Basics of Game Theory

Transcription

1 Basics of Game Theory Giacomo Bacci and Luca Sanguinetti Department of Information Engineering University of Pisa, Pisa, Italy April - May, 2010 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

2 Outline Economic applications Cournot model of duopoly Bertrand model of duopoly Stackelberg model of duopoly Collusion between Cournot duopolists Cournot competition under asymmetric information G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

3 Economic applications Common problems Consider a group of firms in an industry competing for the market. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

4 Economic applications Common problems Consider a group of firms in an industry competing for the market. Many though questions to answer... How does the outcome depends on the firms output, firms nature, cost functions and the number of firms? Will the benefits of competition be passed on to customers? Will a reduction in the number of firms generate a less desirable outcome? Many others come to mind... The first economist attempting to answer some of these questions was Cournot (1838). G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

5 Cournot model of oligopoly Introduction Cournot anticipated the notion of Nash s equilibrium in For simplicity, we focus on a particular model of duopoly in which two firms produce the same product in two different quantities, i.e., q 1 and q 2. The goal of each firm is to maximize its profit. For firm 1, we have that u 1(q 1, q 2) = q 1P(Q) C 1(q 1) where P(Q) is the inverse demand function and C 1(q 1) is its cost while Q = q 1 + q 2 is the firms total output. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

6 Cournot model of duopoly Problem formulation The question we pose ourselves is simply: What are the optimal quantities chosen simultaneously maximizing profits? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

7 Cournot model of duopoly Problem formulation The question we pose ourselves is simply: What are the optimal quantities chosen simultaneously maximizing profits? Assume each firm s cost per unit is the same (no variable costs) C 1(q 1) = c q 1 with 0 c < a. If some fixed costs are present, then j C 1(q 1) = 0 if q 1 = 0 f + c q 1 if q 1 > 0 with 0 c < a and f > 0. For simplicity, in the sequel we concentrate on the first case. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

8 Cournot model of duopoly Problem formulation Suppose the inverse demand function is given by P(Q) = ( a Q if Q < a 0 if Q a with a being the price at which customers are willing to pay the product. P(Q) a Q Under the above circumstances, a response can be easily found. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

9 Cournot model of duopoly Game formulation Translate the problem into a strategic form. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

10 Cournot model of duopoly Game formulation Translate the problem into a strategic form. Players: firms 1 and 2 Strategies: quantities q i [0, + ), i = {1,2} Utilities: each firm s profit is given by j q1 (a c Q) u 1 (q 1, q 2) = cq 1 if Q a if Q > a j q2 (a c Q) u 2 (q 1, q 2) = cq 2 if Q a if Q > a G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

11 Cournot model of duopoly Firm s profit Firm s profit as a function of q 1 for a given q 2 with a = 100 and c = q = 0 2 q 2 = 20 q 2 = 60 q = G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

12 Cournot model of duopoly Nash equilibrium Nash equilibria are such that u i `q i, q\i ui `qi, q \i q i 0. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

13 Cournot model of duopoly Nash equilibrium Nash equilibria are such that u i `q i, q\i ui `qi, q \i q i 0. The best response procedure is used to find Nash equilibria, i.e., qi = arg max u i `qi, q\i. 0 q i < G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

14 Cournot model of duopoly Nash equilibrium Consider firm 1 and assume q 2 is given. Then, with q1 = arg max u 1 (q 1, q2) 0 q 1 < u 1 (q 1, q 2) = q 1 (a c q 1 q 2). G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

15 Cournot model of duopoly Nash equilibrium Consider firm 1 and assume q 2 is given. Then, with q1 = arg max u 1 (q 1, q2) 0 q 1 < u 1 (q 1, q 2) = q 1 (a c q 1 q 2). How to solve the problem? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

16 Cournot model of duopoly Nash equilibrium When q 2 < a c, take the derivate with respect to q 1 and set it to zero. This yields a c q 2 2q 1 = 0. The best quantity q 1 is then given by q 1 = a c q 2. 2 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

17 Cournot model of duopoly Nash equilibrium When q 2 < a c, take the derivate with respect to q 1 and set it to zero. This yields a c q 2 2q 1 = 0. The best quantity q 1 is then given by q 1 = a c q 2. 2 On the other hand, when q 2 a c the maximum is achieved for q 1 = 0 since in these circumstances the profit becomes a negative-decreasing function. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

