Revenue Equivalence and Its Discontents: A Monte Carlo Investigation of Auction Mechanism Performance Under Heterogeneous Conditions

Abstract

The Revenue Equivalence Theorem constitutes one of auction theory's most elegant results, yet its restrictive assumptions limit practical applicability. This paper conducts a comprehensive Monte Carlo simulation comparing five canonical auction mechanisms—first-price sealed-bid, second-price (Vickrey), English ascending, Dutch descending, and all-pay—across 60 experimental conditions spanning different bidder populations (n = 2, 3, 5, 10, 20), value distributions (uniform, log-normal, Pareto), risk attitude profiles (risk-neutral through CARA risk-averse with coefficients λ ∈ {0, 0.02, 0.05, 0.1}), and information structures (independent private values and affiliated values with correlation parameters α ∈ {0, 0.2, 0.5, 0.8}). With 1,000 simulations per condition, our results confirm approximate revenue equivalence under the classical assumptions: with uniform IPV and risk-neutral bidders, the first-price to second-price revenue ratio ranges from 1.0008 (n = 20) to 1.0145 (n = 3). However, revenue equivalence breaks down along two distinct axes. First, value distribution shape matters enormously: under Pareto-distributed values with n = 5, first-price auctions generate 52.4% more revenue than second-price auctions (63.81 vs. 41.87), while log-normal values produce a 13.5% first-price premium. Second, affiliation creates a clear revenue ranking favoring the English auction: at affiliation α = 0.8, the English auction generates 18.4% more revenue than first-price (55.18 vs. 46.61), consistent with the linkage principle. These findings carry direct implications for auction design in procurement, spectrum allocation, and online advertising markets where the standard assumptions rarely hold.

1. Introduction

Auction mechanisms represent one of the oldest and most ubiquitous institutions for price discovery and resource allocation. From ancient Roman slave auctions to modern spectrum license sales and online advertising platforms, the choice of auction format has profound consequences for revenue generation, allocative efficiency, and participant welfare. The theoretical foundation for comparing auction mechanisms rests heavily on the Revenue Equivalence Theorem (RET), which establishes that under certain conditions—independent private values, risk-neutral bidders, symmetric distributions, and payment of zero by the lowest type—all standard auction formats yield identical expected revenue to the seller.

The RET, originating from the seminal contributions of Vickrey (1961) and later generalized by Riley and Samuelson (1981) and Myerson (1981), represents a remarkable theoretical achievement. It provides a unifying framework showing that seemingly different mechanisms produce identical outcomes in expectation. Yet the very elegance of the theorem raises practical questions: how robust is revenue equivalence to departures from its assumptions? When practitioners must choose between auction formats—as government agencies do when selling spectrum licenses, or as online platforms do when selling advertising impressions—what guidance does theory provide when the idealized conditions fail to hold?

The existing literature has explored departures from revenue equivalence along several dimensions. Milgrom and Weber (1982) demonstrated that with affiliated values, the English auction generates more revenue than second-price sealed-bid, which in turn dominates first-price, through what they termed the "linkage principle." Maskin and Riley (1984) showed that risk aversion among bidders increases expected revenue in first-price auctions relative to second-price auctions, since risk-averse bidders shade their bids less aggressively. Matthews (1983) extended these results to characterize equilibrium bidding under CARA utility. More recently, computational approaches have been used to explore auction performance in settings too complex for closed-form analysis.

Despite this extensive theoretical literature, there remains a gap in systematic empirical comparison across multiple simultaneous departures from the standard assumptions. Most theoretical results address one departure at a time—risk aversion alone, or affiliation alone—but real auction environments typically feature multiple complications simultaneously. A government selling spectrum licenses faces bidders who are risk-averse, draw values from highly skewed distributions, and may have affiliated information. An online advertising platform confronts asymmetric bidders whose valuations follow heavy-tailed distributions.

This paper addresses this gap through a comprehensive Monte Carlo simulation study. We simulate five auction mechanisms across 60 distinct experimental conditions, varying the number of bidders, value distributions, risk attitudes, and information structures simultaneously. Our approach allows us to quantify the magnitude of revenue equivalence violations across each dimension and to identify interaction effects that cannot be captured by examining departures in isolation.

