Essays in Fair Allocation Rules

Abstract

This thesis studies fair allocation rules and its associated problems: what they are, how to implement them, and how to apply them in economic environments. In Chapter 1, we introduce the value-free (v-f) reductions, which are operators that map a coalitional game played by a set of players to another similar game played by a subset of those players. We propose properties that v-f reductions may satisfy, we provide a theory of duality for them, and we characterize several v-f reductions (among which the value-free version of the reduced games proposed by Hart and Mas-Colell, 1989, and Oishi et al., 2016). Unlike reduced games, introduced to characterize values in terms of consistency, v-f reductions are not defined in reference to values. However, a v-f reduction induces a value. We characterize v-f reductions that induce the Shapley value, the stand-alone value, and the Banzhaf value. We connect our approach to the theory of implementation. Finally, we show that our new approach is a useful tool to provide new characterizations of values in terms of consistency, and we present new characterizations of the Banzhaf and the stand-alone values. In Chapter 2, we introduce two mechanisms that implement the Shapley value and the equal surplus value, respectively. The main feature of both mechanisms is that multiple proposers put forth allocation plans simultaneously. The implementation of a plan requires both consensus among proposers and acceptance of respondents. In case of disagreement among proposers, we use the bidding procedure introduced by Perez-Castrillo and Wettstein (J. Econ. Theory 100: 274-294, 2001), which facilitates a buyout of one proposer in each round. Then the difference between two values comes down to how proposers negotiate with respondents. In Chapter 3, we define the proportional ordinal Shapley (the POSh) solution, an ordinal concept for pure exchange economies in the spirit of the Shapley value. Our construction is inspired by Hart and Mas-Colell’s (1989) characterization of the Shapley value with the aid of a potential function. The POSh exists and is unique and essentially single-valued for a fairly general class of economies. It satisfies individual rationality, anonymity, and properties similar to the null-player and null-player out properties in transferable utility games. Moreover, the POSh is immune to agents’ manipulation of their initial endowments: It is not D-manipulable and does not suffer from the transfer paradox. Finally, we construct a bidding mechanism à la Pérez-Castrillo and Wettstein (2006) that implements the POSh in subgame perfect Nash equilibrium for economies where agents have homothetic preferences and positive endowments.

Date of Award	7 Jul 2021
Original language	English
Supervisor	Jesus David Perez Castrillo (Director)