18 Cournot model of duopoly Nash equilibrium Consider firm 2. Then, we have that with q2 = arg max u 1 (q1, q 2) 0 q 2 < u 2 (q 1, q 2) = q 2 (a c q 1 q 2) from which using the same arguments it follows that 8 < q 2 = : 1 2 (a c q 1) if q 1 < a c 0 if q 1 a c G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

19 Cournot model of duopoly Nash equilibrium The Nash equilibrium is found looking for the solution of the following system 8 >< >: q 1 = a c q 2 2 q 2 = a c q 1 2 if q 2 < a c if q 1 < a c G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

20 Cournot model of duopoly Nash equilibrium The Nash equilibrium is found looking for the solution of the following system 8 >< >: q 1 = a c q 2 2 q 2 = a c q 1 2 if q 2 < a c if q 1 < a c Solving the above system yields from which it follows that q 1 = q 2 q c = a c 3 u 1 (q 1, q 2) = u 2 (q 1, q 2) = (a c)2. 9 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

21 Cournot model of duopoly Nash equilibrium Nash equilibria can also be found graphically. Assume firm 1 s strategy satisfies q 1 < a c, firm 2 s best response is When q 1 a c, it turns out that R 2(q 1) = a c q1. 2 R 2(q 1) = 0. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

22 Cournot model of duopoly Nash equilibrium R 2(q 1) a c 2 0 a c q 1 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

23 Cournot model of duopoly Nash equilibrium If q 2 < a c, then firm 1 s best response is R 1(q 2) = a c q2. 2 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

24 Cournot model of duopoly Nash equilibrium If q 2 < a c, then firm 1 s best response is R 1(q 2) = a c q2. 2 When q 2 a c, it follows that R 1(q 2) = 0. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

25 Cournot model of duopoly Nash equilibrium q 2 a c 0 a c 2 R 1(q 2) G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

26 Cournot model of duopoly Nash equilibrium The Nash equilibria are given by the intersection points. q 2 a c a c 2 a c 3 0 a c a c a c 3 2 q 1 A unique Nash equilibrium exists. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

27 Cournot model of duopoly Asymmetric costs What does it happen if c i(q i) = c i? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

28 Cournot model of duopoly Asymmetric costs What does it happen if c i(q i) = c i? Following the same arguments yields 8 < R 1(q 2) = : 1 (a c1 q2) if q2 a c1 2 0 otherwise and 8 < R 2(q 1) = : 1 (a c2 q1) if q1 a c2 2 0 otherwise G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

29 Cournot model of duopoly Asymmetric costs If c 1 < 1 (a + c2) then 2 q 2 a c1 a c2 2 a c1 0 a c2 2 q 1 A unique Nash equilibrium exists given by q1 = 1 3 (a 2c1 + c2) and q 2 = 1 (a 2c2 + c1) 3 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

30 Cournot model of duopoly Asymmetric costs If c 1 1 (a + c2) then 2 q 2 a c2 2 a c1 a c1 0 a c2 2 q 1 A unique Nash equilibrium exists given by q1 = 0 and q2 = 1 (a c2) 2 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

31 Cournot model of monopoly Problem formulation and solution What if is there only one firm in the market (monopoly)? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

32 Cournot model of monopoly Problem formulation and solution What if is there only one firm in the market (monopoly)? The market price is now given by P(Q) = a q G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

33 Cournot model of monopoly Problem formulation and solution What if is there only one firm in the market (monopoly)? The market price is now given by P(Q) = a q A single-user optimization problem arises. The firm is better off producing yielding the following monopoly profit q = a c 2 q m u(q m) = (a c)2. 4 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

34 Cournot model Duopoly vs. monopoly Assume q 1 = q 2 = q m/2 = (a c)/4. Each firm s profit is then given by u 1 (q 1, q 2) = u 2 (q 1, q 2) = 1 8 (a c)2. The above profit exceeds the Nash equilibrium profit, i.e., 1 8 (a c)2 > 1 9 (a c)2. Both firms would be better off cooperating. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

35 Cournot model Duopoly vs. monopoly Assume q 1 = q 2 = q m/2 = (a c)/4. Each firm s profit is then given by u 1 (q 1, q 2) = u 2 (q 1, q 2) = 1 8 (a c)2. The above profit exceeds the Nash equilibrium profit, i.e., 1 8 (a c)2 > 1 9 (a c)2. Both firms would be better off cooperating. Why do they not cooperate at the equilibrium? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

36 Cournot model of duopoly Duopoly vs. monopoly The problem is that q i = q m/2 is not the best response to q \i = q m/2! G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

