Topological Social Choice

Existence and uniqueness of topological social choice is a class of problems which consider preference aggregates and generalize Arrow's impossibility theorem.

We consider:

  • Set of preferences that is a topologized as a space $X$, (such as preferred prices, budget allocation ratios, or relative rankings of candidates).
  • Population of $n$ agents, each with a fixed preference.
  • The state of that population's preferences is an $n$-tuple of points $\xi \in X^n$.

The conversion of individual (local) choice into a single global choice is via social choice map $\Xi : X^n \to X$ with constraints:

  1. Continuity: small shifts in local preferences have small impact on the aggregate global preference.
  2. Unanimity: identity on the grand diagonal in $X^n$, so that a unanimous preference is accepted.
  3. Anonymity: $\Xi$ is invariant under the permutation on the factors of $X^n$

The question of existence of a choice map parallels existence of a Nash equilibrium in game theory. Here, instead of universal existence, there is a near-universal non-existence.



Topics in Topological Data Analysis

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