Independent private values model

This model assigns independent private valuation types to the bidders. Symmetry adds identical type distributions and bidder roles. These assumptions determine interim winning probabilities from the valuation distribution in a monotone symmetric equilibrium.

The independent private values model is symmetric when bidders have identical roles and draw their valuations independently from the same distribution. For revenue equivalence in the usual single-object setting, bidders additionally have risk neutrality and quasilinear utility. Under a strictly increasing symmetric equilibrium bidding rule, a type $v$ wins with probability $F(v{)}^{n-1}$.

// Target: mathematical-optimization.bigb

For two independent uniform valuations on $[0,1]$, optimal posted price revenue is $2/(3\sqrt{3})$, attained at $p=1/\sqrt{3}$. An English auction without reserve earns the expected minimum, $1/3$. With reserve $r\in [0,1]$, revenue is $r\mathrm{Pr}(\text{exactly one valuation exceeds }r)$ plus the expected smaller valuation when both exceed $r$, namely $1/3+r^2-4r^3/3$. It is maximized at $r=1/2$, giving $5/12$. Different reserve-dependent allocation probabilities explain why this differs from the no-reserve revenue without violating revenue equivalence.