Time-Varyingness in Auction Breaks Revenue Equivalence

TL;DR

This work studies whether revenue equivalence between first-price and second-price auctions persists when item values evolve over time. It develops a continuous-time learning framework in which bidders update a time-varying value-distribution parameter $\boldsymbol{\theta}^*(t)$, characterized by the basis value $v_m(t)$ and value interval $\Delta v(t)$, and compares long-run payoffs under time-varying conditions. The main finding is that revenue equivalence can fail in the long run, with the direction of inequivalence determined by the correlation between $v_m(t)$ and $\Delta v(t)$; positive correlation favors first-price bidding, negative correlation favors second-price, and zero correlation restores equivalence. The authors derive results for uniform and log-normal distributions and validate via periodic and random-environment experiments, highlighting implications for auction design under non-stationary environments. This provides a theoretical basis for anticipating mechanism choice in dynamic markets and motivates further empirical validation in real-world auctions.

Abstract

Auction is applied for trade with various mechanisms. A simple but practical question is which mechanism, typically first-price or second-price auctions, is preferred from the perspective of bidders or sellers. A celebrated answer is revenue equivalence, where each bidder's equilibrium payoff is proven to be independent of auction mechanisms (and a seller's revenue, too). In reality, however, auction environments like the value distribution of items would vary over time, and such equilibrium bidding cannot always be achieved. Indeed, bidders must continue to track their equilibrium bidding by learning in first-price auctions, but they can keep their equilibrium bidding in second-price auctions. This study discusses whether and how revenue equivalence is violated in the long run by comparing the time series of non-equilibrium bidding in first-price auctions with those of equilibrium bidding in second-price auctions. We characterize the value distribution by two parameters: its basis value, which means the lowest price to bid, and its value interval, which means the width of possible values. Surprisingly, our theorems and experiments find that revenue equivalence is broken by the correlation between the basis value and the value interval, uncovering a novel phenomenon that could occur in the real world.

Figures (5)

• Figure 1: Overview of learning in time-varying auctions. (A). This study considers time-varying environments, such as periodic and random ones. How bidding changes over time under such time-varying environments is plotted for first-price (orange) and second-price (gray) auctions. In second-price auctions, truthful bidders automatically keep their optimal bidding regardless of the environmental change. In first-price auctions, however, bidders should track the optimal bidding by learning. (B). A value distribution (e.g., uniform distribution) is characterized by its basis value and its value interval. (C). This study demonstrates that the correlation between the basis value and the value interval determines how revenue equivalence is broken. The positive correlation results in bidders preferring first-price auctions (Thm. \ref{['thm_inequivalence_1st']}), while the negative correlation results in bidders preferring second-price auctions (Thm. \ref{['thm_inequivalence_2nd']}). The upper-right distributions are examples of uniform and log-normal distributions that provide positive/negative correlation.
• Figure 1: The experiments for a counterexample in which it cannot be determined whether bidders receive higher expected payoffs in the first-price auction than in the second-price auction. How to see each panel is the same as Fig. \ref{['F02']}. The methods and parameters for all the experiments are the same as Fig. \ref{['F02']}. Both (A) and (B) consider the same set of the value distributions $\boldsymbol{\theta}^{*}\in\{\boldsymbol{\theta}^{(-,-)},\boldsymbol{\theta}^{(-,+)},\boldsymbol{\theta}^{(+,-)},\boldsymbol{\theta}^{(+,+)}\}$, where we defined $\boldsymbol{\theta}^{(-,-)}:=(10,20)$, $\boldsymbol{\theta}^{(-,+)}:=(10,30)$, $\boldsymbol{\theta}^{(+,-)}:=(20,30)$, and $\boldsymbol{\theta}^{(+,+)}:=(20,40)$. (A). The cyclic transition of $\boldsymbol{\theta}^{(-,-)}\to \boldsymbol{\theta}^{(-,+)}\to \boldsymbol{\theta}^{(+,+)}\to \boldsymbol{\theta}^{(+,-)}\to \boldsymbol{\theta}^{(-,-)}\cdots$. (B). The reversed transition of $\boldsymbol{\theta}^{(-,-)}\to \boldsymbol{\theta}^{(+,-)}\to \boldsymbol{\theta}^{(+,+)}\to \boldsymbol{\theta}^{(-,+)}\to \boldsymbol{\theta}^{(-,-)}\cdots$.
• Figure 2: The experiments for periodic uniform distributions. The left panels show how the uniform distribution switches between two states. The center panels show the time series of the learned basis value ($x(t)$: orange) and the true basis value ($v_{{\rm m}}(t)$: gray). The right panels show the time series of the time-average payoffs in the first-price auction ($\bar{w}^{{\rm 1st}}(T)$: red) and in the second-price auction ($\bar{w}^{{\rm 2nd}}(T)$: gray) within the range of $0\le t\le T$. In all the experiments, we set the population as $n=10$, the Runge-Kutta fourth-order method of Eq. \ref{['dotx']} with the step size of $10^{-3}$ and accelerated $2\times 10^{3}$. The case of A considers $\boldsymbol{\theta}^{*}(t)\in\{(10,20),(20,40)\}$, B does $\boldsymbol{\theta}^{*}(t)\in\{(10,20),(20,30)\}$, and C does $\boldsymbol{\theta}^{*}(t)\in\{(10,30),(20,30)\}$.
• Figure 3: The experiments for random uniform distributions. The left panels show possible uniform distributions generated by Eqs. \ref{['Langevin_m']} and \ref{['Langevin_M']}. The meaning of the center and right panels is the same as Fig. \ref{['F02']}. The methods and parameters for all the experiments, too. In all the cases of A-C, we set $(\bar{v}_{{\rm m}},\bar{v}_{{\rm M}})=(20,40)$. The case of A considers $(a_{{\rm m}},a_{{\rm M}})=(5,10)$ in Eqs. \ref{['Langevin_m']} and \ref{['Langevin_M']}, B does $(a_{{\rm m}},a_{{\rm M}})=(5,5)$, and C does $(a_{{\rm m}},a_{{\rm M}})=(5,0)$.
• Figure 4: The experiments for periodic log-normal distributions. The left panels show how the log-normal distribution switches between two states. The meaning of the center and right panels is the same as Fig. \ref{['F02']}. To simulate the dynamics of Eq. \ref{['dotx']}, we numerically calculate all the integrals by discretizing the space of $0\le v\le 20$ with $400$ meshes. The methods and parameters for all the experiments, too. The case of A considers $\boldsymbol{\theta}^{*}(t)\in\{(1/4,1/4), (1/2,1/2)\}$, B does $\boldsymbol{\theta}^{*}(t)\in\{(1/4,1/2),(1/2,1/4)\}$.

Theorems & Definitions (6)

• Definition 1: Uniform distribution
• Theorem 1: Revenue inequivalence by positive correlation
• Theorem 2: Revenue inequivalence by negative correlation
• Theorem 3: Revenue equivalence by no correlation
• Definition 2: Log-normal distribution
• Lemma 1: Equivalence between total ascent and descent