APPROVAL VOTING, REPRESENTATION, & LIQUID DEMOCRACY

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1 APPROVAL VOTING, REPRESENTATION, & LIQUID DEMOCRACY Markus Brill TU Berlin Based on joint work with: Haris Aziz, Vincent Conitzer, Edith Elkind, Rupert Freeman, Svante Janson, Martin Lackner, Jean-Francois Laslier, Dominik Peters, Piotr Skowron, & Toby Walsh (and on disjoint work of Luis Sánchez Fernández et al.)

2 APPROVAL VOTING (PROPORTIONAL) REPRESENTATION LIQUID DEMOCRACY

3 3 OUTLINE Representative committees approval-based committee (ABC) rules representation axioms Apportionment methods Representative rankings Liquid Democracy

4 4 SETTING set of candidates C set of voters N = {1,, n} each voter i approves a subset of candidates A i C Goal: select a set W of W = k winners (the committee)

5 5 WHAT S WRONG WITH APPROVAL VOTING? Approval Voting (AV): select the k candidates with the highest approval score Problem: dictatorship of the majority 51% vote {c 1, c 2,, c k } W= {c 1, c 2,, c k } toy example: 4 candidates c 1, c 2, c 3, c 4, n = 9 voters, k = W = 3 5 x {c 1, c 2, c 3 } 4 x {c 4 }

6 6 APPROVAL-BASED COMMITTEE RULES Minimax-AV: minimize the maximal (Hamming) distance to ballots [Brams, Kilgour & Sanver 2007] Toy example (k=3) 5 x {c 1, c 2, c 3 } 4 x {c 4 } Satisfaction-AV: maximize total satisfaction i N sat(i,w), where sat(i,w) = A i W / A i [Brams & Kilgour 2014] equivalent to equal and even cumulative voting Greedy-AV: sequentially pick candidates that are approved by most non-represented voters

7 7 THIELE S METHODS Proportional Approval Voting (PAV): maximize total score i N score(i,w), where score(i,w) = 1 + 1/2 + 1/ / W A i [Thiele 1895, Simmons 2001] sequential version: start with empty set and add candidates sequentially, greedily optimising the total score in each step reverse sequential version: start with full set C and delete candidates sequentially, greedily optimising the total score in each step generalizations: replace (1, 1/2, 1/3, ) with arbitrary weights

8 8 THIELE S METHODS: EXAMPLE Example due to Tenow: 3 candidates c 1, c 2, c 3 n = 44 voters: 12 x {c 1, c 2 } 12 x {c 1, c 3 } 10 x {c 2 } 10 x {c 3 } method k=1 k=2 optimal c 1 c 2, c 3 sequential c 1 c 1,c 2 or c 1,c 3 reverse sequential c 2 or c 3 c 2, c 3

9 9 PHRAGMÉN S RULE Proposed by Swedish mathematician Lars Edvard Phragmén ( ) Sequential load balancing procedure: at each step, a candidate c is chosen and one unit of load is distributed among the approvers of c goal: make distribution of loads as even as possible (i.e., maximal load as small as possible)

10 10 PHRAGMÉN S RULE: EXAMPLE 6 candidates c 1, c 2, c 3, c 4, c 5, c 6 k = W = 4 n = 7 voters: 3 x {c 1, c 2, c 3, c 4 } 1 x {c 5, c 6 } 1 x {c 1, c 2, c 3 } 1 x {c 1, c 2 } 1 x {c 1, c 6 }

11 VOTER 1 VOTER 2 VOTER 3 VOTER 4 VOTER 5 VOTER 6 VOTER 7 {c 1,c 2,c 3,c 4 } {c 5,c 6 } {c 1,c 2,c 3 } {c 1,c 2 } {c 1,c 6 }

12 Committee: c /6 1/6 1/6 1/6 1/6 1/6 VOTER 1 VOTER 2 VOTER 3 VOTER 4 VOTER 5 VOTER 6 VOTER 7 {c 1,c 2,c 3,c 4 } {c 5,c 6 } {c 1,c 2,c 3 } {c 1,c 2 } {c 1,c 6 }

13 Committee: c1 c /6 1/6 1/6 1/6 1/6 1/6 VOTER 1 VOTER 2 VOTER 3 VOTER 4 VOTER 5 VOTER 6 VOTER 7 {c 1,c 2,c 3,c 4 } {c 5,c 6 } {c 1,c 2,c 3 } {c 1,c 2 } {c 1,c 6 }

