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Albert Einstein 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 Paradox .................................... 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 Majority Rule ................................... 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
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