= Uniform private-value first-price bidding equilibrium
{title2=$\beta(v)=\frac{n-1}{n}v$}
For $n\geq2$ <risk-neutral> bidders with independent uniform valuations on $[0,1]$, this rule is a symmetric <Bayesian Nash equilibrium>. If opponents use $\beta(v)=av$, a bid $b\in[0,a]$ has expected utility $(v-b)(b/a)^{n-1}$. Its derivative has the sign of $(n-1)v-nb$, so with $a=(n-1)/n$ the best bid is $av$. Bids above $a$ guarantee winning but only increase payment. The type-$v$ equilibrium utility is $v^n/n$.
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