PHIL 444 (Groups & Choices): Final Study Guide

Elise Woodard University of Michigan, Ann Arbor

Contents

1 Concept Review 3 1.1 Approaches to Multiple Equilibria ...... 3 1.2 Approval Voting ...... 3 1.3 Arrow’s Theorem ...... 3 1.4 Axelrod’s Tournament(s) ...... 3 1.5 Backward Induction ...... 4 1.6 Battle of the Sexes ...... 4 1.7 Borda Count ...... 4 1.8 Centipede ...... 5 1.9 Chicken ...... 5 1.10 Collectively Stable Strategy ...... 5 1.11 Common Knowledge ...... 6 1.12 Conciliationism ...... 6 1.13 Condorcet Jury Theorem ...... 6 1.14 Condorcet ...... 6 1.15 Correlation in Evolutionary Game Theory (EGT) ...... 6 1.16 Cumulative Voting ...... 6 1.17 Cut the Cake ...... 7 1.18 Discount Factor ...... 7 1.19 Equilibrium in Belief ...... 7 1.20 Evolutionary Dynamics ...... 7 1.21 Gibbard-Saterwaite Theorem ...... 7 1.22 Grand Junction Voting ...... 7 1.23 Instant Runoff Voting ...... 8 1.24 May’s Theorem ...... 8 1.25 Method of Pairwise Comparison ...... 8 1.26 Nash Equilibrium ...... 8 1.27 One Peak Voting ...... 8 1.28 Pavlov (Simpleton) ...... 8 1.29 Plurality Voting ...... 9 1.30 Properties of Tit-For-Tat (Axelrod) ...... 9 1.31 Prisoner’s Dilemma ...... 9 1.32 Prisoner’s Dilemma with a Twin ...... 9 1.33 Properties of Voting Systems ...... 9 1.33.1 Anonymity ...... 9 1.33.2 Condorcet Criterion ...... 10 1.33.3 Condorcet Loser Criterion ...... 10 1.33.4 Independence of Irrelevant Alternatives ...... 10 1.33.5 ...... 10 1.33.6 Monotonicity ...... 10

1 1.33.7 Non-Manipulability (Honesty) ...... 10 1.33.8 Neutrality ...... 10 1.33.9 Non-Dictatorship ...... 10 1.33.10 Transitivity ...... 10 1.33.11 Unanimity ...... 11 1.33.12 Unrestricted Domain ...... 11 1.34 Rationalizable Action ...... 11 1.35 Ratifiable Choice ...... 11 1.36 Salient Equilibrium (Focal Point) ...... 11 1.37 Schelling on Segregation ...... 11 1.38 Social Preference Ranking ...... 11 1.39 Steadfast ...... 12 1.40 Subgame Perfect Equilibrium ...... 12 1.41 Tit-For-Tat (Copy Cat) ...... 12

2 1 Concept Review

1.1 Approaches to Multiple Eqilibria

Lecture Week 5 Problem: in non-zero sum games, how should we choose between multiple equilibria? Three approaches: 1. Psychological: There is no solution within game formal theory. The solution is foundin contingent facts about human psychology. 2. Refinement: We can identify ‘structural’ features of equilibria that show some them to be more “rational” than others. 3. Epistemic: We need a theory of how people reason themselves into making various choices. We can only be able to tell where the players will end up if we know something about their ‘path to equilibrium.’ Players who start in different epistemic states will settle at different equilibria.

1.2 Approval Voting

Lecture Week Unlimited: Voters vote for as many candidates as they like without giving a ranking. 11 Limited: Voters vote for N candidates without giving a ranking A vote is supposed to mean that a candidate is ‘good enough’ or ‘acceptable.’ The candidate judged ‘acceptable’ by the most voters wins.

1.3 Arrow’s Theorem

Lecture Week No method of extracting social preferences from voter preferences satisfies all the following con- 12; SEP ditions: page/Elise’s notes • Unrestricted Domain: It applies to any set of ballots that the electorate might offer

• Transitivity of Social Preference: A ≥s B and B ≥s C implies A ≥s C • Unanimity: If ballots rank A above B then the social ordering does too • Independence of Irrelevant Alternatives: The social preferences for B over A is deter- mined entirely by the relative positions of A over B on all the ballots • No dictator: There is no individual who’s preference ranking coincides with the social rank- ing no matter what other individuals in the electorate might prefer.

1.4 Axelrod’s Tournament(s)

Lecture Week 7 Strategies are paired up w/ each other a large number of times. Strategy w most accured utiles at the end wins. Here every strategy plays every other strategy the same number of times. Strategies recall what happened in past rounds. Random mistakes are banned.

