Balanced Voting. Kamali Wickramage. based on joint work with Hans Gersbach. ETH Zurich. June, 2014

Transcription

1 Balanced Voting Kamali Wickramage based on joint work with Hans Gersbach ETH Zurich June, 2014 H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

2 Introduction Balanced Voting is.. a new voting mechanism particularly suitable for making fundamental societal decisions. - typically some voters are very passionate about the issue, while others are not - decision making happens in multiple stages Main Result: If the voting body is sufficiently large....and the stakes are high for some voters.. balanced voting (BV) is superior to simple majority voting (SM), storable votes (ST) and minority voting (MV) from the utilitarian welfare perspective. H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

3 Content Motivation Related Literature The Idea Model Equilibria Welfare Comparison Extensions and Implications H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

4 Motivation Majority voting is used almost exclusively. But.. it has many inherent deficiencies. - Tyranny of the majority - Agents cannot express their strength of preference H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

5 Related Literature Storable Votes (Casella, 2005) - Use the vote in the current period or store it for future use. - Possible to concentrate votes on decisions where stakes are high. Minority Voting (Fahrenberger and Gersbach, 2010) - Losers of first period receive exclusive voting rights in the second. - Protects individuals from repeatedly belonging to the minority. Qualitative Voting (Hortala-Vallve, 2007) - Distribute votes over a series of binary issues simultaneously. - Captures voters strength of preferences over concurrent decisions. H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

6 The Setting Stage 1 : Fundamental Direction A e.g. Nuclear Power B No Nuclear Power Stage 2 : Variations of Fundamental Direction X A 1 X A 2 X B 1 X B 2 e.g. New Nuclear Power Plants Energy Efficiency with Existing Nuclear Power Plants More Renewable Energy More Hydro Power H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

7 Voting Rules Simple Majority Voting Storable Votes Minority Voting Balanced Voting participate abstain n N-n participate n abstain N-n Stage 1: N n N n Stage 2: 1 N L N 2 2 N L n 2 N N-n 1 2 N L N L - The proposal with a simple majority of votes wins in each stage. - Ties are broken by flipping a fair coin. - If no agent applies or qualifies to vote in a particular stage, all agents vote. N-n H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

8 The Model A society of N (N 3) individuals Utility of individual i: U i (x j Ω ) where Ω {A, B}, j {1, 2}, Preferences are privately observed prior to the start of voting Agents are inclined towards a fundamental direction strongly (SI) with probability p or weakly (WI) with probability 1 p Number of SI agents: N s Number of WI agents: N w N s + N w = N Number of losers in stage 1: N L Equal probability of preferring A or B, x 1 A or x2 A, x1 B or x2 B H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

9 Utilities Discrete Utilities: U i (x j Ω ) = 1 + H for some Ω {A, B} & j {1, 2} U i (xω k i SI iff ) = 0 + H for k {1, 2}, j k U i (x j Ω ) = 1 for some j {1, 2}, Ω {A, B}, Ω Ω U i (xω k ) = 0 for k {1, 2}, k j U i (x j Ω ) = 1 + ɛ for some Ω {A, B} & j {1, 2} U i (xω k i WI iff ) = 0 + ɛ for k {1, 2}, j k U i (x j Ω ) = 1 for some j {1, 2}, Ω {A, B}, Ω Ω U i (xω k ) = 0 for k {1, 2}, k j Main Assumption H >> 1 > ɛ > 0 Note: H and ɛ are exogenously given. H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

10 Probability of Winning If n individuals vote, the probability of winning for one individual, if all other individuals vote sincerely is, P(n) = 1 ( ) n 1 2 n n H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

11 Analysis of Equilibria We look for symmetric, perfect Bayesian equilibria in pure strategies. Unique Equilibrium (under BV) I) If H > 1 and ɛ ɛ crit (N) for some critical value ɛ crit (N), (i) Every i SI participates in the first stage while all i WI abstain. (ii) All votes in both stages are cast sincerely. II) The critical value on ɛ is given by a [ P(N) ɛ crit 1 2 (N) := min 1 P(N), 4Λ(N, 1) Φ(N, 2) 3 ]. 2 a If n participate in the first stage, Λ(N, n) and Φ(N, n) give the probability of winning in the second stage for absentees and losers of the first stage, respectively. H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

12 Analysis of Equilibria Unique Equilibrium (under SM) All agents vote sincerely in both stages. Unique Equilibrium (under ST) All i SI vote in the first stage while all i WI vote in the second, if H Hst crit (N) := 1 P(N) P(N) 1 2 ɛ ɛ crit (N). Unique Equilibrium (under MV) Individuals vote sincerely in both stages. H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

13 Critical Thresholds on H and ɛ H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

14 Utilitarian Welfare Comparison BV vs. SM W BV W SM H M(N, p)ɛ + C(N, p) 1 BV vs. ST W BV W ST N N st (p) BV vs. MV W BV > W MV Preferred Voting Scheme Group BV vs. SM BV vs. ST BV vs. MV WI when N s = 0 Indifferent Indifferent BV WI when 0 < N s < N BV 2 ST BV 2 SI when N s = N SM ST Indifferent SI when 0 < N s < N BV 3 BV BV 1 M(N, p) and C(N, p) are positive for all N [3, 10000] and p [0, 1] 2 for sufficiently low ɛ 3 for sufficiently high H H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

15 Main Result Proposition For the ranges N [3, 10000] and p [0, 1], BV is superior to SM, ST and MV with respect to utilitarian welfare if the following conditions are satisfied: H max{hst crit (N), M(N, p) ɛ + C(N, p)}, ɛ ɛ crit (N), N N st (p). H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

16 Extensions and Implications Additional procedural rules To decide whether an issue is fundamental or not To avoid the voting body being very small When the composition of the society changes across stages For acceptance of the voting rule Extensions Publicly observable preference (strategic implications) Deliberation before voting Alternative models - non-fundamental decisions, continuous distribution of preferences, unequal probabilities to prefer A over B or its variations, etc. Extension to several stages H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17

17 References Brighouse, H., Fleurbaey, M. (2010). Democracy and Proportionality, Journal of Politial Philosophy, 18(2), Casella, A., (2005). Storable votes, Games and Economic Behavior, 51, Cox, G., (1990). Centripetal and centrifugal incentives in electoral systems, American Journal of Political Sceince, 34, Coleman, J., (1966). The possibility of a social welfare function, American Economic Review, 56, Fahrenberger, T., Gersbach, H., (2010). Minority voting and long-term decisions, Games and Economic Behavior, 69 (2), Gerber, E., Morton, R., Rietz, T., (1998). Minority representation in multimember districts, American Political Science Review, 92 (1), Hortala-Vallve, R., (2007). Qualitative voting, Discussion paper, Economics Series Working Papers No. 320, University of Oxford. Philipson, T., Snyder, J., (1996). Equilibrium and efficiency in an organized vote market, Public Choice, 89, Sawyer, J., MacRae, D., (1962). Game theory and cumulative voting in Illinois: , American Political Science Review, 56 (4), H. Gersbach & K. Wickramage (ETH Zurich) Balanced Voting June, / 17