37 Cournot model of duopoly Duopoly vs. monopoly The problem is that q i = q m/2 is not the best response to q \i = q m/2! Indeed, when q 2 = q m/2 we have that» u 1 (q 1, q2) 3 = q 1 (a c) q1 4 from which it follows that q1 = 3 8 (a c) > 1 (a c). 4 But if q 1 = 3(a c)/8 then q2 = 5 16 (a c) > 1 (a c). 4 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

38 Cournot model of duopoly Duopoly vs. monopoly q 2 a c 0 a c q 1 The procedure converges to the Cournot Nash equilibrium. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

39 Bertrand model of duopoly Problem formulation Assume now that firms choose prices rather than quantities (Bertrand s ). Consider two firms selling differentiated products in the same market and using two different prices, i.e., p 1 and p 2. The goal of each firm is to maximize its profit. For firm 1, we have that u 1(q 1, q 2) = q 1p 1 C 1(q 1) where q 1 is now a function of both p 1 and p 2. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

40 Bertrand model of duopoly Problem formulation Assume q 1 = a p 1 + b p 2 where b > 0 reflects the effect of firm 2 s product. As in the Cournot model, assume no fixed costs of production C 1(q 1) = c q 1 with 0 c < a. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

41 Bertrand model of duopoly Game formulation Its strategic formulation takes the form. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

42 Bertrand model of duopoly Game formulation Its strategic formulation takes the form. Players: firms 1 and 2 Strategies: prices p i [0,+ ), i = {1, 2} Utilities: each firm s profit. For firm 1, we have that u 1 (p 1, p 2) = q 1(p 1, p 2) (p 1 c) where q 1 = a p 1 + b p 2. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

43 Bertrand model of duopoly Nash equilibrium The price p 1 is the best response to p 2 if p 1 = arg max u 1(p 1, p 2) = arg max (a p 1 + b p 2)(p 1 c) 0 p 1 < 0 p 1 < G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

44 Bertrand model of duopoly Nash equilibrium The price p 1 is the best response to p 2 if p 1 = arg max u 1(p 1, p 2) = arg max (a p 1 + b p 2)(p 1 c) 0 p 1 < 0 p 1 < The solution is found as follows du 1(p 1, p 2) dp 1 p1 =p 1 = a 2p 1 + b p 2 + c = 0. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

45 Bertrand model of duopoly Nash equilibrium The price p 1 is the best response to p 2 if p 1 = arg max u 1(p 1, p 2) = arg max (a p 1 + b p 2)(p 1 c) 0 p 1 < 0 p 1 < The solution is found as follows du 1(p 1, p 2) dp 1 p1 =p 1 = a 2p 1 + b p 2 + c = 0. Paralleling the steps for p 2, it is found that Nash equilibria have to satisfy 8 >< p 1 = 1 (a + c + b 2 p 2) >: p 2 = 1 (a + c + b 2 p 1) G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

46 Bertrand model of duopoly Nash equilibrium The solution is found to be p i = a + c 2 b, i = 1,2 from which it follows that b 2 (otherwise the prices would be negative). What happens if b = 0? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

47 Bertrand model of duopoly Nash equilibrium The solution is found to be p i = a + c 2 b, i = 1,2 from which it follows that b 2 (otherwise the prices would be negative). What happens if b = 0? The Bertrand equilibrium yields the monopoly quantity since no player interaction exists. p i = a + c 2, i = 1,2 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

48 Stackelberg model of duopoly Problem formulation Stackelberg (1934) proposed a dynamic model of duopoly a leader moves first, and a follower moves second. Assume for simplicity the firms choose quantities (Cournot model). The fundamental difference is that here the firms choices are sequential rather than simultaneous. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

49 Stackelberg model of duopoly Problem formulation The timing of the Stackelberg model is as follows: firm 1 chooses a quantity q 1 0; firm 2 observes q 1 and then chooses a quantity q 2 0; the payoff of firm i {1, 2} is given by the profit function u i `qi, q \i = qi (P(Q) c) where P (Q) = a q i q \i and c < a is the constant marginal cost (fixed costs are zero). G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

50 Stackelberg model of duopoly Game formulation Translate the above problem into an extensive game. Players: the leader (firm 1) and the follower (firm 2); Terminal histories: (q 1, q 2), with q 1, q 2 0; Player function: P( ) = leader, P(q 1) = follower; Payoffs: u i `qi, q \i = qi (p c) = q i `a c q i q \i G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

