**Spring 2016**

This course provides an introduction to noncooperative game theory and its application within political science. The first half cover basic tools with applications: games in strategic and sequential form, Nash equilibrium, subgame-perfect equilibrium, and Bayesian equilibrium. The second half of the course will treat important special cases and complications, including bargaining, repeated games, learning or evolution in games, and communication games.

## Course Outline and Approximate Schedule

## 0. Introduction

Read: Osbrone, Chapter 1 and sections 4.1.3, 4.12

- Game Theory
- Elements of strategic form
- The nature of "payoffs"

## 1. Strategic Games and Nash Equilibrium

Covers chapters 2-4 in Osborne textbook

Assignment for Thursday 1/21: read and try all problems in Chapter 2, through at least p. 40

Real-life experiment: guessing 2/3 of the average (from Exercise 34.1)

Assignment for Thursday 1/28: finish Chapter 2; start section 3.3.

**Homework assignments:**

**due Tuesday 1/26**: problems 34.3 (public solution given), 42.2, 48.1**due Tuesday 2/2**: problems 44.1, 49.1, 50.2**due Tuesday 2/9**: problems 74.1, 79.2, 89.1 (public solution given), 91.1**due Tuesday 2/16:**problems 114.2, 127.2, 132.3, 141.1

Through Feb. 4 in class: We'll cover sections 3.3, 3.4, and most of 3.5

Feb 9, 11: Chapter 4

- 4.1.1 and 4.1.2 (we've already discussed the other material in 4.1, but you might want to review this, along with 4.12)
- at least through 4.6 by Tuesday 2/9
- through 4.11 by Tuesday 2/16 (no need to work problems in Sec. 4.11)
- Articles mentioned in class (not required):
- George Tsebelis, "The Abuse of Probability In Political Analysis: The Robinson Crusoe Fallacy."
*American Political Science Review*, Vol. 83, No. 1 (Mar., 1989), pp. 77-91. Click here to obtain via JSTOR. - Thomas R. Palfrey and Howard Rosenthal, "A Strategic Calculus of Voting."
*Public Choice*, Vol. 41, No. 1 (1983), pp. 7-53. Click here to obtain via JSTOR. - John F. Nash, "Equilibrium Points in n-Person Games."
*Proceedings of the National Academy of Sciences*, Vol. 36, No. 1 (Jan. 15, 1950), pp. 48-49. Click here to obtain via JSTOR.

- George Tsebelis, "The Abuse of Probability In Political Analysis: The Robinson Crusoe Fallacy."

**Review and first exam: Feb 16, 18**

(no help session Feb. 19)

## 2. Sequential Games and Subgame-Perfect Equilibrium

Covers chapter 5-7 in Osborne textbook; approximately Feb 23-Mar. 3

For Feb. 23: chapter 5 (all; we'll get at least through the first part of 5.5)

For Feb. 25: Finish exercises in 5.5; Sections 6.1 and 6.3

**HW due Tuesday Mar. 1:**

- problems 163.2, 173.3, 183.2, 187.1
- In problem 173.3, ignore the portion of the problem beginning with "Consider variants of the game..." up through the end.
- save until next week: 196.2

For Mar. 1: Sections 7.1, 7.3, 7.4

Slide show: Backward induction with simultaneous moves (pdf)

For Mar. 3: Section 7.6, 7.7

**HW due Tuesday Mar. 8:**

- problem 196.2
- problems 221.1, 221.2
- problem 228.1

**Not required:** Does the ultimatum game indicate innate altruism? Evidently not: the variations, especially the double-blind dictator game, examined in this paper:

- Elizabeth Hoffman, Kevin McCabe, Keith Shachat, Vernon Smith, "Preferences, Property Rights, and Anonymity in Bargaining Games."
*Games and Economic Behavior*, Vol. 7, Issue 3 (Nov. 1994), pp. 346-380. Click here to obtain via Science Direct on-line.

**Not required:** A paper on cross-cultural variation in experiments based on the ultimatum game: this project, the ultimatum game (along with some related games) was administered in a variety of cultures, including relatively primitive ones. A nice summary reference, followed by many pages of short comments by numerous other scholars, is

- Joseph Henrich et al., "'Economic Man' in Cross-cultural Perspective: Behavioral Experiments in 15 Small-scale Societies."
*Behavioral and Brain Sciences*vol. 28 (2005), pp. 795-855. Click here to obtain via Cambridge Journals on-line.

Kenneth Binmore's commentary (pp. 817-818) is especially pertinent.

