Optimal Auctions

Game Theory · Axiom Academy

Myerson's Theory of Revenue Maximization Through Virtual Valuations and Reserve Prices The key insight in optimal auction design is transforming bidders' true valuations into virtual valuations . For a bidder with valuation v drawn from distribution F(v) with density f(v), the virtual valuation is: The virtual valuation adjusts for the fact that awarding the item to a bidder prevents extracting revenue from higher types of that bidder. The term 1-F(v)/f(v) represents the information rent needed to incentivize truthful reporting. Myerson showed that revenue-maximizing auctions should allocate to the bidder with the highest non-negative virtual valuation . This naturally introduces a reserve price r where φ(r) = 0. For uniform distributions F ~ U[0,1], the optimal reserve is r* = 1/2. This means the seller refuses to sell if all bids are below 1/2, even though some bidders value the item positively! Myerson's Lemma characterizes all Bayesian incentive compatible mechanisms. A mechanism is truthful if and only if: (1) Monotonicity: The allocation probability x_i(v_i) is non-decreasing in bidder i's reported value. (2) Payment Identity: Payments equal the expected virtual surplus plus a boundary term. This determines the payment rule uniquely from the allocation rule. 4. Revenue Maximization vs Efficiency

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