51 Stackelberg model of duopoly Subgame perfect equilibrium Apply the backward-induction procedure. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

52 Stackelberg model of duopoly Subgame perfect equilibrium Apply the backward-induction procedure. Compute first firm 2 s reaction to an arbitrary quantity q 1 by firm 1: R 2(q 1) = arg max 0 q 2 < u 2(q 1, q 2) = arg max 0 q 2 < q 2 (a c q 1 q 2). G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

53 Stackelberg model of duopoly Subgame perfect equilibrium Apply the backward-induction procedure. Compute first firm 2 s reaction to an arbitrary quantity q 1 by firm 1: R 2(q 1) = arg max 0 q 2 < u 2(q 1, q 2) = arg max 0 q 2 < q 2 (a c q 1 q 2). This yields the same result obtained for the strategic Cournot game, i.e., provided that q 1 < a c. R 2(q 1) = a c q1 2 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

54 Stackelberg model of duopoly Subgame perfect equilibrium Apply the backward-induction procedure. Compute first firm 2 s reaction to an arbitrary quantity q 1 by firm 1: R 2(q 1) = arg max 0 q 2 < u 2(q 1, q 2) = arg max 0 q 2 < q 2 (a c q 1 q 2). This yields the same result obtained for the strategic Cournot game, i.e., provided that q 1 < a c. R 2(q 1) = a c q1 2 The difference is that R 2(q 1) is now firm 2 s reaction to firm 1 s observed quantity. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

55 Stackelberg model of duopoly Subgame perfect equilibrium Firm 1 knows that q 1 will be met with reaction R 2(q 1). Then, firm 1 s problem in the first stage of the game amounts to solving q 1 = arg max 0 q 1 < = arg max 0 q 1 < u 1 (q 1, R 2(q 1)) = arg max 0 q 1 < q 1 a c q1. 2 q 1 [a c q 1 R 2(q 1)] G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

56 Stackelberg model of duopoly Subgame perfect equilibrium Firm 1 knows that q 1 will be met with reaction R 2(q 1). Then, firm 1 s problem in the first stage of the game amounts to solving q 1 = arg max 0 q 1 < = arg max 0 q 1 < u 1 (q 1, R 2(q 1)) = arg max 0 q 1 < q 1 a c q1. 2 q 1 [a c q 1 R 2(q 1)] Solving the above problem yields q 1 = a c 2. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

57 Stackelberg model of duopoly Subgame perfect equilibrium Then, in the Stackelberg game we have that q 1 = a c 2 and q 2 = a c 4. Firm 1 s profit results given by u 1(q1, q2) = 1 (a c)2 8 while firm 2 s profit is u 1(q 1, q 2) = 1 16 (a c)2. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

58 Stackelberg model of duopoly Stackelberg vs. Cournot By contrast in the strategic Cournot game each firm produces qc = 1 (a c) 3 and each profit is given by u 1 (q c, q c) = u 2 (q c, q c) = (a c)2. 9 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

59 Stackelberg model of duopoly Stackelberg vs. Cournot Then, firm 1 is better off in the subgame perfect equilibrium, i.e., u 1 (q 1, q 2) = (a c)2 8 > u 1 (q c, q c) = (a c)2. 9 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

60 Stackelberg model of duopoly Stackelberg vs. Cournot Then, firm 1 is better off in the subgame perfect equilibrium, i.e., u 1 (q 1, q 2) = (a c)2 8 > u 1 (q c, q c) = (a c)2. 9 On the other hand, firm 2 is always worse off u 2 (q 1, q 2) = (a c)2 16 < u 2 (q c, q c) = (a c)2. 9 Differently from single-player problems, in multi-player problems more information can make a player worse off. What is the impact of this information? Just try to remove it. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

61 Stackelberg model of duopoly Stackelberg vs. Cournot Consider firm 2 chooses q 2 without observing q 1. What happens now? Firm 2 has to form a belief. If firm 2 supposes that firm 1 will choose its Stackelberg quantity, then q 2 = (a c)/4. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

62 Stackelberg model of duopoly Stackelberg vs. Cournot Consider firm 2 chooses q 2 without observing q 1. What happens now? Firm 2 has to form a belief. If firm 2 supposes that firm 1 will choose its Stackelberg quantity, then q 2 = (a c)/4. Assume firm 1 knows that firm 2 has this belief, then it prefers to choose its best response to (a c)/4. This yields q1 = 3 (a c) 8 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