**Not required:** Using techiques from evolutionary game theory, this paper shows how "rational" players might arrive at a subgame-imperfect equilibrium:

- John Gale, Kenneth G. Binmore, and Larry Samuelson, "Learning To Be Imperfect: The Ultimatum Game."
*Games and Economic Behavior*, Vol. 8, Issue 1 (1995), Pages 56-90. Click here to obtain online.

## 3. Imperfect Information

For Mar 8, 10: sections 10.1 through 10.5

Illustration: weak sequential equilibrium vs. perfect Bayesian equilibrium

**HW due Tuesday Mar. 22:**

- problems 331.1, 335.1; and...
- problem 3: The Gift Game (not in Osborne)
- Player 1 is either a Friend type (probability p) or an Enemy type (probability 1-p), and knows his own type. Player 1 may give a gift (G) or not give a gift (N) to Player 2. If a gift is offered, then Player 2, knowing p but not knowing Player 1's actual type, must decide whether to accept (A) or reject (R) it. Payoffs are as follows:
- If no gift is offered: 0 to both players, regardless of 1's type.
- If a Friend's gift is accepted, each receives 1; if rejected, Player 1 gets -1 and Player 2 gets 0.
- If an Enemy's gift is accepted, Player 1 gets 1 and Player 2 -1; if rejected, Player 1 gets -1 and Player 2 0.

- Find the game's weak sequential equilibria (WSE), being sure to specify the beliefs in every WSE assessment.
- HINT: Let f = Prob(Friend plays G) and e = Prob(Enemy plays G). Can there be pooling equilibria, that is, ones in which f = e? Can there be separating or semi-separating equilbria, that is, ones in which f and e are different?

- Player 1 is either a Friend type (probability p) or an Enemy type (probability 1-p), and knows his own type. Player 1 may give a gift (G) or not give a gift (N) to Player 2. If a gift is offered, then Player 2, knowing p but not knowing Player 1's actual type, must decide whether to accept (A) or reject (R) it. Payoffs are as follows:

**Spring Break, Mar 14-18**

**Review and second exam: Mar. 22-24**

## 4. More Specialized Tools

(About one week for each subtopic below; five weeks total)

### 4.1 Communication

Strategic information transmission: Osborne: sections 10.7-10.9

**HW due Tuesday Apr. 5:**

- problems 342.1, 350.2

Cheap talk and coordination:

**For Tuesday Apr 5: read**

- Joseph Farrell, "Cheap Talk, Coordination, and Entry."
*RAND Journal of Economics*, Vol. 18, No. 1 (Spring, 1987), pp. 34-39. Click here to obtain online via JSTOR.

**HW due Tuesday Apr. 12:**

- Consider the 2-player, symmetric Hawk-Dove game in which the payoffs are as follows: u(D,D) = 0 for both players; u(H, H) = -1 for both players; and u(H,D) = X > 1 for the H-player and -Z, with 0 < Z < 1, for the D-player. Conduct the same analysis as in Farrell's paper for the game in which Hawk-Dove is preceded by one period of communication, and assess substantively the role of communication in outcomes of this game.

Reference concerning a method of perfect coordination in the Farrell setting: Robert J. Aumann and Michael B. Maschler, *Repeated Games with Incomplete Information*. MIT Press (1995). (The relevant material is in Chapter 5, "Repeated Games of Incomplete Information: An Approach to the Non-Zero-Sum Case," which originated as a 1968 report on a grant project to the U.S. Arms Control and Disarmament Agency.)

**Class will not meet on Thursday Apr. 7 due to the Midwest Political Science Association meetings.** We will make this up by holding a final review session during the reading period at the end of the semester.

### 4.2 Repeated Games

April 12 & 14: material from Osborne

- chapter 14 in detail
- chapter 15: read enough to understand the statement of Proposition 458.2

**HW due Tuesday Apr. 19:**

- problem 429.1, 431.1 and the following "exercise 3":
- Exercise 3: For an infinitely repeated PD game, define the following "Adjusted Tit for Tat" (ATFT) strategy.
- For each player, the strategy defines actions in, and movements between, three "states":
**Coop**(mutual cooperation),**Pun**(punish the opponent's deviation), and**Apol**(apologize for own deviation). A player begins the game in state**Coop**. - In state
**Coop**, a player is to play C; in state**Pun**, a player is to play D; in state**Apol**, a player is to play C. - Following a pair of actions in state
**Coop**,- the player moves to state
**Pun**if she played C while her opponent played D; - the player moves to state
**Apol**if she played D while her opponent played C; - otherwise the player remains in state
**Coop**.

