**Spring 2020**

## Course calendar and assignments

##### jump to CURRENT assignment

### Week 2 Jan. 22

Introduction to Evolutionary Game Theory

- Larry Samuelson, “Evolution and Game Theory.”
*Journal of Economic Perspectives*, Vol. 16, No. 2 (Spring, 2002), pp. 47-66. A very gentle introduction to ESS and dynamics. Click here to obtain from JSTOR - Required reading: Larry Samuelson,
*Evolutionary Games and Equilibrium Selection*(MIT Press, 1997), Chapter 2. Available in the shared readings folder.- In the subsection beginning on p. 52 ("Neutrally Stable Strategies") you need not pause to verify Remark 2.2. The point is simply that although none of the Nash equilibria are ESS, some are NSS.
- You may skip the subsections beginning on p. 54, p. 57, and p. 59.
- Presenters:
- Evolutionarily Stable Strategies: Afiq
- discrete-time replicator dynamic: Jeremy
- continuous-time replicator dynamic: Tony

- Homework: ESS and replicator dynamics in a Hawk-Dove game. Before class, please upload your answers to the shared turn-in folder.

### Week 3 Jan. 29

Behavioral Game Theory and Evolution: Rejection in the Ultimatum game

- John Gale, Kenneth G. Binmore, and Larry Samuelson, “Learning To Be Imperfect: The Ultimatum Game.”
*Games and Economic Behavior*Vol. 8 (1995), pp. 56-90. Use the copy in the shared readings folder, which marks such things as:- Erratum: on p. 75, Fig. 1(b), the column labels on the strategic form of the Ultimatum Minigame should be Y and N, rather than H and L.
- Don't worry too much about the proofs to Propositions 2 and 3

- Presenters
- Sections 1 and 2 (Yeon)
- Section 5 (Ryan)
- Section 6 (Ben)

- (We'll all discuss Sections 3-4 as well.)
- Homework: an ultimatum mini-game.
- Some software for plotting phase diagrams such as these:
- Mirzaev et al., "egtplot: A Python Package for Three-Strategy Evolutionary Games."
*Journal of Open Source Software*Vol 3, No. 26 (2018), p. 735 (4 pages). Online. - Gebele and Staudacher, "Using EvolutionaryGames." (11/26/2017) Vignettes for their R package, EvolutionaryGames. Online.
- Izquierdo, Izquierdo, and Sandholm, "EvoDyn-3s: A Mathematica computable document to analyze evolutionary dynamics in 3-strategy games."
*SoftwareX*Vol. 7 (Jan.-June 2018), pp. 226-233. Online.

- Mirzaev et al., "egtplot: A Python Package for Three-Strategy Evolutionary Games."

### Week 4 Feb. 5

Modeling Beliefs: The Common Priors Assumption

- Stephen Morris, “The Common Prior Assumption in Economic Theory.”
*Economics and Philosophy*Vol. 11, no. 2 (1995), pp. 227–53. A general nontechnical, but sophisticated, survey of the issues. Skim 4.3, 4.4, 5.2, 5.3. Available from Morris’s website - Robert Aumann, “Agreeing to Disagree”
*Annals of Statistics*Vol. 34 No. 6 (Nov. 1976), pp. 1236-39. Click here to obtain via JSTOR. Try to understand what basic claim this paper makes, about the inability of players in a game of incomplete information to "agree to disagree." - Robert Aumann, “Correlated equilibrium as an expression of Bayesian Rationality.”
*Econometrica*Vol 55 No. 1 (Jan. 1987), pp 1-18. Click here to obtain via JSTOR.- Maybe a good idea to read through the discussion in sections 4-6 (pp. 8-17) before going back and trying to decipher Aumann's formal model. sections 2-3 (pp. 3-8).
- Of central interest to us: Sec. 5's remarks about the common priors assumption (CPA).
- The overall result here seems related to the claim made in Aumann (1976), recommended above.