63 Stackelberg model of duopoly Stackelberg vs. Cournot Consider firm 2 chooses q 2 without observing q 1. What happens now? Firm 2 has to form a belief. If firm 2 supposes that firm 1 will choose its Stackelberg quantity, then q 2 = (a c)/4. Assume firm 1 knows that firm 2 has this belief, then it prefers to choose its best response to (a c)/4. This yields q1 = 3 (a c) 8 Firm 2 knows this and then prefers to choose its best response to 3(a c)/8, i.e., q2 = 5 (a c) 16 Iterating this procedure, we end up with the Cournot equilibrium! G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

64 Collusion between Cournot duopolists Problem formulation Consider again a Cournot model for duopoly without a leader and a follower. If q i = q c = (a c)/3 u i `qi, q \i uc = 1 (a c)2 9 while if q i = q m/2 = (a c)/4 u i `qi, q \i u m 2 = 1 8 (a c)2. Although 1 8 (a c)2 > 1 (a c)2 9 the absence of cooperation leads both firms to choose q i = q c. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

65 Collusion between Cournot duopolists Problem formulation How to force cooperation in the Cournot model? Playing the game repeatedly! Friedman (1971) showed that cooperation could be achieved in an infinitely repeated game by using trigger strategies. The original application was for a Cournot duopoly. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

66 Collusion between Cournot duopolists Trigger strategy Consider the infinitely repeated game based on the Cournot stage-game. Assume both firms have the same discount factor γ. Compute the values of γ for which the following trigger strategy is a subgame-perfect equilibrium. Start producing q m/2. At time t, produce q m/2 if both firms have produced q m/2 in the t 1 previous periods; otherwise produce the Cournot quantity. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

67 Collusion between Cournot duopolists Trigger strategy If firm \i is going to produce q m/2 at time t, then the quantity that maximizes firm i s profit at time t is qd = 3 (a c) 8 where the subscript d stands for deviation. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

68 Collusion between Cournot duopolists Trigger strategy If firm \i is going to produce q m/2 at time t, then the quantity that maximizes firm i s profit at time t is qd = 3 (a c) 8 where the subscript d stands for deviation. If q i = q d and q \i = q m/2 u i `qi, q \i ud = 9 64 (a c)2. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

69 Collusion between Cournot duopolists Trigger strategy The trigger strategy is a subgame perfect equilibrium provided that u m 2 (1 γ) u d + γ u c. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

70 Collusion between Cournot duopolists Trigger strategy The trigger strategy is a subgame perfect equilibrium provided that u m 2 (1 γ) u d + γ u c. The above inequality holds true for γ 9/17. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

71 Collusion between Cournot duopolists Modified trigger strategy How can cooperation be enforced for γ < 9/17? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

72 Collusion between Cournot duopolists Modified trigger strategy How can cooperation be enforced for γ < 9/17? Compute for any γ the most profitable quantity q γ the firms can produce if they both play trigger strategies. To compute this quantity, let us consider the following trigger strategy: Start producing q γ. At time t, produce q γ if both firms have produced q γ in the t 1 previous periods; otherwise produce the Cournot quantity. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

73 Collusion between Cournot duopolists Modified trigger strategy If both firms play q γ u i `qi, q \i uγ = q γ (a c 2q γ). G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

74 Collusion between Cournot duopolists Modified trigger strategy If both firms play q γ u i `qi, q \i uγ = q γ (a c 2q γ). If one deviates, the deviation quantity is now q d = (a c q γ)/2 with payoff u i `qi, q \i u d = 1 4 (a c q γ) 2. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

75 Collusion between Cournot duopolists Modified trigger strategy As before, the following inequality has to be satisfied u γ (1 γ) u d + γ u c. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

76 Collusion between Cournot duopolists Modified trigger strategy As before, the following inequality has to be satisfied u γ (1 γ) u d + γ u c. Solving with respect to γ produces qγ = 9 5γ (a c). 3(9 γ) G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

77 Collusion between Cournot duopolists Modified trigger strategy As before, the following inequality has to be satisfied u γ (1 γ) u d + γ u c. Solving with respect to γ produces qγ = 9 5γ (a c). 3(9 γ) How does q γ vary with γ? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

78 Collusion between Cournot duopolists Modified trigger strategy How does q γ vary with γ? qγ = 9 5γ (a c) 3(9 γ) q * a c 3 a c 2 0 9/17 Clearly, for γ = 0 the Cournot strategy is played. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

