Introduction To Game Theory
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An Elementary Introduction to Game Theory • Ritesh Pandey
1. Strategic interaction among economic agents : Classical Analysis Game Theoretic analysis 2. Uses of Game theory: Parlor Games, Political Negotiation and Economic behavior.
3. Our focus will be on Two - Person games with a finite no. of strategies. 4. The Concept of a Payoff Matrix. 5. Dominant Strategy: If there is one optimal choice of strategy for a player irrespective of the other’s choice. 6. If a DS exists, it will be the equilibrium outcome of the game.
Player B Left
Right
Top
1,2
0,1
Bottom
2,1
1,0
Player A
7. Usually, dominant strategy equilibria
are not that frequent. 8.But we can be less strict. 9. A Nash Equilibrium: Rather than require that A’s choice be optimal for all choices of B, we can just require that it be optimal only for optimal choices of B.
Player B Left
Right
Top
2,1
0,0
Bottom
0,0
1,2
Player A
10. We say that a pair of strategies is a Nash equilibrium if A’s choice is optimal given B’s choice, and B’s choice is optimal given A’s choice.
11. Note that neither knows beforehand the other person’s choice 12. He only has a set of Expectations. 13. Problems with Nash equilibrium concept: 1. Non-unique Nash equilibria may exist. 2. No Nash equilibria may exist.
Player B Left
Right
Top
0,0
0,-1
Bottom
1,0
-1,3
Player A
No Nash Equilibria exist here. (Pure Strategy)
14. Of course this was a pure-strategy case. 15. We can have a mixed strategy case where each player merely has a probability of playing a choice. For e.g., each could choose with a probability of 50% for either choice and land up with an expected payoff. 16. It can be shown that for games of the kind we are considering we can always have a Nash equilibrium.
Player B Confess
Deny
Confess
-3,-3
0,-6
Deny
-6,0
-1,-1
Player A
The Prisoner’s dilemma. ( also, arms control, cheating in a cartel )
Repeated Games: You have a chance to punish for bad behavior. You have a chance to establish a reputation for co-operation and thereby encouraging the other player to do the same. Viable only when an infinite no. of rounds are allowed.
Sequential games: One Player gets to choose first. Scenario changes entirely in terms of equilibria.
Player B Left
Right
Top
1,9
1,9
Bottom
0,0
2,1
Player A
A,B Player B chooses Player A chooses
Top
Left L e f Right t Left
Bottom Player B chooses
Right
1,9
1,9
0,0
2,1
A Game of Entry Deterrence Incumbent Chooses Entrant chooses
Stay Out
Fight L e f Don’t t fight Fight
Enter Incumbent chooses
Don’t fight
Entrant,Incumbent 1,9
1,9
0,2
2,1
Column Left
Right
Top
2,1
0,0
Bottom
0,0
1,2
Row
Girl Action
Art
Action
2,1
0,0
Art
0,0
1,2
Boy
The battle of the sexes
Player B Confess
Deny
Confess
-3,-3
0,-6
Deny
-6,0
-1,-1
Player A
The prisoner’s dilemma
USSR Refrain
Build
Refrain
4,4
1,3
Build
3,1
2,2
US
An arms race
Boy2 Swerve
Straight
Swerve
0,0
-1,1
Straight
1,-1
-2,-2
Boy 1
Chicken
Keeper Defend Swerve left
Defend Straight right
Kick Swerve left
50,-50 0,0
80,-80 -1,1
Kick Straight right
90,-90 1,-1
20,-20 -2,-2
Striker Boy 1
Penalty point in soccer
Games of co-existence Evolutionary Stable Strategy(ESS) Astonishingly, EESS equilibria are also Nash equilibria.
Dog 2 Hawk
Dove
Hawk
-2,-2
4,0
Dove
0,4
2,2
Dog 1
Hawk-Dove Game
Games of Commitment: Till now simultaneous moves. Now Sequential. In such Games, the important issue is commitment. Commitment must be IRREVERSIBLE & OBSERVABLE
Scorpion Chooses Frog
Carry
chooses
Sting L e f Refraint
Frog,Scorpion -10,5
5,3
Refuse 0,0 The Frog & The Scorpion Story
Scorpion Chooses Frog
Carry
chooses
Sting L e f Refraint
Frog,Scorpion -10,2
5,3
Refuse 0,0 The Smart Frog & The Scorpion
Hostage Chooses Kidnapper
Release
chooses
Identify L e f Refraint
Kidnapper,Hostage -5,5
5,3
Kill -3,-10 The Kindly Kidnapper
Dominant Pig
Subordinate Pig
Don’t press lever
press lever
Don’t press lever
0,0
4,1
press lever
0,5
2,3
When Strength is Weakness: Pigs Pressing Levers
Younger Generation Support
Refrain
Save
3,-1
1,0
Squander
2,-1
-2,-2
Older Generation
Savings & Social Security
Young Choose Old
Squander
choose
Support L e f Refraint Support
Old,Young -5,5
5,3
-3,-10
Save Young Choose Refrain The Savings Game:Extended Form
1,0
Kidnapper,Hostage Client Chooses Contractor chooses
Extort
Charge actual cost The Hold-Up Problem
Give in L e f Find a t painter
1300,0
0,-100
0,1300
1. Strategic interaction among economic agents : Classical Analysis Game Theoretic analysis 2. Uses of Game theory: Parlor Games, Political Negotiation and Economic behavior.