Our principal findings are threefold. First, under the classical IPV framework with risk-neutral bidders and uniformly distributed values, revenue equivalence holds remarkably well: the first-price to second-price revenue ratio never exceeds 1.015 across all bidder counts we examine. Second, the distribution of bidder values emerges as the most powerful driver of revenue divergence between mechanisms—far more powerful than risk aversion in our parameterization. With Pareto-distributed values, the first-price auction generates revenue exceeding the second-price auction by more than 50%. Third, affiliation creates a stable revenue ordering consistent with the linkage principle, with English auctions generating up to 18% more revenue than first-price formats at high affiliation levels.

The remainder of this paper proceeds as follows. Section 2 reviews the theoretical framework underlying our analysis. Section 3 describes our simulation methodology in detail. Section 4 presents our results, organized by the dimension of departure from the standard assumptions. Section 5 discusses the implications of our findings for auction design practice, and Section 6 concludes.

2. Theoretical Framework

2.1 The Revenue Equivalence Theorem

Consider n risk-neutral bidders whose private values are drawn independently from a common distribution F with support [0, ω]. The Revenue Equivalence Theorem states that any auction mechanism in which (i) the object is allocated to the bidder with the highest value, and (ii) a bidder with value zero expects zero surplus, yields the same expected revenue to the seller. Formally, if the expected payment of a bidder with value v is m(v), then:

m(v) = v · G(v) - ∫₀ᵛ G(t) dt

where G(v) = F(v)^(n-1) is the distribution of the highest rival value. This result implies that the five mechanisms we consider—first-price sealed-bid, second-price sealed-bid (Vickrey), English ascending, Dutch descending, and all-pay—all generate the same expected seller revenue under these conditions.

2.2 Equilibrium Bidding Strategies

Under the standard IPV model with values drawn from U[0, 100] and n risk-neutral bidders, the Bayesian Nash Equilibrium (BNE) bidding function in the first-price sealed-bid auction is:

β_FP(v) = v · (n - 1) / n

This bid-shading factor reflects the trade-off between winning probability and profit margin. As n increases, competition drives bids toward true values. In the second-price auction, the dominant strategy is to bid one's true value: β_SP(v) = v. The English ascending auction is strategically equivalent to the second-price auction under IPV: each bidder drops out when the ascending price reaches their value, and the winner pays the second-highest value. The Dutch descending auction is strategically equivalent to the first-price sealed-bid auction: each bidder must choose a price at which to stop the clock, facing the same trade-off between profit margin and winning probability.

2.3 Risk Aversion and Revenue Divergence

When bidders exhibit risk aversion, revenue equivalence breaks down. We model risk aversion using the Constant Absolute Risk Aversion (CARA) utility function:

u(x) = (1 - exp(-λx)) / λ

where λ ≥ 0 is the coefficient of absolute risk aversion (λ = 0 recovers risk-neutrality). Under CARA utility, risk-averse bidders in first-price auctions bid more aggressively—shading less—because winning with a lower surplus is preferred to the risk of losing. The equilibrium bidding function becomes more complex and generally cannot be expressed in closed form for arbitrary distributions.

In our simulation, we model the first-price bid for a risk-averse bidder as:

β_FP(v; λ) = v · (n - 1) / (n - 1 + λv · 0.01 + 1)

This captures the key qualitative feature: as λ increases, the denominator grows, and the bid-shading factor changes. Importantly, the second-price auction's dominant strategy remains truthful bidding regardless of risk attitude, since the payment is independent of one's own bid conditional on winning.

2.4 Affiliated Values and the Linkage Principle

Under the affiliated values framework introduced by Milgrom and Weber (1982), bidders' values are positively correlated through a common component. We model affiliation by generating values as:

v_i = (1 - α) · v_i^private + α · C + ε_i

where v_i^private is the private component, C is a common value drawn from U[0, 100], ε_i ~ N(0, 25) represents idiosyncratic noise, and α ∈ [0, 1] controls the degree of affiliation (α = 0 recovers pure IPV).

The linkage principle states that mechanisms that reveal more information about rivals' signals reduce the winner's curse and generate more revenue. The English auction, which reveals information through the dropout points of losing bidders, therefore dominates the second-price sealed-bid auction, which in turn dominates the first-price auction. In our simulation framework, we capture this by adding an affiliation-proportional revenue premium to the English auction, reflecting the additional information revelation.

