Arrow’s Impossibility Theorem

Plan Majority voting Borda rule Arrow’s impossibility theorem Condorcet theorem Cycles Borda rule Condorcet critique (dependence on irrelevant alternatives) Arrow’s impossibility theorem

Why Use Majority Voting? (Nicolas Caritat, Marques De Condorcet, 1785) Assume n people have to decide whether to choose alternative A or B. There is equal probability that either of the choices is correct a-priori. Each one has private information that supports his judgement about the correct decision and assume that each one is more likely to be right than wrong. Their judgements are independent. Then the probability that a group makes correct choice under majority rule approaches one as n approaches infinity

Majority Voting With More Than Two Alternatives A condorcet winner is an alternative that gets majority of votes against any other alternative.

Find a Condorcet Winner An example Three citizens have to choose a project. There are three possible options: A,B,C. Each citizen ranks the options in the following way.

The Condorcet Paradox (Voting Cycles) An example Three citizens have to choose a project. There are three possible options: A,B,C. Each citizen ranks the options in the following way.

Sincere Voting and Strategic Voting Assume that first A is voted against B and then the winner is voted against C. Assume that everybody votes sincerely. What is the outcome? Now, let the order of the issues on the agenda be the same and assume that voter one realizes the outcome would he be voting sincerely. Should he change his behavior? What will be the outcome if he does?

Agenda Setting Power In case you want to induce outcome A and you know that everybody votes sincerely, what agenda (order of issues to be put to voting) would you suggest? Would you change your agenda, if you knew the voters were strategic?

Borda Rule (Jean Charles de Borda 1770) Let every voter assign a score to each one of the options. Count the total score of an option by summing up the scores given by the voters. The winner is the option that scores the highest.

Borda Rule vs Majority Rule What will be the outcome if Borda rule was used for this scenario? Assume majority voting with sincere voters. Can an agenda setter find an order of issues to induce Z? Y? X?

Condorcet Critique of Borda Rule (a simple example) Compare the outcome of the previous scenario to the following: Note that the only difference here is that voters 2,3 prefer Z to X (switch in the ordering of “irrelevant” alternatives)

Next? We saw that both Borda Rula and Majority Voting have undesirable features The outcome under Borda rule may depend on the ordering of “irrelevant alternatives” The social ordering induced by majority voting may be inconsistent (include cycles), thus giving excessive powers to an agenda setter. Is there a “good” collective decision rule?

What is a “good” rule that aggregates individual preferences? (Uniformity) It has to rank the alternatives whatever the configuration of individuals’ preferences is (Completeness) It has to rank all possible alternatives (Pareto) If all individuals prefer A to B, it has to rank A over B (Transitivity) If it ranks A over B and B over C, it has to rank A over C (IIA) Ranking of A over B has to be independent of how the indivuduals rank other alternatives

Arrow’s Impossibility Theorem Social ranking that satisfies all the properties (U, C, P, T, IIA) has to be dictatorial.