3 It does not matter that one strategy beats another when they play. All that matters is theaggregate score. TFT does well despite the fact it ties or loses to every opponent.

1.5 Backward Induction

Lecture Week 2 Method that can sometimes be used to find equilibria of extensive form games of perfect informa- tion. To use BI, it must be true that every terminal choice node, all players can identify a unique action that the player(s) choosing at that node will surely select. Assumes all future play will be rational. How BI can fail: • Indifference on other players’ part • Lack of perfect information See slides for examples.

1.6 Battle of the Sexes

Lecture Week 2 Background: Rowena and Columba want to see a concert. She likes opera, he likes chamber music. They want to be together whatever they hear. They have no prior agreement aboutwhat they will do, and act independently.

opera quartet Opera 3,2 0,0 Quartet 0,0 2,3

What are the equilibria? Start with row. If column goes to opera, then row will also prefer to go to opera (3>0). If column goes to the quartet, then row also prefers to go to quartet (2>0). Same reasoning applies when figuring out what column will want to do. In short, they want to coordinate. Compare: stoplights, coordinating on what side of road we drive on! Equilibria: (O, o), (Q, q), (.4O + .6Q, .6o + .4q) Pure equilibria bolded:

opera quartet Opera 3,2 0,0 Quartet 0,0 2,3

1.7 Borda Count

Lecture Week Give each candidate 1 point for every ballot on which they appear as the first choice, 2 points for 11 every ballot on which they appear as the 2nd choice, 3 points for every ballot on which they appear as the 3rd choice, and so on

4 Add up the points Candidate w/ fewest number of points wins.

1.8 Centipede

See Lecture Week 2 & Joyce, “The Role of Incredible Beliefs” Centipede is an extensive form game where two players take turns choosing to take a slightly larger share of an increasing pot (defect) or to pass the pot to the other player (cooperate). The payoffs are such that if one cooperates but then the other player defects, one would getaslightly smaller share of the pot than if they had previously defected. In the example below, the pot increases by $2 each time. The paradox is that a rational player that obeys the Common Knowledge Assumptions and satisfies backwards reasoning will defect on the first try! (and thus get 2.) This just shows how strong the common knowledge assumptions are. (Empirical studies showthat most of us do not defect immediately.)

Figure 1: Centipede Game 1 C 2 C 1 C 2 C… 1 C 2 C 100, 100

D D D D D D

2, 0 1, 3 4, 2 3, 5 100, 98 99, 101

Here’s a video: https://www.youtube.com/watch?v=S3pdNJ-69Z8

1.9 Chicken

Lecture Week 2 Rowena and Columba are driving at high speed toward one another. Neither wants to be the ‘chicken’ who swerves (right) when the other goes straight, but neither wants to crash. They have no prior agreement about what they will do, and act independently.

chicken hero Chicken 2,2 1,3 Hero 3,1 0,0

Equilibria: (H, c), (C, h), (.5C + .5H, .5c + .5h) (Pure equilibria bolded)

1.10 Collectively Stable Strategy

Lecture Week 8 A strategy is collectively stable when a population of individuals that are all playing it cannot be invaded by a ‘mutant’ strategy. This happens when the mutant strategy’s expected payoff isless than the average expected payoff for the population at large.

5 1.11 Common Knowledge

Lecture Week Common Knowledge Assumptions: 2; Joyce, “The 1. All players are rational Role of ‘Incredible’ 2. All players understand structure of game Beliefs” 3. All players know what the others believe & want 4. All players know (a)-(c) 5. All players know (d) Etc., ad infinitum

1.12 Conciliationism

Lecture Week The disagreement of others should typically cause one to be much less confident in one’s belief 13; Christensen than one would be otherwise — at least when those people seem like epistemic peers (2009)

1.13 Condorcet Jury Theorem

Lecture Week i“f voters (a) face two options, (b) vote independently of one another, (c) vote their judgment of 13; Anderson what the right solution to the problem should be (i.e., they do not vote strategically), and (d) have, (2006) on average, a greater than 50% probability of being right, then, as the number of voters approaches infinity, the probability that the majority vote will yield the right answer approaches 1.” (Anderson, pp. 10-11)

1.14 Condorcet Paradox

Lecture Week Suppose that Voter 1 preferences A > B > C, Voter 2 prefers B > C > A, and Voter 3 prefers C > 1; AOS ch.12 A > B. Then the group prefers A to B (2:1), B to C (2:1), but C to A (2:1). These preferencesare intransitive/cyclic.

1.15 Correlation in Evolutionary Game Theory (EGT)

Lecture Week 9 One strategy S is positively/negatively correlated with another strategy S* when the average num- ber of times a given S interacts with an S* is greater/less than the total number of S*s in the popu- lation. One strategy S is positively/negatively self-correlated when the average number of times a given S interacts with another S is greater/less than the total number of Ss in the population minus 1

1.16 Cumulative Voting

Lecture Week If there are N candidates each voter gets N votes that she can apportion to candidates as she sees 11 fit. A candidate may receive more than one vote from a given voter.

6 1.17 Cut the Cake

Lecture Week A simple bargaining game where we each independently specify a proportion of a cake between 0 8–9 and 1. If our ‘asks’ sum to 1 or less we both get what we ask for. If our ‘asks’ sum to more than 1 the whole cake is wasted. Any pair of asks (x, y) that sum to strictly less than 1 is not a Nash equilibrium. Only (x, y) pairs that are Nash equilibrium are efficient solutions: they use up all the cake.

1.18 Discount Factor

Lecture Week 8 Suppose that before two strategies can play PD a coin is tossed. The game is only played if the coin lands heads. The coin can have any bias toward heads between 0 and 1. This isthe discount factor The probability of 2 strategies playing n games against one another is wn. If discount factor is closer to 0, the players will play relatively few times; if it’s closer to 1, they will probably play many. w Expected # of meetings = (1−w)2

1.19 Eqilibrium in Belief

Lecture Week 6 One interpretation of mixed equilibria (Aumann:) mixing probabilities reflect players degree of belief (credence) about given player’s action

1.20 Evolutionary Dynamics

Lecture Week 9 Worst to Best: The lowest scoring strategies have their numbers cut in half. New players are split evenly among the highest scoring strategies. Worse to Better: The lowest scoring strategies have their numbers cut in half. New players are split evenly among all other strategies equally. Bayes Dynamics: Each strategy sees its numbers increase or decrease in direct proportion to the degree to which its uses exceed/fall short of the status quo, i.e. the average score in the population.

1.21 Gibbard-Saterwaite Theorem

Lecture Week Every deterministic voting system with at least three candidates that satisfies Unrestricted Domain 12 and Unanimity is either Manipulable or Dictatorial.

1.22 Grand Junction Voting

Lecture Week aka Bucklin Voting 12 1st round: If any candidate gets a majority of the first-place votes that candidate wins 2nd round, if needed: Add up all first and second place votes for each candidate. If any candi- date’s first+second place votes adds up to more than half the electorate, then the candidate with

7 the most first+second place votes wins. 3rd round, if needed: Add up all first, second, and third place votes each candidate gets. Ifthis= more than half the electorate, then candidate w most 1st-3rd place votes win Continue until you have a winner.

1.23 Instant Runoff Voting

Lecture Week Arrive at a winner by repeatedly deleting candidates that are “least preferred” in the sense of being 11 at the top of the fewest ballots

1.24 May’s Theorem

Lecture Week 9 For elections with only two candidates, Majority Rule is the only voting system that satisfies: • Anonymity • Neutrality • Monotonicity • Unanimity

1.25 Method of Pairwise Comparison

Lecture Week Compare every pair of candidates. Give the winner a point and the loser nothing. For ties, each 11 gets ¹⁄₂ point. The candidate with the most points wins.

1.26 Nash Eqilibrium

In a Nash Equilibrium, each player’s choice is a best response to what all the other players do— No player would want to/have an incentive to change her choice/strategy if she were told the choices/strategies of all other players. A Nash Equilibrium is a combination of 2 conditions (AOS, 131): 1. Each player is choosing a best response to what he believes the other player will do in the game. 2. Each player’s beliefs are correct. The other players are doing just what everyone thinks they are doing.

1.27 One Peak Voting

Lecture Week See ex. on Week 12, slide 16. Contrast Two Peak Voting. 12

1.28 Pavlov (Simpleton)

Lecture Week 7 If other player cooperated on the previous turn repeat your play from that turn. If she defected on the previous turn, reverse your play from that turn.

8 1.29 Plurality Voting

Lecture Week Candidate with most 1st place votes wins (need not be majority) 11

1.30 Properties of Tit-For-Tat (Axelrod)

Lecture Week 7 Nice: Cooperates on the 1st round Clear: Easy to understand and implement Provocable: Will defect after being defected upon Forgiving: not a grudger Non-envious: doesn’t try to lower the other player’s score

1.31 Prisoner’s Dilemma

Lecture Weeks defect cooperate 9–10; Week 1 Defect -9, -9 0, -10 Cooperate -10, 0 -1, -1

Defect strictly dominates for each player: each individual does better (1 fewer year in prison) by defecting whatever the other does. Pareto Superiority: (Cooperate, cooperate) is better for each player than (Defect, defect). Soif a player on their own could choose (C, c) the double cooperate outcome, they would, but they cannot! Note what CDT vs. EDT says Evidential : One should always choose the action that gives one the best evi- dence for thinking that the world is in a desirable state. Causal Decision Theory: One should always choose the action that is likely to causally promote the most desirable results.

