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# Game theory 2

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### Game theory 2

1. 1. GAME THEORY
2. 2. Game Theory• What is Game Theory?• What is strategic environment?• What are the main assumptions in Game Theory?• How are games represented?• What are the elements of Games ?
3. 3. Game representations Matrix form Extensive form (aka normal form aka strategic form) player 2’s strategy Up 1, 2 Up, Up Up, Down D, D Down , Up Up Coca Cola Down 3, 4 Up 1, 2 1, 2 3, 4 3, 4 player 1’sPepsi strategy Up 5, 6 Down 5, 6 7, 8 5, 6 7, 8 Down Coca Cola Down 7, 8
4. 4. Dominated Strategy• What is a dominated strategy?
5. 5. Rules of the Game• The strategic environment – Players – Strategies – Payoffs• The assumptions – Rationality – Common knowledge• The rules – Timing of moves – Informational conditions
6. 6. Formal definitionDefinitions• Let Ai be the set of actions available for player I• a = (a1, a2, …, ai,…) be an action profile: An action foreach player in the game.• write (ai, a-i) if i chooses ai, other players according to a• ui (ai , a-i ): payoff for player i from playing action ai andothers playing a-i
7. 7. Prisoner’s Dilemma
8. 8. Prisoner’s Dilemma Not Confess ConfessNot Confess -2, -2 -5, -1Confess -1, -5 -3,-3
9. 9. Comments• Simultaneous actions does not imply taking actions at the same time.• Rationality implies knowing the structure of the game.• It does not mean that there is coordination on beliefs / cooperation• Rationality is an assumption not necessary in reality.
10. 10. Comments• (Not Confess, Not Confess) is a pareto optimal outcome• This is common knowledge• Will Not Confess be chosen by both?• Depends upon their beliefs about the actions of other players
11. 11. Prisoner’s Dilemma Not Confess ConfessNot Confess -2, -2 -5, -1Confess -1, -5 -3,-3
12. 12. NASH EQUILIBRIUM
13. 13. Nash EquilibriumNash Equilibrium:• A set of strategies, one for each player, such that each player’s strategy is a best response to others’ strategiesEverybody is playing a best response• No incentive to unilaterally change my strategy
14. 14. Nash Equilibrium• a* =( a* 1 , a* 2 , ……) = =( a* i , a* -i)is a Nash equilibrium if for every player i and everyaction, ai that is element of Ai: ui(a*) ≥ ui (ai , a-i*)where ui is the payoff function representing thepreferences of player I
15. 15. L C RU 8, 3 0, 4 4,4M 8,5 1,5 5,3D 3,7 0,1 2,0
16. 16. L RL 1,-1 -1, 1R -1, 1 1,-1
17. 17. Coordination Games Opera MovieOpera 2,1 0,0Movie 0,0 1,2
18. 18. SUV Price Wars Discount No DiscountDiscount 3,3 8, 0No Discount 0, 8 5,5
19. 19. Games of Chicken• Entry into small markets Firm 2 Stay Swerve Stay -50 , -50 100 , 0 Firm 1 Swerve 0 , 100 50 , 50
20. 20. Stag Hunt Game• Each can individually hunt a stag or hunt a hare together.• You can hunt a stag alone.• You can only hunt a hare when you hunt it together.• Hunting a hare alone means no dinner!
21. 21. Stag Hunt Game Hare StagHare 3,3 -1, 0Stag 0, -1 0,0
22. 22. Games of AssuranceJoint research ventures – Each firm may invest \$50,000 into an R&D project – Project succeeds only if both invest – If successful, each nets \$75,000 – Stag Hunt Firm 2 \$50K \$0 \$50K 75 , 75 -50 , 0 Firm 1 \$0 0 , -50 0 , 0
23. 23. Nash Equilibrium• May or may not have to exist in pure strategies.• Can be multiple in a single game.