Costless honesty in voting B. Dutta and J.-F. Laslier
Social Choice and Welfare conference, Moscow, 2010
B. Dutta and J.-F. Laslier
Costless honesty in voting
Introduction Lexicographic preference for honesty: honesty as a secondary motive of choice.
B. Dutta and J.-F. Laslier
Costless honesty in voting
Introduction Lexicographic preference for honesty: honesty as a secondary motive of choice. When the individual is to announce a preference, honesty requires that the agent states his sincere preference. Suppose that an individual who would derive the same utility from a sincere and a non-sincere statement prefer the sincere one. Dutta and Sen (2010) show that as soon as one of the individuals is “partially honest” all social choice correspondences satisfying No Veto Power can be Nash-implemented.
B. Dutta and J.-F. Laslier
Costless honesty in voting
Introduction Lexicographic preference for honesty: honesty as a secondary motive of choice. When the individual is to announce a preference, honesty requires that the agent states his sincere preference. Suppose that an individual who would derive the same utility from a sincere and a non-sincere statement prefer the sincere one. Dutta and Sen (2010) show that as soon as one of the individuals is “partially honest” all social choice correspondences satisfying No Veto Power can be Nash-implemented. In votes, often, any single vote cannot change the result, a case of interest for studying the implication of lexicographic preference for sincerity.
B. Dutta and J.-F. Laslier
Costless honesty in voting
Sincere voting Jury setting: voting according to personal information. Laslier and Weibull (2009) : a slight preference for honesty implies sincerity at equilibrium.
B. Dutta and J.-F. Laslier
Costless honesty in voting
Sincere voting Jury setting: voting according to personal information. Laslier and Weibull (2009) : a slight preference for honesty implies sincerity at equilibrium. Single-name plurality voting: voting for a preferred candidate. Borda rule: the ballot shows no contradiction with her true preference. Approval voting: if the voter approves a candidate x she also approves all candidates she strictly prefers to x.
B. Dutta and J.-F. Laslier
Costless honesty in voting
Sincere voting Jury setting: voting according to personal information. Laslier and Weibull (2009) : a slight preference for honesty implies sincerity at equilibrium. Single-name plurality voting: voting for a preferred candidate. Borda rule: the ballot shows no contradiction with her true preference. Approval voting: if the voter approves a candidate x she also approves all candidates she strictly prefers to x. In this paper, restrict attention to k-approval rules. k-approval : a voter approves at least one, but not more than k candidates. k = 1 corresponds to plurality voting. If the number of candidates is m, then k = m is approval voting, and k = m − 1 will also be considered as approval voting. B. Dutta and J.-F. Laslier
Costless honesty in voting
The model I = {1, ..., n} : set of voters. X : set of candidates. ui : VNM utility function. Strict preferences: if x 6= y then ui (x) 6= ui (y ), and ui (x) > ui (y ) denoted xPi y . m = #X : number of candidates. Bk (X ) : set of nonempty subsets of X with at most k candidates. The k-approval rule:
B. Dutta and J.-F. Laslier
Costless honesty in voting
The model I = {1, ..., n} : set of voters. X : set of candidates. ui : VNM utility function. Strict preferences: if x 6= y then ui (x) 6= ui (y ), and ui (x) > ui (y ) denoted xPi y . m = #X : number of candidates. Bk (X ) : set of nonempty subsets of X with at most k candidates. The k-approval rule: Each player has Bk (X ) as her strategy set. Given B = (B1 , ..., Bn ) ∈ (Bk (X ))I the score of x: s(x, B) = #{i ∈ I : x ∈ Bi }. Outcome of B : set of candidates with highest scores: φ(B) = {x ∈ X : ∀y ∈ X , sx (B) ≥ sy (B)}. 1 P Fair draw : ui (Y ) = #Y x∈Y ui (x). B. Dutta and J.-F. Laslier
Costless honesty in voting
Hequilibrium refinement Ballot Bi ∈ Bk (X ) is sincere if, for all x, y ∈ X , if y ∈ Bi and xPi y then x ∈ Bi . Under the k-rule, voter has k different sincere ballots. The k-approval voting game with lexicographic sincerity:
B. Dutta and J.-F. Laslier
Costless honesty in voting
Hequilibrium refinement Ballot Bi ∈ Bk (X ) is sincere if, for all x, y ∈ X , if y ∈ Bi and xPi y then x ∈ Bi . Under the k-rule, voter has k different sincere ballots. The k-approval voting game with lexicographic sincerity: a profile of individual rankings i of strategy profiles. I
If ui (φ(B)) < ui (φ(B 0 )) then B ≺i B 0 .