**read to get the claim:**Faruk Gul, “A Comment on Aumann’s Bayesian View.”*Econometrica*Vol. 66 No. 4 (Jul. 1998), pp. 923-27. Click here to obtain via JSTOR. Try to get the gist of Gul's basic characterization and criticism of Aumann 1987.- Robert J. Aumann, “Common Priors: A Reply to Gul.”
*Econometrica*, Vol. 66, No. 4 (Jul., 1998), pp. 929-938. Click here to obtain via JSTOR. Helpful (at least partial) clarification of the discussion that has been somewhat mysterious from Aumann (1976) through Gul (1998). Do the best you can with the core formal treatment in Section 4. - Presenters
- Aumann (1976): Reese
- including HW problems 2 & 3

- Aumann (1987) except section 5: Ben
- including HW problem 1

- Aumann (1987) section 5 and Aumann (1998) clarification: Jeremy

- Aumann (1976): Reese
- Homework: Problems on common prior beliefs. These problems offer elementary applications of the ideas of correlated equilibrium, information partitions, and common prior beliefs. Given some basic definitions, you could probably benefit from trying the exercises prior to doing the readings. So here are some basic definitions:
- A
*correlated equilibrium*is a probability distribution over outcomes (that is, over pure-strategy profiles) of a normal-form game. The probability distribution is equivalent to a set of instructions to players about how to condition their actions on the result of some publicly observed randomization, such as a public coin flip. It is an "equilbrium" in that, for each possible outcome of the public randomization, each player, assuming that all other players will adhere to the instructions, maximizes her expected payoff by also adhering to the instructions.- A correlated equilibrium of a normal-form game G is a subgame perfect equilibrium in the extensive-form game consisting of (1) Nature chooses the result of the public randomization, and then (2) the players, knowing that result, play a single iteration of G.

- Let S be any nonempty set. A
*partition*of S is a set P = {S_{1}, ... S_{k}} of nonempty subsets of S that is (1) mutually exclusive and (2) exhaustive. That is, (1) every pair of subsets has empty intersection; and (2) the union of the subsets is equal to S.- Example: S = {1, 2, 3, 4} and P = { {1}, {2, 3, 4} }.
- We can refer to the subsets making up the partition as the "elements," "atoms," "members," or "components" of the partition.

- Let Ω be a state space, the set of possible values of a random variable. An
*information partition*for an individual describes how much that individual -- i, say -- gets to learn in advance about the true outcome ω, in terms of a partition P_{i}of Ω. Specifically, individual i will receive a signal indicating which partition element S_{ij}in P_{i}contains ω. - Bayesian updating according to an information partition: Suppose i has prior beliefs described by a probability mass function p(ω) on Ω. Consider each element S
_{ij}of i's information partition P_{i}as an event in Ω, having probability f(S_{ij}). Then upon receiving her signal that the true value ω is in S_{ij}, the individual can use Bayes's rule to update her beliefs to f( ω | f(S_{ij}) ).

- A

### Week 5 Feb. 12

Network games

- Andrea Galeotti, Sanjeev Goyal, Matthew O. Jackson, Fernando Vega-Redondo, and Leeat Yariv, "Network Games."
*Review of Economic Studies*, Vol. 77, Issue 1 (Jan. 2010), pp. 218-44. Click here to obtain via JSTOR- In 3.3 The Bayesian games read carefully to understand the setup; not proofs
- Thereafter we won't delve much into the details of proofs, but will try to understand results

- Matthew O. Jackson and Asher Wolinsky, "A Strategic Model of Social and Economic Networks."
*Journal of Economic Theory*, vol 71, No. 1 (1996), pp 44-74. Available in the shared readings folder.- Through sec. 3.1.2 (ending on p. 54) only, including all examples.

- Homework: Problems on network games. You will need to read at least part-way into the assigned articles before addressing these problems. Before class, please upload your answers to the shared turn-in folder.
- Presenters
- Galeotti examples and all material before 3.3: Afiq
- Include HW #1,2

- Galeotti remainder of paper, set-up and results only: Yeon
- Assigned portion of Jackson & Wolinsky: Tony
- Include HW #3

- Galeotti examples and all material before 3.3: Afiq

Possible unfamiliar math terminology from Galeotti:

- Increasing/decreasing differences and strategic complements/substitutes (226-27); degree complementarity (231); refer back to EBdM & Ashcroft paper on "Monotone Comparative Statics"
- Support of a distribution (p. 230): supp(f) is the set of x-values for which f(x) > 0.
- First-order stochastic dominance ("FOSD"): a cdf F first-order stochastically dominates another cdf G if F(t) ≤ G(t) for every t.
- Intuitively, if X~F and Y~G, this means X tends to be larger than Y.
- Lemma: Let X,Y be random variables with distributions F,G. If F first-order stochastically dominates G, then Eu(X) ≥ Eu(y) for any increasing function u.

