Theorem Entry Fees And Reserve Prices In IPV Auctions
theorem staged

Entry Fees And Reserve Prices In IPV Auctions

Theorem (MSZ Thm 12.23, Cor 12.24; Krishna Ch 2.5). In a symmetric IPV auction (first-price or second-price), a positive reserve price $r > 0$ that excludes all bidders with valuation below $r$ strictly increases expected revenue compared to the no-reserve auction, provided the seller's valuation $v_0$ satisfies $v_0 < r^* = \psi^{-1}(0)$, where $\psi(t) = t - (1 - F(t))/f(t)$ is the virtual valuation and $r^*$ is the optimal reserve price. An entry fee $e$ paid by each participating bidder is, in the symmetric IPV setting, payoff- and revenue-equivalent to a reserve price $r(e)$ that solves $r - e = \mathbb{E}[\text{net expected payoff at type } r]$.

Setup

  • Symmetric IPV environment as in [[mechanism_design.auction.bayesian.single_item_framework]]: types $t_i \sim F$ i.i.d. on $[0, \omega]$ with continuous positive density $f$.
  • Reserve price $r$: the auctioneer commits to not sell the object for less than $r$. In first-price, bidders below $r$ drop out (or equivalently bid $0$ and lose); bidders above $r$ bid in equilibrium conditional on not facing a binding reserve. In second-price, the price paid by the winner is $\max(r, \text{second-highest bid})$.
  • Entry fee $e$: each bidder who participates pays a fixed fee $e \ge 0$ regardless of outcome. Bidders enter iff their expected surplus net of $e$ is nonnegative.

Optimal reserve (regular case)

When the virtual-valuation function $\psi(t) = t - (1 - F(t))/f(t)$ is monotone non-decreasing (the regular case of [[mechanism_design.myerson.optimal_auction]]), the revenue-maximising reserve in any standard auction is $$ r^* \;=\; \psi^{-1}(\max(v_0, 0)), $$ where $v_0$ is the seller's own valuation. Setting $r = r^*$ optimises revenue for the auctioneer with valuation $v_0$.

This is the auction-side specialisation of [[mechanism_design.myerson.reserve_price]] — the optimal mechanism in the symmetric IPV setting is just the second-price auction (or its revenue-equivalent first-price counterpart) with reserve $r^*$.

Entry fee ↔ reserve price equivalence

A reserve price $r$ and an entry fee $e$ are equivalent in the following sense:

  • An entry fee $e$ induces a participation threshold $\hat t(e)$: a bidder enters iff their expected interim surplus $U(t) - e \ge 0$, i.e. iff $t \ge \hat t(e)$ where $\hat t$ is the unique solution of $U(\hat t(e)) = e$.
  • A reserve price $r$ induces the threshold $\hat t = r$ directly: bidders below $r$ cannot win.
  • For each entry fee $e$ there is a unique reserve $r(e)$ producing the same participation threshold, and the two regimes generate the same expected revenue.

This equivalence breaks under risk aversion or budget constraints, which is one motivation for the entry-fee formulation in some practical settings.

Why it matters

The reserve-price result is the prototypical example of how a small amount of seller commitment power dramatically improves revenue: even without optimising over arbitrary mechanisms (as Myerson's general optimal auction does), simply adding a reserve to a standard auction already extracts a substantial portion of the optimal revenue.

The entry-fee variant captures real-world auction conventions (e.g. bidder pre-qualification, deposit requirements) and gives the auctioneer a second instrument to fine-tune participation.

Lean port (deferred)

Planned Lean modules: EconCSLib/Auction/Reserve.lean and EconCSLib/Auction/EntryFee.lean.

Planned declarations:

  • SecondPriceAuctionWithReserve (mechanism with reserve).
  • secondPriceWithReserve_isDSIC (truth-telling remains DSIC).
  • optimalReserve (closed-form $r^*$ under regularity).
  • secondPriceWithReserve_revenue_ge_secondPrice_under_regularity.
  • entryFee_reserve_equivalence (revenue equivalence theorem).

Dependencies: virtual valuation and Myerson regularity (already in EconCSLib/MechanismDesign/Auction/Myerson.lean); the IPV single-item framework ([[mechanism_design.auction.bayesian.single_item_framework]]); the revenue-equivalence theorem ([[mechanism_design.myerson.revenue_equivalence]]). Tracked in the MSZ Ch.12 auction gap review.

References

  • [MSZ Chapter 12, Thm 12.23 and Cor 12.24] Maschler, Solan, and Zamir, Game Theory. Reserve prices and entry fees in IPV auctions.
  • [Krishna, Chapter 2, Section 2.5] Vijay Krishna, Auction Theory, 2nd ed.. Reserve prices in first-price and second-price auctions.
  • [Myerson 1981] Roger Myerson, "Optimal Auction Design", Math. Oper. Res. 6(1):58–73. Derivation of the optimal reserve via virtual valuations.
  • [Riley-Samuelson 1981] John Riley and William Samuelson, "Optimal Auctions", AER 71(3):381–392. Reserve prices and revenue improvement.

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