Lecture 4

1. Lecture 4 Social Choice Readings: Shepsle, Analyzing politics, chapter 3

2. Group Preferences and group choice • Suppose individuals have rational preferences (i.e. their preferences are complete, transitive) • They have to undertake a decision that affects the entire group • Does the group have preferences that are rational?

3. 3 people, 3 options

4. Group choice • Unanimity? • First preference majority? • Majority voting The method of majority voting requires that, for any pair of alternatives A and B, A is preferred by the group to B if the number of group members who prefer A to B exceeds the number of group members that prefer B to A • Round-robin tournament • Movie vs. Pub • Movies vs.. Stadium • Pub vs.. stadium • Are group preferences rational?

5. 3 people, 3 options

6. Round-robin tournament • Movie vs. Pub • Movies vs. Stadium • Pub vs. stadium • Are group preferences rational?

7. Individual and group preferences • Even if individuals have rational preferences, the group preferences may not satisfy rationality • In the previous example, the group preferences do not satisfy transitivity • As a consequence, in each pair wise comparison of alternatives we obtain that a different majority coalition is formed in favour of an alternative • In other words, group preferences “cycle”

8. Condorcet winner • When group preferences are rational, then a unique winner emerges in pair wise comparisons • The alternative that beats all the others in any pair wise comparison is the so called Condorcet winner • When group preferences are not transitive, there is no Condorcet winner

9. Distributive politics and cycling majority • One important example where cycling majority can arise is the when individuals have to agree on the sharing of resource • Example: budget allocation across constituencies or different type of individuals • This is an example of “divide the dollar game” where typically group preferences are not transitive • For any allocation supported by a majority, there is an alternative allocation preferred by a different majority • Hence, in a divide the dollar game, there is no Condorcet winner

10. Agenda power and voting rules • Divide-the-dollar type of situations are common in distributive politics and simple majority voting would result in a complete legislative impasse • In practice, legislative deadlock is prevented adopting rules that restrict the set of alternatives of which the vote can take place • Agenda power to some member • Defeated alternatives are eliminated • Limited number of voting rounds If an individual has agenda power (for example he can decide the order according to which proposal are put forward for the vote) and the number of voting rounds is limited, the order according to which proposals are put up for vote has a crucial impact on the outcome that will be approved! The agenda setter can manipulate the agenda to have his most preferred outcome implemented

11. Desirable properties of mechanisms of collective decision making • The previous example shows how restricting the set of alternatives of which the vote can take place, cycling can be avoided • Are those restrictions desirable? • What could be some “minimal properties” that a mechanism of collective decision making should satisfy?

12. Arrow’s theorem: Assumptions • Minimal properties • Rationality of Individuals + • Universal Admissibility (Condition U) • Pareto Optimality (Condition P) • Independence from Irrelevant Alternatives (Condition I) • Nondicatorship (condition D)

13. Arrow’s theorem: Assumptions • Let i be an individual belonging to group G and let A be the set of alternatives over which G must take a decision • Rationality of i: preference over the alternatives in A are complete and transitive • Universal Admissibility (Condition U): each individual i can adopt any strong or weak and transitive preference ordering over the alternatives in A • Pareto Optimality (Condition P): If every member of the group G prefers alternative a over alternative b, then the group preference must reflect a preference for a over b • Independence from Irrelevant Alternatives (Condition I): If alternatives a and b are ranked in a specific way by every individual and this ranking does not change, then the group ranking of a with respect to b should not change. This must be true even if individual preferences of other irrelevant alternatives are introduced. • Nondicatorship (condition D): there is no individual i belonging to the group whose own preferences dictates the group preferences

14. Arrow’s Theorem • There exists no mechanism for translating the preferences of rational individuals into coherent group preferences that simultaneously satisfy conditions U, P, I and D • In other words, any mechanism generating a group choice that satisfies U, P and I is either dictatorial or incoherent • Hence, the Arrow Impossibility theorem highlights the existence of a fundamental trade-off in social choice: trade-off between rationality and the concentration of power

15. Majority voting and Arrow’s Theorem • Arrow’s theorem applies to any mechanism of collective decision making, hence it applies to majority voting • Remember the previous divide-the-dollar game • Either cycling (rationality violated) • Or give agenda power to a group member (non-dictatorship violated)

16. Majority voting and Arrow’s Theorem • However, we have seen that in some cases individual rational preferences can be aggregated into coherent group preferences via majority voting (remember the example where a Condorcet winner exists) • How is this possible? • The Arrow theorem states some very general conditions implying that aggregation of individual preferences cannot guarantee group coherence in all situations • However, if we impose some restrictions (i.e. we relaxed some of the minimal assumptions behind the theorem) then we can obtain group coherence

17. Questions • Mark, Jim and Sarah won a lottery price of £400 and Mark proposes to share the price equally. Would Mark’s proposal be approved by the majority? If not, how would they split the price? • Discuss whether majority voting is a mechanism of aggregation of individual preferences that can lead to coherent group preferences