This course is an introduction to some of the prominent choice-theoretic models used in political science. The main objective of the course is to introduce the fundamentals of noncooperative game theory. After about eight weeks of work on game theory and applications, we will survey social choice theory and the spatial theory of voting. Finally, we will read and discuss an important critical evaluation of rational-choice theory and some of the responses it has generated.

A game is a model in which decision makers choose actions based on (1) their preferences over outcomes and (2) their beliefs about the choices of other decision makers, where outcomes are the consequences of choices made by more than one decision maker. As a glance at the leading journals will demonstrate, game theory has become ubiquitous in political science. As we will be studying it, game theory is about developing a good model for a given real-world situation and applying the appropriate technique ("equilibrium concept") to the model to generate solutions, which in turn are used to generate predictions of real-world behavior.

Social choice theory is a general approach to problems of collective decision making. While game theory seeks to uncover predictions of the outcomes of particular classes of choice situations, social choice theory emphasizes the characteristics of the outcomes of collective decisions made under generic conditions. Most of social choice theory is concerned with the problem of aggregating individual preferences into collective preference orderings. Many of the findings of social choice theory are negative, showing that certain desirable properties of choice situations yield outcomes that lack some other desirable properties.

Spatial voting theory is two things: a variety of cooperative game theory used in the study of voting in committees, and a variety of noncooperative game theory coupled with a theory of voting behavior used in the study of electoral competition. We will study the second of these. The text we will use for this section both introduces the standard models in spatial voting theory and proposes a substantial revision to the theory by incorporating a notion of ideology into the decisions of voters and candidates.

This course has no prerequisites. In the section on game theory, none of the models we will study requires much math; when we need some mathematical knowledge, it will be introduced in class. The text on spatial voting uses matrix notation and calculus, but we will not work much with the mathematics of these models. All of the models we will study require the use of mathematical notation. Most of the material here is cumulative and therefore unforgiving to people who are reluctant to ask questions when they are confused. On one or two occasions I will introduce models that use simple calculus; you will be asked to understand what is going on but not to reproduce it in exams or assignments.

You will give four presentations in class, write a paper, complete some problem sets, and write a take-home exam that will consist mostly of writing. The exam might include some problems in game theory.

Grading (approximate): Assignments 1, 2, and 3: 10% each Problem sets: 20% total
Assignment 4: 20% Exam: 20%
Class participation: 10%

The texts are Morrow, Game Theory for Political Scientists
Gibbons, Game Theory for Applied Economists, optional
Hinich and Munger, Ideology and the Theory of Political Choice
Green and Shapiro, Pathologies of Rational Choice Theory
Critical Review, vol. 9 nos. 1-2, Winter-Spring 1995

Schedule of classes (see also the list of assignments at the end of the syllabus)

1. January 15 Morrow, ch. 1-2

The theory of utility used in standard models of consumer behavior is not equipped to deal with risk. The utility functions used in standard consumer models are ordinal relations on choice sets and are invariant to any monotonic transformation. Von Neumann-Morgenstern utility theory incorporates risk into utility functions, so that the functions contain more than simply ordinal-level information about preferences (von Neumann and Morgenstern 1943). Von Neumann-Morgenstern utility is the foundation of most game-theoretic models. Von Neumann-Morgenstern utility functions are invariant only to positive linear transformations, not to arbitrary monotonic transformations. Because the utility payoffs used in game theory are von Neumann-Morgenstern utilities, they include all of the information related to the players’ attitudes toward risk. This means, among other things, that in solving a game we do not consider the effects of different attitudes toward risk, e.g. risk aversion, on the outcome; this information is already included in the players’utility functions. Another implication is that if we multiply a player’s utility by 100, or subtract 4.67 from it, or otherwise subject it to a positive linear transformation, nothing in the analysis of the game changes, including the predicted outcome. Think back to the section in your quantitative methods courses on variables and scales (there was probably a section on categorical variables, ordinal scales, interval scales, and ratio scales). The ordinary utility function of consumer theory is clearly an ordinal scale. What is the von Neumann-Morgenstern utility function: an ordinal scale, an interval scale, a ratio scale, or something else?

