Theorem Auction Comparisons Under Risk Aversion
theorem staged

Auction Comparisons Under Risk Aversion

Theorem (MSZ Thm 12.25, 12.26, 12.29; Krishna Ch 4). Under symmetric independent private values, the risk-neutral revenue equivalence ([[mechanism_design.myerson.revenue_equivalence]]) between first-price, second-price, and all-pay auctions breaks down once bidders are risk averse. The standard ordering of expected revenue under bidder risk aversion is:

$$ \mathbb{E}[R_{\mathrm{FP}}] \;\ge\; \mathbb{E}[R_{\mathrm{SP}}] \;=\; \mathbb{E}[R_{\mathrm{AP}}], $$

with strict inequality for any non-trivial risk aversion. First-price extracts strictly more revenue than second-price, while all-pay and second-price coincide. Conversely, risk-averse bidders strictly prefer second-price to first-price.

Setup

  • Symmetric IPV environment as in [[mechanism_design.auction.bayesian.single_item_framework]]: types $t_i \sim F$ i.i.d. on $[0, \omega]$.
  • Each bidder has a strictly increasing, concave utility function $u : \mathbb{R} \to \mathbb{R}$ (e.g. CARA: $u(x) = -e^{-\rho x}$ with $\rho > 0$, or CRRA: $u(x) = x^{1-\gamma}/(1-\gamma)$ with $\gamma \in (0, 1)$).
  • Each bidder maximises expected utility of net payoff (value minus payment if winning, zero if losing).

Key driver

In first-price, bid shading below truthful is bounded by the uncertainty about the marginal opponent's bid: - A higher bid increases the probability of winning (good) at the cost of a higher payment if winning (bad). - Risk aversion raises the marginal value of winning relative to the marginal cost of paying more — the bidder prefers a more certain, smaller, payoff over a riskier larger one. - Hence risk-averse bidders shade less than risk-neutral bidders, bidding closer to their true valuation.

The result is that the first-price equilibrium bid function $\beta^{\mathrm{FP}}_u(t)$ under risk aversion lies strictly above the risk-neutral counterpart from [[mechanism_design.auction.bayesian.symmetric_first_price_equilibrium]]: $\beta^{\mathrm{FP}}_u(t) > \beta^{\mathrm{FP}}_{\text{rn}}(t)$ for every interior type $t$. Higher bids ⇒ higher seller revenue.

In second-price (and all-pay), bidders' optimal strategies do not depend on $u$: truth-telling in second-price and the deterministic "pay your bid no matter what" structure in all-pay both leave the strategy invariant. The expected revenue is therefore unchanged from the risk-neutral case.

Bidder welfare comparison

The same logic reverses for bidder preferences: - Risk-averse bidders strictly prefer the deterministic "pay only on losing" structure of second-price to the lottery-like outcomes of first-price. - Quantitatively, ex-ante expected utility from second-price exceeds that from first-price (under symmetric IPV and any concave $u$).

Why it matters

  • Auction design under risk aversion. A revenue-maximising seller facing risk-averse bidders should prefer first-price over second-price — the reverse of the risk-neutral revenue-equivalence prediction.
  • Identification of risk attitudes. Comparing observed first-price bids to the risk-neutral benchmark $\beta^{\mathrm{FP}}_{\text{rn}}$ identifies the curvature of bidder utility from auction data.
  • Mechanism-design extensions. Risk aversion creates a wedge between expected revenue (seller objective) and expected utility (bidder objective), motivating optimal mechanisms that exploit this wedge (Maskin-Riley 1984).

Lean port (deferred)

Planned Lean module: EconCSLib/Auction/RiskAversion.lean.

Planned declarations:

  • RiskAverseUtility (utility-function wrapper with concavity and strict monotonicity).
  • symmetricFirstPriceBid_riskAverse (equilibrium bid function under risk aversion).
  • firstPrice_revenue_ge_secondPrice_under_riskAversion (revenue comparison theorem).
  • bidder_prefers_secondPrice_under_riskAversion (welfare comparison).

Dependencies: a Lean treatment of concave utility (likely via Mathlib's ConvexOn / ConcaveOn); the risk-neutral first-price equilibrium ([[mechanism_design.auction.bayesian.symmetric_first_price_equilibrium]]); the all-pay equilibrium ([[mechanism_design.auction.bayesian.all_pay_equilibrium]]). Tracked in the MSZ Ch.12 auction gap review.

References

  • [MSZ Chapter 12, Thm 12.25, 12.26, 12.29] Maschler, Solan, and Zamir, Game Theory. Revenue and welfare comparisons under risk aversion.
  • [Krishna, Chapter 4] Vijay Krishna, Auction Theory, 2nd ed.. Risk aversion in auctions.
  • [Maskin-Riley 1984] Eric Maskin and John Riley, "Optimal Auctions with Risk Averse Buyers", Econometrica 52(6):1473–1518. Optimal mechanism design under risk aversion.
  • [Holt 1980] Charles Holt, "Competitive bidding for contracts under alternative auction procedures", Journal of Political Economy 88(3):433–445. Original risk-aversion revenue comparison.

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