Consistent Probabilistic Social Choice

Transcription

1 Consistent Proilisti Soil Choie Felix Brndt (with F. Brndl nd H. G. Seedig) University of Oxford, Novemer 2015

2 Preliminries Finite set of lterntives A A is not fixed Liner preferene reltions L (A) Frtionl preferene profiles R Δ(L (A)) = R A Δ(L (A)) denotes the ( A!-1)-dimensionl unit simplex impliitly ssumes nonymity nd homogeneity will only onsider rtionl frtions Frtionl olletive preferene R(x,y) = R( ) L (A): x y Mjority mrgin gr(x,y) = R(x,y) - R(y,x) (Wek) Condoret winner x with gr(x,y) 0 for ll y ½ ⅓ ⅙ {,,} R(,)=⅚ gr(,)=⅔ Consistent Proilisti Soil Choie 2 Felix Brndt

3 Proilisti Soil Choie Funtions A proilisti soil hoie funtion (PSCF) f mps preferene profile R R A to non-empty suset of Δ(A). f is (upper hemi-)ontinuous (ontinuity) f(r) is onvex set (onvexity) R R {x,y} nd R(x,y)=1 imply f(r) = {x} (unnimity) {R R A : f(r) =1} is dense in R A (deisiveness) Non-proilisti SCFs re PSCFs where, for ll R R A, f(r)=δ(x) for some X A. The xioms we propose for PSCFs oinide with lssi xioms for the speil se of non-proilisti SCFs. Consistent Proilisti Soil Choie 3 Felix Brndt

4 Rndom Dittorship One gent is piked uniformly t rndom nd his most preferred lterntive is implemented s the soil hoie. RD(R) = { R( ) mx(a) } L (A) RD is (single-vlued) PSCF. It stisfies ontinuity, onvexity, unnimity, nd deisiveness. ½ ⅓ ⅙ RD(R) = {⅚ + ⅙ } Consistent Proilisti Soil Choie 4 Felix Brndt

5 Popultion-Consisteny... p p... Whenever two disjoint eletortes gree on lottery, this lottery should lso e hosen y the union of oth eletortes. Consistent Proilisti Soil Choie 5 Felix Brndt

6 Popultion-Consisteny ½ ½ ½ ½ ¼ ¼ ½ R R ½ R + ½ R ½ + ½ ½ + ½ ½ + ½ f(r) f(r ) f(λr + (1-λ)R ) strong popultion-onsisteny requires equlity (not only inlusion) whenever left-hnd-side is non-empty first proposed y Smith (1973), Young (1974), Fine & Fine (1974) lso known s reinforement (Moulin, 1988) vrints used y Fishurn, Merlin, Myerson, Sri, et. Consistent Proilisti Soil Choie 6 Felix Brndt

7 Composition-Consisteny Deomposle preferene profiles re treted omponent-wise. Consistent Proilisti Soil Choie 7 Felix Brndt

8 Composition-Consisteny ⅓ ⅙ ½ ½ ½ ⅓ ⅔ R R A R B ½ + ⅓ + ⅙ ½ + ½ ⅔ + ⅓ f(r A ) f(r B) = f(r) Lffond, Lslier, nd Le Breton (1996) Cloning-onsisteny preursors: Arrow nd Hurwiz (1972), Mskin (1979), Moulin (1986), Tidemn (1987) Consistent Proilisti Soil Choie 8 Felix Brndt

9 Non-Proilisti Soil Choie Chevlier de Bord Mrquis de Condoret All soring rules stisfy popultion-onsisteny. (Smith 1973; Young, 1974) No Condoret extension stisfies popultion-onsisteny. (Young nd Levenglik, 1978) Mny Condoret extensions stisfy ompositiononsisteny. (Lffond et l., 1996) No Preto-optiml soring rule stisfies ompositiononsisteny. (Lslier, 1996) Theorem: There is no SCF tht stisfies popultiononsisteny nd omposition-onsisteny. But: These two xioms uniquely hrterize PSCF. Consistent Proilisti Soil Choie 9 Felix Brndt

