UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Bargaining and Negotiation Review Midterm3/23 2/23

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.

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

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

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

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

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

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|q) = 5q EP 1 (F|q) = 3-3q q* = 3/8 EP 2 (O|p) = 3p EP 2 (F|p) = 5-5p p* = 5/8 NE = {(1, 1); (0, 0); (5/8, 3/8)} ); (0, 0); (5/8, 3/8 Game 6. Equalizers

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

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)

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

Existence of Nash Equilibrium Prisoner’s DilemmaButton-Button Battle of the Sexes GAME 3.GAME 2.GAME 6. See Gibbons, pp 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)

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

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

Bargaining Bargaining Games We Play a Game Credibility Subgame Perfection Alternating Offers and Shrinking Pies

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 involves a combination of common as well as conflicting interests. The central issue in all bargaining games is credibility: the strategic use of threats, bluffs, and promises.

Bargaining Games P P 1 Disagreement point 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. Divide a Dollar P 1 = x; P 2 = 1-x.

We Play a Game PROPOSER RESPONDERPlayer # ____ Offer $ _____ Accept Reject

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 20 Mean = $ Offers > 0 Rejected 1 Offer < 1.00 (20%) Accepted (3/6/00)

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 33 Mean = $ Offers > 0 Rejected 1 Offer < $1 (20%) Accepted (2/28/01)

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 37 Mean = $ Offers > 0 Rejected* 3 Offers < $1 (20%) Accepted (2/27/02) * 1 subject offered 0

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 12 Mean = $ Offers > 0 Rejected 0 Offers < 1.00 (20%) Accepted (7/10/03)

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 17 Mean = $ Offers > 0 Rejected 0 Offers < 1.00 (20%) Accepted (3/10/04)

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

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 16 Mean = $ Offers > 0 Rejected 1 Offer < 1.00 (20%) Accepted (2/25/08)

The Ultimatum Game OFFERS REJECTED ACCEPTED N = 131 Mean = $ Offers > 0 Rejected 6/26 Offers < 1.00 (20%) Accepted Pooled data

The Ultimatum Game P 1 P /9 4/4 25/27 2/2 3/3 20/28 13/15 N = 131 Mean = $ Offers > 0 Rejected 6/26 Offers < 1.00 (20%) Accepted Pooled data 6/7 3/17

The Ultimatum Game P 1 P What is the lowest acceptable offer? 9/9 4/4 25/27 2/2 3/3 20/28 13/15 N = 131 Mean = $ Offers > 0 Rejected 6/26 Offers < 1.00 (20%) Accepted Pooled data 6/7 3/17

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)

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) How would you advise Proposer? What do you think would happen if the game were repeated?

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

Next Time 3/2 Bargaining Problems & (some) Solutions Gibbons, Ch 2: Gintis, Ch 5: