Theorem First-Price Fails DSIC
✓ theorem proved

First-Price Fails DSIC

Theorem. The first-price auction ([[mechanism_design.auction.basic.first_price_mechanism]]) is not dominant-strategy incentive compatible: for every bidder $i$ and every bid $b_i$, there exists an opposing bid profile that makes $b_i$ strictly worse than some other bid for $i$.

Formally: assuming the bid space U contains a positive element ($\exists a : U.\; 0 < a$), no constant bid is weakly dominant in the strategic-game game v for any valuation profile $v$.

Proof

sketch

Given a candidate bid $b_i$ for bidder $i$, choose a positive $a > 0$ and construct opponents' bids at the level $b_i - 2a$. Then:

  • Bidding $b_i$, bidder $i$ wins and pays $b_i$, yielding payoff $v_i - b_i$.
  • Bidding $b_i - a$, bidder $i$ still wins (since $b_i - a > b_i - 2a$) but pays $b_i - a$, yielding payoff $v_i - (b_i - a) = v_i - b_i + a$.

The deviation $b_i \mapsto b_i - a$ raises payoff by exactly $a > 0$, so $b_i$ cannot be weakly dominant.

Lean form

  • no_dominant_strategy v i bi ha : ¬ IsWeaklyDominant (game v) i bi states that no bid $b_i \in U$ is a weakly dominant strategy for $i$ in the first-price game, under the bid-space-has-positives hypothesis ha.
  • mechanism_not_isDSIC lifts this to the mechanism-level negation ¬ Auction.FirstPrice.mechanism.isDSIC Auction.FirstPrice.utility.

Why it matters

The first-price auction is the canonical example of a non-truthful single-item mechanism. Equilibrium analysis in first-price requires moving to the Bayesian setting and computing a symmetric strictly-increasing Bayes–Nash equilibrium of the form $b^*(t) = E[Y_1 \mid Y_1 < t]$, where $Y_1$ is the maximum opponent valuation ([[mechanism_design.auction.bayesian.symmetric_first_price_equilibrium]]). This contrasts sharply with the second-price auction ([[mechanism_design.auction.basic.second_price_dsic]]), where truthful bidding is dominant without any distributional assumption.

The first-price/second-price contrast is the simplest concrete instance of the dominant-strategy vs. Bayes–Nash gap that motivates much of mechanism-design theory.

References

  • [AGT, Chapter 9, Section 9.3.1] Nisan, Roughgarden, Tardos, and Vazirani, Algorithmic Game Theory. Non-truthfulness of first-price as motivation for VCG.
  • [Krishna, Chapter 2, Section 2.3] Vijay Krishna, Auction Theory, 2nd ed.. Strategic analysis of first-price auctions.
  • [MFoGT, Chapter 12] Maschler, Solan, and Zamir, Game Theory. First-price auction discussion in the private-values chapter.

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