Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2...
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Transcript of Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2...
![Page 1: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/1.jpg)
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Topic 3
Games in Extensive Form
![Page 2: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/2.jpg)
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A. Perfect Information Games in Extensive Form
. 1
RaiseFold
1
22
Fold Fold
Fold Raise
Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
![Page 3: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/3.jpg)
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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.
![Page 4: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/4.jpg)
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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.
![Page 5: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/5.jpg)
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A. Perfect Information Games in Extensive Form
1’s Instructions(Fold,Fold)
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 6: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/6.jpg)
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A. Perfect Information Games in Extensive Form
1’s Instructions(Raise,Fold)
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 7: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/7.jpg)
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A. Perfect Information Games in Extensive Form
1’s Instructions(Raise,Raise)
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 8: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/8.jpg)
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A. Perfect Information Games in Extensive Form
1’s Instructions(Fold,Raise)
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 9: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/9.jpg)
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A. Perfect Information Games in Extensive Form
Strategies:In this game player 1 has 4 pure strategies1. (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.
![Page 10: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/10.jpg)
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A. Perfect Information Games in Extensive Form
2’s Instructions(Fold,Fold)
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 11: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/11.jpg)
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A. Perfect Information Games in Extensive Form
2’s Instructions(Raise,Fold)
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 12: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/12.jpg)
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A. Perfect Information Games in Extensive Form
2’s Instructions(Raise,Raise)
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 13: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/13.jpg)
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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.
![Page 14: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/14.jpg)
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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.
![Page 15: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/15.jpg)
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B. Imperfect Information Games in Extensive Form
2 sees 1’s action 1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(0,0)
![Page 16: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/16.jpg)
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B. Imperfect Information Games in Extensive Form
2 does not see 1’s action1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(0,0)
Information Set
![Page 17: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/17.jpg)
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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.
![Page 18: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/18.jpg)
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B. Imperfect Information Games in Extensive Form
Information Sets can be used to describe situations where players move simultaneously:
![Page 19: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/19.jpg)
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B. Imperfect Information Games in Extensive Form
Simultaneously 1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(0,0)
Information Set
![Page 20: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/20.jpg)
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C. Nash Equilibrium and Subgame Perfection
At a Nash equilibrium each player’s strategy is a best response to the other players’ strategies:
![Page 21: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/21.jpg)
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C. Nash Equilibrium
A Nash Equilibrium. 1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)(-2,-2)
![Page 22: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/22.jpg)
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C. Nash Equilibrium
A Nash Equilibrium.
Checking that 1’s strategy is a best responseIf she folds she gets -1.
1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)(-2,-2)
![Page 23: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/23.jpg)
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C. Nash Equilibrium
A Nash Equilibrium.
Checking that 1’s strategy is a best responseIf she raises she gets -2.
1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)(-2,-2)
![Page 24: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/24.jpg)
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C. Nash Equilibrium
A Nash Equilibrium.
Checking that 2’s strategy is a best responseIf she changes her action on the right her payoff is
unaltered.
1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)(-2,-2)
![Page 25: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/25.jpg)
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C. Nash Equilibrium
A Nash Equilibrium.
Checking that 2’s strategy is a best responseIf she changes her action on the right her payoff is
unaltered.
1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)(-2,-2)
![Page 26: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/26.jpg)
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C. Nash Equilibrium and Subgame Perfection
A Nash Equilibrium.
Checking that 2’s strategy is a best responseIf she changes her action on the left her payoff goes
down.
1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)(-2,-2)
![Page 27: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/27.jpg)
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C. Nash Equilibrium and Subgame Perfection
A Nash Equilibrium.
Checking that 2’s strategy is a best responseIf she changes her action on the left her payoff goes
down.
1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)(-2,-2)
![Page 28: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/28.jpg)
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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.
![Page 29: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/29.jpg)
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C. Nash Equilibrium
A Non-Credible Threat.1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)(-2,-2)
2 is making a threat herethat she would not carry out .
![Page 30: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/30.jpg)
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C. Nash Equilibrium
A Non-Credible Threat.1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)(-2,-2)
2 is making a threat herethat she would not carry out .
![Page 31: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/31.jpg)
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C. Nash Equilibrium
A Non-Credible Threat.
Once player 1 realizes this she is better off raising too
1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)(-2,-2)
2 is making a threat herethat she would not carry out .
![Page 32: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/32.jpg)
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C. Nash Equilibrium
A Non-Credible Threat.
Once player 1 realizes this she is better off raising too
1
RaiseFold
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1) (-2,-2)
2 is making a threat herethat she would not carry out .
![Page 33: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/33.jpg)
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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).
