Game Theory

3/3/09 8:31 AM Reading assignment: chapter 9.6, 12 Quiz monopoly, game theory, ___ theory Game theory, analyzing the actions of other firms I. Dominant strategy equilibrium There is no dominant strategy Dependent on other players Other players can change their strategy So you would change our strategy II. A Nash Equilibrium: is a set of strategies, one for each player in the game, such that, no player, taking his/her rivals? N.E. strategies as given, has a unilateral incentive to deviate from his/her N.E. strategy. John Nash (1950) proved that all noncooperative games have a Nash equilibrium III. Prisoner?s Dilemma 2 people commit crime, police catch them, circumstantial evidence Quiet, Stone wall, or Confess Years in jail Unique Nash Equilibrium: (confess, confess) dominant strategy Incentive to deviate if one person stonewalls Equilibrium is not he same as efficient Stonewall, stonewall Paredo Superior Confess, Confess N.E., though a lousy result Trying to maximize their own payoff IV. Multiple Nash Equilibria ?Battle of the Sexes? story: man and women in love, want to send Friday night together on a date choices: ballet, boxing Pure strategy Nash Equilibria Nash equilibrium 1: (Ballet, Ballet) If deviate, ay off would go from 2/3 to 1/1 Nash Equilibrium 2: (Boxing, Boxing) elements of conflict and coordination V. ?Matching Pennies Game? Game of pure conflict zero sum game No pure strategy Nash Equilibrium* * Mixed strategy N.E. randomization process if other person is random, your strategy is to be random also VI. Oligopoly Cournot (1838): Systematically analyzed Cournot Model Small number of firms with identical costs producing a homogeneous/ identical product All firms in industry simultaneously choose output to maximize their profits Q (aggregate output)= q1 + q2 + q3 +? + qn p= f(Q) : inverse demand function A duopoly example of Cournot?s model n = 2 firms Q= q1 + q2 P= f(Q) = f (q1 + q2) Suppose inverse demand curve is linear: P= a - bQ = a ? bq1 ? bq2 Each firm has a constant marginal cost of production equal to $C Assume a > c Max q1 FIRM #1 Price = (a ? bq1 ? bq2) x q1 ? c x q1 Max q2 FIRM #2 Price = (a ? bq1 - - bq2) x q2 ? c x q2 Strategic dependence Need to anticipate what the other firm will likely do. Their profits are a function of each other Need to solve both problems Results: Nash Equilibrium is q1* = q2* = 1/3 (a-c) b Q courot= 2/3 (a-c) b P cournot = a-b Q cournot = 1/3 a + 2/3 c > c because a> c results in a market clearing price which is greater than the marginal cost How do Cournot?s results compare to perfect competition ad Monopoly Perfect competition a-bQ = c a-c = bQ (a-c) = Q* b Perfect competition is bigger When you just have a few firms, less aggregate output from industry. Cournot price is greater than perfect competition Monopoly Marginal revenue & marginal cost P= a ? bQ MR= a ? 2bQ MC= c MR = MC: a ? 2bQ =c a-c = 2bQ - Q mon = ½ (a-c) b Monopolist is getting the higher price than Q monopoly < Q cournot < Q* perfect competition P monopoly > P cournot > P= MC VII. Bertrand (1883) Says Cournot?s model is not realistic He has wrong strategic variable- quantity Firms choose their prices and sell as much as they can with those prices Firms simultaneously choose their prices If one firm has a lower price they will get the entire market demand If 2 firms are choosing the same price, they each get ½ of the market demand If firms are choosing prices simultaneously, will we get a duplication of Cournot?s results?? Is that a Nash Equilibrium to the game? NO! would want to deviate from strategy If firm 1 chose P cournot I would choose P=MC, choose a slightly lower price, b/c would get entire market demand Game Theory 3/3/09 8:31 AM 3/3/09 8:31 AM P1, P2 Stone wall confess Stone wall -1, -1 -6, 0 Confess 0, -6 -5, -5 P1, P2 Ballet Boxing Ballet 3, 2 1, 1 Boxing 1, 1 2, 3 P1, P2 Heads Tails Heads 1, -1 -1, 1 Tails -1, 1 1, -1