There is another way to compute the symmetric BNE in increasing strategy. Let $U(v)$ denote the expected utility of a player in equilibrium when his type is $v$: Given that the bidding strategy is increasing, a player with type $0$ will get the good with probability zero. Thus he/she must bid zero and $U(0) = 0$. For any other $v > 0$, the probability ...


Value, Competition and Exploitation by Cogliano, Flaschel, Franke, Fröhlich and Veneziani, Classical Political Economics and Modern Capitalism by Tsoulfidis and Tsaliki provide formalizations of classical/Marxian approach, you can find them on libgen.

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