Geometry of anonymous binary social choices that are strategy-proof

TL;DR

This paper characterizes strategy-proof, anonymous binary social choice functions using a geometric representation on a grid, providing a new way to understand their structure and a Python implementation.

Contribution

It introduces a geometric framework for describing strategy-proof, anonymous social choice functions and links it to previous quota-based representations.

Findings

Every such function can be described by a sequence of segment lengths.

The geometric description involves alternating horizontal and vertical segments.

A Python code for implementing these functions is provided.

Abstract

Let $V$ be society whose members express preferences about two alternatives, indifference included. Identifying anonymous binary social choice functions with binary functions $f=f(k,m)$ defined over the integer triangular grid $G=\{(k,m)\in \mathbb{N}_0\times\mathbb{N}_0 : k+m\le |V|\} $, we show that every strategy-proof, anonymous social choice function can be described geometrically by listing, in a sequential manner, groups of segments of G, of equal (maximum possible) length, alternately horizontal and vertical, representative of preference profiles that determine the collective choice of one of the two alternatives. Indeed, we show that every function which is anonymous and strategy-proof can be described in terms of a sequence of nonnegative integers $(q_1, q_2, \cdots, q_s)$ corresponding to the cardinalities of the mentioned groups of segments. We also analyze the connections between our present representation with another of our earlier representations involving sequences of majority quotas. A Python code is available with the authors for the implementation of any such social choice function.