Assignment 5.
ECON2141
4 questions, due 5:00pm, Friday, 18 October, 2024.
1. The Expropriation ''Game.'' (15 points)
First they came for the socialists and I did not speak out because I was not a socialist, then they came for the trade unionists and I did not speak out because I was not a trade unionist, then they came for the Jews and I did not speak out because I was not a Jew but then they came for me and there was no one left to speak for me. (Martin Niemoller, 1892 - 1984.)
There are three people on an island: A, B, and C. Each round they vote on whether to keep on the island the person whose name is last alphabetically. If a person is voted off, their wealth is expropriated and so their payoff is zero. Only those left on the island can vote. The game ends in the first round in which a person is not voted off, and the payoff of those remaining on the island is 1=k, where k is the number left on the island. So in the first round the three people vote whether or not to throw C off. The game proceeds to the second round, if and only if C is thrown off in which case the remaining two individuals vote whether or not to throw B off. If at least fifty percent of the voters vote to throw a person off then that person is thrown off. Everyone must vote. Abstaining is not permitted. Finally, assume voters vote as if pivotal. In particular, this means a person who is the subject of a vote will always vote against being thrown off the island.
(a) (2 points) Find the subgame perfect equilibrium and show that in this equilibrium C is not voted off the island.
(b) (3 points) Find the subgame perfect equilibrium of the same game but starting now with six people on the island: A, B, C, D, E, and F:
(c) (5 points) Find the subgame perfect equilibrium of a similar game starting with six people on the island: A, B, C, D, E, and F, but in which a (supermajority) vote of at least two-thirds is required for a person to be voted o§ the island. Compare this solution with the one you worked out for part (b).
(d) (5 points) Find the subgame perfect equilibrium of a game starting with fifteen people
{A; B; C; D; E; F; G; H; I; J; K; L; M; N; O}
in which a (supermajority) vote of at least two-thirds is required for a person to be voted off the island.
2. Information and Nuclear Safety. (25 points) Consider the following game involving two real players and a chance move by 'nature'. America and Russia have the nuclear capability to destroy each other. 'Nature' tosses a fair coin so that with probability 2/1 America moves first and Russia moves second, and with probability 2/1 Russia moves first and America moves second. For now, assume that both countries observe nature's choice so they know whether they are first or second. The country who moves first decides whether to 'fire' its missiles or to 'wait'. If it fires, the game ends: the country who fired gets a payoff of -1, and the other country gets -4. If the first country waits, then the second country gets to move. It too must decide to 'fire' or to 'wait'. If it 'fires' then the game ends, it gets -1 and the other country gets -4. If it 'waits' then both countries get 0. Assume that each country maximizes its expected payoff.
Treat this as one game, rather than as two different games. Figure 1 is an extensive form. (game tree) representing this game. The first payoff refers to America. The second payoff refers to Russia. There are no payoffs for 'nature'.
Figure 1
(a) (5 points) What makes this a game of perfect information? Write down the definition of a strategy in an extensive form. game, and identify the possible strategies for America and for Russia in this game.
(b) (5 points) Find and explain any pure-strategy subgame-perfect equilibria (SPE), making clear what constitutes a subgame. Are there any Nash equilibria which are not SPE?
(c) (5 points) Now suppose that neither Russia nor America observes the move by nature, or each other's move. That is, should a country be called upon to move, it does not know whether it is the first mover or whether it is the second mover and the other country chose 'wait'. Again, treat this as one game. Draw a game tree similar to Figure 1 but for this new game indicate clearly which nodes are in the same information sets.
(d) (5 points) Identify the possible strategies for America and for Russia in the game from part (c). Find and explain carefully two 'symmetric' pure-strategy SPEs in this game that have very different outcomes.
(e) (5 points) Now suppose that America can observe the move by nature and also (when it is the second mover) Russia's move. Russia knows what America can observe, but, as before, Russia can observe neither natureís nor America's move. Draw the game tree for this game. Argue whether you think the world is a safer place or a more dangerous place now that America is better informed than Russia. That is, compare the SPE of this game with the SPE of the game of parts (b) and (d).
3. A Strategic Promise. (15 points) Suppose Robert Jenrick and James Cleverly are the remaining two candidates for the UK Conservative Party leadership contest to replace the outgoing leader Rishi Sunak. The winner will be determined by a vote by the party membership. Robert Jenrick is currently leading in the opinion polls. The two candidates simultaneously choose whether or not to go ''negative'' in their campaigns. The party does best when neither candidate goes negative and it is particularly costly costly for the leading candidate (Jenrick) to go negative. For concreteness, letting ''+'' denote the choice of not going negative and ''-'' denote the choice of going negative, suppose the payoff matrix is as follows:
(a) (3 points) Find the Nash equilibrium for this game.
