Characterizations of the $$\textbf{u}$$-prenucleolus by dually-$$\textbf{u}$$-essential coalitions
将带效用函数的TU博弈理论推广到对偶博弈，定义了对偶u-本质联盟，并证明它们能刻画u-平衡博弈的u-预核仁。
Abstract 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- $$\textbf{u}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> -essential coalitions and show that they characterize the $$\textbf{u}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> -prenucleolus of $$\textbf{u}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> -balanced games. Additionally, we demonstrate that the intersection of $$\textbf{u}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> -essential and dually- $$\textbf{u}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> -essential coalitions also forms a characterization set for the $$\textbf{u}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> -prenucleolus, provided that the $$\textbf{u}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> -least-core is a proper subset of the $$\textbf{u}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> -core.