I
If ui (φ(B)) = ui (φ(B 0 )) and Bi is sincere for i and Bi0 is not, then B ≺i B 0 .
I
If ui (φ(B)) = ui (φ(B 0 )) and Bi are both sincere or both non-sincere for i, then B ≈i B 0 .
B. Dutta and J.-F. Laslier
Costless honesty in voting
Hequilibrium refinement Ballot Bi ∈ Bk (X ) is sincere if, for all x, y ∈ X , if y ∈ Bi and xPi y then x ∈ Bi . Under the k-rule, voter has k different sincere ballots. The k-approval voting game with lexicographic sincerity: a profile of individual rankings i of strategy profiles. I
If ui (φ(B)) < ui (φ(B 0 )) then B ≺i B 0 .
I
If ui (φ(B)) = ui (φ(B 0 )) and Bi is sincere for i and Bi0 is not, then B ≺i B 0 .
I
If ui (φ(B)) = ui (φ(B 0 )) and Bi are both sincere or both non-sincere for i, then B ≈i B 0 .
Hequilibrium (equ. with honesty): a strategy profile B such that no i ∈ I has a deviation Bi0 s.t. B ≺i (B−i , Bi0 ). Any hequilibrium is a Nash equilibrium of the normal form game defined by u without lexicographic preference for sincerity. B. Dutta and J.-F. Laslier
Costless honesty in voting
Results: Existence (1) Result 0. For any k, if n ≥ 3, without sincerity there are many Nash equilibria . Any alternative can be supported. Proof. All voters vote for the same alternative x and only for x. A Nash equilibrium as soon as n ≥ 3 (Quite general !).
B. Dutta and J.-F. Laslier
Costless honesty in voting
Results: Existence (1) Result 0. For any k, if n ≥ 3, without sincerity there are many Nash equilibria . Any alternative can be supported. Proof. All voters vote for the same alternative x and only for x. A Nash equilibrium as soon as n ≥ 3 (Quite general !). Result 1. (1) For k = 1, for any odd number of voters n ≥ 5 there exists a profile with no hequilibrium. (m) b Example 1: a c d
(1) c b a d
(1) d a b c
(m + 1) a b c d
B. Dutta and J.-F. Laslier
Costless honesty in voting
Results: Existence (2)
(m) b a c d
(1) c b a d
(1) d a b c
(m + 1) a b c d
Proof: For k = 1, if everyone votes sincerely, then a wins one vote ahead of b, and the second voter should vote b rather than c. If the best score is m + 2, someone not voting sincerely implies that another alternative has score m + 1. If s(a) = m + 2 and s(x) = m + 1 then one x voter should vote sincerely. So s(a) = m + 1 and the m + 2 other voters vote for the same x. All the x voters prefer x to a, which is impossible in this profile.
B. Dutta and J.-F. Laslier
Costless honesty in voting
Results: Existence (3) (2) If n = m ≥ 3 and k = m − 1 there exists a preference profile with no hequilibrium. 1 n ... Example 2: 4 3 2
2 1 ... 5 4 3
3 2 ... 6 5 4
... ... ... ... ...
i −1 i −2 ... i +2 i +1 i
... ... ... ... ...
n−1 n−2 ... 2 1 n
n n−1 ... 3 2 1
The utility profile is such that for voters 1, .., n − 1, there is a huge gap in utility between the last ranked alternative and the others, whereas for the individual n the gap is between the best alternative for him and all the others.
