Three essays in collective choice theory

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Virginia Tech


This dissertation contains three essays at the frontier of social choice theory and the theory of games.

In the first essay, we consider the problem of dividing a fixed quantity of a perfectly divisible good among n individuals with single-peaked preferences.

We show that the properties of Strategy-proofness, Efficiency, and either Anonymity or No Envy, together characterize a unique solution which we call the uniform allocation rule: everyone gets his best choice within the limits of an upper and a lower bound that are common to all individuals and determined by the feasibility constraint.

We further analyze the structure of the class of all strategy-proof allocation rules.

The second essay explores the idea of Population Monotonicity in the framework of cooperative games. An allocation scheme for a cooperative game specifies how to allocate the worth of every coalition. It is population monotonic if each player's payoff increases as the coalition to which he belongs grows larger.

We show that, essentially, a game has a population monotonic allocation scheme (PMAS) if and only if it is a positive linear combination of monotonic simple games with veto control. A dual characterization is also provided.

Sufficient conditions for the existence of a PMAS include convexity and "increasing average marginal contributions". If the game is convex, its (extended) Shapley value is a PMAS.

The third essay considers the problem of two individuals who must jointly choose one from a finite set of alternatives.

We argue that more consensus should not hurt: the closer your preferences are to mine, the better I should like the selected alternative. Two classes of Pareto optimal choice rules -- called generalized maximin" and "choosing-by-veto" rules -- are shown to satisfy this principle. If we strengthen Pareto Optimality along the lines of Suppes' grading principle, the only choice rules satisfying our condition are "simple" maximin rules.