![Page 34: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/34.jpg)
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D. Backwards Induction
Example
Start at the last move and figureout what is optimal there
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 35: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/35.jpg)
35
D. Backwards Induction
Example
Start at the last move and figureout what is optimal there
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 36: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/36.jpg)
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D. Backwards Induction
Example
Do all last moves
1
RaiseFold
1
22
Fold Fold Raise
(0,0)(-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 37: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/37.jpg)
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D. Backwards Induction
Example
Do all last moves
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 38: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/38.jpg)
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D. Backwards Induction
Example
Now do the second last move.
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 39: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/39.jpg)
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D. Backwards Induction
Example
Now do the second last move.
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 40: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/40.jpg)
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D. Backwards Induction
Example:
Finally do the first move
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 41: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/41.jpg)
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D. Backwards Induction
Example:
Finally do the first move
1
RaiseFold
1
22
Fold Fold Raise
(0,0) (-1,1)
Raise
(1,-1)
(-1,1) (2,-2)
RaiseFold
![Page 42: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/42.jpg)
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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.
![Page 43: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/43.jpg)
43
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.
![Page 44: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/44.jpg)
44
1
E. Subgame Perfection
Example 1
RaiseFold
2
L R
(1,1)
(1,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 45: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/45.jpg)
45
1
E. Subgame Perfection
A (Bad) NE 1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 46: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/46.jpg)
46
1
E. Subgame Perfection
A (Bad) NE:
The move of player2 is not credibleit always prefersL to R if she getsto move!
1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 47: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/47.jpg)
47
1
E. Subgame Perfection
A (Bad) NE:
But cannot dobackwards inductionsometimes 1 prefers‘a’ sometimes ‘b’.
1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 48: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/48.jpg)
48
1
E. Subgame Perfection
Where are the subgames?
1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 49: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/49.jpg)
49
1
E. Subgame Perfection
Here
1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 50: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/50.jpg)
50
1
E. Subgame Perfection
And here
1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 51: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/51.jpg)
51
1
E. Subgame Perfection
Let’s look at thelast subgameand make the players play aNE on it.
1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 52: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/52.jpg)
52
1
E. Subgame Perfection
It hasonly 1 NE,because player2 always prefersL to R.
1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 53: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/53.jpg)
53
1
E. Subgame Perfection
Oncewe have solved thislast game it is easyto figure outwhat player 1 willdo.
1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 54: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/54.jpg)
54
1
E. Subgame Perfection
Oncewe have solved thislast game it is easyto figure outwhat player 1 willdo.
1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 55: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/55.jpg)
55
1
E. Subgame Perfection
This is thesubgame perfectequilibrium.
1
RaiseFold
2
L R
(1,1)
(2,2) (-1,1) (0,1)
ba
1
b a
(1/2,0)
![Page 56: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/56.jpg)
56
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.
![Page 57: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/57.jpg)
57
E. Subgame Perfection
1
YX
2
b
1
L R
(1,3)(0,0) (0,0)
a
1
b a
(2,2)
1
L R
(3,1) (0,0) (0,0)
a
1
b a
(2,2)
2
b
IF 1 plays right the players then play a coordination gamewhich has 3 NE’s.
![Page 58: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/58.jpg)
58
E. Subgame Perfection
1
YX
2
b
1
L R
(1,3)(0,0) (0,0)
a
1
b a
(2,2)
1
L R
(3,1) (0,0) (0,0)
a
1
b a
(2,2)
2
b
IF 1 plays right the players then play a coordination gamewhich has 3 NE’s.
![Page 59: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/59.jpg)
59
E. Subgame Perfection
1
YX
2
b
1
L R
(1,3)(0,0) (0,0)
a
1
b a
(2,2)
1
L R
(3,1) (0,0) (0,0)
a
1
b a
(2,2)
2
b
IF 1 plays right the players then play a coordination gamewhich has 3 NE’s.
![Page 60: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/60.jpg)
60
E. Subgame Perfection
1
YX
2
b
1
L R
(1,3)(0,0) (0,0)
a
1
b a
(2,2)
1
L R
(3,1) (0,0) (0,0)
a
1
b a
(2,2)
2
b
IF 1 plays X the players then play the battleof the sexes which also has 3 NE’s.
![Page 61: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/61.jpg)
61
E. Subgame Perfection
1
YX
2
b
1
L R
(1,3)(0,0) (0,0)
a
1
b a
(2,2)
1
L R
(3,1) (0,0) (0,0)
a
1
b a
(2,2)
2
b
IF 1 plays X the players then play the battleof the sexes which also has 3 NE’s.
![Page 62: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/62.jpg)
62
E. Subgame Perfection
1
YX
2
b
1
L R
(1,3)(0,0) (0,0)
a
1
b a
(2,2)
1
L R
(3,1) (0,0) (0,0)
a
1
b a
(2,2)
2
b
IF 1 plays X the players then play the battleof the sexes which also has 3 NE’s.