2. January 22 Morrow, ch. 3-4

The strategic (or "normal") form is one way of representing games. Many of the games used as heuristic devices or as stage games of a multi-stage or repeated game are easy to represent in the strategic form. As we will see, dynamic games, in which the sequence of action matters, can also be represented by the strategic form, but the extensive form will generally be more meaningful. Components of a strategic-form game are the players, the actions (strategies) available to each player, and the outcomes (utility payoffs) to each player for each combination of strategies. A solution of the game tells us which actions are taken by each player and the payoff that each player achieves.

A strictly dominated strategy for a player is one that gives the player a strictly lower payoff than any other strategy, regardless of the strategies of other players. If we eliminate all strictly dominated strategies and arrive at a unique solution to the game, the game is dominance-solvable. Prisoners’ dilemma is an important example of a dominance-solvable game in strategic form.

The set of best-response strategies for a player is the set of strategies that give the player the highest payoff possible for each combination of other players’ strategies. When every player plays a best-response strategy given the strategies of the other players, the result is a Nash equilibrium, the central solution concept in game theory. Stated differently, a Nash equilibrium is a strategy combination (and associated outcome) in which no player can achieve a higher payoff by deviating unilaterally. A mixed strategy is a probability distribution over the strategies of a player. Every finite game in strategic form has a Nash equilibrium, possibly in mixed strategies. Do mixed strategies make sense? What might it mean for a person to "play a mixed strategy" in a real-world decision context?
 

3. January 29 Gibbons, ch. 1 (optional, skipping the calculus if necessary)

Cournot competition is the basic model of oligopoly in economics. Cournot predates Nash by about a century. Note that the Cournot equilibrium is a Nash equilibrium. Alternative models of oligopolistic competition introduce a dynamic structure to the game (Stackelberg) or use price rather than quantity as the choice variable (Bertrand), yielding different outcomes. Think about ways in which the Cournot model, or something like it, might be used in the study of politics. What are the basic features of the Cournot model (that is: if we strip away the idea that it is a model of oligopoly, what are the main features of the game)?

A coalition-proof equilibrium is (roughly) a Nash equilibrium in which no group of players can deviate together and thereby increase their payoffs. A correlated equilibrium is an equilibrium in which players correlate their strategies with publicly observable random events. In some games, correlated equilibrium permits players to achieve higher payoffs than (plain-old) Nash equilibrium. A question that is worth thinking about is: do coalition-proof equilibrium and correlated equilibrium make sense? How can their use be justified, i.e. are there conditions in which they are more convincing as solution concepts than Nash equilibrium? Is Nash equilibrium a convincing solution concept? What problems are there with Nash equilibrium?

4. February 5 Morrow, ch. 5, excluding the section on Rubinstein’s bargaining model

Gibbons, pp. 55-64, 115-130 (optional, skipping the calculus if necessary

· continue to work on assignment 2

The extensive form permits explicit analysis of dynamic games. The simplest solution concept for extensive-form games is backward induction. Any finite extensive-form game of complete and perfect information can be solved by backward induction (if there are no "ties;" i.e. if every combination of actions yields a distinct outcome). Extensive-form games can also be solved by Nash equilibrium.

Analysis of extensive-form games reveals that some Nash equilibria involve incredible threats (irrational behavior off the equilibrium path). Subgame perfection is a stronger equilibrium concept than Nash equilibrium in that it requires rational behavior everywhere, both on and off the equilibrium path. (Selten’s (1965) development of the concept of subgame perfection, along with Harsanyi’s (1967-68) analysis of games of incomplete information (or "Bayesian games") radically changed the focus of game-theoretic analysis. Many of the developments since the late 1960s have dealt with behavior off the equilibrium path. Get used to thinking about off-equilibrium-path behavior now, and you will find it easier to grasp the concepts of perfect Bayesian equilibrium and sequential equilibrium, coming up in a couple of weeks).

An information set is a collection of nodes between which a player cannot distinguish (every player knows which information set (collection of nodes) has been reached, but the player with the move does not know which node in the information set has been reached. To analyze extensive-form games we use the concept of behavior strategies, which prescribe an action at each information set for each history of the game (mixed strategies become unwieldy in extensive-form games). A strategic-form game can have multiple extensive-form representations. An extensive-form game has only one strategic-form representation. Does this mean that the strategic form is fundamental, or that the extensive form is fundamental? Why does it matter?