Suppose now Cleverly can announce ''I promise not to go negative.'' Suppose further, such a promise by Cleverly changes the payoffs of the game. In particular if he breaks his promise it is very costly for Cleverly and benefits Jenrick. However, if Cleverly keeps his promise, he gets a reward. Finally, Jenrick incurs a cost if he goes negative after a promise by Cleverly not to do so. For concreteness suppose should Cleverly make a promise not to go negative, the payoff matrix for their simultaneous choice of whether or not to go negative becomes:
(b) (12 points) Find the subgame perfect equilibrium of the game in which Cleverly first decides whether or not to promise not to go negative and then the two candidates decide simultaneously whether or not to go negative in their campaigns. Be careful to specify distinctly, what are: (i) the subgame perfect equilibrium strategy profile of the two players; (ii) the equilibrium play of the game; and, (iii) the equilibrium payoffs for the two players.
4. Satellite(-TV) Wars (45 points)
Two satellite TV operators, Ozstar (O) and Wolftel (W), compete in the same market and play the following game. At the beginning of the year, they each simultaneously announce whether they will continue to provide their pay-TV service for the year or close down.
If an operator announces it will be continuing to provide its satellite TV service that year, it incurs a fixed cost of $50 million. This cost does not depend on how many subscribers it eventually has, but it is sunk once it has announced it is to continue operating for that year.
If an operator announces it is closing down it does not incur the fixed cost and it does not incur any operating cost. However, the announcement to close down is irreversible and the operator cannot resume service in future years.
If both operators decide to close down then they each get a zero payoff and no satellite service is supplied or sold in that market for the current year and every year thereafter.1
If both operators decide to continue providing its satellite TV service then, since viewers in the market regard the two services as essentially equivalent, a potential subscriber will only ever sign up to at most one service and if he does decide to subscribe will choose the one with the lower annual subscription fee. In the case where their subscription fees are equal he will randomly choose between them (with equal probability).
This results in fierce price competition for subscribers, leading both operators to charge a subscription fee for the year of $100 that is equal to their common and constant marginal cost of providing one year's worth of service to a subscriber. Hence in this case subscription fees generate no variable profit to offset the fixed cost of $50 million, leading both to experience a loss of $50 million (that is, a payoff of -$50 million).
If only one operator decides to close down, then the remaining operator becomes a monopolist provider of satellite TV. The maximum profit per year (that is, payoff per year) it can earn is $200 million ($250 operating profit less the $50 million fixed cost). It achieves this by setting a subscription fee of $600 for one year of service.
For parts (a), (b) and (c) assume the game ends after only one year, no matter what actions the two satellite TV operators take.
(a) [5 point] Derive the unique Nash equilibrium for the subgame following Ozstar announc-ing it will be operating this year and Wolftel announcing it is closing down.
(b) [5 points] In the subgame following both Ozstar and Wolftel announcing they will be operating their satellite TV services this year, explain why the unique Nash equilibrium is for both to set their subscription fees equal to $100.
(c) [10 points] Using your answers from parts (a) and (b), find all the subgame perfect equilibria of the one-year game. [HINT: Do not forget about the equilibrium that entails both operators randomizing over their choice between deciding to operate this year or deciding to close down.]
For the rest of the question, assume the game now has an infinite horizon and both satellite TV operators have a common discount factor 0:8. Notice that the payoff from having the market to oneself is equal to the net present value of a profit stream of $200 million every year, that is,
(d) [15 points] Construct the symmetric 'war-of-attrition' stationary equilibrium for this infinite horizon game.
Denote by swat the strategy you were asked to derive in part (d).
(e) [10 points] Show that both operators playing the following strategy constitutes a subgame perfect equilibrium in which along the equilibrium path in every year both announce they will operate and both set a subscription fee equal to $600.
Description of strategy
Year 1: Announce 'we will operate' and then set subscription fee equal at $600 (irrespective of what your competitor has announced it will do).
Year t: (where t > 1)
IF, in all previous years, both operators have announced they will operate and have set their subscription fees at $600, or in some previous year, the other operator ceased operating;
THEN, announce 'we will operate' and set subscription fee at $600;
ELSE, play the stationary 'war-of-attrition' strategy swat.