Uniform private-value first-price bidding equilibrium ($\beta (v)=\frac{n-1}{n}v$)

For $n\ge 2$ 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 $v^n/n$.