5. February 12     Morrow, ch. 9 and the section in ch. 5 on Rubinstein’s bargaining model
                               Gibbons, pp. 68-71, 82-102 (optional)

Repeated game models are everywhere, or were once everywhere, in political science. Friedman (1971) did some of the first work on this subject in the area of pure game theory. Nicholson (1972) employed a 2-player infinitely repeated prisoners’-dilemma-type game to model oligopolistic competition. Taylor (1976, 1987) is responsible for the introduction of the itrerated PD model into political science and for a generalization of the model to N players. Axelrod (1984) devised a computer tournament that resembled an iterated PD. The iterated PD is still used quite a lot, but it has fallen from favor in the eyes of many as a result of the folk theorem. The various folk theorems (e.g., Fudenberg and Maskin 1986) prove that under some conditions, infinitely repeated games have an uncountably infinite number of equilibria.

The Rubinstein bargaining model, which is not an infinitely repeated game but is an infinite-horizon multi-stage game with observed actions, is the central model in noncooperative bargaining theory. In its simplest form, it has the remarkable property that bargainers never actually bargain, but agree immediately on the game’s unique subgame-perfect equilibrium. In solving this model we make use of an important feature of some games, stationarity, which permits the simple solution of what appears to be very complicated game. What might account for real-world bargaining situations in which players make offers that are rejected and countered? What useful modifications could be made to the infinitely repeated PD or the Rubinstein bargaining model?

6. February 19     Morrow, ch. 6
                                Gibbons, pp. 143-163 (optional, skipping the calculus if necessary)
                                Gibbons, pp. 173-183, 210-232 (optional)

· games of incomplete information

· Bayes's rule
· Bayesian Nash equilibrium
· perfect Bayesian equilibrium

· continue working on assignment 3

Games of imperfect or incomplete information require a stronger equilibrium concept than subgame perfection. Since players make decisions from nonsingleton information sets (information sets containing more than one decision node), the equilibrium concept must include information about their beliefs: beliefs about the choices made by others (in games of imperfect information) and beliefs about the characteristics of the game (action sets and payoffs of other players, the sequence of events, etc.) in games of incomplete information. Solving these games involves imposing reasonable restrictions on the beliefs that players can have, requiring rational behavior in light of these beliefs, and applying an equilibrium concept that includes information about both beliefs and strategies.

A Bayesian Nash equilibrium is a combination of beliefs and strategies such that beliefs are derived from Bayes's rule and players choose best-response strategies to each other's strategies given their beliefs. The solution concept is not much used in dynamic games since it fails to rule out some unreasonable equilibria. Perfect Bayesian equilibrium is a stronger equilibrium concept requiring rational behavior at every information set (similar to subgame perfection) and beliefs derived from Bayes's rule wherever possible. "Wherever possible" is an important qualification: at information sets that are reached with zero probability in equilibrium, Bayes's rule cannot be used to calculate players' beliefs. Therefore, perfect Bayesian equilibrium requires some ad hoc restrictions on out-of-equilibrium beliefs and/or behavior.

7. February 26 Morrow, ch. 7

· perfect Bayesian equilibrium again
· sequential equilibrium
· continue working on assignment 3

Perfect Bayesian equilibrium is not fully satisfactory as an equilibrium concept: it does not completely pin down beliefs off the equilibrium path, and it does not always rule out unreasonable equilibria. Sequential equilibrium resolves these problems by requiring that strategies and beliefs exhibit a particular form of consistency. A sequential equilibrium is a pair (s , m ) of strategies and beliefs that exhibit sequential rationality (rational behavior at every information set) and consistency in the sense that (1) beliefs m are derived from the strategies s and other features of the game via Bayes’s rule and (2) the equilibrium (s , m ) is the limit of a sequence (s ⁿ, m ⁿ) of totally mixed strategies. The final requirement means that the sequence leading up to the equilibrium places a positive probability on every action of every player. The intuition behind this idea is that players make decisions with "trembling hands;" i.e., they might make mistakes that lead unexpectedly to play off the equilibrium path. Since each action has positive probability, each information set is reached with positive probability (in the sequence leading up the limit, though not necessarily in the limit). This means that beliefs can be derived using Bayes's rule at every point in the game.

Sequential equilibrium has its drawbacks and its detractors. Sequential equilibrium is not invariant to the addition of irrelevant actions to the game tree, and therefore it depends crucially on the way in which the game tree is constructed (the latter, though, could be said of most aspects of game theory). More limiting is the problem that arises in games with infinite (e.g., continuous) strategy sets. Since an infinite number of actions cannot all have positive probability (why not?), sequential equilibrium becomes intractable or impossible to apply in such games.