1.32 Prisoner’s Dilemma with a Twin

Lecture Weeks PD where my action and the other player’s actions are correlated effects of a common cause (e.g. 9–10 bc we’re twins) See slides for more.

1.33 Properties of Voting Systems

Lecture Week All in Lecture Week 11 unless otherwise stated. 11

1.33.1 Anonymity

The identities of voters do not matter. If two voters change ballots, the overall outcome staysthe same.

9 1.33.2 Condorcet Criterion

Always choose the candidate that beats every other candidate in head-to-head elections (if there is such a candidate).

1.33.3 Condorcet Loser Criterion

Never choose the candidate that loses to every other candidate in head-to-head elections (if there is such a candidate).

1.33.4 Independence of Irrelevant Alternatives

If X wins an election against a given set of candidates, then X should also win against any proper subset of those candidates (with the same voters). 2nd formulation: The social preference ranking between two candidates should depend only on how they are ranked relative to each other on the ballots. It should not matter where other candidates are ranked.

1.33.5 Majority Rule

If one candidate captures an absolute majority of the first-place votes, then that candidate wins.

1.33.6 Monotonicity

Moving a candidate up in some individual’s ballot should never turn her from a winner into a loser. Moving her down in some individual’s ballot should never turn her from a loser into a winner.

1.33.7 Non-Manipulability (Honesty)

Voters should not have an incentive to submit a ballot that does not reflect their true preference ranking among the candidates.

1.33.8 Neutrality

Voters should not have an incentive to submit a ballot that does not reflect their true preference ranking among the candidates.

1.33.9 Non-Dictatorship

Lecture Week There is no individual who’s preference ranking coincides with the social ranking no matterwhat 12 other individuals in the electorate might prefer.

1.33.10 Transitivity

≥ ≥ ≥ Lecture Week A s B and B s C implies A s C 12

10 1.33.11 Unanimity

If everyone prefers candidate X to candidate Y, then candidate Y should not win.

1.33.12 Unrestricted Domain

Lecture Week Way of extracting social preference ranking applies to any set of ballots that the electorate might 12 offer.

1.34 Rationalizable Action

An act A is rationalizable when there exists an assignment of probabilities over profiles of other player’s acts relative to which A maximizes expected utility.

1.35 Ratifiable Choice

Lecture Week “Choose for the person you expect to be once you have chosen.” 10 V(B | dA) = the news value of act B once you decide to do act A. A is ratifiable if and only if V(A | dA) > V(B | dA). Maxim of Ratifiability: Only ratifiable acts may be chosen. “Ratificationism requires performance of the chosen act, A, to have at least as high an expected desirability as any alternative act B on the hypothesis that one’s final decision will be to perform A.” (Jeffrey)

1.36 Salient Eqilibrium (Focal Point)

Lecture Week A focal point equilibrium is one that stands out as particularly salient to all players, and that all 6, AOS ch. 4 players realize is particularly salient to others. (Salience is a psychological notion. And the notion of a focal point is due to Thomas Schelling.) Focal points may depend on cultural, historical, linguistic, etc. background

1.37 Schelling on Segregation

Lecture Week “segregation can result as the equilibrium of a game in which each household chooses where to 6, AOS ch. 9 live, even when they all have a measure of racial tolerance.” See AOS ch.9 and Lectures Notes Week 6 for Schelling’s analysis in terms of tipping points.

1.38 Social Preference Ranking

Lecture Week Represents preferences of group 11 How to determine social preference ranking w voting methods: 1. Start with a set of candidates and ballots, one for each person. Use the Method to choose a winner. Put them at the top of the social preference ranking.

11 2. Eliminate the winner from all the ballots, keeping the rankings of the other candidates oth- erwise the same. Use the Method to choose a winner. Put them 2nd in the social raking. Continue in this way until there are no candidates left.

1.39 Steadfast

Lecture Week One may typically, or not infrequently, maintain confident in face of disagreement 13; Christensen (2009) 1.40 Subgame Perfect Eqilibrium

Lecture Week 5 Definition: An equilibrium is subgame perfect if an only if it remains an equilibrium in every subgame of the game. General Principle: Rationality require that agents (who obey the common knowledge assump- tions) should never end up at a subgame imperfect equilibrium.

1.41 Tit-For-Tat (Copy Cat)

AOS, ch. 3 Strategy in iterated prisoner’s dilemma: cooperate in first period and then mimic rival’s action from previous period

12