B. Dutta and J.-F. Laslier
Costless honesty in voting
Results: Existence (4)
Sketch of proof: Note for k = m − 1 the best-response correspondence contains sincere ballots, thus in an hequilibrium for k = m − 1, voters vote sincerely. In the counter example above, there is no sincere equilibrium for k = m − 1. There may be an hequilibrium for k < m − 1. Example for k = 2, m = 5 let the votes be 1, 21,3, 43, and 3 (non-sincere) for the last voter. (3) Same example extended, adding bad alternatives, gives same conclusion for all m ≥ 3 for all k there is a number of voters n and a utility profile... (Take n = k + 1 and add bad alternatives to the example above)
B. Dutta and J.-F. Laslier
Costless honesty in voting
Existence (5)
Remark: The existence question for 1 < k < n − 1 is open. Remark: Any sincere ballot profile with no tie is a hequilibrium in a replicated economy. (The score differences are 3 or more.) Because the set of such ballot profiles expands with k, larger ks define more flexible rules according to hequilibrium.
B. Dutta and J.-F. Laslier
Costless honesty in voting
Pareto efficiency Result 2. For k = m − 1, if y is chosen and x Pareto dominates y , then x is chosen. This is not true for k < m − 1. Counter-example with k = 2 votes and n = 4 voters: 2 x y Example 3: y’ . . z
2 z . . x y y
with votes {y , y 0 } and {z}. Then the outcome is {y , y 0 , z} but x Pareto dominates y . Note that inefficient outcomes can be chosen with others at hequilibrium even for k = m − 1. B. Dutta and J.-F. Laslier
Costless honesty in voting
Outcome efficiency
Result 3. An hequilibrium lottery can be Pareto-dominated by another hequilibrium lottery. Example for any k > 1: 2 x Example 4: y . z
2 z y . x
Then voting {x} and {z} is an hequilibrium. It is Pareto-dominated by voting {x, y } and {z, y } that gives the outcome {y }.
B. Dutta and J.-F. Laslier
Costless honesty in voting
Sincerity
Result 4. For k = m − 1, at hequilibrium, they vote sincerely. Proof: By definition, a hequilibrium is in pure strategies, thus randomness only comes from ties. It follows that the best response correspondences everywhere contains sincere ballots. The result follows easily. Result 5. For k < m − 1, this is not true. (See example 3 above.)
B. Dutta and J.-F. Laslier
Costless honesty in voting
Condorcet-consistency Can the Condorcet winner be supported? Result 6. If k = m − 1 and n ≥ 3, yes.
B. Dutta and J.-F. Laslier
Costless honesty in voting
Condorcet-consistency Can the Condorcet winner be supported? Result 6. If k = m − 1 and n ≥ 3, yes. Sketch of proof. Let everyone vote down to c, the CW, except those who rank it last. Let those vote for their best only. This is a sincere profile, where c wins. To prove that it is an hequilibrium one just need to prove that it is Nash...
B. Dutta and J.-F. Laslier
Costless honesty in voting
Condorcet-consistency Can the Condorcet winner be supported? Result 6. If k = m − 1 and n ≥ 3, yes. Sketch of proof. Let everyone vote down to c, the CW, except those who rank it last. Let those vote for their best only. This is a sincere profile, where c wins. To prove that it is an hequilibrium one just need to prove that it is Nash... Result 7. If k < m − 1, not always. b a Example 4: [c] d
d [c] a b
a b [c] d
n = 3 voters, m alternatives (with m − 2 of them in the component c) and a is a Concorcet winner. The utility of the second player is such that u2 (c) >> u2 (a). B. Dutta and J.-F. Laslier
Costless honesty in voting
