1. UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Bargaining and Negotiation Review Midterm3/21 2/28

2. Nonzero-sum Games Examples: Bargaining Duopoly International Trade

3. Nonzero-sum Games The Essentials of a Game Eliminating Dominated Strategies Best Response Nash Equilibrium Duopoly: An Application Solving the Game Existence of Nash Equilibrium Properties and Problems See: Gibbons, Game Theory for Applied Economists (1992): 1-51.

4. The Essentials of a Game 1. Players: We require at least 2 players (Players choose actions and receive payoffs.) 2. Actions: Player i chooses from a finite set of actions, S = {s 1,s 2,…..,s n }. Player j chooses from a finite set of actions T = {t 1,t 2,……,t m }. 3A. Payoffs: We define P i (s,t) as the payoff to player i, if Player i chooses s and player j chooses t. We allow that P i (s,t) + P j (s,t) = 0. 4. Information: What players know (believe) when choosing actions. NONZERO-SUM

5. Eliminating Dominated Strategies 1,0 1,20,1 0,3 0,12,0 1,0 1,2 0,3 0,1 1,0 1,2 L M R TBTB R is strictly dominated by M, so the game can be reduced to Now, B is strictly dominated by T... TBTB T (T, M)

6. Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 4,4 S1S2S3S1S2S3 T 1 T 2 T 3 A dominated strategy will never be played by a rational player.

7. Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 4,4 S1S2S3S1S2S3 T 1 T 2 T 3 A strategy can be dominated by a mixture of other strategies … T 3 is dominated by a mixture of T 1 and T 2.

8. Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 4,4 S1S2S3S1S2S3 T 1 T 2 T 3 A strategy can be dominated by a mixture of other strategies … Now S 3 is dominated by a mixture of S 1 and S 2.

9. Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 4,4 S1S2S3S1S2S3 T 1 T 2 T 3 A strategy can be dominated by a mixture of other strategies … Now S 3 is dominated by a mixture of S 1 and S 2.

10. Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. -3 0 -10 -1 5 2 -2 -4 0 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 6,6 S1S2S3S1S2S3 S1S2S3S1S2S3 Does either player have a dominant strategy? A dominated strategy? T 1 T 2 T 3

11. Best Response Best Response Strategy: a strategy, s*, is a best response strategy, iff the payoff to (s*,t) is at least as great as the payoff to (s,t) for all s. -3 0 -10 -1 5 2 -2 -4 0 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 6,6 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3 Player 1’s best responses are in red: Underline the highest value in each column.

12. Best Response Best Response Strategy: a strategy, s*, is a best response strategy, iff the payoff to (s*,t) is at least as great as the payoff to (s,t) for all s. -3 0 -10 -1 5 2 -2 -4 0 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 6,6 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3 Player 2’s best responses are in green: Underline the highest value in each row.

13. Nash Equilibrium Best Response Strategy: a strategy, s*, is a best response strategy, iff the payoff to (s*,t) is at least as great as the payoff to (s,t) for all s. -3 0 -10 -1 5 2 -2 -4 0 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 6,6 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3 Nash Equilibrium: a set of best response strategies (one for each player), (s*, t*) such that s* is a best response to t* and t* is a b.r. to s*. NE = {(S 3,T 3 )}

14. Nash Equilibrium -3 0 -10 -1 5 2 -2 -4 0 4,4 2,3 1,5 3,2 1,1 0,0 5,1 0,0 3,3 S1S2S3S1S2S3 S1S2S3S1S2S3 Nash equilibrium need not be efficient. T 1 T 2 T 3

15. Nash Equilibrium -3 0 -10 -1 5 2 -2 -4 0 1,1 0,0 0,0 0,0 1,1 0,0 0,0 0,0 1,1 S1S2S3S1S2S3 S1S2S3S1S2S3 Nash equilibrium need not be unique. A COORDINATION PROBLEM What is the effect of repeated play? T 1 T 2 T 3

16. Nash Equilibrium -3 0 -10 -1 5 2 -2 -4 0 1,1 0,0 0,0 0,0 1,1 0,0 0,0 0,0 3,3 S1S2S3S1S2S3 S1S2S3S1S2S3 Multiple and inefficient equilibria. T 1 T 2 T 3

17. Duopoly Consider a market in which two identical firms can produce a good with marginal cost = $2 per unit (assume no fixed cost). Market demand can be described by: P(rice) = 8 – Q(uantity) Where Q is total industry output (Q = q 1 + q 2 )

18. Duopoly Consider a market in which two identical firms can produce a good with marginal cost = $2 per unit (assume no fixed cost). For each firm, Profit (  ) = Total Revenue – Total Cost = Pq – 2q Each firm will choose a level of output q, to maximize its profit, taking into account what it expects the other firm to produce.

