rules: (Dixit and Skeath, ch 14) Recall parkland provision decision: Assume - n=10; - total cost of proposed parkland=38; - if provided, each pays equal share = 3.8 - there are two groups of individuals in society: 8 have α=, 1 1 and remaining 2 have α = 16. 2 So: efficient to provide public good, since sum of benefits (40) > total cost (38). Suppose held a referendum on provision; would parkland be provided (with this financing scheme)? Depends.on voting scheme used. Possible schemes? Page 1 of 8
General: classifiy vote aggregation methods by number of options or candidates at any time: three types: A. Binary: two alternatives - if only two, aggregate votes by majority rule; - if more than two: sequence of pairwise votes, winner by majority rule; - types? 1. Condorcet method: round robin; overall winner is the one who defeats all others in pairwise contests. 2. successive elimination (various indices) B. Plurative methods: simultaneously consider multiple options 1. plurality rule: - each voter has one vote; - candidate with most votes wins; - plurality may be less than majority Page 2 of 8
2. Borda count: - voters rank order all options - assign points to each, based on own ranking - if 3 alternatives, then most preferred gets 3 points, least preferred gets 1 - sum points across voters - highest wins 3. Approval voting: - vote for any, and all, of which approve; - winner(s): highest number of votes C Mixed methods - multistage - combine binary and plurative paradoxes: 1. Condorcet Paradox: intransitivity of social ordering - best known (among economists) - may be no winner - that is, no alternative which is successful against all others: Page 3 of 8
Example: suppose three individuals (1,2,3) choosing one of three options (A,B,C); preferences are - Ind'l 1: A B C - Ind'l 2: B C A - Ind'l 3: C A B Results? A wins in (A,B); B wins in (B,C); C wins in (A,C) - social ordering: A B C A 2. Reversal Paradox: - from Borda count: when slate of candidates changes after votes, and new vote held - need at least 4 candidates (A,B,C,D) - Suppose 7 voters 1 2 3 4 5 6 7 A D A B D D B B A B C A A C C B C D B B D D C D A C C A - With all four options, winner is A: A: 2*4 + 3*3 + 2*1=19 B: 2*4 + 2*3 + 3*1=15 C: 2*3 + 2*2 + 3*1=13 D: 3*4 + 2*2 + 2*1=18 Page 4 of 8
Now: suppose discovered that C not valid alternative - vote again, with A,B,D on ballot (same voters): 1 2 3 4 5 6 7 A D A B D D B B A B D A A D D B D A B B A Now, winner is D: A: 2*3 + 3*2 + 2*1=14 B: 2*3 + 2*2 + 3*1=13 D: 3*3 + 2*2 + 2*1=15 (eliminated an irrelevant alternative?) 3. Agenda Paradox: - binary elections - winner goes on, loser doesn't - matters who meets whom first. 4. Different methods give different outcomes Example: 100 voters, in 3 distinct groups: 40: A B C 25: B C A 35: C B A Page 5 of 8
Who wins election? a) plurality: A b) Borda count: each voter assigns numbers 1,2,3 to candidates - 3 indicating "most preferred" A obtains 40x3 + 60x1 = 180 B " 25x3 + 75x2 = 225 C " 35x3 + 25x2 + 40x1 = 195 Hence B wins. c) Majority run-off: use first round to eliminate one alternative (lowest number of votes); second round pairs top two from first round. Here, C wins. Is there a "best", reasonable voting rule? No. Page 6 of 8
Arrow's "impossibility theorem" Six criteria for aggregation of preferences 1. complete ranking 2. transitive ranking 3. Pareto property: unanimous preferences within population reflected in social ranking 4. ranking not independent of preferences of individuals in soc'y 5. ranking not dictatorial - not the reflection of the preferences of one individual 6. independent of irrelevant alternatives Theorem? no such ranking exists. How to choose between flawed mechanisms? One criterion is manipulability - how easy is it to affect outcome by strategic voting - not voting in accord with own preferences, to produce a result which is more in accord with preferences. Page 7 of 8
Consider a voting game in which 3 players (denoted 1, 2, and 3) are deciding among three alternatives (A, B, and C). Alternative B is the "status quo" and alternatives A and C are "challengers". In the first stage, players choose which of the two challengers should be considered; they do this by casting votes for either A or C, with the majority choice being the winner and abstentions not allowed. In the second stage, players vote between the status quo (B) and the winner of the first stage, with majority rule again determining the winner. The players care only about the alternative that is finally selected. The payoffs are u ( A) = u ( B) = u ( C) = 2; 1 2 3 u ( B) = u ( C) = u ( A) = 0; 1 2 3 u ( C) = u ( A) = u ( B) = 1. 1 2 3 Suppose that at each stage each player votes for the alternative they most prefer as the final outcome. a) What would the outcome be? Do these strategies constitute a Nash equilibrium? Page 8 of 8