Every equilibrium concept has some deficiencies. These days, most papers in political science that use incomplete-information game models employ sequential equilibrium as the solution concept if the action sets are finite and pefect Bayesian equilibrium if the action sets are infinite. There are also several refinements of these equilibrium concepts; e.g., the intuitive criterion, divine equilibrium, and universally divine equilibrium. There are, indeed, dozens of equilibrium concepts that we do not have time to discuss in this course. The idea behind doing game theory well is not to write down a game, apply the fanciest solution concept you know of, and blindly insist upon the real-world implications of the equilibrium. It becomes important to think about how the real-world situation can best be captured in a model. Sequential equilibrium, Nash equilibrium, etc., are not direct implications of rationality. They are behavioral hypotheses that might be good or bad approximations to reality.

Still, before surrendering to the popular notion that advanced game-theoretic solution concepts are invalid because they require ultra-rational behavior on the part of partially rational people: note that sequential equilibrium takes pains to assume that people might mistakenly choose the "wrong" strategies, indeed that the whole point of sequential equilibrium is to depict people as imperfect decision-makers. In light of that, what real-world arguments might be used to justify sequential equilibrium as a predictive device?

8. March 5     Morrow, ch. 8 ,Gibbons (optional, with the usual caveat): sections 2.1.C, 2.2.B, 2.2.C, 2.3.C, 2.3.E, 4.2.D

· presentations 3

· some applications

The models we have looked at have numerous political applications: agency discretion, deterrence, collective action and public-goods provision, international macroeconomic and trade policy, interest-group competition, etc. This week’s survey of applications is a small slice of what has been done. One implication of this is that the first step in developing a game-theoretic model of your own should be a review of the relevant literature to see how others have approached similar problems. Usually, you will find something to build on; building on existing work will make your work easier and, probably, better.

9. March 12 Spring break

· think about what happens to sequential equilibrium when players have continuous strategy sets
-- or--
· think about the Monte Hall paradox

Main points of week 9:

Spring break is an excellent time to think about conditional probability. What would we need to be able to do in order to construct a sequential equilibrium of a game with continuous strategy sets (i.e., what is the analog of the consistency requirement in a game where players choose, say, a point on the interval [0,1] rather than choosing either 0 or 1 but nothing in between)? If that doesn’t sound like a good way to spend your vacation, consider instead the Monte Hall paradox: a game-show host named Monte Hall presents you with three closed doors and tells you that behind one of the doors there is a new car. Behind the other two doors are goats, to which cars are assumed to be preferable. He gives you the opportunity to choose one of the doors and win whatever is behind it. Suppose you choose door number 1. Before opening the door to show you what is behind it, Monte Hall opens door number 2 to reveal a goat. Then he gives you the opportunity to change your decision, choosing to win whatever is behind door number 3 instead of (your original choice) door number 1. Should you switch to door number 3? Why or why not?

10. March 19 Hinich and Munger, ch. 1-3

It is easy to prove the existence of a socially (Pareto-) optimal equilibrium in markets for private goods; that such an equilibrium results from the pursuit of self-interest is one way of restating the central point of Smith’s Wealth of Nations. A meaningful socially optimal equilibrium in an economy with public goods is generally nonexistent. One way of stating this is through the maximization problem and associated discussion on pages 29-36 in Hinich and Munger. Another way is through an axiomatic system such as Arrow’s (1951). A social choice function is a function that maps a collection of individual preferences into a social preference ordering. Arrow’s general possibility result ("Impossibility Theorem") demonstrates that any social choice function that exhibits certain desirable properties can be dictatorial, i.e. it can allow a single individual to be decisive in society’s choice over every pair of alternatives. Stated alternatively, Arrow’s theorem demonstrates that nondictatorial social choice mechanisms can lead to intransitive social preference orderings. For simple majority voting, the situation is even more unpromising: with no constraints on agendas, outcomes of majority decisions on multiple-dimensional choice sets are, by some standard, arbitrary.