19. Duopoly Each firm’s profit maximizing level of output, q*, is a function of the other firm’s output. q13 q13 6 q 2 P = 8 - Q Demand Condition  1 = Total Revenue – Total Costq 1 - 2q 1 = 6q 1 - q 1 2 - q 2 q 1 FOC: 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2 

20. Duopoly Each firm’s profit maximizing level of output, q*, is a function of the other firm’s output. q13 q13 6 q 2 P = 8 - Q  1 = Pq 1 - 2q 1 +q 2 )]q 1 - 2q 1 = 6q 1 - q 1 2 - q 2 q 1 FOC: 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2  Q = (q 1 + q 2 )

21. Duopoly Each firm’s profit maximizing level of output, q*, is a function of the other firm’s output. q13 q13 6 q 2 P = 8 - (q 1 +q 2 )  1 = Pq 1 - 2q 1 = [8 - (q 1 +q 2 )]q 1 - 2q 1 = 6q 1 - q 1 2 - q 2 q 1 = 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2  substitute for P in profit function

22. Duopoly Each firm’s profit maximizing level of output, q*, is a function of the other firm’s output. q13 q13 6 q 2 P = 8 - (q 1 +q 2 )  1 = Pq 1 - 2q 1 = [8 - (q 1 +q 2 )]q 1 - 2q 1 = 6q 1 - q 1 2 - q 2 q 1 = 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2  d1d1 dq1dq1 FOC:

23. Duopoly A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *. q 1 q 1 * = 2 q 2 * = 2q 2 q 2 * = 3 - 1/2q 1 q 1 * = 3 - 1/2q 2

24. Duopoly A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *. q1q1* q1q1* q 2 * q 2 Is this the best they can do? If Firm 1 reduces its output while Firm 2 continues to produce q 2 *, the price rises and Firm 2’s profits increase.

25. Duopoly A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *. q1q1* q1q1* q 2 * q 2 Is this the best they can do? If Firm 2 reduces its output while Firm 1 continues to produce q 1 *, the price rises and Firm 1’s profits increase.

26. Duopoly A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *. q1q1* q1q1* q 2 * q 2 Is this the best they can do? There are a range of outcomes to the SW that make both firms better off.

27. Duopoly Consider a market in which two identical firms can produce a good with marginal cost = $2 per unit (assume no fixed cost). Market demand can be described by: P(rice) = 8 – Q(uantity) Where Q is total industry output (Q = q 1 + q 2 ) Assume that each firm can only choose a discrete quantity 0, 1, 2, 3, 4. What is the matrix form of this game? What is the best response for Firm 1, if it thinks Firm 2 will produce 3 units of output? If it thinks Firm 2 will produce 4 units of output? Find the Nash equilibrium in the market.

28. Duopoly P = 8 - Q Q = q 1 + q 2  i = (P - 2)q i q 1 q 2 P  1  2 0 0 8 0 0 1 0 7 5 0 2 0 6 8 0 3 0 5 9 0 4 0 4 8 0 1 1 6 4 4 2 1 5 6 3 3 1 4 6 2 4 1 3 4 1 2 2 4 4 4 3 2 3 3 2

29. Duopoly q 1 q 2 P  1  2 0 0 8 0 0 1 0 7 5 0 2 0 6 8 0 3 0 5 9 0 4 0 4 8 0 1 1 6 4 4 2 1 5 6 3 3 1 4 6 2 4 1 3 4 1 2 2 4 4 4 3 2 3 3 2 0,00,5 0,8 0,9 0,8 5,04,4 3,6 2,6 1,4 8,06,3 4,4 2,3 0,0 9,06,2 3,2 0,0 -3,-4 8,04,10,0 -4,-3 -8,-8 0123401234 01 2 3 4

