The Distortion of Cardinal Preferences in Voting Ariel D. Procaccia and Jeffrey S. Rosenschein

Lecture outline Introduction to Voting Distortion –Definition and intuition –Discouraging results Misrepresentation –Definition and intuition –Results Conclusions Introduction Distortion MisrepresentationConclusions

What is voting? n voters and m candidates. Each voter expresses ordinal preferences by ranking the candidates. Winner of election determined according to a voting rule. –Plurality. –Borda. Applications in multiagent systems (candidates are beliefs, schedules [Haynes et al. 97], movies [Ghosh et al. 99]). Introduction Distortion MisrepresentationConclusions

Got it, so what’s distortion? Humans don’t evaluate candidates in terms of utility, but agents do! With voting, agents’ cardinal preferences are embedded into space of ordinal preferences. This leads to a distortion in the preferences. Introduction Distortion MisrepresentationConclusions

Distortion illustrated utility rank c2 c3 c utility rank c3 c1 c2 Introduction Distortion MisrepresentationConclusions

Distortion defined (informally) Candidate with max SW usually not the winner. –Depends on voting rule. Informally, the distortion of a rule is the worst-case ratio between the maximal SW and SW of winner. Introduction Distortion MisrepresentationConclusions

Distortion Defined (formally) Each voter has preferences u i = ; u i j = utility of candidate j. Denote u j =  i u i j. Ordinal prefs denoted by R i. j R i k = voter i prefers candidate j to k. An ordinal pref. profile R is derived from a cardinal pref profile u iff: 1.  i,j,k, u i j > u i k  j R i k 2.  i,j,k, u i j = u i k  j R i k xor k R i j  (F,u) = max j u j /u F(R). Introduction Distortion MisrepresentationConclusions

An unfortunate truth F = Plurality. argmax j u j = 2, but 1 is elected. Ratio is 9/ utility rank c2 c c2 2 1 utility rank c1 c2 2 1 utility rank c2 c c2 c1 c2 c1 Introduction Distortion MisrepresentationConclusions

Distortion Defined (formally) Each voter has preferences u i = ; u i j = utility of candidate j. Denote u j =  i u i j. Ordinal prefs denoted by R i. j R i k = voter i prefers candidate j to k. An ordinal pref. profile R is derived from a cardinal pref profile u iff: 1.  i,j,k, u i j > u i k  j R i k 2.  i,j,k, u i j = u i k  j R i k xor k R i j.  (F,u) = max j u j /u F(R).  n m (F)=max u  (F,u). –S.t.  j u i j = K. Introduction Distortion MisrepresentationConclusions

An unfortunate truth Theorem:  F,  3 2 (F)> utility rank c2 c c2 2 1 utility rank c1 c2 2 1 utility rank c2 c c2 c1 c2 c1 Introduction Distortion MisrepresentationConclusions

Scoring rules – a short aside Scoring rule defined by vector  =. Voter awards  l points to candidate l’th-ranked candidate. Examples of scoring rules: –Plurality:  = –Borda:  = –Veto:  = Introduction Distortion MisrepresentationConclusions

Distortion of scoring rules – the plot thickens F has unbounded distortion if there exists m such that for all d,  n m (F)>d for infinitely many values of n. Theorem: F = scoring protocol with  2  1/(m-1)  l  2  l. Then F has unbounded distortion. Corollary: Borda and Veto have unbounded distortion. Introduction Distortion MisrepresentationConclusions

An alternative model So far, have analyzed profiles u s.t.  i,  j u i j =K. Weighted voting: voter with weight K counts as K identical voters.  j u i j =K i. Voter i has weight K i. Define  n m (F) analogously to previous def. Theorem: For all F, n1, m,  n1 m ≤  n1 m, and there exists n2 s.t.  n1 m ≤  n2 m. Corollary: For all F,  3 2 (F)>1. Corollary: F has unbounded   F has unbounded . Introduction Distortion MisrepresentationConclusions

Introducing misrepresentation A voter’s misrepresentation w.r.t. l’th ranked candidate is  i j = l-1. Denote  j =  i  i j. Misrep. can be interpreted as (restricted) cardinal prefs. –e.g. u i j = m -  i j - 1.  n m (F)=max R (  F(R) /min j  j ). Introduction Distortion MisrepresentationConclusions

Misrepresentation illustrated 19:00 18:00 17:00 16:00 15:00 14:00 13:00 12:00 11:00 10:00 9:00 19:00 18:00 17:00 16:00 15:00 14:00 13:00 12:00 11:00 10:00 9:00 19:00 18:00 17:00 16:00 15:00 14:00 13:00 12:00 11:00 10:00 9:00 19:00 18:00 17:00 16:00 15:00 14:00 13:00 12:00 11:00 10:00 9:00 Introduction Distortion MisrepresentationConclusions

Misrepresentation of scoring rules Borda has misrepresentation 1. –Denote by l i j candidate j’s ranking in R i. –j’s Borda score is  i (m-l i j )=  i (m-  i j -1)=n(m-1)-  i  i j =n(m-1)-  j –j minimizes misrep.  j maximizes score. –Borda has undesirable properties. Scoring protocols with  = 1 are fully characterized in the paper. Theorem: F is a scoring rule. F has unbounded misrep. iff  1 =  2. –Corollary: Veto has unbounded misrep. Introduction Distortion MisrepresentationConclusions

The Maximin rule For two candidates j,k, denote by N(j,k) the number of voters who prefer j to k. Maximin elects max j min k N(j,k). –Condorcet consistent. Claim:  n m (Maximin)  1.62 (m-1). Proof: –Let R. W.l.o.g. 1 = argmin j  j, 2 = Maximin(R). –Let d be candidate 2’s Maximin score. –Denote c = (3-5 1/2 )/2. Introduction Distortion MisrepresentationConclusions

The Maximin rule continued We distinguish two cases. Case 1: d > cn. –At least cn voters prefer candidate 2 to 1. –Worst case: (1-c)n voters rank 1 first and 2 last, cn voters rank 2 first and 1 second. –  2 /  1  (1-c)/c (m-1)  1.62 (m-1). Case 2: d  cn. –Candidate 1’s maximin score  d. –  candidate s.t. 1 is ranked higher by  cn. –At least (1-c)n don’t rank 1 first. –  2 /  1  n(m-1)/(1-c)n  1.62 (m-1). Introduction Distortion MisrepresentationConclusions

Summary of misrepresentation results Voting RuleMisrepresentation Borda= 1 VetoUnbounded Plurality= m-1 Plurality w. Runoff= m-1 Copeland  m-1 Bucklin  m Maximin  1.62 (m-1) STV  1.5 (m-1) Introduction Distortion MisrepresentationConclusions

Computational issues discussed in paper, but exact characterization remains open. Distortion may be an obstacle for applying voting in multiagent systems. If prefs are constrained, still an important consideration. –In scheduling example with m=3, in STV there might be 3 times as much conflicts as in Borda. Introduction Distortion MisrepresentationConclusions