S 2,2-1, x c C x r, 1 0,0

Transcription

1 Problem Set 5 1. There are two players facing each other in the following random prisoners dilemma: S C S, -1, x c C x r, 1 0,0 With probability p, x c = y, and with probability 1 p, x c = 0. With probability p, x r = y, and with probability 1 p, x r = 0. Assume y >. i. What are the players types in this game? Define what a Bayesian Nash equilibrium is for this particular example. Sketch all the possible strategic forms that might be realized from the random payoffs. The row player has two types, x r = 0 and x r = y, and the column player has two types, x c = 0 and x c = y. A Bayesian Nash equilibrium is a strategy S or C for each player-type x r = 0, x r = y, x c = 0 and x c = y so that no player-type has an incentive to change its strategy. S C S C S C S C S, -1, 0 C 0, -1 0,0 S, -1, y C y, -1 0,0 S, -1, y C 0, -1 0,0 S, -1, 0 C y, -1 0,0 ii. Show that always choosing C is a Bayesian Nash equilibrium of the game. The proposed equilibrium is: x r = 0 C x r = y C x c = 0 C x c = y C Let s check first that the row player doesn t want to deviate. Suppose the row player draws x r = 0. Then the expected payoff (given the proposed strategies for the other types) to confess is p0+(1 p)0 = 0. The expected payoff (given the proposed strategies for the other types) to silent is p( 1)+(1 p)( 1) = 1. So confess is better than silent, and there is no profitable deviation. Suppose the row player draws x r = y. Then the expected payoff (given the proposed strategies for the other types) to confess is p0+(1 p)0 = 0. The expected payoff (given the proposed strategies for the other types) to silent is p( 1) + (1 p)( 1) = 1. So confess is better than silent, and there is no profitable deviation. Since this work is the same for the column player, neither player has a profitable deviation, and this is a Bayesian Nash equilibrium. iii. Show when a Bayesian Nash equilibrium exists where players remain silent if they draw x i = 0 and confess if they draw x i = y.

2 The proposed equilibrium is: x r = 0 S x r = y C x c = 0 S x c = y C Let s check that the row player doesn t want to deviate. Suppose the row player draws x r = 0. Then the expected payoff (given the proposed strategies for the other types) to confess is p0+(1 p)0 = 0. The expected payoff (given the proposed strategies for the other types) to silent is (1 p)+p( 1). The expected payoff (given the proposed strategies for the other types) to confess is (1 p)0 + p0. So as long as (1 p) p 0, the x r = 0-type row player will want to remain silent. This is true if p /3. Now suppose the row player draws x r = y. Then the expected payoff (given the proposed strategies for the other types) to silent is (1 p)+p( 1). The expected payoff (given the proposed strategies for the other types) to confess is (1 p)y + p(0). Then since y >, (1 p)y > (1 p) p, so confessing is always better than remaining silent. Then as long as p /3, none of the player-types have an incentive to deviate from the proposed strategies, and this is a Bayesian Nash equilibrium of the game. iv. How does your answer for part iii depend on y? Explain briefly. The y only comes up when considering whether or not to confess, given that you ve drawn a y. Since, in that scenario, it is a strictly dominant strategy to confess, the value of y is actually irrelevant to the answer (as long as it is greater than ).. There are two firms in a Cournot market. One, however, firm a, has an excellent market research division, and the other, firm b, does not. So firm a has better information than firm b about what market demand will be in any given period. In particular, the price is given by p(q a,q b ) = A+ q a q b The quantity  is observed perfectly by firm a through market research, but not by firm b. Firm b, however, knows that with probability p, it takes the value  = A, and with probability 1 p, it takes the value  = 0. The firms have no costs. i. What are the players types in this game? Define what a Bayesian Nash equilibrium is for this particular example. i. Player a has two types,  = A and  = 0, and firm b has one type (firm b is just firm b). A Bayesian Nash equilibrium is a strategy q A, q 0 or q b for each player-type A, 0 or b so that no player-type has an incentive to change its strategy. ii. Solve for a Bayesian Nash equilibrium. How do the equilibrium strategies depend on p? Does p appear in firm a s strategy; explain why this is interesting, since firm a perfectly observes Â.