10 Mximl Lotteries Germin Krewers Krewers (1965) nd Fishurn (1984) Peter C. Fishurn redisovered y Lffond et l. (1993), Felsenthl nd Mhover (1992), Fisher nd Ryn (1995), Rivest nd Shen (2010) Extend gr to lotteries: gr(p,q) = x,y p(x) q(y) gr(x,y) frtionl olletive preferenes over lotteries p is mximl lottery, p ML(R), if gr(p,q) 0 for ll q Δ(A). proilisti Condoret winner lwys exists due to Minimx Theorem (v. Neumnn, 1928) Set of profiles with unique mximl lotteries is open nd dense. set of profiles with multiple mximl lotteries is negligile lwys unique for odd numer of voters (Lffond et l., 1997) generlized uniqueness onditions y Le Breton (2005) Consistent Proilisti Soil Choie 10 Felix Brndt

11 Exmples Two lterntives p(x) 1 0 Mximl lotteries 0 ½ 1 gr n e interpreted s symmetri zero-sum gme. Mximl lotteries re mixed minimx strtegies The unique mximl lottery is 3/5 + 1/5 + 1/5. R(x,y) p(x) 1 0 Rndom dittorship 0 ½ 1 R(x,y) Consistent Proilisti Soil Choie 11 Felix Brndt

12 Min Result Theorem: A PSCF f stisfies popultion-onsisteny nd omposition-onsisteny iff f=ml. Proof struture: Composition-onsisteny implies neutrlity. Two-lterntive hrteriztion (vi three-lterntive profiles) Condoret-onsisteny round uniform profile f ML. - Assume for ontrdition tht f yields lottery tht is not mximl. - Construt Condoret profile in whih uniform lottery is returned. - Derive density violtion. ML f. - For ny vertex of the set of mximl lotteries in profile, onstrut sequene of profiles tht onverges to the originl profile nd whose unique mximl lotteries onverge to the originl mximl lottery. - Apply ontinuity nd onvexity. Consistent Proilisti Soil Choie 12 Felix Brndt

13 Two-Alterntive Proof Lemm: Let f e omposition-onsistent PSCF nd A={x,y}. p f(r) with p x,y implies f(r)= Δ(A). Proof: Let r=r(x,y). r 1-r x y y x Hene, λ 2 x + (1-λ 2 )y f(r). r 1-r x x y Repeted pplition, ontinuity, nd onvexity imply the sttement. As onsequene, RD violtes omposition-onsisteny. y x x 1 1-r λx + (1-λ)y λ 2 x + λ(1-λ)x + (1-λ)y λ 2 x + (1-λ 2 )y x y y x Consistent Proilisti Soil Choie 13 Felix Brndt

14 Two-Alterntive Proof (td.) p(x) 1 Simple mjority rule Rndom dittorship 0 0 r ½ 1 R(x,y) Consistent Proilisti Soil Choie 14 Felix Brndt

15 Remrks Independene of xioms popultion, not omposition: rndom dittorship RD omposition, not popultion: mximl lotteries vrint ML 3 ML lmost lwys stisfy strong popultion-onsisteny Composition-onsisteny n e wekened to loningonsisteny when lso requiring Condoret-onsisteny RD stisfies loning-onsisteny ML lso stisfy gend-onsisteny (Sen s α nd γ) Axioms imply Fishurn s C2 (pirwiseness) s well s Condoret-onsisteny. Consistent Proilisti Soil Choie 15 Felix Brndt

16 Remrks (td.) Possile non-proilisti interprettion of outomes s frtionl division (e.g., udget division, time shres) Axioms re eqully nturl. Preto-dominted lterntives lwys get zero proility in every mximl lottery. In ft, ML is even SD-effiient (Aziz et l., 2012). ML does not require symmetry, ompleteness, or even trnsitivity of preferenes. Rndom dittorship requires unique mximum. In ssignment domin, ML re known s populr mixed mthings (Kvith et l., 2011). ML n e effiiently omputed vi liner progrmming. Consistent Proilisti Soil Choie 16 Felix Brndt

17 Mximl Lotteries Rndom Seril Dittorship Bord s Rule popultion-onsisteny only for strit prefs gend-onsisteny loning-onsisteny even omposition-onsisteny Condoret-onsisteny (wek SD-) strtegyproofness wek group-strtegyproofness (wek SD-) prtiiption even for groups even very strongly (SD-) effiieny only for strit prefs otherwise only ex post effiient omputility #P-omplete in P for strit prefs Consistent Proilisti Soil Choie 17 Felix Brndt

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