Jerry S. Kelly

Chapter 1 Impossibility theorems in the arrovian framework

Given a set of outcomes that affect the welfare of the members of a group, K.J. Arrow imposed the following five conditions on the ordering of the outcomes as a function of the preferences of the individual group members, and then proved that the conditions are logically inconsistent:- The social choice rule is defined for a large family of assignments of transitive orderings to individuals.- The social ordering itself is always transitive.- The social choice rule is not dictatorial. (An individual is a dictator if the social ordering ranks an outcome x strictly above another outcome y whenever that individual strictly prefers x to y.)- If everyone in the group strictly prefers outcome x to outcome y, then x should rank strictly above y in the social ordering.- The social ordering of any two outcomes depends only on the way that the individuals in the group order those same two outcomes.The chapter proves Arrows theorem and investigates the possibility of uncovering a satisfactory social choice rule by relaxing the conditions while remaining within the Arrovian framework, which is identified by the following five characteristics:- The outcome set is unstructured.- The society is finite and fixed.- Only information about the ordering of the outcome set is used to convey information about individual welfare.- The output of the social choice process is an ordering of the outcome set.- Strategic play by individuals is not considered.

NP-completeness of some problems concerning voting games

SummaryThe problem of confirming lower bounds on the number of coalitions for which an individual is pivoting is NP-complete. Consequently, the problem of confirming non-zero values of power indices is NP-complete. The problem of computing the Absolute Banzhaf index is #P-complete. Related problems for power indices are discussed.

Almost all social choice rules are highly manipulable, but a few aren't

Explores, for several classes of social choice rules, the distribution of the number of profiles at which a rule can be strategically manipulated. In this paper, we will do comparative social choice, looking for information about how social choice rules compare in their vulnerability to strategic misrepresentation of preferences.

Information and preference aggregation

Abstract. We investigate the implications of relaxing Arrows independence of irrelevant alternatives axiom while retaining transitivity and the Pareto condition. Even a small relaxation opens a floodgate of possibilities for nondictatorial and efficient social choice.

t or 1 minus t. That Is the Trade-Off

A social welfare function f is Arrowian if it has transitive values and satisfies Arrows independence axiom. For any fraction t and any Arrowian f, either there will be some individual who dictates on a subset containing at least the fraction t of outcomes, or at least the fraction 1 minus t of the pairs of outcomes have their social ranking fixed independently of individual preference. And for any Arrowian f, there is a set containing a large fraction of the citizens whose preferences are never consulted in determining the social ranking of a large fraction of the pairs of alternatives. Copyright 1993 by The Econometric Society.

Social choice and computational complexity

Abstract Mays conditions characterizing simple majority voting are used to determine what sets of properties force social choice rules to be computable and what sets of properties still permit uncomputable rules. The techniques used also allow discussion of degrees of computational complexity within the class of computable rules.

The Voting Paradox

We now have a fairly clear sense of exactly what simple majority voting is and what a set of characteristic properties for that rule looks like. It is extremely important for you to see that simple majority voting has only been defined for the case of exactly two alternatives. Most of the problems with simple majority voting arise in a context of more than two alternatives and all interesting contexts have more than two alternatives. Sometimes this is disguised as when we are asked to make a final choice between two party-selected candidates. But the social choice process for filling office starts with many potential candidates and includes many preliminary narrowing procedures: decisions to run or not run, nominating petitions, primaries, withdrawals, conventions,.... So let’s explore what happens when we increase the number of alternatives to three or more, labeled x, y, z ....

Non-monotonicity does not imply the no-show paradox

For a feasible set X of exactly three alternatives, Moulin [2] presents a resolute social choice rule satisfying the Condorcet condition, but which does not exhibit the no-show paradox or the twin paradox. He goes on to prove that when there are more than three alternatives no such example exists: The Condorcet condition implies both the no-show paradox and the twin paradox. Nurmi [3] asks a related question: Must every non-monotonic social choice rule exhibit the noshow paradox? This note provides a negative answer for jX jb 3. Let N 1⁄4 f1; 2; . . .g be the set of individuals, with LðX Þ denoting the set of strong orders on X. A resolute social choice rule g maps pairs ðJ; aÞ to X, where J is a non-empty subset of N and a A LðX Þ is a profile of preferences for J. A rule g exhibits the no-show paradox if there is a pair ðJ; aÞ with jJj > 1, and an element i A J, such that aðiÞ has gðJ; aÞ ranked below gðJnfig; ajJnfigÞ where ajJnfig is the restriction of a to Jnfig. Rule g exhibits the twin paradox if there is a pair ðJ; aÞ with jJj > 1, and elements i; j A J, such that aðiÞ 1⁄4 að jÞ and aðiÞ has gðJ; aÞ ranked below gðJnf jg; ajJnf jgÞ. Rule g exhibits the reinforcement paradox if there is a partition fJ1; J2g of J and profile a such that gðJ1; ajJ1Þ 1⁄4 gðJ2; ajJ2Þ but gðJ; aÞ is di¤erent from this common alternative. Now we show the existence of non-monotonic rules that do not exhibit the no-show paradox.

Weak independence and veto power

Abstract Weak independence (WI) prevents x from socially ranking above y at profile p if y ranks above x at profile p′ and each individual ordering of {x,y} is the same at p as at p′. If f is transitive-valued and satisfies WI and Pareto then someone has veto power.

Super Arrovian Domains with Strict Preferences

Given