- Stochastic affiliation: a strong form of positive correlation (which, recall, = Cov(X,Y)/(Var X Var Y)) between random variables. X and Y are affiliated if their joint pdf f is log-supermodular.
- Intuitively: conditional on a high (low) outcome of X, the prob of a high (low) Y is increased. Tells us a lot about the conditional distribution.
- For twice-differentiable f, this means the mixed second partial of ln(f) is nonnegative.
- If x ≤ x’ and y ≤ y’ then affiliation implies f(x’, y) f(x, y’) ≤ f(x’, y’) f(x, y); with some rearrangement, this means the likelihood ratio is increasing in x (MLRP)

- Handy online reference for FOSD (p. 2) and affiliation (p. 10-11): Dan Quint, lecture notes from Advanced Micro Theory (U of Wisc.; Sept 23, 2008).

### Week 6 Feb. 19

Participation: Voting, Protest and Collective Action

- Thomas R. Palfrey and Howard Rosenthal, “Voter Participation and Strategic Uncertainty.”
*American Political Science Review*, Vol. 79, No. 1 (Mar., 1985), pp. 62-78. Click here to obtain via JSTOR - Susanne Lohmann, “A Signaling Model of Informative and Manipulative Political Action.”
*American Political Science Review*Vol. 87, No. 2 (Jun., 1993), pp. 319-333. Click here to obtain via JSTOR - Homework: Contributions to a threshold public good. Before class, please upload your answers to the shared turn-in folder.
- Try these problems prior to doing the readings -- they're designed to prepare you for what is to come.
- An additional hint: problems 2 and 3 ask you to "Give a formula that determines the equilibrium..." The formula in question will involve the binomial probability function, and might be impossible to solve explicitly for the strategy parameters describing the equilibrium.

Presenters:

- Palfrey and Rosenthal:
- Reese: HW problems; set up; state the complete information result (from earlier P&R paper), pp. 64-66
- Afiq: Results—concentrate on Theorem 1 & what’s involved in proving it. At least set up and state the subsequent results.

- Lohmann:
- Jeremy: Model and Proposition 1 (322-24)
- Tony: Props. 2 & 3 (324-28)

### Week 7 Feb. 26

Behavioral Game Theory II: Limits of Strategic Thinking

- Colin F. Camerer, Teck-Hua Ho and Juin-Kuan Chong. “A Cognitive Hierarchy Model of Games.” Quarterly Journal of Economics Vol. 119, No. 3 (Aug., 2004), pp. 861-898. Click here to obtain via JSTOR
- in Sec. IV.A. of their article, Camerer et al. report on a number-guessing game and analyze the data of numerous experiments (Table II). In August 2015 the
*New York Times*column "The Upshot" presented an identical number-guessing game as a no-prize contest for its readers. The 56,000 responses are nicely graphed in this article and analyzed in terms of Camerer et al.'s scheme of "k-step" thinking. (The original statement of the Times contest was here.)

- in Sec. IV.A. of their article, Camerer et al. report on a number-guessing game and analyze the data of numerous experiments (Table II). In August 2015 the
- James Andreoni and John H. Miller, "Rational Cooperation in the Finitely Repeated Prisoner’s Dilemma: Experimental Evidence."
*Economic Journal*vol. 103, issue 418 (May 1993) pp. 570-585. Click here to obtain via JSTOR. - Available in the shared readings folder.

Homework: Strategy in number-guessing games. You can do this before starting the readings. Before class, please upload your answers to the shared turn-in folder.

Presenters:

- Yeon: Camerer. Hierarchy of rationality. How the different experiments test this. What they find.
- Ben: Andreoni & Miller. What factors distinguish their various designs? Conclusion re: laboratory vs. theory.

