A Study of Approval voting on Large Poisson Games

Transcription

1 A Study of Approval voting on Large Poisson Games Ecole Polytechnique Simposio de Analisis Económico December 2008 Matías Núñez () A Study of Approval voting on Large Poisson Games 1 / 15

2 A controversy over Approval Voting In large elections, Plurality voting is the most applied voting rule. Plurality voting allows the voter to vote for one candidate and the candidate with the most votes wins the election. There is a literature that advocates the use of Approval Voting, a voting rule similar to Plurality voting but a higher degree of flexibility. Approval voting allows each voter to vote for as many candidates as he wishes by giving at most one point to each candidate. The candidate with the most votes wins the election. Matías Núñez () A Study of Approval voting on Large Poisson Games 2 / 15

3 Large Poisson Games Standard methods do not lead to a clear-cut conclusion to solve this controversy. Indeed, strategic voting theory uses Nash equilibrium to compare the properties of voting rules. The main problem is the lack of predictive power of Nash equilibrium in voting environments. If every voter votes for the same candidate, this is an equilibrium as no voter can increase his payoff by unilaterally deviating. However, these techniques are not useful to differentiate between one-shot voting rules. Matías Núñez () A Study of Approval voting on Large Poisson Games 3 / 15

4 Large Poisson Games Myerson s Large Poisson Games try to solve this problem by introducing beliefs (=probability distribution) as far as the relative chances of winning the election of the different candidates are concerned. The underlying idea: self-consistent beliefs. Voters anticipate that a pair of candidates is the most likely one to be in contention for victory. Voters vote given this belief and their votes generate again a belief. If the initial and the posterior belief coincide, this is an equilibrium. Matías Núñez () A Study of Approval voting on Large Poisson Games 4 / 15

5 A simple example. There are 3 candidates: a,b and c. In a large election with 10 million voters, there are two types of voters: T1 and T2. 6 Million 4 Million a a b c c b T1 T2 Let us assume that the common belief of voters is such that candidates b and c are the most likely winners. Matías Núñez () A Study of Approval voting on Large Poisson Games 5 / 15

6 A simple example: Plurality Voting There are 3 candidates: a,b and c. In a large election with 10 million voters, there are two types of voters: T1 and T2. 6 Million 4 Million a a b c c b T1 T2 Let us assume that the common belief of voters is such that candidates b and c are the most likely winners. Under Plurality voting, T1-voters vote for b and T2-voters vote for c. Their votes generate again the belief that candidates b and c are the most likely winners. This is an equilibrium, due to the wasted-vote effect. Matías Núñez () A Study of Approval voting on Large Poisson Games 6 / 15

7 A simple example: Approval Voting There are 3 candidates: a,b and c. In a large election with 10 million voters, there are two types of voters: T1 and T2. 6 Million 4 Million a a b c c b T1 T2 Let us assume that the common belief of voters is such that candidates b and c are the most likely winners. Under Approval voting, T1-voters vote for a and b and T2-voters vote for b and c. Their votes do not generate again the belief that candidates b and c are the most likely winners. This is not an equilibrium! Due to its flexibility, Approval voting reduces the impact of the wasted-vote effect. Matías Núñez () A Study of Approval voting on Large Poisson Games 7 / 15

8 Our contribution The previous example summarizes Myerson (2002) s conclusion: approval voting leads to better preference aggregation than other voting rules in simple voting situations. Our work shows that Approval Voting does not always have good properties on Large Poisson Games. Indeed, it does not always lead voters to the correct belief about the relative chances of winning the election of the different candidates. A candidate who is ranked first by more than half of the voters need not be the Winner of the election under AV. Matías Núñez () A Study of Approval voting on Large Poisson Games 8 / 15

9 Approval Voting does not always lead to good preference aggregation There are 3 candidates: a,b and c. In a large election with 10 million voters, there are two types of voters: T1, T2 and T3. 1 Million 6 Million 3 Million a b c c a a b c b T 1 T 2 T 3 Let us assume that the common belief of voters is such that candidates a and c are the most likely winners. Matías Núñez () A Study of Approval voting on Large Poisson Games 9 / 15

10 Approval Voting does not always lead to good preference aggregation There are 3 candidates: a,b and c. In a large election with 10 million voters, there are two types of voters: T1, T2 and T3. 1 Million 6 Million 3 Million a b c c a a b c b T 1 T 2 T 3 Let us assume that the common belief of voters is such that candidates a and c are the most likely winners. Under Approval voting, T1-voters vote for a, T2-voters vote for b and a and T3-voters vote for c. Their votes generate again the belief that candidates a and c are the most likely winners. This is an equilibrium! Matías Núñez () A Study of Approval voting on Large Poisson Games 10 / 15

11 Approval Voting does not always lead to good preference aggregation There are 3 candidates: a,b and c. In a large election with 10 million voters, there are two types of voters: T1, T2 and T3. 1 Million 6 Million 3 Million a b c c a a b c b T 1 T 2 T 3 In this equilibrium, Candidate a wins the election and candidate b is the preferred by more than half of the population. Approval voting does not always lead to good preference aggregation! Matías Núñez () A Study of Approval voting on Large Poisson Games 11 / 15

12 Why do these bad equilibria arise? K represents the set of candidates A voter is characterized by his type t that determines his preferences over K : u t = (u t (k)) k K. Number of voters: x P(n), The number of voters is equal to v with probability e n nv v!. Number of voters with type t: x t P(nr(t)), Number of voters who choose ballot c: x(c) P(nτ(c)) with τ(c) = t r(t)σ(c t) and σ(c t) stands for the strategy of voters with type t. Two main properties stated by Myerson: x(c) and x(c ) are independent. (Independent ballots) Strategies depend uniquely on the type t. (Common public information) Matías Núñez () A Study of Approval voting on Large Poisson Games 12 / 15

13 Why do these bad equilibria arise? A ballot is a list of the candidates a voter approves of. C k is the set of ballots that approve candidate k. The score s(k) of a candidate is the number of votes candidate k gets: the sum of the votes each ballot c C k gets. s(k) = c C k x(c) P(n c C k τ(c)) The scores of the candidates are correlated!: this is a source of problems in Large Poisson Games. Matías Núñez () A Study of Approval voting on Large Poisson Games 13 / 15

14 Summary Poisson Games are a framework to study large elections and information manipulation. Myerson (2002) suggests that AV is more robust to this information manipulation than other voting rules by comparing simple voting situations. However, we show that the equilibria that remain under AV are not always desirable. Indeed, a candidate preferred by the majority of voters need not be the winner of the election. Besides, it can be the case that the Condorcet Winner does not win the election in any of the equilibria of the game. Matías Núñez () A Study of Approval voting on Large Poisson Games 14 / 15

15 Summary Laslier (2006) implements a different framework to study the role of information over the strategic properties of approval voting. In his framework, the size of the population is constant and voters are uncertain with regards to the scores of the candidates (a vote is wrongly recorded with positive probability). He shows that, given every candidate gets a strictly positive share of votes, under AV there is a unique equilibrium where The Condorcet Winner wins the election (whenever it exists) Voters optimal strategies are sincere. In his framework, scores are independent random variables. Matías Núñez () A Study of Approval voting on Large Poisson Games 15 / 15