3 The expected payoff for type A is the expected payoff for type 0 is the expected payoff for type b is π A = (A+A q A q b )q A π 0 = (A+0 q 0 q b )q 0 π b = p{(a+a q A q b )q b }+(1 p){(a+0 q 0 q b )q b } Now, to be in equilibrium, each type must be maximizing its expected payoff, given the strategiesadoptedbytheothertypes. Sowemaximizeeachpayoffwithrespecttowhatthattype controls, and then solve all the equations at the same time to find a profile of strategies where no one wants to change what they are doing (they are all mutually maximizing in expectation). Then π A = 3A q A q b = 0 q A = 3A q b q A π 0 = A q 0 q b = 0 q 0 = A q b q 0 π b q b = p{a+a q A q b }+(1 p){a+0 q 0 q b } = 0 Now, substituting the first two results into the third gives { p 3A 3A q { b q b }+(1 p) A A q } b q b = 0 This equation only has q b s in it, so we can solve for that player-type s equilibrium strategy: q b = pa+(1 p)a 3 Substituting this into the equations above for q 0 and q A yields ( A pa+(1 p) A ) q0 3 = ( 3A pa+(1 p) A ) qa = 3 So p appears in every player s strategy, despite the fact that p only appears in π b, not π 0 or π A. This is what happens when we study equilibria: Since b strategy depends on p, and 0 s and A s strategies depend on what they believe b will do, their strategies end up depending on p as well. So all of the equilibrium strategies depend on p. p? iii. How does firm b s equilibrium strategy depend on p? How does firm a s strategy depend on

4 q b p = A > 0 so raising p (the probability that demand is high, Â = A) makes firm b produce more; anticipating high demand and more profits on average, they produce more. However, in equilibrium, q A = 3A q b q0 = A q b The only way that qa and q 0 depend on p is through q b. So if q b goes up, both of these must go down. So anticipating that firm b will produce more if p goes up, both of a s types reduce their production. iv. Would firm a ever want to share its information with firm b? (Explain briefly in words how you could use the model to answer this question; no extra work is needed, unless you just want to know the answer.) First, I would compute the profits of both firms using the work above. Second, I would compute their profits when firm a shares the information with firm b, and firm b maximizes (this is equivalent to the regular Cournot game). Third, I would compare the firms expected profits in both cases, and see if they are both better off when a shares the information. Since knowing the truth will make firm b produce less on average, it s hypothetically possible that it is profitable or unprofitable for either player. If firm b backs down in the low demand state and produces more in the high demand state, this could be better on average for firm a. On the other hand, firm a might make more money by being better informed, and revealing information to firm b would reduce its advantage. 3. There are two firms deciding whether or not to enter a market. If they both enter, they become duopolists and make low (or negative) profits, while if only a single firm enters, it becomes a profitable monopolist. If both firms fail to enter, they get payoffs of zero. However, their costs of entry are private information. The payoffs for the game are given by the strategy form below, where E is Enter, D is Don t Enter, e r is the privately known entry cost of the row player, and e c is the privately known entry cost of the column player: E D E 1-e r,1 e c -e r,0 D 0, - e c 0,0 With probability p, a firm is high-cost and has entry cost 1.5. With probability 1 p, a firm is low-cost and has entry cost 1. The firm s entry cost is private information, known only to it. i. What are the players types in this game? Define what a Bayesian Nash equilibrium is for this particular example. Sketch all the possible strategic forms that might be realized from the random payoffs.