### Week 8 Mar. 4

Candidate Entry in Electoral Competition

- Palfrey, Thomas R., “Spatial equilibrium with entry.”
*Review of Economic Studies*Vol 51 Issue 1 (Jan. 1984), 139-156. Click here to obtain via JSTOR. - Martin J. Osborne and Al Slivinski, “A Model of Political Competition with Citizen-Candidates.”
*Quarterly Journal of Economics*Vol. 111, Issue 1 (Feb. 1996), pp. 65–96. Click here to obtain via JSTOR.- We will focus on Section III, the results concerning plurality elections.

**Homework exercise:**

- For Example 1 in the Palfrey article (p. 142-43) use the model’s terms W
_{A}, W_{B}, W_{C}etc. to verify the claims about the election outcomes. That is, verify that- For all θ
_{C}between 1/4 and 3/4, V_{C}= 1/4 - C always loses (that is, for θ
_{C}< 1/4, V_{C}is no more than than 1/4 ) - If C enters randomly between 1/4 and 3/4, then each of A and B wins with probability 1/2 and has an expected vote total of 3/8.

- For all θ
- Before class, please upload your answers to the shared turn-in folder.

Presenters:

- Ryan: Palfrey
- Benjamin: Osborne & Slivinski

##### ******************************************

Spring Break (extended to two weeks by WU)

-- no class Mar. 11 or Mar. 18

******************************************

### Week 9 Mar. 25

Institutions and Coalitions as Equilibria

- skim to get the general idea: Dennis Epple and Michael H. Riordan, “Cooperation and Punishment under Repeated Majority Voting.”
*Public Choice*Vol. 55, No. 1/2 (Sep., 1987), pp. 41-73. Click here to obtain via JSTOR. - Kathleen Bawn,"Constructing ‘Us': Ideology, Coalition Politics, and False Consciousness."
*American Journal of Political Science*43 (1999): 303-34. Click here to obtain via JSTOR. - Randall L. Calvert, "Rational Actors, Equilibrium, and Social Institutions." In Jack Knight and Itai Sened, eds.,
*Explaining Social Institutions*(University of Michigan Press, 1995; revised for paperback edition, 1998). Available in the shared readings folder.

"Presenters" (technical précis)

- Jeremy: Calvert paper
- Reese: Bawn paper

Non-presenters can write their prose descriptions about either one of these papers. DON'T focus too much on the lengthy applications-and-conclusions sections at the ends of both these papers.

- Both the technical précis and the prose descriptions should be turned in to the new shared file Class Materials for Circulation by noon Wednedsay.
- Then, all students should download and read these in preparation for our 3:00 class discussion.

Homework: Repeated Majority Voting. You can do this before starting the readings. Before class, please upload your answers to the shared turn-in folder.

### Week 10 Apr. 1

Global Games

- Stephen Morris and Hyun Song Shin, “Global Games: Theory and Applications.” (2003) In Dewatripont, Hansen, and Turnovsky, eds.,
*Advances in Economics and Econometrics*(Cambridge Univ. Press, 2003), pp. 56-114. Sections 1 and 2, pp. 56-77. Available in the shared readings folder. - Carles Boix and Milan W. Svolik, “The Foundations of Limited Authoritarian Government: Institutions, Commitment, and Power-sharing in Dictatorships'”
*Journal of Politics*Vol. 75, No. 2 (Apr. 2013), pp. 300-316. Click here to obtain via JSTOR.

Presenters: Note that we will begin by discussing the homework problem; in your techical précis you may assume this material has been digested, and use its results without explanation if convenient.

- Tony: Boix and Svolik
- Ben: Morris & Shin sections 2.1 (detail) and 2.2 (describe and relate generalizations)

For non-presenters, your prose description may take either the Boix and Svolik article or the whole of Section 2 in Morris and Shin as its topic.

- Both the technical précis and the prose descriptions should be turned in to the new shared file Class Materials for Circulation by noon Wednedsay.
- Then, all students should download and read these in preparation for our 3:00 class discussion.