79 Cournot competition under asymmetric information Problem formulation Consider again a Cournot duopoly model with p i(q) = p i `qi, q \i = ( a Q, Q < a 0, Q a The payoff function for both firms is equal to u i `qi, q \i = qi (p i c i) and the cost for firm 1 is c 1(q 1) = c (and this is known to both firms) while firm 1 knows only that firm 2 s marginal cost c 2(q 2) is given by c 2(q 2) = ( c H, c L, with probability θ with probability 1 θ G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

80 Cournot competition under asymmetric information Nash equilibrium The Nash equilibrium is computed as it were a three-player game. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

81 Cournot competition under asymmetric information Nash equilibrium The Nash equilibrium is computed as it were a three-player game. If firm 2 cost is c H, it will choose q 2(c H) to solve q 2(c H) = arg max 0 q 2 < = arg max 0 q 2 < u 2 (q 1, q 2) (a q 1 q 2 c H) q 2 = a ch q 1 2 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

82 Cournot competition under asymmetric information Nash equilibrium The Nash equilibrium is computed as it were a three-player game. If firm 2 cost is c H, it will choose q 2(c H) to solve q 2(c H) = arg max 0 q 2 < = arg max 0 q 2 < u 2 (q 1, q 2) (a q 1 q 2 c H) q 2 = a ch q 1 2 If firm 2 cost is c L instead, it will choose q 2(c L) to solve q 2(c L) = arg max 0 q 2 < = arg max 0 q 2 < u 2 (q 1, q 2) (a q 1 q 2 c L) q 2 = a cl q 1 2 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

83 Cournot competition under asymmetric information Nash equilibrium Firm 1 chooses q 1 to solve the following optimization problem q 1 = arg max 0 q 1 < = arg max 0 q 1 < u 1 (q 1, q 2) θ [(a q 1 q 2(c H) c) q 1] + (1 θ) [(a q 1 q 2(c L) c) q 1] = θ (a c q 2(c H)) + (1 θ) (a c q 2(c L)) 2 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

84 Cournot competition under asymmetric information Nash equilibrium Firm 1 chooses q 1 to solve the following optimization problem q 1 = arg max 0 q 1 < = arg max 0 q 1 < u 1 (q 1, q 2) θ [(a q 1 q 2(c H) c) q 1] + (1 θ) [(a q 1 q 2(c L) c) q 1] = θ (a c q 2(c H)) + (1 θ) (a c q 2(c L)) 2 The Nash equilibrium is found solving the system 8 q1 >< = θ (a c q 2(c H)) + (1 θ) (a c q2(c L)) 2 q2(c H) = a ch q 1 2 >: q2(c L) = a cl q 1 2 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

85 Cournot competition under asymmetric information Nash equilibrium It turns out that q 1 = a 2c + θ ch + (1 θ) cl 3 q 2(c H) = a 2cH + c 3 q 2(c L) = a 2cL + c θ (ch cl) 6 θ (ch cl) 6 G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

86 Cournot competition under asymmetric information Nash equilibrium It turns out that q 1 = a 2c + θ ch + (1 θ) cl 3 q 2(c H) = a 2cH + c 3 q 2(c L) = a 2cL + c θ (ch cl) 6 θ (ch cl) 6 Compare this solution to the Cournot equilibrium under complete information with unequal costs c 1 and c 2, i.e., q1 a 2c1 + c2 = 3 q2 a 2c2 + c1 = 3 For example, how about q 2(c H) and q 2(c L)? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

87 Cournot competition under asymmetric information Nash equilibrium It is easily seen than q 2(c H) > q 2 while q 2(c L) < q 2. Why? G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

88 Cournot competition under asymmetric information Nash equilibrium It is easily seen than q 2(c H) > q 2 while q 2(c L) < q 2. Why? This occurs because firm 2 not only tailors its quantity to its cost but also responds to the fact that firm 1 cannot do so. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53

89 Cournot competition under asymmetric information Nash equilibrium It is easily seen than q 2(c H) > q 2 while q 2(c L) < q 2. Why? This occurs because firm 2 not only tailors its quantity to its cost but also responds to the fact that firm 1 cannot do so. For instance, assume firm 2 s cost is high. The best quantity q 2 is higher than that in the complete-information case because firm 2 knows that firm 1 will produce a quantity that maximizes its expected profit. This expected profit is smaller than that firm 1 would produce if it knew firm 2 s cost to be high. G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, / 53