Finally do the first move 1 RaiseFold Raise (0,0) (-1, 1 ) Raise (1,-1) ( - 1,1) (2,-2) Raise Fold 41

D. Backwards Induction Notes: 1.This process works for all finite perfect information games. 2.Proves the existence of a NE in such games (Zermelo’s Theorem) – Chess Draughts etc. 3.It is a generalization of dynamic programming. 4.BUT backwards induction will not work in games with information sets. 5.So Selten generalized the idea of Backwards induction to create Subgame perfection. 42

E. Subgame Perfection Subgame perfect equilibrium divides the game tree up into subgames (that is, parts of the tree that can be considered separately). It requires that the player’s strategies are a Nash equilibrium on every subgame. Again these can be found by working backwards, taking the last independent game and finding strategies that are a Nash equilibrium. Then taking the next last independent game and so on. Backwards induction always finds a subgame perfect equilibrium. 43

1 E. Subgame Perfection Example 1 RaiseFold 2 LR (1,1) (1,2)(-1,1)(0,1) b a 1 ba (1/2,0) 44

1 E. Subgame Perfection A (Bad) NE 1 RaiseFold 2 LR (1,1) (2,2)(-1,1)(0,1) b a 1 ba (1/2,0) 45

1 E. Subgame Perfection A (Bad) NE: The move of player 2 is not credible it always prefers L to R if she gets to move! 1 RaiseFold 2 LR (1,1) (2,2)(-1,1)(0,1) b a 1 ba (1/2,0) 46

1 E. Subgame Perfection A (Bad) NE: But cannot do backwards induction sometimes 1 prefers ‘a’ sometimes ‘b’. 1 RaiseFold 2 LR (1,1) (2,2)(-1,1)(0,1) b a 1 ba (1/2,0) 47

1 E. Subgame Perfection Where are the subgames? 1 RaiseFold 2 LR (1,1) (2,2)(-1,1)(0,1) b a 1 ba (1/2,0) 48

1 E. Subgame Perfection Here 1 RaiseFold 2 LR (1,1) (2,2)(-1,1)(0,1) b a 1 ba (1/2,0) 49

1 E. Subgame Perfection And here 1 RaiseFold 2 LR (1,1) (2,2)(-1,1)(0,1) b a 1 ba (1/2,0) 50

1 E. Subgame Perfection Let’s look at the last subgame and make the players play a NE on it. 1 RaiseFold 2 LR (1,1) (2,2)(-1,1)(0,1) b a 1 ba (1/2,0) 51

1 E. Subgame Perfection It has only 1 NE, because player 2 always prefers L to R. 1 RaiseFold 2 LR (1,1) (2,2)(-1,1)(0,1) b a 1 ba (1/2,0) 52

1 E. Subgame Perfection Once we have solved this last game it is easy to figure out what player 1 will do. 1 RaiseFold 2 LR (1,1) (2,2)(-1,1)(0,1) b a 1 ba (1/2,0) 53

1 E. Subgame Perfection Once we have solved this last game it is easy to figure out what player 1 will do. 1 RaiseFold 2 LR ( 1,1) (2,2)(-1,1)(0,1) b a 1 ba (1/2,0) 54

1 E. Subgame Perfection This is the subgame perfect equilibrium. 1 RaiseFold 2 LR ( 1,1) ( 2,2) (-1,1)(0,1) b a 1 ba (1/2,0) 55

E. Subgame Perfection Backwards induction will usually give you a unique solution to a game (it will not be unique if there are ties in the players’ payoffs). As the subgames have many Nash equilibria, however, a game can have many subgame perfect equibria. Example. 56

E. Subgame Perfection 1 YX 2 b 1 L R (1,3) (0,0) a 1 ba (2,2) 1 LR (3,1)(0,0) a 1 ba (2,2) 2 b IF 1 plays right the players then play a coordination game which has 3 NE’s. 57

E. Subgame Perfection 1 YX 2 b 1 L R (1,3) (0,0) a 1 ba (2,2) 1 LR (3,1)(0,0) a 1 ba (2,2) 2 b IF 1 plays right the players then play a coordination game which has 3 NE’s. 58

E. Subgame Perfection 1 YX 2 b 1 L R (1,3) (0,0) a 1 ba (2,2) 1 LR (3,1)(0,0) a 1 ba (2,2) 2 b IF 1 plays right the players then play a coordination game which has 3 NE’s. 59

E. Subgame Perfection 1 YX 2 b 1 L R (1,3) (0,0) a 1 ba (2,2) 1 LR (3,1)(0,0) a 1 ba (2,2) 2 b IF 1 plays X the players then play the battle of the sexes which also has 3 NE’s. 60

E. Subgame Perfection 1 YX 2 b 1 L R (1,3) (0,0) a 1 ba (2,2) 1 LR (3,1)(0,0) a 1 ba (2,2) 2 b IF 1 plays X the players then play the battle of the sexes which also has 3 NE’s. 61

