Deriving the first-price shading factor

HardProbability Expected Value Market Making & Betting Games~6m

In a first-price sealed-bid auction there are 5 bidders with values i.i.d. Uniform $(0,1)$ . In the symmetric equilibrium each bidder shades to bid the expected highest rival value, conditional on their own value being the top. Your value is $v = 0.90$ . Derive the equilibrium bid (to two decimals). You are not given the bid function - construct it.

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Conditional on winning, each of the $n-1$ rivals is Uniform $(0,v)$ .
$E[\max$ of $m$ Uniform $(0,v)] = \tfrac{m}{m+1}v$ ; here $m=4$ .

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0.72 (± 0.01)

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Deriving the first-price shading factor

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