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Introduction to Game Theory

Introduction to Game Theory. Lecture 5: Extensive Games with Perfect Information. Today’s Plan. Review Focal Points Predictions in the case of multiple Nash equilibria Games with perfect information and complete information Extensive games with compete information Nash equilibrium

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Introduction to Game Theory

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  1. Introduction to Game Theory Lecture 5: Extensive Games with Perfect Information

  2. Today’s Plan • Review • Focal Points • Predictions in the case of multiple Nash equilibria • Games with perfect information and complete information • Extensive games with compete information • Nash equilibrium • Subgame perfect Nash equilibrium • Backward induction

  3. Review • Simultaneous games with complete information • Iterative elimination of dominated strategies • Nash equilibrium • Mixed strategy Nash equilibrium • Today: • Multiplicity of Nash equilibria • Driver vs. Pedestrian • Simultaneous vs. sequential decision making • Cournot’s duopoly vs. Stackelberg’s duopoly

  4. Focal Points (Schelling) • Schelling (1960) • Think about the following game and assume only pure strategies. • You will be randomly coupled with one person from your cohort. • Both of you would like to meet but cannot communicate with each other. Both of you can choose whatever place in the Prague and whatever time. • If you decide for the same time and place you win. If not you loose. • Please write down the place and time.

  5. Meeting – Results • What is the Nash equilibrium in this game? • What is the ex-ante probability that one of these equilibriums is actually observed? • What have you submitted? • Did any couple reached Nash equilibrium? • “Any feature of equilibrium that draws attention to itself, making it stand out among the equilibria will tend to produce self-fulfilling expectations that this salient equilibrium will result.”

  6. Example 2 • You are assigned with other player in random • Both of you are asked to submit number between 0 and 1 included. • If numbers you have submitted sum up to one you win, if they do not you loose. • Is there any Nash equilibrium achieved?

  7. Cars and Broken Traffic Lights • Two cars simultaneously approach crossroad. • Both want to pass crossroad as soon as possible. • The least preferable result is crash. • Model the situation in pure strategies. • What are the Nash equilibria? • Submit your bids if I tell you that Player 1 should go and Player 2 should stop. (Equivalent of the right hand rule) • Third party – no effect on players’ payoffs • Makes one of the equilibriums special.

  8. Extensive Games with Complete Inf • Dynamic game • Players move sequentially. • Order can depend on the actions that has been taken. • Players are aware of moves taken before their decisions. • Players have complete information.

  9. Entrant vs. Incumbent • Albert considers to build a new supermarket across Tesco in Narodni Trida. • Albert chooses either to build a supermarket or to quit the idea. • Tesco, after it observes Albert’s decision, can either fight or acquiesce. • Payoffs are public information

  10. NE – Normal Form Representation • Such action profile that non of the players can do better by unilateral deviation • Two Nash equilibria: (Out, Fight) and (In, Acq.) • “Fight” is not a credible threat in equilibrium (Out, Fight) • How does Albert know that Tesco plays “Fight” when Albert plays “In”? “Fight” is not optimal strategy once “In” has been played by Albert “subgame”.

  11. Extensive Form Representation • Tree specifies: • Players • Every player’s opportunity to make decision • Options when making decision • Knowledge of preceding action when making decision • Payoffs for every possible combination of moves OUT IN 0 2 Fight Acq. -3 -1 2 1

  12. Strategy • What happens if somebody changes his decision? • Need to know how the payoffs are affected by the change. • Player’s complete plan of the actions that specifies the action that player takes in every node he makes decision. • Player’s 1 strategies: Left, Right • Player’s 2 strategies: (Left, Left), (Left, Right), (Right, Left), (Right, Left) • Strategy profile 1 Left Right 2 2 Left Left Right. Right 2 1 0 2 3 0 1 3

  13. Subgame (Perfect and Complete I.) • Very informal definition: • Part of the game that follows single decision node. • Contains all the nodes that follow the nodes that subgame contains. • The entire game is subgame to itself • How many?

  14. Subgame • Very informal definition: • Part of the game that follows single decision node. • Contains all the nodes that follow the nodes that subgame contains. • The entire game is subgame to itself • How many? Not a subgame!!!

