Political Science 272/372: Problem Set 1

Political Science 272: Practice Set 1
Spring 2003
Tuesday and Thursday, 9:40-11:00
Randall Stone, Associate Professor of Political Science, University of Rochester
Office Hours: Thurs., 1-3:00 Harkness Hall 306
273-4761

NOTE: THIS IS A SET OF PRACTICE QUESTIONS. This is not the problem set that is due on March 7, but it may help you to solve those problems.

The answers are available at www.courses.rochester.edu/stone/PSC272/practice set1-answers.html

Write your answers neatly on a separate sheet of paper (attach as many sheets as necessary). Show all calculations, game diagrams, etc. No credit will be given for answers that just show the final result, but partial credit will be given for getting most of the way to an answer. Please write your name on your work and staple the pages.

1. Strategic-form games.
Draw a strategic form game for each of the following situations, using (high) numbers to represent (high) ordinal utilities. There are two players {Row, Column} and each has two possible strategies {C,D}. Find and circle all the Nash equilibria in pure strategies and all dominant strategies.

(a) U(R): CC>CD>DC>DD; U(C): DC>CC>DD>CD
(b) U(R): DC>CC>DD>CD; U(C): DC>CC>DD>CD
(c) U(R): CC>DC>DD>CD; U(C): CC>DC>DD>CD
(d) U(R): DC>CC>CD>DD; U(C): DC>CC>CD>DD
(e) U(R): CC>DD>CD>DC; U(C): DD>CC>DC>CD

2. Lotteries.
Two states, Austria and Prussia, are involved in negotiations at Olmuetz (1850) after Austria has made demands upon Prussia regarding the state of Hanover. Austria has three options: (i) withdraw its demands, in which case the situation remains unchanged---the status quo (SQ) prevails; (ii) attack Prussia, in which case Austria wins with some probability p; and (iii) issue an ultimatum (in which case Austria estimates that Prussia will resist with probability q). If Prussia resists, war occurs, and Austria again wins with probability p (i.e. there are no advantages to offense or defense). If Prussia fails to attack, it must acquiesce, or submit to Austria's demands. Assume the following utilities for the Austria: U(SQ) = 5, U(Win)= 6, U(Lose) = 0, U(Prussia acquiesces)= 10.

(a) Express Austria's expected utility of attacking immediately (the war lottery)
(b) Express Austria's expected utility of issuing an ultimatum
(c) Given the utilities above and assuming the second state will attack with probability q = .5, what probability of winning a war will make Austria prefer to attack immediately? Hint: use the formulas you constructed for (a) and (b).

3. Extensive and strategic form games.
In 1853 Russia and Turkey became involved in a dispute over Christian holy places in Palestine, and Britain, France and Austria (and Sardinia, but never mind that) faced the decision of whether to support Turkey in what became the Crimean War. Assume that each country had only a binary choice (fight, don't fight). The following describes the interests of the players, as well as they can be reconstructed in hindsight:
France: Preferred to fight Russia if supported by Britain or Austria, and preferred to have the support of both, but preferred Britain to Austria as an ally if forced to choose. Preferred to back down if forced to defend Turkey alone.
Britain: Preferred to fight if supported by France; preferred not to fight if France refused to fight or if Austria joined the war against Russia.
Austria: Preferred not to join the war under any circumstances.

(a) Write this as an extensive-form game in which France, then Britain, and then Austria must decide whether to fight or back down. What is the unique SPE?
(b) Now assume that the three countries make their choices simultaneously. Write this as a strategic-form game, and find all the Nash equilibria.
(c) (Extra credit): How would the strategic form be different if we assumed that the countries made their choices sequentially (France, then Britain, then Austria)?

4. Repeated games.
Assume that two players find themselves in a Prisoner's Dilemma (i.e., DC>CC>DD>CD), and each attaches the following cardinal utilities to the respective outcomes: U(CC)=R, U(DC)=T, U(CD)=S, U(DD)=0, where R, T>0 and S<0 (so T>R>0>S). R stands for Reward, T for Temptation, and S for Sucker. Now, assume that the game is infinitely repeated, and that each actor discounts future payoffs, using discount factor d .

(a) Find the discount factor necessary to prevent a player from defecting if both players play grim trigger strategies: cooperate in the first period; always continue to cooperate if the other player cooperates, and always defect if the other player has ever defected.
(b) Find the discount factor necessary to prevent a player from defecting if both players play tit-for-tat strategies: cooperate in the first period; always cooperate thereafter if the other player has cooperated in the previous period, and always defect if the other player defected in the last period.
(c) Are these pairs of strategies Nash equilibria? Which of these pairs of strategies is subgame perfect? Why (and why not)?