This paper revisits the n-player rent-seeking contest with homogeneous valuations and increasing returns. Our main result says that, for any <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40505_2025_301_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="143" /> </InlineMediaObject> <EquationSource Format="TEX">$m\in \{2,\ldots ,n-1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, there are threshold values <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40505_2025_301_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="178" /> </InlineMediaObject> <EquationSource Format="TEX">$1<R_(m)<R^(m)\le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mmultiscripts> <mi>R</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <msup> <mi>R</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> for the Tullock parameter R such that a pure strategy equilibrium with m active players exists if and only if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40505_2025_301_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="147" /> </InlineMediaObject> <EquationSource Format="TEX">$R\in [R_(m),\,R^(m)]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mmultiscripts> <mi>R</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.166667em" /> <msup> <mi>R</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. Among other things, this observation leads to a simple characterization of the values of R for which the n-player contest has a unique pure strategy equilibrium.

On asymmetric equilibria in rent-seeking contests with strictly increasing returns

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2025年10月1日

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Christian Ewerhart

摘要

This paper revisits the n-player rent-seeking contest with homogeneous valuations and increasing returns. Our main result says that, for any $m\in \{2,\ldots ,n-1\}$ $m \in {2, \dots, n - 1}$ , there are threshold values $1<R_*(m)<R^*(m)\le 2$ $1 < R_{*} (m) < R^{*} (m) \leq 2$ for the Tullock parameter R such that a pure strategy equilibrium with m active players exists if and only if $R\in [R_*(m),\,R^*(m)]$ $R \in [R_{*} (m), R^{*} (m)]$ . Among other things, this observation leads to a simple characterization of the values of R for which the n-player contest has a unique pure strategy equilibrium.