  15. Subgame Perfect Nash Equilibrium • Strategy profile that in any subgame no player cannot do better by unilateral deviation. • (Out,Fight) is not subgame perfect equilibrium since Tesco can do better in red subgame by playing Acq. in “red” subgame • (In, Acq.) is subgame perfect equilibrium • Subgame Perfect Nash Equilibrium is Nash equilibrium as well. OUT IN 0 2 Fight Acq. -3 -1 2 1

  16. Backward Induction • Players anticipate actions of subsequent players. • Players deduce for each of their option the actions of subsequent players and chooses the option that yields the most preferable outcome. • Start solution from the last subgames. • Continue backwards. • Results in subgame perfect Nash equilibria. • Backward induction keeps equilibria only with credible threat.

  17. Backward Induction • In the third node, Player 1 would chose “LLL”. P1 R L P2 2, 0 RR LL P1 1, 1 LLL RRR 0, 2 3, 0

  18. Backward Induction • In the lowest node, P1 would chose “LLL”. • In the middle node, P2 knows that by playing RR he gets 0. Plays “LL”. P1 R L P2 2, 0 RR LL P1 1, 1 LLL RRR 0, 2 3, 0

  19. Backward Induction • In the lowest node, P1 would chose “LLL”. • In the middle node, P2 knows that by playing RR he gets 0. He plays “LL”. • In the top node, P1 knows that by playing R she gets 1. She plays L. P1 R L P2 2, 0 RR LL P1 1, 1 LLL RRR 0, 2 3, 0

  20. Backward Induction • In the lowest node, P1 would chose “LLL”. • In the middle node, P2 knows that by playing RR he gets 0. He plays “LL”. • In the top node, P1 knows that by playing R she gets 1. She plays L. P1 R L P2 2, 0 RR LL P1 1, 1 LLL RRR 0, 2 3, 0

  21. Backward Induction - SPNE • In the lowest node, P1 would chose “LLL”. • In the middle node, P2 knows that by playing RR he gets 0. He plays “LL”. • In the top node, P1 knows that by playing R she gets 1. She plays L. • SPNE – ((L,LLL),LL) P1 R L P2 2, 0 RR LL P1 1, 1 LLL RRR 0, 2 3, 0

  22. Tesco vs. Albert Revised • Albert enhanced his model by considering pricing strategy as well. Three decision making levels: • Albert stays out or built a supermarket • Tesco fights by lowering the price or acquiesces by keeping prices high. • Albert observes Tesco’s decision and either lower the price or keep it high.

  23. Tesco vs. Albert Revised • Albert plays: • High if he gets opportunity in the lowest-left node. • Is indifferent in lowest-right node. OUT IN 0 3 Low High. Low Low High High -1 0 0 2 1 1 1 2

  24. Tesco vs. Albert Revised • Albert plays: • High if he gets opportunity in the lowest-left node. • Is indifferent in lowest-right node. • Tesco is • Indifferent between High and Low if Albert plays High in lowest-right node • Plays Low if Albert plays Low in lowest-right node. OUT IN 0 3 Low High. Low Low High High -1 0 0 2 1 1 1 2

  25. Tesco vs. Albert Revised • Albert plays: • High if he gets opportunity in the lowest-left node. • Is indifferent in lowest-right node. • Tesco is • Indifferent between High and Low if Albert plays High in lowest-right node • Plays Low if Albert plays Low in lowest-right node. • Albert plays: • In given that Tesco Plays High • Is indifferent given that Tesco plays Low. OUT IN 0 3 Low High. Low Low High High -1 0 0 2 1 1 1 2

  26. Tesco vs. Albert Revised • What a mess!!! • What are the SPNE? • ((Out,HL),Low) • ((Out,HH),Low) • ((In,HH),Low) • ((In,HL),Low) • ((In,HL),High) • ((In,HH),High) OUT IN 0 3 Low High. Low Low High High -1 0 0 2 1 1 1 2

  27. Proportional Envy-free Allocation • Two persons (she and he) divide a pie • Assumptions: • Pie is homogenous and of size one • Everybody can precisely cut the cake (no mistakes) and everybody can precisely measure the size of the cake he gets. • More is better. • How to cut the pie such that division is envy-free?

  28. Proportional Envy-free Allocation • I cut, you chose • She cuts and he chooses the part or vice-versa. • Assume she cuts and he chooses the part • She decides about the sizes of two parts, he decides which he consumes • Tree – p-size of smaller piece 1-p,p Small p she Big he p,1-p

  29. Proportional Envy-free Allocation • He all the time chooses bigger part • She knows it. Given that her payoff is all the time the size of smaller part, she maximizes p. • SPNE : p=1/2 1-p,p Small p she Big he p,1-p

  30. Summary • Dynamic Games • Extensive form presentation • Nash equilibrium • Subgame perfect Nash equilibrium • Backward induction

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