- the player moves to state
- Following a pair of actions in state
**Pun**- the player moves to state
**Coop**if her opponent played C, regardless of her own action - otherwise, the player remains in state
**Pun**

- the player moves to state
- Following a pair of actions in state
**Apol**- the player moves to state
**Coop**if she played C, regardless of her opponent's action - otherwise, the player remains in state
**Apol**.

- the player moves to state

- For each player, the strategy defines actions in, and movements between, three "states":

3(a): Use the description of ATFT above to construct a strategy diagram in the style of those on p. 427.

3(b): Suppose the underlying PD game has the payoffs shown in Figure 429.1, with discount factor \delta. Use the methods of section 14.10.3 to derive the conditions on x, y, and \delta under which (ATFT, ATFT) is a subgame perfect equilibrium profile. Hint: In applying the methods of section 14.10.3, rather than "history ending in (C,D)," etc., use the possible pairs of player states (**Apol**, **Pun**, and **Coop**) as the situations in which to define possible deviations by a player from ATFT. Assuming that the player's opponent will always do what ATFT prescribes in all three states, and show that no single deviation by the player can produce a higher payoff than adhering to ATFT.

### 4.3 Legislative Bargaining

Read for Apr. 19: relevant material in Osborne: section 16.1

Read for Apr. 21:

- David P. Baron and John A. Ferejohn, "Bargaining in Legislatures."
*American Political Science Review*, Vol. 83, No.4 (Dec. 1989), pp. 1181-1206. Click here to obtain via JSTOR. - You can omit the sections on "Amendments: A Simple Open Rule" and "The Choice of an Amendment Rule," pp. 1195-99.

**HW due Tuesday Apr. 26:**

- problem 473.1
- Problem 2: Consider a Baron-Ferejohn-style legislative bargaining game with three legislators, in which all players have an equal probability of recognition. Suppose that Player 1's strategy says that, whenever recognized to propose, she should always allocate zero to Player 2. Likewise, Player 2 always proposes zero for Player 3, and Player 3 always allocates zero to Player 1. Could such a strategy profile be a SPE? Explain.
- Problem 3: Consider a Baron-Ferejohn-style legislative bargaining game with three legislators, in which Player 3 is never recognized to propose (that is, p
_{3}= 0), and Players 1 and 2 are recognized with equal probability (p_{1}= p_{2}= 1/2). Suppose that the players use strategies in which Player 1, if recognized, allocates a positive amount to Player 2 and zero to Player 3 with probability 1; and Player 2 does likewise, allocating a positive amount to Player 1 and zero to Player 3. Could such a strategy profile be an SPE? Explain.

Read for Tuesday Apr. 26:

- Jeffrey S. Banks and John Duggan, "A Bargaining Model of Collective Choice."
*American Political Science Review*Vol 93, No. 1 (Mar. 2000), pp 73-88. Click here to obtain via JSTOR. - Concentrate on the formulation of the model, and on examples 2, 3, and 4, pp. 73-78.
- I found these Technical Notes on the Banks & Duggan paper that I had made from a previous course. They might help clarify mathematical definitions that you can't quite recall.

### 4.4 Monotone Comparative Statics

Read for Thursday, Apr. 28:

- Scott Ashworth and Ethan Bueno de Mesquita, "Monotone Comparative Statics for Models of Politics."
*American Journal of Political Science*, Vol. 50, No. 1 (Jan. 2006), pp. 214-231. Click here to obtain via JSTOR.

**HW to be discussed at Monday May 2 help session (not turned in):**

**Problem 1.**Are any additional assumptions necessary for this game to satisfy the condition of strategic complementarities:

- Each action set is compact;
- Each payoff function is continuous in actions and in all parameters;
- Each payoff function has increasing differences in all pairs of actions and in each action paired with each parameter.

**Problem 2.**Suppose that, in place of the payoff function specified by Osborne, we simply assumed that the game has strategic complementarities.

- (a.) What could we then conclude about Nash equilibria in this game?
- (b.) How does this compare to the conclusions Osborne drew from his original example?
- (c.) What are some important properties of Osborne's payoff functions that were NOT necessary in order to get the results in part (a)?

For future reference on a more general approach to monotone comparative statics, consult this classic paper:

- Paul Milgrom and Chris Shannon, "Monotone Comparative Statics ."
*Econometrica*, Vol. 62, No. 1, (Jan., 1994), pp. 157-180. Click here to obtain via JSTOR.

**Review session Monday May 2, 1:00 pm (in our usual location)**

**Exam out May 2; return by Friday May 6**