Homework: A Global Game. You can do this problem before starting the readings. The game and analysis are adapted from section 8.2 of Gehlbach, *Formal Models of Domestic Politics*, which you may feel free to consult. Before class, please upload your answers to the shared turn-in folder.

### Week 11 Apr. 8

Evolution of Preferences

- Recommended for Informal Overview
- Larry Samuelson, “Introduction to the Evolution of Preferences.” Journal of Economic Theory Vol 97 Issue 2 (Apr. 2001), pp. 225-230. Available in the shared readings folder.

- Kaushik Basu, “Notes on Evolution, Rationality, and Norms,”
*Journal of Institutional and Theoretical Economics*152 (1996): 739-50. Available in the shared readings folder.- In Section 3, Basu claims that in the Hawk-Dove-Man game, the mixed strategy p (half H, half D) is an ESS as it was in the original Hawk-Dove game. I believe that's false, without a very specific and weird assumption about what "Man" (that is, strategy R) does when he is matched up with a p-playing opponent. Does this effect Basu's ultimate conclusion?

- Rajiv Sethi and E. Somanathan, "Preference Evolution and Reciprocity."
*Journal of Economic Theory*Vol 97 Issue 2 (Apr. 2001), pp. 273-297. Available in the shared readings folder.- Note apparent typo: in expression (7) p. 280 ">" should be "<"
- Once they get into proving their Propostions, Sethi and Somanathan neglect to point out very clearly where they are relying on the material payoff π and where they rely on the actual player payoffs u. Be sure you understand the roles of both functions. Is their overall approach the same as Basu's?

Techical précis presenters:

- Reese: Basu
- Afiq: Sethi & Somanathan

Non-presenters may turn in prose reviews/reaction papers either on Basu or on Sethi & Somanathan.

- Both the technical précis and the prose descriptions should be turned in to the new shared file Class Materials for Circulation by noon Wednedsay.
- Then, all students should download and read these in preparation for our 3:00 class discussion.

In place of a **homework problem**, ALL students should provide an extra half-page (with your technical précis or review) describing the most important differences between the two main papers.

### Week 12 Apr. 15

Fixed Points and Supermodularity

- John Nachbar, "Fixed Point Theorems." WUStL Dept of Economics online notes, 17 pages (Oct. 10, 2017). Available in the shared readings folder.
- You don't have to master the proofs at this point, but definitely go through this carefully enough to appreciate the different fixed point theorems and the different environments in which they apply. Especially important to us: Brouwer's, Kakutani's, and Tarski's theorems.
- Skip Section 2.2 Homeomorphisms and the Fixed Point Property.
- Don’t get too wrapped up in the proof of Theorem 3 (a version of the Brouwer theorem).
- Skip Section 2.5 The Eilenberg-Montgomery Fixed Point Theorem.

- Paul Milgrom and Chris Shannon, "Monotone Comparative Statics."
*Econometrica*, Vol. 62, No. 1, (Jan., 1994), pp. 157-180. Click here to obtain via JSTOR.- Skip the subsection on “Dissection” (166-169) and all of Sec. 4.
- Special attention to the definitions at the beginnings of Sec. 2, 3, 5; and to quasisupermodular functions (p. 162)
- In Section 2, don't worry about Theorems 1, 2, 3; special attention to Theorem 4 and its corollaries.
- In Section 3, special attention to Theorems 5, 6
- In Section 5, special attention to Theorems 12, 13

Prerequisite concepts, some of which are briefly defined in Milgrom-Shannon:

**partial ordering**(bottom of p. 158, and footnote 2)- A relation between pairs of elements of a set that is transitive, reflexive, and antisymmetric.
- Example: weak set inclusion ("is a subset of") on a set of sets.
- A complete order, such as greater-than-or-equal-to on a set of numbers, which has those three properties, and is also complete rather than merely partial, is also considered a "partial ordering".

**upper semicontinuity**. Suppose f is a real-valued function.- Formally, f is upper semi-continuous at x
_{0}if for every y > x_{0}there is a neighborhood U of x_{0}such that f(x) < y for every x in U. - Less formally, f has discontinuities but only of the following type: f may jump up at a point x
_{0}, but whenever it does, its value exactly at x_{0}is the "upper" value at the jump. - A continuous function is also considered to be upper semi-continuous.

