UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Bargaining and Negotiation Review Midterm3/19 2/27

Nonzero-sum Games Examples: Bargaining Duopoly International Trade

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.

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 )

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.

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 q 2 q 1 FOC: 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2 

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 q 2 q 1 FOC: 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2  Q = (q 1 + q 2 )

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 q 2 q 1 = 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2  substitute for P in profit function

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 q 2 q 1 = 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2  d1d1 dq1dq1 FOC:

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

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.

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.

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.

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.

Duopoly P = 8 - Q Q = q 1 + q 2  i = (P - 2)q i q 1 q 2 P  1 

Duopoly q 1 q 2 P  1  ,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,

Duopoly q 1 q 2 P  1  q 1 q 1 (q 2 ) q 2 (q 1 ) q2q2 Best Response Functions NE

Duopoly q 1 q 2 P  1  ,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,

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

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.

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