<p>Baryshnikov presented a remarkable algebraic topology proof of Arrow’s impossibility theorem trying to understand the underlying reason behind the numerous proofs of this fundamental result of social choice theory. We continue this program, but focusing on combinatorial topology arguments that do not use advanced mathematics, providing a very intuitive geometric reason for Arrow’s impossibility under domain restrictions. We present a geometric proof for the basis case of two voters, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and three alternatives, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|X |=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, based on the index lemma, that counts the absolute number of times that a closed curve in the plane travels around a point. This yields a characterization of the domain restrictions that allow non-dictatorial aggregation functions and, as a consequence, Baryshnikov’s conjecture relating such domains with contractible spaces is revealed as untrue. It also exposes the geometry behind prior pivotal arguments to Arrow’s impossibility. We explain why the basis case of two voters, is where this interesting geometry happens, by giving a simple proof that this case implies Arrow’s impossibility for any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$|X |\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and any finite <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

A combinatorial topology approach to Arrow’s impossibility theorem

期刊论文
发表日期：

2025年6月4日

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Sergio Rajsbaum,
Armajac Raventós-Pujol

摘要

Baryshnikov presented a remarkable algebraic topology proof of Arrow’s impossibility theorem trying to understand the underlying reason behind the numerous proofs of this fundamental result of social choice theory. We continue this program, but focusing on combinatorial topology arguments that do not use advanced mathematics, providing a very intuitive geometric reason for Arrow’s impossibility under domain restrictions. We present a geometric proof for the basis case of two voters, $n=2$ $n = 2$ , and three alternatives, $|X |=3$ $| X | = 3$ , based on the index lemma, that counts the absolute number of times that a closed curve in the plane travels around a point. This yields a characterization of the domain restrictions that allow non-dictatorial aggregation functions and, as a consequence, Baryshnikov’s conjecture relating such domains with contractible spaces is revealed as untrue. It also exposes the geometry behind prior pivotal arguments to Arrow’s impossibility. We explain why the basis case of two voters, is where this interesting geometry happens, by giving a simple proof that this case implies Arrow’s impossibility for any $|X |\ge 3$ $| X | \geq 3$ and any finite $n\ge 2$ $n \geq 2$ .