- Formally, f is upper semi-continuous at x
**topology**: A topology on a set is the collection of its subsets that are designated as "open sets".- The collection must meet certain properties to qualify: for example, every countable union and every finite intersection of open sets must also be open.
- Definitions of convergent sequences, boundaries, continuous functions, and other concepts can be stated in their most general forms in terms of relationships among open sets. These are known as "topological definitions" of those properties. Topological definitions are general in the sense that they don't necessarily depend on notions of distance or length; but in most applications that we encounter, distance or length is natural, and there may be no reason to avoid them.
- In the language of topology, a "neighborhood" of a point x is any set containing an open set that in turn contains x. So x is definitely not on a boundary but rather interior to the neighborhood. When a concept of distance is involved, an "open ball" centered at x is a trusty example of a neighborhood.

**compactness**(p. 159, including footnote 3):- a set S is compact if every infinite sequence of points within S has a limit point also within S.
- (A limit point is just the limit of a convergent subsequence; and, by implication, this definition requires that any sequence in S must have a convergent subsequence.)

- For sets of real numbers or real vectors, this is equivalent to the set being closed (includes its boundary) and bounded (doesn't run off to infinity in any direction).
- Compactness is important to us because an upper semi-continuous function (such as a payoff function) on a compact, convex set (such as a strategy set) always attains a maximum on that set. Otherwise, who knows?
**Compact lattice**(mentioned on p. 175) just means a lattice whose set of elements is compact in some relevant topology. In most cases of interest, we are just talking about the usual topology of open and closed sets for real numbers or vectors.- In homework problems 1 and 3, the lattice is compact under the usual topology.
- In homework problem 2, because the set is finite, it turns out to be easy to define a topology on it under which it too is a compact lattice---just designate every subset as "open".

- a set S is compact if every infinite sequence of points within S has a limit point also within S.

Written summaries to circulate

- Afiq, Reese: Your mostly-prose response papers should focus on the Nachbar notes: the idea of fixed point theorems, and the setting and statements of the Brouwer, Kakutani, and Tarski theorems.
- Tony: technical précis focusing on the main developments in Sec. 2 of Milgrom and Shannon: development of lattices, quasisupermodularity, and monotone comparative statics.
- Yeon: technical précis focusing on the main developments in Sec. 3 of Milgrom and Shannon, especially on Theorems 5 and 6.
- Ben: technical précis focusing on the results of section 5
- Then, all students should download and read these in preparation for our 3:00 class discussion.

Homework: Lattices. To get you thinking productively about the sort of animal under study here. These questions use the definitions in the first few pages of Section 2 in Milgrom and Shannon. Before class, please upload your answers to the shared turn-in folder.

### Week 13 Apr. 22

Democracy, Elections, and Backsliding

- Andrew Little, Joshua A. Tucker, and Tom LaGatta, "Elections, Protest, and Alternation of Power."
*Journal of Politics*Vol. 77, No. 4 (Oct. 2015), pp. 1142-1156. Click here to obtain via JSTOR. - Zhaotian Luo and Adam Przeworski, "Democracy and Its Vulnerabilities: Dynamics of Democratic Backsliding." Working paper (Nov. 24, 2019). Click here to obtain online via SocArXiv Papers.

Techical précis presenters:

- Afiq: Little et al.
- Reese, Yeon: Luo & Przeworski

Non-presenters may turn in prose reviews/reaction papers on either article.

- Then, all students should download and read these in preparation for our 3:00 class discussion.

In place of a **homework problem**, ALL students should provide an extra half-page (with your technical précis or review) describing the most important differences between the two assigned papers.

### Note on due date for final paper

If you turn in your final paper before the morning of Sunday, May 3, I promise to hand in your grade on time, that is by end-of-business Thursday, May 7. Students turning in papers between those two dates may receive a temporary Incomplete, which I will resolve as my first priority (that is, within a few days).

Papers turned in after that, but before August 1, will be graded ASAP, and the recorded Incomplete will be replaced before the Fall semester begins. For papers turned in after August 1, I will finalize the course grade as soon as practical, but no promises. The Fall semester gets very busy.