E. Subgame Perfection 1 YX 2 b 1 L R (1,3) (0,0) a 1 ba (2,2) 1 LR (3,1)(0,0) a 1 ba (2,2) 2 b IF 1 plays X the players then play the battle of the sexes which also has 3 NE’s. 62

E. Subgame Perfection 1 YX 2 b 1 L R (1,3) (0,0) a 1 ba (2,2) 1 LR (3,1)(0,0) a 1 ba (2,2) 2 b IF 1 plays X the players then play the battle of the sexes which also has 3 NE’s. 63

E. Subgame Perfection In a finite game,as a Nash equilibrium always exists so does a subgame perfect equilibrium. But again there are certain games for which subgame perfection is not going to work and we are going to need to rule out non-credible actions. Here is an example… 64

E. Subgame Perfection This game has no subgames, but so the above NE is also a subgame perfect equilibrium, but it appears that 2 is making a non-credible threat…. 1 R M 2 2 aab (2,2) (0,1) b (3,1) (1,3) (0,0) L 65

F. Perfect Bayesian Equilibrium To remedy the problem with the game above we need another notion of equilibrium… Perfect Bayesian equilibrium. 66

F. Perfect Bayesian Equilibrium If you are going to do something like Backwards induction in this game, then you are going to need to decide what action player 2 will take. 1 R M 2 2 a ab (2,2) (0,1) b (3,1) (1,3) (0,0) L 67

F. Perfect Bayesian Equilibrium If you are going to do something like Backwards induction in this game, then you are going to need to decide what action player 2 will take. Must give player 2 beliefs 1 R M 2 2 a ab (2,2) (0,1) b (3,1) (1,3) (0,0) L   68

F. Perfect Bayesian Equilibrium Can now evaluate player 2’s expected payoff from each action Action a = 2  +1(1-  )=1+  1 R M 2 2 a ab (2, 2 ) (0,1) b (3, 1 ) (1,3) (0,0) L   69

F. Perfect Bayesian Equilibrium Can now evaluate player 2’s expected payoff from each action Action b = 1  +0(1-  )=  1 R M 2 2 a ab (2,2) (0, 1 ) b (3,1) (1,3) (0, 0 ) L   70

F. Perfect Bayesian Equilibrium Action a is always better than action, so player 2 should play a. 1 R M 2 2 a ab (2,2) (0, 1 ) b (3,1) (1,3) (0, 0 ) L   71

F. Perfect Bayesian Equilibrium Hence 1 should play R. 1 R M 2 2 a ab (2,2) (0, 1 ) b (3,1) (1,3) (0, 0 ) L   72

F. Perfect Bayesian Equilibrium At an equilibrium we want 2’s beliefs to be consistent with 1’s actions (rational expectations). So at an equilibrium we will have  =0. 1 R M 2 2 a ab (2,2) (0, 1 ) b (3,1) (1,3) (0, 0 ) L   73

F. Perfect Bayesian quilibrium To find Perfect Bayesian equilibria we equip players with beliefs at each information set at which they must take a move. We require their actions at that set to be optimal given their beliefs this is called “Sequential Rationality”. We then work backwards through the tree. Given the sequentially rational actions we find for the players we then check that beliefs are consistent given these actions. (Rational expectations). 74

F. Perfect Bayesian Equilibrium A Signalling Game 0 22 (0,0)(-1,1)(1,2)(0,0) /2 T TWW aa aa bbb b (1,2)(0,0) (-1,1) 75

F. Perfect Bayesian Equilibrium A Signalling Game We must give player 2 two lots of beliefs (sometimes these are called assessments) (0,0)(-1,1)(1,2)(0,0) /2 T TWW aa aa bbb b (1,2)(0,0) (-1,1)   76

F. Perfect Bayesian Equilibrium A Signalling Game 2 plays a if: . 2 plays a if:  (0,0)(-1,1)(1,2)(0,0) /2 T TWW aa aa bbb b (1,2)(0,0) (-1,1)   77

aa a F. Perfect Bayesian Equilibrium A Signalling Game Both types of 1 play T 0 22 (0,0)(-1,1)(1,2)(0,0) /2 T TWW abbb b (1,2)(0,0) (-1,1)   78

aa a F. Perfect Bayesian Equilibrium A Signalling Game Both types of 1 play T So rational expectations says , but e is arbitrary – must have  (0,0)(-1,1)(1,2)(0,0) /2 T TWW abbb b (1,2)(0,0) (-1,1)   79

F. Perfect Bayesian Equilibrium At a Perfect Bayesian Equilibrium (PBE) we can choose beliefs at unused information sets in any way we want. Two interesting types of PBE in signalling games: 1. Pooling equilibria: All types choose the same, identical action. 2. Separating equilibria: No two types choose the same action. 80