![Page 63: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/63.jpg)
63
E. Subgame Perfection
1
YX
2
b
1
L R
(1,3)(0,0) (0,0)
a
1
b a
(2,2)
1
L R
(3,1) (0,0) (0,0)
a
1
b a
(2,2)
2
b
IF 1 plays X the players then play the battleof the sexes which also has 3 NE’s.
![Page 64: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/64.jpg)
64
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…
![Page 65: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/65.jpg)
65
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
RM
22
a a b
(2,2) (0,1)
b
(3,1)
(1,3)
(0,0)
L
![Page 66: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/66.jpg)
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F. Perfect Bayesian Equilibrium
To remedy the problem with the game above we need another notion of equilibrium…
Perfect Bayesian equilibrium.
![Page 67: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/67.jpg)
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F. Perfect Bayesian Equilibrium
If you are goingto do somethinglike Backwardsinduction in thisgame, then youare going to needto decide what action player 2 will take.
1
RM
22
a a b
(2,2)(0,1)
b
(3,1)
(1,3)
(0,0)
L
![Page 68: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/68.jpg)
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F. Perfect Bayesian Equilibrium
If you are goingto do somethinglike Backwardsinduction in thisgame, then youare going to needto decide what action player 2 will take.
Must give player 2 beliefs
1
RM
22
a a b
(2,2)(0,1)
b
(3,1)
(1,3)
(0,0)
L
p 1-p
![Page 69: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/69.jpg)
69
F. Perfect Bayesian Equilibrium
Can now evaluate player 2’s expected payoff from each action
Action a = 2p+1(1-p)=1+p
1
RM
22
a a b
(2,2)(0,1)
b
(3,1)
(1,3)
(0,0)
L
p 1-p
![Page 70: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/70.jpg)
70
F. Perfect Bayesian Equilibrium
Can now evaluate player 2’s expected payoff from each action
Action b = 1p+0(1-p)=p
1
RM
22
a a b
(2,2) (0,1)
b
(3,1)
(1,3)
(0,0)
L
p 1-p
![Page 71: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/71.jpg)
71
F. Perfect Bayesian Equilibrium
Action a is always better than action, so player 2 should
play a.
1
RM
22
a a b
(2,2) (0,1)
b
(3,1)
(1,3)
(0,0)
L
p 1-p
![Page 72: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/72.jpg)
72
F. Perfect Bayesian Equilibrium
Hence 1 should play R.
1
RM
22
a a b
(2,2) (0,1)
b
(3,1)
(1,3)
(0,0)
L
p 1-p
![Page 73: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/73.jpg)
73
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 p=0.
1
RM
22
a a b
(2,2) (0,1)
b
(3,1)
(1,3)
(0,0)
L
p 1-p
![Page 74: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/74.jpg)
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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).
![Page 75: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/75.jpg)
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F. Perfect Bayesian Equilibrium
A SignallingGame
0
22
(0,0) (-1,1)(1,2) (0,0)
11
2 2
1/2 1/2
T TW W
a a a ab b b b
(1,2)(0,0) (0,0)(-1,1)
![Page 76: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/76.jpg)
76
F. Perfect Bayesian Equilibrium
A SignallingGame
We must give player 2 two lots of beliefs (sometimes these
are called assessments).
0
22
(0,0) (-1,1)(1,2) (0,0)
11
2 2
1/2 1/2
T TW W
a a a ab b b b
(1,2)(0,0) (0,0)(-1,1)
1-pp 1-ee
![Page 77: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/77.jpg)
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F. Perfect Bayesian Equilibrium
A SignallingGame
2 plays a if: >1/3p . 2 plays a if: >2/3e .
0
22
(0,0) (-1,1)(1,2) (0,0)
11
2 2
1/2 1/2
T TW W
a a a ab b b b
(1,2)(0,0) (0,0)(-1,1)
1-pp 1-ee
![Page 78: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/78.jpg)
78
aaa
F. Perfect Bayesian Equilibrium
A SignallingGame
Both types of 1 play T
0
22
(0,0) (-1,1)(1,2) (0,0)
11
2 2
1/2 1/2
T TW W
ab b b b
(1,2)(0,0) (0,0)(-1,1)
1-pp 1-ee
![Page 79: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/79.jpg)
79
aaa
F. Perfect Bayesian Equilibrium
A SignallingGame
Both types of 1 play TSo rational expectations says =1/2p ,but e is arbitrary – must have >2/3e .
0
22
(0,0) (-1,1)(1,2) (0,0)
11
2 2
1/2 1/2
T TW W
ab b b b
(1,2)(0,0) (0,0)(-1,1)
1-pp 1-ee
![Page 80: Topic 3 Games in Extensive Form 1. A. Perfect Information Games in Extensive Form. 1 RaiseFold 1 2 2 Raise (0,0) (-1,1) Raise (1,-1) (-1,1)(2,-2) 2.](https://reader036.fdocuments.in/reader036/viewer/2022081513/56649f165503460f94c2b9a7/html5/thumbnails/80.jpg)
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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.