Impartial-culture asymptotics a central limit theorem for - PowerPoint PPT Presentation

Voting rules Manipulation Coalition size Asymptotics Impartial-culture asymptotics a central limit theorem for manipulation of elections Geoffrey Pritchard ∗ , Mark Wilson University of Auckland March 20, 2009 Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Voting rules One of m candidates must be elected by n voters. How much information to ask the voters for? favourite candidate less info . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (e.g. plurality) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . preference order approval set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . more info . . . . . . . . . . candidate ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Preference-order rules Each voter has one of the m ! possible preference orders (types, opinions). A full profile specifies the type of each voter. A voting situation specifies only the number of voters of each type this is all we need, if the voting rule treats voters symmetrically (anonymously). e.g. 3 candidates, 6 preference orders 6 � N = ( N 1 , N 2 , N 3 , N 4 , N 5 , N 6 ) , with N i = n i =1 Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Scoring (positional) voting rules A candidate gets w i points when a voter ranks him in i th place; 1 = w 1 ≥ w 2 ≥ · · · ≥ w m = 0 . Example (3 candidates): abc acb bac bca cab cba number of voters N t : 2 2 0 3 1 0 For w = (1 , 1 2 , 0) (Borda’s rule), a wins. For w = (1 , 1 , 0) (anti-plurality rule), c wins. Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Probabilistic voter behaviour IAC: all voting situations are equally likely to occur. For large n , our voting situation is approximately uniformly distributed on a simplex. Probabilities → volumes of convex bodies... IC: voters have independent, uniform random types. For large n , our voting situation is approximately (multivariate) normally distributed. Central Limit Theorem, here we come... Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics IC asymptotics Voting situation m ! + √ n ( m ! − 1) 1 / 2 n ∼ : Z t , Z t ∼ N (0 , 1) N t m ! The voter types are about equally numerous. Scoreboard � 1 / 2 w + √ n σ w � m � ( Z α − ¯ | α | = N t σ t ( α ) ∼ n ¯ : Z ) m − 1 t The scores tend to be nearly equal. Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Tied scores Ignore the possibility of tied scores. P (any ties) → 0 as n → ∞ . Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Manipulation Logical possibility of manipulation: some coalition of voters can improve the result (for themselves) by voting insincerely. Ignores counterthreats Ignores complexity IC is very manipulable: P (L.P.M.) → 1 as n → ∞ for all scoring rules except anti-plurality. Minimum manipulating coalition size MCS ( ∞ if not L.P.M.) Study the distribution of this random variable. Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Recruiting a manipulating coalition Our coalition will contain (for each type t ): x t voters (sincerely) of type t ; y t voters who insincerely vote t ; � � x t = y t . t t Post-manipulation score of α is � | α | + ( y t − x t ) σ t ( α ) . t Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Manipulation: an integer linear program Minimum manipulating coalition size MCS = min β Q 1 ( β ), where Q 1 ( β ) = min x , y Σ t x t s.t. | β | + Σ t ( y t − x t ) σ t ( β ) ≥ | α | + Σ t ( y t − x t ) σ t ( α ) ∀ α � = β t x t = t y t Σ Σ ≥ 0 y t 0 ≤ x t ≤ N t x t , y t integer For IC and large n , we’ll want x t ∼ √ n , but N t ∼ n , so the last two constraints will very rarely matter. Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Phantom voters Let Q 2 = min. coalition size without the last two constraints. Now we can recruit non-existent voters, of any types we please. Example (3 candidates): abc acb bac bca cab cba number of voters N t : 2 2 0 3 1 0 Borda scores: | a | = 4 . 5, | b | = 4, | c | = 3 . 5. Regular manipulation: Q 1 ( b ) = ∞ . Everybody who prefers b to a already ranks b top, a bottom. Relaxed manipulation: Q 2 ( b ) = 1. One cba could do it (by voting bca ). To make b sole winner, 1 . 00001 such voters would suffice. But this example is misleading . . . Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Phantom voters don’t hurt Theorem. Relaxing makes manipulation easier, but not by much. P ( | Q 1 ( β ) − Q 2 ( β ) | ≤ K ) → 1 as n → ∞ , where K depends only on the voting rule. Coalition sizes Q i ( β ) ∼ √ n , so allowing phantom voters really hasn’t made much difference. Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Phantom-voter manipulation is well-behaved Theorem. Second-placegetter has smallest phantom manipulating coalition. min β Q 2 ( β ) = Q 2 ( b ) . (Only the constraint x t ≥ N t could have given another candidate a smaller one.) Theorem. Minimal phantom coalition for b consists only of types . . . ba . . . (They can insincerely put b first and a last.) Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics Phantom-voter manipulation is well-behaved Theorem. Second-placegetter has smallest phantom manipulating coalition. min β Q 2 ( β ) = Q 2 ( b ) . (Only the constraint x t ≥ N t could have given another candidate a smaller one.) Theorem. Minimal phantom coalition for b consists only of types . . . ba . . . (They can insincerely put b first and a last.) Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics An even simpler linear program Recruit z i phantom voters of types ranking b in i th place, a in ( i + 1)st place. Consider Q = min z Σ i z i s.t. i (1 − w i + w i +1 ) z i ≥ | a | − | b | ( b catches up to a ) Σ i (1 − w i ) z i ≥ n ¯ w − | b | ( b above average) Σ z i ≥ 0 Theorem. These two constraints are enough! Q = Q 2 ( b ) ( ≈ MCS ) . Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics

Voting rules Manipulation Coalition size Asymptotics A two-variable linear program Take the dual linear program: two variables only. Q = max { ( | a | − n ¯ w ) λ + ( n ¯ w − | b | ) µ : ( λ, µ ) ∈ M w } where the feasible set M w = { ( λ, µ ) : 0 ≤ λ ≤ µ and w i +1 λ + (1 − w i ) µ ≤ 1 ∀ i } depends only on the voting rule. The random coefficients ( | a | − n ¯ w , n ¯ w − | b | ) : ∼ bivariate normal Geoffrey Pritchard ∗ , Mark Wilson Impartial-culture asymptotics