<p>We extend the theory of TU-games with utility functions, which is a generalization of TU-games with restricted cooperation, to include dual games. By using the theory of dual games, we define dually-<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6549_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{u}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">u</mi> </math></EquationSource> </InlineEquation>-essential coalitions and show that they characterize the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6549_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{u}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">u</mi> </math></EquationSource> </InlineEquation>-prenucleolus of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6549_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{u}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">u</mi> </math></EquationSource> </InlineEquation>-balanced games. Additionally, we demonstrate that the intersection of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6549_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{u}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">u</mi> </math></EquationSource> </InlineEquation>-essential and dually-<InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6549_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{u}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">u</mi> </math></EquationSource> </InlineEquation>-essential coalitions also forms a characterization set for the <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6549_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{u}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">u</mi> </math></EquationSource> </InlineEquation>-prenucleolus, provided that the <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6549_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{u}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">u</mi> </math></EquationSource> </InlineEquation>-least-core is a proper subset of the <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6549_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{u}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">u</mi> </math></EquationSource> </InlineEquation>-core.</p>

Characterizations of the $\textbf{u}$-prenucleolus by dually-$\textbf{u}$-essential coalitions

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2025年3月31日

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Zsófia Dornai,
Miklós Pintér

摘要

We extend the theory of TU-games with utility functions, which is a generalization of TU-games with restricted cooperation, to include dual games. By using the theory of dual games, we define dually- $\textbf{u}$ $u$ -essential coalitions and show that they characterize the $\textbf{u}$ $u$ -prenucleolus of $\textbf{u}$ $u$ -balanced games. Additionally, we demonstrate that the intersection of $\textbf{u}$ $u$ -essential and dually- $\textbf{u}$ $u$ -essential coalitions also forms a characterization set for the $\textbf{u}$ $u$ -prenucleolus, provided that the $\textbf{u}$ $u$ -least-core is a proper subset of the $\textbf{u}$ $u$ -core.

Characterizations of the \(\textbf{u}\)-prenucleolus by dually-\(\textbf{u}\)-essential coalitions

摘要