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The fundamentals of game theory, including Nash equilibrium

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- 1. An introduction to gametheoryToday: The fundamentals ofgame theory, including Nashequilibrium
- 2. Today Introduction to game theory We can look at market situations with twoplayers (typically firms) Although we will look at situations whereeach player can make only one of twodecisions, theory easily extends to three ormore decisions
- 3. Who is this?
- 4. John Nash, the person portrayedin “A Beautiful Mind”
- 5. John Nash One of the earlyresearchers ingame theory His work resultedin a form ofequilibriumnamed after him
- 6. Three elements in every game Players Two or more for most games that areinteresting Strategies available to each player Payoffs Based on your decision(s) and thedecision(s) of other(s)
- 7. Game theory: Payoff matrix A payoffmatrixshows thepayout toeachplayer,given thedecision ofeachplayerAction C Action DAction A 10, 2 8, 3Action B 12, 4 10, 1Person1Person 2
- 8. How do we interpret this box? The first number ineach box determinesthe payout forPerson 1 The second numberdetermines thepayout for Person 2ActionCActionDActionA10, 2 8, 3ActionB12, 4 10, 1Person1Person 2
- 9. How do we interpret this box? Example If Person 1chooses Action Aand Person 2chooses Action D,then Person 1receives a payoutof 8 and Person 2receives a payoutof 3ActionCActionDActionA10, 2 8, 3ActionB12, 4 10, 1Person1Person 2
- 10. Back to a Core Principle:Equilibrium The type of equilibrium we are lookingfor here is called Nash equilibrium Nash equilibrium: “Any combination ofstrategies in which each player’s strategyis his or her best choice, given the otherplayers’ choices” (F/B p. 322) Exactly one person deviating from a NEstrategy would result in the same payout orlower payout for that person
- 11. How do we find Nashequilibrium (NE)? Step 1: Pretend you are one of the players Step 2: Assume that your “opponent” picks aparticular action Step 3: Determine your best strategy (strategies),given your opponent’s action Underline any best choice in the payoff matrix Step 4: Repeat Steps 2 & 3 for any other opponentstrategies Step 5: Repeat Steps 1 through 4 for the otherplayer Step 6: Any entry with all numbers underlined is NE
- 12. Steps 1 and 2 Assume thatyou arePerson 1 Given thatPerson 2choosesAction C,what isPerson 1’sbestchoice?ActionCAction DActionA10, 2 8, 3Action B 12, 4 10, 1Person1Person 2
- 13. Step 3: Underlinebest payout,given thechoice of theother player ChooseAction B,since12 > 10 underline 12ActionCAction DActionA10, 2 8, 3Action B 12, 4 10, 1Person1Person 2
- 14. Step 4 Nowassumethat Person2 choosesAction D Here,10 > 8 Chooseandunderline10Action C ActionDAction A 10, 2 8, 3Action B 12, 4 10, 1Person1Person 2
- 15. Step 5 Now,assume youare Person 2 If Person 1chooses A 3 > 2 underline 3 If Person 1chooses B 4 > 1 underline 4Action C Action DActionA10, 2 8, 3ActionB12, 4 10, 1Person1Person 2
- 16. Step 6 Whichbox(es) haveunderlinesunder bothnumbers? Person 1chooses Band Person2 choosesC This is theonly NEAction C Action DAction A 10, 2 8, 3Action B 12, 4 10, 1Person1Person 2
- 17. Double check our NE What ifPerson 1deviatesfrom NE? Couldchoose Aand get 10 Person 1’spayout islower bydeviating Action C Action DAction A 10, 2 8, 3Action B 12, 4 10, 1Person1Person 2
- 18. Double check our NE What ifPerson 2deviatesfrom NE? Couldchoose Dand get 1 Person 2’spayout islower bydeviating Action C Action DAction A 10, 2 8, 3Action B 12, 4 10, 1Person1Person 2
- 19. Dominant strategy A strategy isdominant if thatchoice isdefinitely madeno matter whatthe otherperson chooses Example:Person 1 has adominantstrategy ofchoosing BAction C Action DAction A 10, 2 8, 3Action B 12, 4 10, 1Person1Person 2
- 20. New example Suppose inthis examplethat twopeople aresimultaneously going todecide on thisgameYes NoYes 20, 20 5, 10No 10, 5 10, 10Person1Person 2
- 21. New example We will gothrough thesame steps todetermine NEYes NoYes 20, 20 5, 10No 10, 5 10, 10Person1Person 2
- 22. Two NE possible (Yes, Yes) and(No, No) areboth NE Although (Yes,Yes) is themore efficientoutcome, wehave no way topredict whichoutcome willactually occurYes NoYes 20, 20 5, 10No 10, 5 10, 10Person1Person 2
- 23. Two NE possible When there are multiple NE that arepossible, economic theory tells us littleabout which outcome occurs withcertainty
- 24. Two NE possible Additional information or actions mayhelp to determine outcome If people could act sequentially instead ofsimultaneously, we could see that 20, 20would occur in equilibrium
- 25. Sequential decisions Suppose that decisions can be madesequentially We can work backwards to determinehow people will behave We will examine the last decision first andthen work toward the first decision To do this, we will use a decision tree
- 26. Decision tree in a sequentialgame: Person 1 chooses firstABCPerson1choosesyesPerson1choosesnoPerson 2choosesyesPerson 2choosesyesPerson 2chooses noPerson 2chooses no20, 205, 1010, 510, 10
- 27. Decision tree in a sequentialgame: Person 1 chooses first Given point B,Person 2 willchoose yes(20 > 10) Given point C,Person 2 willchoose no(10 > 5)ABCPerson1choosesyesPerson1choosesnoPerson 2choosesyesPerson 2choosesyesPerson 2chooses noPerson 2choosesno20, 205, 1010, 510, 10
- 28. Decision tree in a sequentialgame: Person 1 chooses first If Person 1 isrational, she willignore potentialchoices thatPerson 2 will notmake Example: Person2 will not chooseyes after Person 1chooses noABCPerson1choosesyesPerson1choosesnoPerson 2choosesyesPerson 2choosesyesPerson 2chooses noPerson 2choosesno20, 205, 1010, 510, 10
- 29. Decision tree in a sequentialgame: Person 1 chooses first If Person 1 knowsthat Person 2 isrational, then shewill choose yes,since 20 > 10 Person 2 makes adecision from pointB, and he willchoose yes also Payout: (20, 20)ABCPerson1choosesyesPerson1choosesnoPerson 2choosesyesPerson 2choosesyesPerson 2chooses noPerson 2choosesno20, 205, 1010, 510, 10
- 30. Summary Game theory Simultaneous decisions NE Sequential decisions Some NE may notoccur if people are rational
- 31. Can you think of ways game theorycan be used in these games?

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