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Department of Economics
ECON 329 - Game Theory
Homework #5
1. Stackelberg Model of duopoly where firms have convex costs (8 points).
A product is produced by only two firms: firm 1 and firm 2. The timing of the game is as follows: firm 1 chooses a quantity q1 ≥ 0, then firm 2 observes q1 and chooses a quantity q2 ≥ 0. The payoff of each firm depends on the market price P and the cost of production Ci. The market price is determined by the following formula: P(Q) = a - Q, where a is a constant number and Q = q1 + q2 is the aggregate quantity produced on the market by both firms. The cost to firm 
of producing qi is Ci(qi)) = q2i. (Note: the only difference between this problem and the standard problem is that instead of Ci(qi) = cqi we have Ci(qi) = q2i).
a) Find the best response function for firm 2 as a function of firm 1’s quantity 
. (2 points).
b) Find the best response function for firm 1, given that firm 1 knows the best response of firm 2. (2 points).
c) What is SPNE of this game? (2 points).
d) What is the outcome of SPNE of this game: equilibrium quantities for firm 1 and 2? (2 point).
2. Resolving conflicts by flipping a fair coin (10 points).
Consider a game of potential conflict between two players for a prize valued at v by both players. The game takes place in two sequential stages, the conflict resolution stage and the conflict stage. In the conflict resolution stage, players 1 and 2 simultaneously decide whether to resolve the conflict by flipping a fair coin (call this strategy “Flip”) or to enter the conflict stage (call this strategy “Conflict”). If both players agree to “Flip”, then the game ends with neither player advancing to the conflict stage. The prize is allocated to each player with probability p1Flip = p2 Flip = 0.5, and players 1 and 2 receive the expected payoffs of E(π1Flip) = E(π2Flip) = v/2. However, if either player refuses to “Flip” (i.e., one or both players choose “Conflict”), then both players advance to the conflict stage. In the conflict stage, both players make irreversible effort expenditures e1 ≥ 0 and e2 ≥ 0 to increase their probabilities of receiving the prize. Players have different conflict capabilities (strengths) a1 > 0 and a2 > 0, so that the stronger player 1 (a1 > a2) can expend the same effort, yet have a higher chance of winning the prize. Specifically, player 1’s probability of winning is p1(e1,e2) = a1e1/(a1e1 + a2e2) and player 2’s probability of winning is p1(e1,e2) = a2e2/(a1e1 + a2e2). The expected payoff in a conflict for player 1 is E(π1Conflict) = p1(e1,e2)v - e1 = va1e1/(a1e1 + a2e2) - e1 and the expected payoff in a conflict for player 2 is E(π2Conflict) = p2(e1,e2)v - e2 = va2e2/(a1e1 + a2e2) - e2.
a) What is the Nash equilibrium of the conflict stage subgame? (2 points). Hint: In the conflict stage both players will maximize their respective payoffs, so just solve the following two First Order Conditions ∂E(π1Conflict)/∂e1 = 0 and E(π2Conflict)/∂e2 = 0 simultaneously, in order to get e1* and e2*.
b) What is the Nash equilibrium payoff in the conflict stage? (2 points). Hint: Plug in the solution e1* and e2* into E(π1Conflict) and E(π2Conflict) to get the Nash equilibrium payoffs.
c) Prove that the weaker player 2 will always choose “Flip.” (2 points). Hint: show that the payoff for player 2 in the case of “Conflict” is always lower than in the case of “Flip”, i.e., E(π2Conflict) < E(π2Flip).
d) What are the needed restrictions on a1 and a2 for the stronger player 1 wanting to choose “Flip.” (2 points). Hint: examine when the payoff for player 1 in the case of “Conflict” is lower than in the case of “Flip” i.e., E(π1Conflict) < E(π1Flip).
e) Friedrich Nietzsche said "Justice originates among those who are approximately equally powerful (...) where there is no clearly recognizable predominance and a fight would mean inconclusive mutual damage (...)." Explain how your findings support this conjecture. (2 point).
3. Commitment problems in conflict resolution (2 points).
In the literature on conflict resolution there is a well-known problem, called a “commitment problem”: in the absence of credible commitment, parties on the less favored side of any proposed resolution face incentives to ignore the resolution. Such commitment problems arise both on a macro level between countries and competing economies, as well as on a micro level between individuals. For example, a country that finds a UN resolution counter to its interests could simply exit the organization and ignore the international community. When examining the “commitment problem” using game theoretic analysis, all Subgame Perfect Nash Equilibria (SPNE) lead to a conflict.
a) Do you think that making commitment costly would help resolving the “commitment problem”? (2 points). Hint: To answer this question, please read the article “Commitment Problems in Conflict Resolution” which was published in the Journal of Economic Behavior and Organization http://www.sciencedirect.com/science/article/pii/S0167268115000281. If you are off campus, you may not have a direct access to the journal, so here is the link to an open access version: https://ideas.repec.org/p/chu/wpaper/13-11.html.