5 The row player has two types, e r = 1 and e r = 1.5, and the column player has two types, e c = 1 and e c = 1.5. A Bayesian Nash equilibrium is a strategy E or D (or later, a mixed strategy σ = pr[e H]) for each player-type e r = 1, e r = 1.5, e c = 1 and e c = 1.5 so that no player-type has an incentive to change its strategy. E D E D E D E D E 0,0 1,0 D 0,1 0,0 E -1/,-1/ 1,0 D 0,1 0,0 E 0,-1/ 1,0 D 0,1/ 0,0 E -1/,0 1/,0 D 0,1 0,0 ii. Show that it is a weakly dominant strategy for a low-cost firm to always enter. For a low-cost (row) firm, the world looks like E D E 0,? 1, 0 D 0,? 0,0 So enter and don t tie if column enters, and entering is strictly better if column doesn t enter. So in any scenario, entering is at least as good as staying out. Therefore, it is a weakly dominant strategy for the (row) player. Since the game treats the row and column player the same, enter is a weakly dominant strategy for the low-type column player as well. iii. Suppose the high-type row player and high-type column player adopt the strategy, Enter. Is this a Bayesian Nash equilibrium, or not? Explain. The proposed equilibrium is e r = 1 E e r = 1.5 E e c = 1 E e c = 1.5 E But then the low-cost entrants always get a payoff of zero, and the high-cost entrants always get a payoff of 1/. By staying out, the high-cost entrants could get a payoff of 0 instead, which is strictly better. Therefore, the high-cost entrants have a strictly profitable deviation, and this is not a Bayesian Nash equilibrium. iv. Suppose the high-type row player adopts the strategy, Enter, and the high-type column player adopts the strategy, Don t Enter. For what values of p is this a Bayesian Nash equilibrium? Why isn t it an equilibrium for all values of p? The proposed equilibrium is e r = 1 E e r = 1.5 E e c = 1 E e c = 1.5 D

6 We know that it is a weakly dominant strategy for the low-cost firms to enter. So we only need to check that e r = 1.5 wants to enter and e c = 1.5 wants to stay out. Suppose e r = 1.5. The expected payoff to D is 0, since staying out always gives a payoff of zero. The expected payoff to E is p(1/) + (1 p)( 1/). Then E is better than D if p(1/)+(1 p)( 1/) 0 or p 1/ (this is because, for high p, the row player will often face a high-cost column player, who stays out). So E is better than D, and the e r = 1.5 type has no profitable deviation. Suppose e c = 1.5. The expected payoff to D is 0. The expected payoff to E is p( 1/) + (1 p)( 1/) = 1/. So D is better than E, and the e c = 1.5 type has no profitable deviation. So the proposed strategies are a Bayesian Nash equilibrium if p 1/. v. For some values of p, there is a mixed strategy equilibrium in which the high-type row and column player both mix over E and D, and the low-type row and column player always enter. Solve for the mixed-strategy equilibrium, and explain how it depends on p. (This can be conceptually difficult, feel free to ask questions in class or send me an if you get stuck.) The proposed equilibrium is Type Action e r = 1 E e r = 1.5 E with probability σ, D with probability 1 σ e c = 1 E e c = 1.5 E with probability σ, D with probability 1 σ As in the previous questions, the low cost types have a weakly dominant strategy to enter. The high cost types, however, are supposed to mix. To solve for mixed strategies, each player chooses his own mix (σ) to make his opponents indifferent over their pure strategies (E and D). The expected payoff of D is 0, since you get that no matter what. The expected payoff of E is (1 p) }{{} Low-cost opponent ( 1/) + p }{{} High-cost opponent σ }{{} opponent enters, given high cost ( 1/)+ (1 σ) (1/) }{{} opponent stays out, given high cost Setting the expected payoff to E equal to the expected payoff from D, we get and solving yields (1 p)( 1/)+p{σ( 1/)+(1 σ)(1/)} = 0 σ = p 1 p This is supposed to be a probability, so as long as p 1 0, the whole thing is positive, or p 1/. Then for p 1/, the strategies

7 e r = 1 E e r = 1.5 E with probability p 1 p e c = 1 E e c = 1.5 E with probability p 1 p are a Bayesian Nash equilibrium of the game.