Myerson Reserve-Price Characterisation
Corollary (MSZ Cor 12.37). In the symmetric IPV selling environment — all bidders draw their types from the same regular prior $F$, with identical value functions — the optimal auction ([[mechanism_design.myerson.optimal_auction]]) is the second-price auction with reserve price $r^* = \psi^{-1}(0)$:
- Solicit sealed bids $b = (b_1, \dots, b_n)$ from all bidders.
- If $\max_i b_i < r^*$: no sale.
- Otherwise: the highest bidder wins and pays $\max\big(r^*, \; \text{second-highest bid}\big)$.
Two notable corollaries
-
Reserve price depends only on the prior, not on the number of bidders. The same $r^* = \psi^{-1}(0)$ is optimal whether there are 2 or 200 bidders, as long as the distribution is unchanged.
-
No-sale is sometimes optimal: when all bids fall below the reserve, the seller intentionally keeps the object. This trades realised revenue against incentivising future-period high bids — a static one-shot reflection of monopoly pricing.
Proof
outline
Specialise the Myerson optimal auction ([[mechanism_design.myerson.optimal_auction]]) to symmetric $F_i = F$:
- By symmetry of $\psi$, $\arg\max_i \psi_i(t_i) = \arg\max_i t_i$ (same monotone $\psi$ across bidders, so argmax is preserved).
- Hence the optimal allocation gives to the highest bidder, subject to $\psi(\text{winner}) \ge 0 \iff \text{winner's bid} \ge r^*$.
- The Myerson payment formula computes the winner's payment as $\max(r^*, \text{second-highest bid})$, matching the second-price structure with reserve.
Lean port (deferred — see #176)
Planned additions to
EconCSLib/MechanismDesign/Auction/MyersonOptimalAuction.lean:
optimalReservePrice(def: $\psi^{-1}(0)$ via implicit-function / monotone-inverse)optimalReservePrice_independent_of_n(the n-independence corollary)myersonOptimalAuction_symmetric_eq_secondPriceWithReserve(the structural equivalence)
Depends on the underlying optimal-auction port (#175).
References
- [MSZ Cor 12.37] Maschler, Solan, Zamir, Game Theory.
- Myerson, R. B. (1981). "Optimal Auction Design". Math. Oper. Res. 6, §6.
- Krishna, V. (2010). Auction Theory, §5.2..