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Introduction to game theory

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The fundamentals of game theory, including Nash equilibrium

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Introduction to game theory

  1. 1. An introduction to gametheoryToday: The fundamentals ofgame theory, including Nashequilibrium
  2. 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. 3. Who is this?
  4. 4. John Nash, the person portrayedin “A Beautiful Mind”
  5. 5. John Nash One of the earlyresearchers ingame theory His work resultedin a form ofequilibriumnamed after him
  6. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 20. New example Suppose inthis examplethat twopeople aresimultaneously going todecide on thisgameYes NoYes 20, 20 5, 10No 10, 5 10, 10Person1Person 2
  21. 21. New example We will gothrough thesame steps todetermine NEYes NoYes 20, 20 5, 10No 10, 5 10, 10Person1Person 2
  22. 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. 23. Two NE possible When there are multiple NE that arepossible, economic theory tells us littleabout which outcome occurs withcertainty
  24. 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. 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. 26. Decision tree in a sequentialgame: Person 1 chooses firstABCPerson1choosesyesPerson1choosesnoPerson 2choosesyesPerson 2choosesyesPerson 2chooses noPerson 2chooses no20, 205, 1010, 510, 10
  27. 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. 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. 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. 30. Summary Game theory Simultaneous decisions  NE Sequential decisions  Some NE may notoccur if people are rational
  31. 31. Can you think of ways game theorycan be used in these games?

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