Mechanism design is an alternative way of looking at social choice problems that emphasizes the problems generated by private information. Mechanism design deals with the construction of game forms that lead to outcomes having desirable properties. A mechanism is a game in which (possibly misleading) messages sent by agents about their private information are used as the basis of an allocation of costs or benefits among the agents. The revelation principle demonstrates that any equilibrium of a mechanism can be replicated by a direct-revelation mechanism in which agents reveal their private information truthfully. Implementation theory deals with the conditions under which allocations having certain desirable properties can be implemented as the equilibrium of some mechanism; i.e. game. The subject is rather advanced, and we will only touch on its fundamentals.

11. March 26 Hinich and Munger, ch. 4-7
 

· the spatial analog and spatial voting theory
· assumptions, ideas, notation
· equilibrium
· ideology in the spatial model
· continue working on assignment 4

Classical spatial voting theory (Downs 1957) depicts candidates as points on the real number line. Voters in this model choose candidates or parties based on Euclidean preferences, meaning that each person votes for the candidate nearest to his or her own ideal point. Given exogenous distributions of voter preferences and exogenous numbers of candidates or parties, it is possible to derive meaningful predictions of candidate competition in single-dimensional issue spaces. The most familiar result is the median voter theorem, which holds that in a two-candidate single-issue competition with a single-peaked distribution of voter preferences, no candidate can win against a candidate who adopts the position of the median voter under majority voting. Generalizing this model to multiple dimensions complicates matters substantially, yielding the result that only under special conditions does a unique equilibrium exist.

Hinich and Munger introduce a notion of ideology in which candidate positions are functions of voters’ perceptions of the candidates in an ideological space. Issue positions then represent "latent" ideological dimensions, which are imperfectly observable. Under some special conditions, the "noise" that characterizes the imperfect perceptibility of candidate positions generates a unique equilibrium in N-dimensional candidate competitions. Evaluate the concept of ideology employed by Hinich and Munger. How much of what is normally meant by "ideology" is captured by this concept? What else might we do to incorporate ideology into a spatial model?

12. April 2 Hinich and Munger, ch. 8-11

· directional voting and probabilistic voting
· interest groups, lobbying, political parties
· continue working on asssignment 4

The classical spatial theory has been augmented by the introduction of alternative behavioral hypotheses: the directional theory of voting and the probabilistic theory of voting. The directional theory of voting assumes that voters are concerned not only with distance between their ideal points and the candidates’ positions, but also with the direction of that distance. The directional model has been tested against the classical model, yielding results that appear to favor the directional model. The probabilistic theory of voting assumes that voters’ decisions are not deterministic, but that candidate behavior can influence the probability of receiving a given voter’s vote. In this section of the reading, Hinich and Munger present the existence theorem that is the central result of their book. What does the existence theorem say (including not only its conclusion but also its assumptions)? How limiting or how realistic are these assumptions?

13. April 9 Green and Shapiro (all)

· pathologies of rational choice theory
· continue working on assignment 4

When Marxism was a more active area of research, it was common to see people defending Marxism on the grounds that some egregious error committed in the theory’s name was an "error of the artifice" rather than an "error of the art." Green and Shapiro advance the view that more than just poor practice on the part of rational choice theorists is responsible for what they perceive as the approach’s throughgoing failure. That failure, they argue, is the consequence of fundamental flaws built into the approach itself. Is Green and Shapiro’s argument an attack on rational choice theory, or on social-scientific explanation in general? What conditions in the real world might be especially hostile to or conducive to the validity of rational-choice explanation? What about "point predictions": an error of the artifice or of the art?

14. April 16 · Critical Review, vol. 9 nos. 1-2, Winter-Spring 1995 (all)

Responses to Green and Shapiro, even when positive, generally point out some error they have made in interpreting or understanding rational choice theory or some aspect of it. The most common objection to Green and Shapiro is that they criticize rational choice theory without comparing it to some alternative form of explanation. Green and Shapiro’s denunciation of rational choice theory seems to presume that there is some superior form of explanation available, but nowhere is it specified in their critique. So, goes this line of argument, suppose that everything Green and Shapiro wrote about rational choice theory were true, but that their criticisms, or worse ones, were truer still as characterizations of all alternative approaches to social-scientific explanation. Most people would agree that rational choice theory would then be the preferable approach. Based on Green and Shapiro’s book, it is impossible to tell whether other approaches are superior to rational choice theory.