14 Committee: c1 c2 c / / /6 1/6 1/6 1/6 1/6 1/6 VOTER 1 VOTER 2 VOTER 3 VOTER 4 VOTER 5 VOTER 6 VOTER 7 {c 1,c 2,c 3,c 4 } {c 5,c 6 } {c 1,c 2,c 3 } {c 1,c 2 } {c 1,c 6 }

15 Committee: c1 c2 c6 c /4 1/4 1/4 1/ / / /6 1/6 1/6 1/6 1/6 1/6 VOTER 1 VOTER 2 VOTER 3 VOTER 4 VOTER 5 VOTER 6 VOTER 7 {c 1,c 2,c 3,c 4 } {c 5,c 6 } {c 1,c 2,c 3 } {c 1,c 2 } {c 1,c 6 }

16 16 REPRESENTATION AXIOMS Intuition: each group of n/k voters deserves one representative group needs to be cohesive, i.e., have at least one candidate in common Similarly: for each l {0,1,,k}, each l cohesive group of l n/k voters deserves l representatives

17 17 REPRESENTATION AXIOMS EJR: extended justified representation 8X N : X `n k and \ i2x A i ` ) (9i 2 X : W \ A i `) PJR: proportional justified representation 8X N : X `n k and \ i2x A i ` ) W \ ([ i2x A i ) ` JR: justified representation n 8X N : X k and \ i2x A i 1 ) W \ ([ i2x A i ) 1

18 18 REPRESENTATION AXIOMS EJR PAV (Thiele s optimal method) not for the sequential version! not for any other weight vector! PJR sequential Phragmén Greedy-AV JR Many rules fail JR: AV, Minimax-AV, Satisfaction-AV,

19 19 OPTIMAL VS SEQUENTIAL RULES optimal" sequential" description solving global optimisation problem greedy approximation, adding/ deleting alternatives one-by-one pro s satisfy representation axioms easy to compute committee monotonic con s computational complexity not committee monotonic no axiomatic guarantees (exception: seq. Phragmén)

20 20 REPRESENTATIVE RANKINGS Same input, but different output: ranking of candidates Sequential committee rules naturally produce rankings When is a ranking representative? require representation axiom (e.g., EJR) to hold for every prefix of the ranking? Problem 1: no sequential rule satisfying EJR is known Problem 2: binary criterion

21 21 PROTECTION OF MINORITIES IN LIQUID DEMOCRACIES The Principles of LiquidFeedback (Behrens, Kistner, Nitsche, Swierczek)

22 22 JUSTIFIABLE DEMAND Idea: cohesive groups can demand to be proportionally represented in the top positions of the ranking Example: if a group makes up 25% of the voters, it has a justifiable demand of 1 w.r.t. the top 4 positions, a justifiable demand of 2 w.r.t. the top 8 positions, and a justifiable demand of 3 w.r.t. the top 12 positions Measure how often (and by how much) justifiable demands are violated for different rules!

23 23 EXPERIMENTAL RESULTS I avg. satisfaction (fraction of justified demand) a subgroup of size n/3 has an average satisfaction of only 50% of their justified demand relative size of subgroup

24 24 EXPERIMENTAL RESULTS II Seq. PAV Rev. seq. PAV 2-geom. RAV seq. Phragmén For details, see forthcoming IJCAI paper Proportional Rankings (available on arxiv)

25 25 OTHER RESEARCH PROBLEMS INSPIRED BY LIQUID DEMOCRACY Delegated voting dealing with delegation cycles, abstentions, Preference elicitation (e.g. participatory budgeting) Desirable properties (e.g. independence of clones) Strategic aspects

26 26 REFERENCES H. Aziz, M. Brill, V. Conitzer, E. Elkind, R. Freeman, and T. Walsh. Justified representation in approval-based committee voting. Social Choice and Welfare (and AAAI 2015). M. Brill, R. Freeman, S. Janson, and M. Lackner. Phragmén s voting methods and justified representation. AAAI M. Brill, J.-F. Laslier, and P. Skowron. Multiwinner approval rules as apportionment methods. AAAI Sánchez-Fernández, N. Fernández, J. A. Fisteus, P. Basanta Val, E. Elkind, M. Lackner, P. Skowron. Proportional Justified Representation. AAAI P. Skowron, M. Lackner, M. Brill, D. Peters, and E. Elkind. Proportional rankings. IJCAI Forthcoming. THANK YOU!