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Formal Theory Minor Field Examination Department of Political Science The Ohio State University September 22, 2008 Instructions: You have four hours to complete this examination. Answer all three questions. You may type and print the answers or write them by hand, whichever you prefer. This is a closed book, closed note exam. 1. Consider a stylized model of an election between an Incumbent and a Challenger, in which the two players simultaneously choose whether to compete in the election or stay out a. The following normal form represents the game for the case of a strong challenger. What are the Nash equilibria to this game? b. The following normal form represents the game for the case of a weak challenger. What are the Nash equilibria to this game? c. Consider a somewhat more complex game in which, prior to the simultaneous choice, Nature chooses whether the challenger is strong (with probability p) or weak (with probability 1 p ). This information is revealed only to the challenger. i. ii. Draw the extensive form of this game. Draw the normal form of this game. d. Find all pure strategy Bayesian perfect equilibria to the game in part (c) for all ranges of p. e. Based on this analysis, if an incumbent knows that one-third of all potential challengers are strong, should she seek to remain in office or should she retire? Why? 2. Suppose that a group of n individuals is engaged in a collective choice problem over the provision level of some public good (e.g., school financing), where the decision will be made by simple majority rule. Let [0, x ] R denote the set of possible levels with common element x. Individual preferences are defined over two parameters, the level of the public good x and money income y, where the latter will reflect the payments an individual has to make towards the financing of the public good. Suppose individual i’s preference over x and y can be represented by a utility function of the form ui ( x, y) vi ( x) y, where the function vi(x) is strictly concave. The cost associated with the level of the public good x is equal to c(x) where c() is increasing and convex. The project is financed by a uniform tax, so that if the level x is selected, each of the n individuals pays c(x)/n. Given this financing scheme, what is the equilibrium level of public good? Justify your answer. 3. For an important question in political science that is relevant to your research, discuss the contributions of formal theory toward answering that question. A complete answer should include: background on the question you are addressing, a history of the formal theoretical models that have been advanced to address the question (with sufficient derivations to make the work clear to formal modelers who have not worked in the particular substantive area), and a discussion of the extent to which the predictions of those formal models have been tested (and with what findings).