1. Motivation: Condorcet Cycles Let people 1, 2 and 3 have to make a decision between options A, B, and C. Suppose they decide that majority voting is a good decision rule. Suppose further that 1’s preferences are A>B>C, 2’s preferences are C>A>B and 3’s preferences are B>C>A. We do stage-wise voting: A beats B because 1 and 2 prefer A to B; C beats A because 2 and 3 prefer C to A; and B beats C because 1 and 3 prefer B to C. Rinse and repeat and A beats B and we are trapped in an endless cycle. This is called a Condorcet Cycle. These can be generated by making each person’s preferences a cyclic permutation of the previous person’s preferences. Arrow generalized the idea of a Condorcet Cycle to show that any voting system with three or more options that takes only ordinal preferences and satisfies transitivity, unanimity and independence of irrelevant alternatives is a dictatorship. A Possible Out? Because aggregating individual’s preferences into a social preference is seen as the basis of democracy and Arrow’s Impossibility Theorem suggests that this is impossible, it is tempting to try to find an out. The most intuitive out is the impoverished information in ordinal preferences. Suppose instead that we had cardinal preferences. Rather than just information on whether person 1 prefers A to B, we also knew that person 1 thought that A was fives times better than B. We can assign intensity of preferences: Person 1: Person 2: Person 3: where the superscripts on the inequality sign represents the multiple by which each option is preferred. Combining, we see that A now beats B and C. Providing cardinal preferences seems like an out. Based on a conjecture by Samuelson (Arrow’s brother- in-law), Kalai and Schmeidler (1977) proved an impossibility theorem with four or more alternatives and cardinal preferences. Arrow’s Impossibility Theorem: I Definition: A transitive preference is a complete ranking of a set of outcomes. A voting system combines the preferences of N individuals into a social preference. Definition: The voting system satisfies unanimity if when every voter strictly prefers α to β, the voting system places α strictly above β. Definition: The voting system satisfies independence of irrelevant alternatives if the relative ranking of α and β is determined solely by the binary preferences of individuals (introducing a third alternative γ does not change the relative ranking of αand β). Definition: The voting system is a dictatorship if there is an individual such that for every pair α and β if that individual strictly prefers α to β, then the voting system places α strictly above β. Will His Vote Matter? A Cardinal Version of Arrow’s Impossibility Theorem Isaac Sorkin ’07 Swarthmore College, Department of Mathematics & Statistics References Geanakoplos, J. 2005. Three Brief Proofs of Arrow’s Impossibility Theorem. Economic Theory. 26: 211-215. Kalai, E. and D. Schmeidler. 1977. Aggregation Procedure for Cardinal Preferences: A Formulation and Proof of Samuelson’s Impossibility Conjecture. Econometrica. 45: 1431-1438. An Impossibility Theorem for Cardinal Preferences Definition: A procedure for aggregating cardinal preferences is a function that maps the set of cardinal preferences of N individuals over the outcome set into one set of cardinal preferences. Definition: Cardinal independence of irrelevant alternatives means that if two people have identical cardinal preferences over some subset of more than three outcomes, then the aggregating procedure maps to the same set of cardinal preferences over that subset of outcomes. Definition: Unanimity is defined as before. Definition: Continuity of the procedure is given by the continuity of representations of a preference relation. Theorem: If the number of alternatives is greater than or equal to four, then a procedure for aggregating cardinal preferences is continuous and satisfies cardinal independence of irrelevant alternatives and unanimity if and only if it is cardinally dictatorial. Proof Strategy: Project the cardinal function to an ordinal space. Identify an ordinal dictator by invoking Arrow’s Theorem. Then show that the ordinal dictator is a cardinal dictator. Step 1: Show that the aggregating procedure satisfies ordinal independence of irrelevant alternatives. Step 2: Show that tthe aggregating procedure satisfies the premises of Arrow’s Theorem, and hence has an ordinal dictator. Step 3: Show that the ordinal dictator is cardinal dictator: i)Show that the projection of the ordinal dictator’s cardinal preferences into ordinal space equals the projection of society’s cardinal preferences into ordinal space; ii) Show that if the ordinal dictator’s preferences are equal over two cardinal profiles, then so are society’s. iii) Show that the cardinal preferences of the ordinal dictator equal society’s cardinal preference. Acknowledgements I want to thank Henry Swift for invaluable aid; this, of course, entirely implicates him in any errors I may have committed. Conclusion I have presented a proof of Arrow’s Theorem followed by a proof of a stronger theorem which shows that the obvious out of introducing intensity of preference does not work. Another possible out is to question whether dictatorship as defined in social choice theory is really so bad, or else to wonder if we must aggregate preferences through voting: perhaps some Quaker consensus building would work! Arrow’s Impossibility Theorem: II Theorem: Any voting system that satisfies transitivity, unanimity, and independence of irrelevant alternatives is a dictatorship. Proof Strategy: Identify a limited dictator (a dictator over one profile (set of preferences)). Show that this limited dictator is a genuine dictator. Step 1 (Extremal lemma): Show that if everyone in society places alternative α at either the top or bottom of his preference ranking then society must as well. Step 2: Show that there exists a voter who at some profile can move α from the top to bottom, or vice- versa (this voter is the limited dictator). Step 3: Show that this voter is a dictator over any pair not involving α. Step 4: Show that this voter is a dictator over every possible pair, and hence a genuine dictator.