UNIT IV: INFORMATION & WELFARE

1. UNIT IV: INFORMATION & WELFARE • Decision under Uncertainty • Externalities & Public Goods • Bargaining Games • Review 11/30

2. Bargaining Whoever offers to another a bargain of any kind, proposes to do this. Give me that which I want, and you shall have this which you want …; and it is this manner that we obtain from one another the far greater part of those good offices we stand in need of. It is not from the benevolence of the butcher, the brewer, or the baker that we expect our dinner, but from their regard to their own interest. -- A. Smith, 1776

3. Bargaining Games • Subgame Perfection • We Play a Game • The Ultimatum Game • Playing Fair • Alternating Offers & Shrinking Pies

4. Bargaining Games Bargaining involves (at least) 2 players who face the the opportunity of a profitable joint venture, provided they can agree in advance on a division between them. Bargaining can be characterize as a game with: • common as well as conflicting interests • incomplete or asymmetric information Bargaining involves the use of threats, bluffs, and promises: the strategic use of information. • (credibility?)

5. Bargaining Games Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or surplus: b = Buyer’s Reservation Price s = Seller’s Reservation Price S = Surplus S = b - s

6. Bargaining Games Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or surplus: S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b 0 50 100 150 200 250 s If s and b are known to both players: How should the surplus be divided? Complete Information

7. Bargaining Games Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, orSurplus: S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b 0 50 100 150 200 250 s i) b > s If b and s are known to both players: How should the surplus be divided? Surplus = 50

8. Mixed Motives: The Bargain Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b = s, we say the price is fully determined, and there is no room for negotiation. S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b 0 50 100 150 200 250 s ii) b = s

9. Mixed Motives: The Bargain Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b < s, there is nothing to gained from the exchange. S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b 0 50 100 150 200 250 s (iii) b < s No “zone of agreement” What happens if information is incomplete?

10. Bargaining Games Buyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or surplus: S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b 0 50 100 150 200 250 s If s is known only to the seller; Buyer has a probability distribution over the “true” value of b. Asymmetric Information

11. Bargaining Games Divide a Dollar P2 1 0 1 P1 Two players have the opportunity to share $1, if they can agree on a division beforehand. Each writes down a number. If they add to $1, each gets her number; if not; they each get 0. Every division s.t. x+(1-x)=1 is a NE. P1= x; P2 = 1-x. Disagreement point

12. Subgame Perfection • Subgame: a part (or subset) of an extensive game, starting at a singleton node (not the initial node) and continuing to payoffs. • Subgame Perfect Nash Equilibrium (SPNE): a NE achieved by strategies that also constitute NE in each subgame. • eliminates NE in which the players threats are not credible. • selects the outcome that would be arrived at via backwards induction.

13. Subgame Perfection Chain Store Game A firm (Player 1) is considering whether to enter the market of a monopolist (Player 2). Player 2 can then choose to fight the entrant, or not. 1 Enter Don’t Enter Fight Don’t Fight 2 (2,2) (0,0) (3,1) Subgame

14. Subgame Perfection Chain Store Game Fight Don’t Enter Don’t 0, 0 3, 1 2, 2 2, 2 1 Enter Don’t Fight Don’t 2 (2,2) (0,0) (3,1) NE = {(E,D), (D,F)}, but Fight for Player 2 is an incredible threat.

15. Subgame Perfection Chain Store Game Fight Don’t Enter Don’t 0, 0 3, 1 2, 2 2, 2 1 Enter Don’t Fight Don’t 2 (2,2) (0,0) (3,1) Subgame Perfect Nash Equilibrium NE = {(E,D), (D,F)}. SPNE = {(E,D)}.

16. We Play a Game PROPOSER RESPONDER Player # ____ Player # ____ Offer $ _____ Accept Reject

17. The Ultimatum Game OFFERS 5 4 3 2 1 0 REJECTEDACCEPTED N = 20 Mean = $1.30 9 Offers > 0 Rejected 1 Offer < 1.00 (20%) Accepted (3/6/00)

18. The Ultimatum Game OFFERS 5 4 3 2 1 0 REJECTED ACCEPTED N = 32 Mean = $1.75 10 Offers > 0 Rejected 1 Offer < $1 (20%) Accepted (2/28/01)

19. The Ultimatum Game OFFERS 5 4 3 2 1 0 REJECTED ACCEPTED N = 38 Mean = $1.69 10 Offers > 0 Rejected* 3 Offers < $1 (20%) Accepted (2/27/02) * 1 subject offered 0

20. The Ultimatum Game OFFERS 5 4 3 2 1 0 REJECTED ACCEPTED N = 6 Mean = $1.93 0 Offers > 0 Rejected 0 Offers < $1 (20%) (8/4/04)

21. The Ultimatum Game OFFERS 5 4 3 2 1 0 REJECTED ACCEPTED N = 12 Mean = $1.90 0 Offers > 0 Rejected 1 Offer < 1.00 (20%) Accepted (3/9/05)

22. The Ultimatum Game OFFERS 5 4 3 2 1 0 REJECTED ACCEPTED N = 131 Mean = $2.25 34 Offers > 0 Rejected 6/26 Offers < 1.00 (20%) Accepted Pooled data

23. The Ultimatum Game P2 5 2.25 0 What is the lowest acceptable offer? 9/9 4/4 2/2 2/2 3/3 2.50 1.00 25/27 N = 131 Mean = $2.25 34 Offers > 0 Rejected 6/26 Offers < 1.00 (20%) Accepted Pooled data 13/15 6/7 20/28 3/17 0 2.755 P1

24. The Ultimatum Game Theory predicts very low offers will be made and accepted. Experiments show: • Mean offers are 30-40% of the total • Mode = 50% • Offers <20% are rare and usually rejected Guth Schmittberger, and Schwarze (1982) Kahnemann, Knetsch, and Thaler (1986) Also, Camerer and Thaler (1995)

25. The Ultimatum Game Theory predicts very low offers will be made and accepted. Experiments show: • Mean offers are 30-40% of the total • Mode = 50% • Offers <20% are rare and usually rejected How would you advise Proposer? What do you think would happen if the game were repeated?