Is it ever true that one approach to explanation is superior to another? What does that mean? Is it possible to write a "Pathologies" for every approach to social-scientific explanation? What lines of criticism would be likely candidates for inclusion in more or less all of these "Pathologies" books? Are they the same ones emphasized by Green and Shapiro?

15. April 23 ·  presentations 4

· concluding thoughts and observations

Axelrod, Robert M. 1984. The Evolution of Cooperation. New York: Basic Books.

Downs, Anthony. 1957. An Economic Theory of Democracy. New York: Harper.

Friedman, James W. 1971. "A Noncooperative Equilibrium for Supergames." Review of Economic Studies 38:1-12.

Fudenberg, Drew and Eric Maskin. 1986. "The Folk Theorem in Repeated Games with Discounting or with Incomplete Information. Econometrica 54:533-556.

Harsanyi, John C. 1967-68. "Games with Incomplete Information Played by ‘Bayesian’ Players." Management Science 14:159-182, 320-324, 486-502.

Nicholson, Michael. 1972. Oligopoly and Conflict: A Dynamic Approach. Toronto: University of Toronto Press.

Osborne, Martin and Ariel Rubinstein. 1994. A Course in Game Theory. Princeton: Princeton University Press

Rasmusen, Eric. 1989. Games and Information: an Introduction to Game Theory. Oxford: Basil Blackwell.

Selten, Reinhardt. 1965. Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit." Zeitschrift für die gesamte Staatswissenschaft 12:301-324.

Taylor, Michael. 1976. Anarchy and Cooperation. New York: Wiley.

Taylor, Michael. 1987. The Possibility of Cooperation. Cambridge: Cambridge University Press.

von Neumann, John and Oskar Morgenstern. 1943. Theory of Games and Economic Behavior. New York: Wiley.
 

Choose some real-world political situation and represent it as a game in extensive form. Carefully explain the choices you have made in constructing the game: the players, the action sets, the sequence of play, the information sets, and the payoffs. Is the game solvable by backward induction? Does it have a Nash equilibrium in pure strategies? In mixed strategies? Does it have a Nash equilibrium that is not subgame perfect? Be prepared to answer questions about how your game would change if different modelling decisions were made. Prepare a presentation of about 10 minutes in length, plus 5 minutes for questions, and present your work in class February 12. If you wish, you can turn in written material to supplement your presentation. In-class handouts or transparencies are welcome.

Locate a journal article that uses a game-theoretic model to analyze a subject of interest to you. Good places to look are the American Political Science Review, International Studies Quarterly, the Journal of Conflict Resolution, the American Journal of Political Science, Rationality and Society, and the Journal of Theoretical Politics. Carefully read the article so that you understand the model and its workings, assumptions, and implications. Come see me to discuss the article and model. Prepare a presentation of about 15 minutes in length, plus additional time for questions, in which you:

-explain the purpose of the article

-explain the model and assumptions

-discuss the equilibrium

-discuss how the model might be altered and what effects the changes would have on the equilibrium.

Deliver your presentation in class on March 5. This assignment will be much more fun for everyone if you choose an article that has a slightly complicated game model—something other than simply a one-shot prisoners’ dilemma, for example. If the paper you choose has features that you can’t decipher, come see me and I will try to help you clear up the confusion.

This assignment recapitulates assignments 1 and 2, but with two main differences. First, you may use any sort of model that suits your needs: strategic form, extensive form, incomplete-information, spatial voting, or what have you. Second, your work should be more like a professional research paper than in the first two assignments, which resemble exercises. Choose a political situation or class of political phenomena that interests you. Briefly survey the main literature on the subject, including not only the formal literature but also informal analysis. Devise a model that captures interesting aspects of the subject you have chosen, preferably in a way that contributes to the literature—a new finding, or a new variable, or a new wrinkle in the game structure, or the addition of incomplete information in a situation that has been viewed as a complete-information problem, etc. Prepare a presentation of 15-25 minutes in which you clearly discuss the subject, the literature, and your model. Deliver the presentation in class on April 23. Be prepared for questions. Hand in a paper that presents the subject, the literature, and your model/research.

Note: this assignment asks you, in effect, to do a research project using the tools we have studied in this course. This is not the same thing as asking you to write a paper that is capable of being accepted for publication by the American Political Science Review. Try to come up with an interesting problem and a game model that puts to work some of what you’ve learned in this course. Try to do so in a way that contributes something new. It need not be revolutionary.