Convergence to Equilibria in Plurality Voting

1. Convergence to Equilibria in Plurality Voting Reshef Meir School of Computer Science and Engineering Hebrew University, Jerusalem, Israel Joint work with Maria Polukarov, Jeffery S. Rosenschein and Nick Jennings To appear in AAAI 2010

2. Content • Example • Voting background • Voting as a normal form game • Iterative voting and convergence • Variations of the game • Results • conclusion

3. CARS C>A>B C>B>A

4. Voting: model CARS a b c d • Set of voters V = {1,...,n} • Voters may be human or machines • Set of Candidates A = {a,b,c...}; |A|=m. • Candidates may also be any set of alternatives, e.g. a set of movies to choose from • Every voter has a private rank over candidates • The ranking is a complete, transitive order • E.g. d>a>b>c 4

5. Voting profiles a b a b a c c c b • The preference order of voter i is denoted by Ri • Denote by R (A) the set of all possible orders on A • Ri is a member of R (A) • The preferences of all voters are called a profile • R = (R1,R2,…,Rn)

6. Voting rules • A voting rule decides who is the winner of the elections • The decision has to be defined for every profile • Formally, this is a function f : R(A)n A

7. The Plurality rule • Each voter selects a candidate • Voters may have weights • The candidate with most votes wins • Tie-breaking scheme • Deterministic: the candidate with lower index wins • Randomized: the winner is selected at random from candidates with highest score

8. CARS Voting as a normal-form game W2=4 a c b W1=3 a b c Initial score: 7 9 3

9. CARS Voting as a normal-form game W2=4 a c b W1=3 a b c Initial score: 7 9 3

10. CARS Voting as a normal-form game W2=4 a c b W1=3 a b c Initial score: 7 9 3

11. Voting as a normal-form game W2=4 a c b W1=3 a b c a > b > c Voters preferences: Nash Equilibria c> a > b

12. Voting in turns • We allow each voter to change his vote • Only one voter may act at each step • The game ends when there are no objections This mechanism is implemented in some on-line voting systems, e.g. in Google Wave

15. Rational moves We assume, that voters only make rational steps, but what is “rational”? • Voters do not know the preferences of others • Voters cannot collaborate with others Thus, improvement steps are myopic, or local .

16. CARS Dynamics • There are two types of good steps that a voter can make C>D>A>B “Better replies”

17. CARS Dynamics • There are two types of good steps that a voter can make C>D>A>B “Best reply” (always unique)

18. Properties of the game Properties of the players Variations of the voting game • Tie-breaking scheme: • Deterministic / randomized • Agents are weighted / non-weighted • Number of voters and candidates • Voters start by telling the truth / from arbitrary state • Voters use best replies / better replies

19. Some games never converge • Initial score = (0,1,3) • Randomized tie breaking W2=3 a c b W1=5 a b c

20. Some games never converge a > b > c Voters preferences: b > c> a W2=3 a c b W1=5 a a a c b bc b b c c c c

21. Some games never converge a > b > c >bc Voters preferences: b > bc> c> a W2=3 a c b W1=5 No equilibrium! a a a c b bc b b c c c c

22. The main question: Under what conditions the game is guaranteed to converge? Also, if it converges, then • How fast? • To what outcome?

23. Is convergence guaranteed? ? X

24. Some games always converge Theorem: Let G be a Plurality game with deterministic tie-breaking. If voters have equal weights and always use best-reply, then the game will converge from any initial state. Furthermore, convergence occurs after a polynomial number of steps.

25. Some games always converge Theorem: Let G be a Plurality game with deterministic tie-breaking. If voters have equal weights and always use best-reply, then the game will converge from any initial state. We will show the proof for the case where voters start by telling the truth

26. Proof of convergence (I) We prove by induction on the following invariants: • The score of the winner never decreases • Each step promotes a less preferable candidate (for the manipulator) • If a voter i “deserts” a candidate, no other voter will ever vote for this candidate (or for any candidate that is better for i)

27. Proof of convergence (II) Base case: • Before the first step, each voter votes for his most preferable candidate, thus (2) holds. • No one will desert the winner, thus (1) also holds. • (3) cannot be violated in a single step.

28. Proof of convergence (III) Suppose that (3) is violated at time t. that is, there is some step at time t’ < t : a c d b st’(winner) > st’(b) > st’(c) + 1

29. (winner) Proof of convergence (IV) and at time t some agent votes for c again This cannot be an improvement step! (unless there had been an earlier violation) a c d b st(winner) ≥ st’(winner) > st’(b) > st’(c) + 1 ≥ st(c) + 1

30. Proof of convergence (V) Now assume (2) is violated. That is, a voter j votes for a more preferable candidate (e.g. a) at time t. Thus there was a step t’<t, where j selected c <ja, since a could not win. Therefore, a cannot win now. st(winner) ≥ st’(winner) > st’(a) +1 ≥ st(a) +1 a c d b

31. Proof of convergence (VI) Finally, a violation of (1) implies a violation of (2), since a voter will not desert the winner for a less preferable candidate. Therefore, if there are no violations until step t, there are no violations in step t+1 We also note, that in this case convergence occurs after at most m-1 steps

32. Results - summary X X X X V V V X V X X V X X X X ? V X X X

33. Conclusions • The “best-reply” seems like the most important condition for convergence • The winner may depend on the order of players (even when convergence is guaranteed) • Iterative voting is a mechanism that allows all voters to agree on a candidate that is not too bad

34. Related work (solution concepts) • A lot of work about strategic behavior by multiple independent voters • Feddersen, Sened, and Wright 1990 (single peak) • Messner & Polborn 2002 (strong equilibrium) • Peleg 2002 • Dhillon and Lockwood 2004 (dominated strategies) and many more… • Crucially, they all assume that full preferences of all voters are known

35. Related work (partial knowledge) • Myerson & Weber (1993) analyzed voting equilibria in a complex model with partial information (polls) and non-atomic voters • Our model is more suitable when there are few voters • Chopra, Pacuit and Parikh (2004) focus on the relations between knowledge and strategic behavior

36. Related work (sequential voting) • Farquharson (1969) analyzed a model where a different issue is voted upon in each turn • Showed how the game can be solved with backward induction • A different model was studied by Airiau and Endriss (2009), where in every step voters choose between the current winner and a suggested alternative • Show sufficient conditions for convergence (of payoffs)

37. Future work • Extend to voting rules other than Plurality • Investigate the theoretic properties of the newly induced voting rule (Iterative Plurality) • Study more far sighted behavior • In cases where convergence in not guaranteed, how common are cycles? • Experimental data

38. Questions?