3. Our focus will be on Two - Person games with a finite no. of strategies. 4. The Concept of a Payoff Matrix. 5. Dominant Strategy: If there is one optimal choice of strategy for a player irrespective of the other’s choice. 6. If a DS exists, it will be the equilibrium outcome of the game.
Player B Left
Right
Top
1,2
0,1
Bottom
2,1
1,0
Player A
7. Usually, dominant strategy equilibria
are not that frequent. 8.But we can be less strict. 9. A Nash Equilibrium: Rather than require that A’s choice be optimal for all choices of B, we can just require that it be optimal only for optimal choices of B.
Player B Left
Right
Top
2,1
0,0
Bottom
0,0
1,2
Player A
10. We say that a pair of strategies is a Nash equilibrium if A’s choice is optimal given B’s choice, and B’s choice is optimal given A’s choice.
11. Note that neither knows beforehand the other person’s choice 12. He only has a set of Expectations. 13. Problems with Nash equilibrium concept: 1. Non-unique Nash equilibria may exist. 2. No Nash equilibria may exist.
Player B Left
Right
Top
0,0
0,-1
Bottom
1,0
-1,3
Player A
No Nash Equilibria exist here. (Pure Strategy)
14. Of course this was a pure-strategy case. 15. We can have a mixed strategy case where each player merely has a probability of playing a choice. For e.g., each could choose with a probability of 50% for either choice and land up with an expected payoff. 16. It can be shown that for games of the kind we are considering we can always have a Nash equilibrium.
Player B Confess
Deny
Confess
-3,-3
0,-6
Deny
-6,0
-1,-1
Player A
The Prisoner’s dilemma. ( also, arms control, cheating in a cartel )
Repeated Games: You have a chance to punish for bad behavior. You have a chance to establish a reputation for co-operation and thereby encouraging the other player to do the same. Viable only when an infinite no. of rounds are allowed.
Sequential games: One Player gets to choose first. Scenario changes entirely in terms of equilibria.
Player B Left
Right
Top
1,9
1,9
Bottom
0,0
2,1
Player A
A,B Player B chooses Player A chooses
Top
Left L e f Right t Left
Bottom Player B chooses
Right
1,9
1,9
0,0
2,1
A Game of Entry Deterrence Incumbent Chooses Entrant chooses
Stay Out
Fight L e f Don’t t fight Fight
Enter Incumbent chooses
Don’t fight
Entrant,Incumbent 1,9
1,9
0,2
2,1
Column Left
Right
Top
2,1
0,0
Bottom
0,0
1,2
Row
Girl Action
Art
Action
2,1
0,0
Art
0,0
1,2
Boy
The battle of the sexes
Player B Confess
Deny
Confess
-3,-3
0,-6
Deny
-6,0
-1,-1
Player A
The prisoner’s dilemma
USSR Refrain
Build
Refrain
4,4
1,3
Build
3,1
2,2
US
An arms race
Boy2 Swerve
Straight
Swerve
0,0
-1,1
Straight
1,-1
-2,-2
Boy 1
Chicken
Keeper Defend Swerve left
Defend Straight right
Kick Swerve left
50,-50 0,0
80,-80 -1,1
Kick Straight right
90,-90 1,-1
20,-20 -2,-2
Striker Boy 1
Penalty point in soccer
Games of co-existence Evolutionary Stable Strategy(ESS) Astonishingly, EESS equilibria are also Nash equilibria.
Dog 2 Hawk
Dove
Hawk
-2,-2
4,0
Dove
0,4
2,2
Dog 1
Hawk-Dove Game
Games of Commitment: Till now simultaneous moves. Now Sequential. In such Games, the important issue is commitment. Commitment must be IRREVERSIBLE & OBSERVABLE
Scorpion Chooses Frog
Carry
chooses
Sting L e f Refraint
Frog,Scorpion -10,5
5,3
Refuse 0,0 The Frog & The Scorpion Story
Scorpion Chooses Frog
Carry
chooses
Sting L e f Refraint
Frog,Scorpion -10,2
5,3
Refuse 0,0 The Smart Frog & The Scorpion
Hostage Chooses Kidnapper
Release
chooses
Identify L e f Refraint
Kidnapper,Hostage -5,5
5,3
Kill -3,-10 The Kindly Kidnapper
Dominant Pig
Subordinate Pig
Don’t press lever
press lever
Don’t press lever
0,0
4,1
press lever
0,5
2,3
When Strength is Weakness: Pigs Pressing Levers
Younger Generation Support
Refrain
Save
3,-1
1,0
Squander
2,-1
-2,-2
Older Generation
Savings & Social Security
Young Choose Old
Squander
choose
Support L e f Refraint Support
Old,Young -5,5
5,3
-3,-10
Save Young Choose Refrain The Savings Game:Extended Form
1,0
Kidnapper,Hostage Client Chooses Contractor chooses
Extort
Charge actual cost The Hold-Up Problem
Give in L e f Find a t painter
1300,0
0,-100
0,1300
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