30. Duopoly q 1 q 2 P  1  2 0 0 8 0 0 1 0 7 5 0 2 0 6 8 0 3 0 5 9 0 4 0 4 8 0 1 1 6 4 4 2 1 5 6 3 3 1 4 6 2 4 1 3 4 1 2 2 4 4 4 3 2 3 3 2 4321043210 01 2 3 4 q 1 q 1 (q 2 ) q 2 (q 1 ) q2q2 Best Response Functions NE

31. Duopoly q 1 q 2 P  1  2 0 0 8 0 0 1 0 7 5 0 2 0 6 8 0 3 0 5 9 0 4 0 4 8 0 1 1 6 4 4 2 1 5 6 3 3 1 4 6 2 4 1 3 4 1 2 2 4 4 4 3 2 3 3 2 0,00,5 0,8 0,9 0,8 5,04,4 3,6 2,6 1,4 8,06,3 4,4 2,3 0,0 9,06,2 3,2 0,0 -3,-4 8,04,10,0 -4,-3 -8,-8 0123401234 01 2 3 4

32. Duopoly In a Nash Equilibrium, no firm can increase its profits by changing its output unilaterally. It is strategically stable or self-enforcing agreement. A Nash Equilibrium does not necessarily maximize joint (or industry) profits. But improving the situation creates an enforcement problem. In general, we observe a tension between –Collusion: maximize joint profits –Competition: capture a larger share of the pie

33. Solving the 2x2 Game T 1 T 2 3. Prisoner’s Dilemma 4. Stag Hunt 5. Chicken 6. Battle of the Sexes S 1 S 2 x 1,x 2 w 1,w 2 z 1,z 2 y 1,y 2

34. T 1 T 2 S 1 S 2 3,3 0,5 5,0 1,1 3. Prisoner’s Dilemma NE = {(S 2,T 2 )} Solving the Game

35. T 1 T 2 S 1 S 2 5,5 0,3 3,0 1,1 4. Stag Hunt (also, Assurance Game) NE = {(S 1,T 1 ), (S 2,T 2 )} Solving the Game

36. T 1 T 2 S 1 S 2 3,3 1,5 5,1 0,0 5. Chicken (also Hawk/Dove) NE = {(S 1,T 2 ), (S 2,T 1 )} Solving the Game

37. O F O F 5,3 0,0 0,0 3,5 6. Battle of the Sexes NE = {(O,O), (F,F)} Solving the Game Find the mixed strategy Nash Equilibrium

38. Solving the Game O F O F 5,3 0,0 0,0 3,5 Let (p,1-p) = prob 1 (O, F ) (q,1-q) = prob 2 (O, F ) Then EP 1 (O) = 5q EP 1 (F) = 3-3q q* = 3/8 EP 2 (O) = 3p EP 2 (F) = 5-5p p* = 5/8 NE = {(1, 1); (0, 0); (5/8, 3/8)} ); (0, 0); (5/8, 3/8 Game 6. Equalizers

39. q OPERA 1 3/8 FIGHT 0 0 5/8 1 p Game 6. q*(p) if p<5/8, then Player 2’s best response is q* = 0 (FIGHT) if p>5/8q* = 1 (OPERA) Solving the Game

40. q OPERA 1 3/8 FIGHT 0 0 5/8 1 p Game 6. q*(p) p*(q) NE = {(1, 1); (0, 0); (5/8, 3/8)} Solving the Game (p, q); (p, q)

41. The Battle of the Sexes OPERA FIGHT 5, 3 0, 0 0, 0 3, 5 efficiency equity bargaining power or skill P1P1 P2P2 NE = {(1, 1); (0, 0); (5/8, 3/8)} Game 6. (0,0) (1,1) (5/8, 3/8) Solving the Game

42. Existence of Nash Equilibrium Prisoner’s DilemmaButton-Button Battle of the Sexes GAME 3.GAME 2.GAME 6. See Gibbons, pp. 50-53. 01 01 0 1 p q10q10 There can be (i) a single pure-strategy NE; (ii) a single mixed-strategy NE; or (iii) two pure-strategy NEs plus a single mixed-strategy NE (for x=z; y=w). (i)(ii)(iii)

43. Problems 1.Indeterminacy: Nash equilibria are not usually unique. 2. Inefficency: Even when they are unique, NE are not always efficient.

44. Next Time 3/7Bargaining Problems & (some) Solutions Gibbons, Ch. 2: 53-82. Hargreaves-Heap, Ch. 4: 11-145.