Theoretical background Coalition manipulability Optimal voting systems Manipulability of voting systems François Durand, Fabien Mathieu, Ludovic Noirie François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Introduction Origin: politics. Applications: any situation of collective choice. Can we trust the electors? If not, is the result stable? François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Table of contents 1 Theoretical background Preferences Gibbard’s theorem 2 Coalition manipulability Definition and examples Simulation results 3 Optimal voting systems Position of the problem Reformulation 4 Conclusion François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Table of contents 1 Theoretical background Preferences Gibbard’s theorem 2 Coalition manipulability Definition and examples Simulation results 3 Optimal voting systems Position of the problem Reformulation 4 Conclusion François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Preferences of one elector m C = fcj gj=1: candidates. Preferred candidate: c1 2 C ! very poor description. 0 1 c1 Strict preference ordering: @ c2 A 2 Sm ! better. c3 But does not describe strength of preferences. Profile 1 Profile 2 c1 c2 c3 François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Preferences of one elector: utility The elector attributes a real number uj called utility to each candidate cj . Profile 1 Profile 2 c1 10 10 c2 5 -5 c3 -10 -10 Von Neumann-Morgenstern theorem (1944) gives sufficient and necessary conditions for such a representation to make sense. m When a utility vector in R is without ties, it induces a strict preference ordering in Sm. François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Preferences of the population n E = fei gi=1: electors. m n Population: an element of (R ) . m n Culture: a probability law over (R ) . Uniform spherical culture: each elector is drawn independently with a uniform distribution over the unit hypersphere. Each elector almost surely has a strict preference order in Sm. n It induces a uniform distribution over (Sm) . François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems 65 40 35 75 60 25 cj is Condorcet winner iff for any other candidate ck , cj is preferred to ck by more than half the electors. No Condorcet winner in the above example: Condorcet paradox. Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Condorcet winner Majority matrix (matrix of duels): Mjk is the number of electors preferring cj to ck . Electors 40 35 25 Duels c1 c2 c3 c1 c2 c3 c1 Preferences c2 c3 c1 c2 c3 c1 c2 c3 François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems 40 75 60 25 cj is Condorcet winner iff for any other candidate ck , cj is preferred to ck by more than half the electors. No Condorcet winner in the above example: Condorcet paradox. Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Condorcet winner Majority matrix (matrix of duels): Mjk is the number of electors preferring cj to ck . Electors 40 35 25 Duels c1 c2 c3 c1 c2 c3 c1 65 Preferences c2 c3 c1 c2 35 c3 c1 c2 c3 François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems 75 25 cj is Condorcet winner iff for any other candidate ck , cj is preferred to ck by more than half the electors. No Condorcet winner in the above example: Condorcet paradox. Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Condorcet winner Majority matrix (matrix of duels): Mjk is the number of electors preferring cj to ck . Electors 40 35 25 Duels c1 c2 c3 c1 c2 c3 c1 65 40 Preferences c2 c3 c1 c2 35 c3 c1 c2 c3 60 François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems cj is Condorcet winner iff for any other candidate ck , cj is preferred to ck by more than half the electors. No Condorcet winner in the above example: Condorcet paradox. Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Condorcet winner Majority matrix (matrix of duels): Mjk is the number of electors preferring cj to ck . Electors 40 35 25 Duels c1 c2 c3 c1 c2 c3 c1 65 40 Preferences c2 c3 c1 c2 35 75 c3 c1 c2 c3 60 25 François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Condorcet winner Majority matrix (matrix of duels): Mjk is the number of electors preferring cj to ck . Electors 40 35 25 Duels c1 c2 c3 c1 c2 c3 c1 65 40 Preferences c2 c3 c1 c2 35 75 c3 c1 c2 c3 60 25 cj is Condorcet winner iff for any other candidate ck , cj is preferred to ck by more than half the electors. No Condorcet winner in the above example: Condorcet paradox. François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Table of contents 1 Theoretical background Preferences Gibbard’s theorem 2 Coalition manipulability Definition and examples Simulation results 3 Optimal voting systems Position of the problem Reformulation 4 Conclusion François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Game form n E = fei gi=1: electors. m C = fcj gj=1: candidates. For i 2 f1;:::; ng; Si a non-empty set: elector ei ’s strategies. Game form: g : S1 × ::: × Sn !C. In the usual two-round system, a strategy is given by: The name of a candidate; For each pair of candidates, the name of one of them. François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Dominant strategies Preference ordering: a total preorder over C (allow ties). A strategy s 2 Si is dominant for elector ei and preference ordering P iff, whatever the other electors do, strategy s for ei produces an outcome at least as high in P as does any other strategy in Si . A game form is straightforward iff for every elector ei and preference ordering P, some strategy is dominant for ei and P. If the game form is not straightforward, it means that even if you have a preference ordering, you may not know what to vote! It may depend on the other electors... François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Dictatorship An elector ei is a dictator for game form g iff, for every possible outcome c 2 g(S1 × ::: × Sn), elector ei has a strategy that ensures c to win, whatever the other electors do. A game form is dictatorial if there is a dictator for g. ei being a dictator doesn’t mean that g depends only on its i-th variable. Simply, ei can choose the winner if she wants to. François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Gibbard’s theorem (1973) Every straightforward game form with at least three possible outcomes is dictatorial. Very general: No symmetry assumption between the electors or candidates. No assumption that the sets of strategies are the same. No assumption about what a “sincere” strategy should be. No assumption that an elector has a preference ordering. Only strong assumption: the game form is deterministic. But cf. Gibbard 1977. François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Definition and examples Coalition manipulability Simulation results Optimal voting systems Table of contents 1 Theoretical background Preferences Gibbard’s theorem 2 Coalition manipulability Definition and examples Simulation results 3 Optimal voting systems Position of the problem Reformulation 4 Conclusion François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Definition and examples Coalition manipulability Simulation results Optimal voting systems Voting system and manipulability m For i 2 f1;:::; mg, σi : R ! Si : sincerity functions. Voting system: game form + sincerity functions. A voting system is subject to manipulation if the sincere strategy is not always dominant. In most systems, we have a cultural idea of sincerity, but not always. Example: each elector votes black or white. If the number of black ballots is odd, candidate c1 wins: otherwise, c2 wins... Gibbard’s theorem ensures that, whatever the sincerity functions are, a voting system that is not subject to manipulation and has at least three possible outcomes is dictatorial. François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems