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We provide new, concise characterizations of the Banzhaf value on a fixed player set employing just the standard dummy player property and one of the collusion properties suggested by Haller (Int J Game Theory 23:261–281, <CitationRef CitationID="CR8">1994</CitationRef>) and Malawski (Int J Game Theory 31:47–67, <CitationRef CitationID="CR11">2002</CitationRef>). Within these...</citationref></citationref>
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