26. The Ultimatum Game How can we explain the divergence between predicted and observed results? • Stakes are too low • Fairness • Relative shares matter • Endowments matter • Culture, norms, or “manners” • People make mistakes • Time/Impatience

27. Subgame Perfection Mini-Ultimatum Game 5,5 0,0 8,2 0,0 A(ccept) 2 H(igh) 1 L(ow) R(eject) Proposer (Player 1) can make High Offer (50-50%) or Low Offer (80-20%).

28. Subgame Perfection Mini-Ultimatum Game 5,5 0,0 8,2 0,0 A(ccept) 2 H(igh) 1 L(ow) R(eject) Subgame Perfect Nash Equilibrium AA RR AR RA H 5,50,0 5,5 0,0 L8,20,0 0,0 8,2 SPNE = {(L,AA)} (H,AR) and (L,RA) involve incredible threats.

29. Subgame Perfection 5,5 0,0 8,2 1,9 2 H 1 L 2 AA RR AR RA H 5,5 0,0 5,5 0,0 L 8,2 1,9 1,9 8,2

30. Subgame Perfection 5,5 0,0 2 H 1 L AA RR AR RA H 5,50,0 5,50,0 L 8,2 1,9 1,98,2 1,9 SPNE = {(H,AR)}

31. Alternating Offer Bargaining Game Two players are to divide a sum of money (S) is a finite number (N) of alternating offers. Player 1 (‘Buyer’) goes first; Player 2 (‘Seller’) can either accept or counter offer, and so on. The game continues until an offer is accepted or N is reached. If no offer is accepted, the players each get zero. A.Rubinstein, 1982

32. Alternating Offer Bargaining Game Two players are to divide a sum of money (S) is a finite number (N) of alternating offers. Player 1 (‘Buyer’) goes first; Player 2 (‘Seller’) can either accept or counter offer, and so on. The game continues until an offer is accepted or N is reached. If no offer is accepted, the players each get zero. 1 (a,S-a) 2 (b,S-b) 1 (c,S-c) (0,0)

33. Alternating Offer Bargaining Game 1 (a,S-a) 2 (b,S-b) 1 (c,S-c) (0,0) S = $5.00 N = 3

34. Alternating Offer Bargaining Game 1 (a,S-a) 2 (b,S-b) 1 (4.99, 0.01) (0,0) S = $5.00 N = 3

35. Alternating Offer Bargaining Game 1 (4.99,0.01) 2 S = $5.00 N = 3 SPNE = (4.99,0.01) The game reduces to an Ultimatum Game

36. Shrinking Pie Game Now consider what happens if the sum to be divided decreases with each round of the game (e.g., transaction costs, risk aversion, impatience). Let S = Sum of money to be divided N = Number of rounds d = Discount parameter

37. Shrinking Pie Game 1 (a,S-a) 2 (b,S-b) 1 (c,S-c) (0,0) S = $5.00 N = 3 d = 0.5

38. Shrinking Pie Game 1 (3.74,1.26)2 (1.25, 1.25) 1 (1.24,0.01) (0,0) S = $5.00 N = 3 d = 0.5

39. Shrinking Pie Game 1 (3.13,1.87)2 (0.64,1.86) 1 (0.63,0.62) 2 (0.01, 0.61) (0,0) 1 S = $5.00 N = 4 d = 0.5

40. Shrinking Pie Game for d = ½ N = 1 (4.99, 0.01) 2 (2.50, 2.50) 3 (3.74, 1.26) 4 (3.13, 1.87) 5 (3.43, 1.57) … … This series converges to (S/(1+d), S – S/(1+d)) = (3.33, 1.67) This pair {S/(1+ d),S-S/(1+ d)} are the payoffs of the unique SPNE. P2 5 1.67 0 2 4 5 3 1 0 3.33 5 P1

41. Shrinking Pie Game Optimal Offer (O*) expressed as a share of the total sum to be divided = [S-S/(1+d)]/S O* = d/(1+d) SPNE = {1- d /(1+ d), d /(1+ d)} Thus both d=1 and d=0 are special cases of Rubinstein’s model: When d=1 (no bargaining costs), O* = 1/2 When d=0, game collapses to the ultimatum version and O* = 0 (+e)

42. Shrinking Pie Game

43. Bargaining Games Bargaining games are fundamental to understanding the price determination mechanism in “small” markets. The central issue in all bargaining games is credibility: the strategic use of threats, bluffs, and promises. When players have incomplete, asymmetric, or private information (e.g., only the seller of a used car knows its true quality and hence its true value) profitable exchanges may be “left on the table.” In such cases, there is an incentive to make oneself credible (e.g., appraisals; audits; “reputable” agents; brand names; lemons laws; “corporate governance”).