There are two firms, A & B. A can choose either of acceptable price (H) or attack price (L) while B can choose either of enter (E) or not-enter (N) after A's decision. The payoff is given as follows: Choices Player A HHLL Player B Ν Ε Ν Ε
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- Does each individual in a prisoners dilemma benefit more from cooperation or from pursuing self-interest? Explain briefly.Sometimes oligopolies in the same industry are very different in size. Suppose we have a duopoly where one firm (Film A) is large and the other film (Film B) is small, as the prisoners dilemma box in Table 10.4 shows. Assuming that both films know the payoffs, what is the likely outcome in this case?Consider the extensive form game portrayed below. The top number at aterminal node is player 1’s payoff, the middle number is player 2’s payoff,and the bottom number is player 3’s payoff.a. Derive the strategy set for each player. (Note: If you do not want to listall of the strategies, you can provide a general description of a player’sstrategy, give an example, and state how many strategies are in thestrategy set.)b. Derive all subgame perfect Nash equilibria. c. Derive a Nash equilibrium that is not a SPNE, and explain why it isnot a SPNE.
- Consider the following coordination game: Player 2P1 Comedy Show Concert Comedy Show 11,5 0,0 Concert 0,0 2,2 a. Find the Nash equilibrium(s) for this game.b. Now assume Player 1 and Player 2 have distributional preferences. Specifically, both people greatly care about the utility of the other person. In fact, they place equal weight on their outcome and the other person’soutcome, ρ = σ = ½. Find the Nash equilibrium(s) with these utilitarianpreferences.c. Now consider the case where Player1 and Player2 do not like each other. Specifically, any positive outcome for the other person is viewed as anegative outcome for the individual, ρ = σ = -1. Find the Nashequilibrium(s) with these envious preferences.Consider the following price game: Firm 1 Firm 2 High Low High 20, 20 12, 24 Low 24, 12 14, 14 Remark: In simultaneous move games (games with rows and columns) theconvention is to write the row player’s payoff first and the column player’spayoff second. (a) What is the Nash equilibrium of this game? Recall that for each playeryou should find the best response to each of the opponents’ strategies andunderline the associated payoff. Then look for a cell where both strategiesare best responses to each other. This is a Nash equilibrium. (b) Does either firm have a dominate strategy (a strategy that is always abest response)?Please no written by hand Two players bargain over how to split $10. Each player i ∈ {1, 2} choose a number si ∈ [0, 10] (which does not need to be an integer). Each player’s payoff is the money he receives. We consider two allocation rules. In each case, if s1 + s2 ≤ 10, each player gets his chosen amount si and the rest is destroyed. 1. In the first case, if s1 + s2 > 10, both players get zero. What are the (pure strategy) Nash equilibria? 2. In the second case, if s1 + s2 > 10 and s1 6= s2, the player who chose the smallest amount receives this amount and the other gets the rest. If s1 + s2 > 10 and s1 = s2, they both get $5. What are the (pure strategy) Nash equilibria? 3. Now suppose that s1 and s2 must be integers. Does this change the (pure strategy) Nash equilibria in either case?
- Consider the location game we covered in Lecture 3. Now assume there arethree players (vendors). As we assumed in the lecture, consumers in each area choosethe closest vendor and if there are multiple closest vendors then these vendors receiveequal share of consumers in the area. Notice Si = {1, 2, 3, ...., 9} for i = 1, 2, 3. Here aresome examples of payoffs: u1(1, 1, 1) = 3, u1(1, 1, 9) = u2(1, 1, 9) = 2.25, u3(1, 1, 9) =4.5, u1(1, 5, 9) = u3(1, 5, 9) = 2.5 and u2(1, 5, 9) = 4. (a) Is s′1 = 1 strictly dominated by s′′1 = 2 for player 1?(b) Is s′1 = 1 weakly dominated by s′′1 = 2 for player 1?(c) Can you find a Nash equilibrium in pure strategies?Consider the strategic form game shown. a. Assume that both players are rational. What happens?b. Assume that both players are rational and that each believes that theother is rational. What happens?c. Find the strategies that survive the ISDS.Paramter y = 0 If ⟨a, d⟩ is played in the first period and ⟨b, e⟩ is played in the second period, whatis the resulting (repeated game) payoff for the row player?
- The prisoners' dilemma is a two-person game illus·trating thata. t he cooperative outcome could be worse for bothpeople than t he Nash equilibrium.b . even if the cooperative outcome is better than theNash equilibrium for one person, it might be worsefor the other.c. even if cooperation is better than the Nash equilibrium,each person might have an incentive not tocooperate.d . rational, seff·interested individuals will naturallyavoid the Nash equilibrium because it is worse forboth of them.12. Consider a game where each player picks a number from 0 to 60. The guess that is closest to half ofthe average of the chosen numbers wins a prize. If several peopleare equally close, then they share theprize. The game theory implies that (A) all players have dominant strategies to choose 0 (B) all players have dominant strategies to choose 30 (C) there is a Nash equilibrium where all players pick 0 (D) there is a Nash equilibrium where all players pick positive numbers 13. Behavioral data in such games suggests that (A) most subjects choose 0; (B) most subjects choose 30; (C) common answers include 30, 15, 7.5, and 0; (D) most subjects use randomization. Can you help me answer number 13 please?