For a real number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1319_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>, a weak game of threats (N, v) consists of a set N of n players and a function <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1319_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$v:2^N\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo>:</mo> <msup> <mn>2</mn> <mi>N</mi> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1319_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="181" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega v(\emptyset )+(1-\omega )v(N)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∅</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1319_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$v(\emptyset )\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∅</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> possibly. It is shown that there exists a unique value with respect to <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1319_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> for weak games of threats that satisfies efficiency, linearity, symmetry and the null player property.

Pseudo-Shapley value for weak games of threats

期刊论文
发表日期：

2025年5月21日

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Daniel Li Li,
Erfang Shan

摘要

For a real number $\omega $ $ω$ , a weak game of threats (N, v) consists of a set N of n players and a function $v:2^N\rightarrow \mathbb {R}$ $v : 2^{N} \to R$ such that $\omega v(\emptyset )+(1-\omega )v(N)=0$ $ω v (\emptyset) + (1 - ω) v (N) = 0$ , where $v(\emptyset )\ne 0$ $v (\emptyset) \neq 0$ possibly. It is shown that there exists a unique value with respect to $\omega $ $ω$ for weak games of threats that satisfies efficiency, linearity, symmetry and the null player property.