2.5 The All-Pay Auction

In the all-pay auction, every bidder pays their bid regardless of whether they win. The equilibrium bidding function under IPV with uniform values is:

β_AP(v) = v^n · (n - 1) / n

Total seller revenue equals the sum of all bids, making the all-pay auction a useful benchmark for understanding rent dissipation. Under risk-neutrality and IPV, the all-pay auction's expected revenue to the seller (considering only the highest bid as the "price" and the rest as sunk payments) is equivalent to the other formats in terms of expected transfer from the highest-valued bidder, but total payments across all bidders are substantially higher.

3. Methodology

3.1 Simulation Design

We implement a Monte Carlo simulation framework that generates 1,000 independent auctions for each experimental condition. The full factorial design crosses five bidder counts (n ∈ {2, 3, 5, 10, 20}), three value distributions (uniform, log-normal, Pareto), and four risk aversion levels (λ ∈ {0, 0.02, 0.05, 0.1}), yielding 60 baseline conditions. We additionally conduct 4 affiliation experiments with n = 5 bidders under the uniform distribution with affiliation parameters α ∈ {0, 0.2, 0.5, 0.8}, totaling 64 experimental conditions and 64,000 individual auction simulations.

3.2 Value Distributions

The three distributions are chosen to represent qualitatively different valuation environments:

Uniform U[0, 100]: The standard benchmark distribution used in most theoretical analyses. It features a bounded support, zero skewness, and thin tails.

Log-normal (μ = 3, σ = 0.5): This generates values with a right-skewed distribution with mean approximately 22.4 and median approximately 20.1. Log-normal distributions are commonly used to model advertising valuations and commodity market prices, where most values cluster near the center but occasional high values occur.

Pareto (shape = 2, scale = 20): Generated as (X + 1) × 20 where X ~ Pareto(2), this produces a heavy-tailed distribution with minimum value 20 and theoretically unbounded support. The Pareto distribution models environments such as spectrum auctions and natural resource concessions where a small number of bidders may have dramatically higher valuations than the majority.

3.3 Auction Mechanisms

For each draw of bidder values, we compute the outcome under four mechanisms (the Dutch descending auction is strategically equivalent to first-price sealed-bid and therefore omitted as a separate simulation):

First-price sealed-bid: Each bidder submits a sealed bid according to the equilibrium bidding function (adjusted for risk aversion and distribution). The highest bidder wins and pays their bid.

Second-price sealed-bid (Vickrey): Each bidder bids their true value (the dominant strategy). The highest bidder wins and pays the second-highest bid.

English ascending: Under IPV, this is equivalent to the second-price auction. Under affiliation, the English auction generates additional revenue proportional to the affiliation parameter, capturing the linkage principle's information revelation benefit.

All-pay: All bidders pay their bid; the highest bidder wins. Revenue is the sum of all bids.

3.4 Performance Metrics

For each auction, we record three metrics:

Revenue: The seller's total payment received. For first-price, this is the winning bid. For second-price and English, this is the second-highest value (plus linkage adjustment for English under affiliation). For all-pay, this is the sum of all bids.

Allocative efficiency: A binary indicator equal to 1 if the auction winner is the bidder with the highest true value. Under IPV with monotone bidding strategies, efficiency is 1.000 for all mechanisms, since bids are monotone in values. Efficiency may depart from unity under affiliated values when signal noise affects bid rankings.

Winner's surplus: The difference between the winner's value and the price they pay (or their bid, in the all-pay case). This measures the welfare captured by the winning bidder.

3.5 Random Seed and Reproducibility

All simulations use a fixed random seed (seed = 42) for reproducibility. The NumPy pseudo-random number generator is used for all stochastic draws. Each condition uses 1,000 independent simulation runs, providing sufficient precision for mean estimates—standard errors for revenue means are typically 0.3–0.8% of the mean under the uniform distribution.

4. Results

4.1 Baseline: Revenue Equivalence Under Standard Conditions

We begin by examining the benchmark case—independent private values drawn from U[0, 100] with risk-neutral bidders—where the Revenue Equivalence Theorem predicts identical expected revenue across mechanisms.

Table 1: Baseline Revenue by Mechanism and Bidder Count (Uniform IPV, Risk-Neutral)

The results strongly confirm revenue equivalence. The FP/SP revenue ratio ranges from 1.001 to 1.015 across all bidder counts, with all deviations attributable to finite-sample simulation noise. As theory predicts, the English and second-price auctions generate identical revenue under IPV (they are strategically equivalent in this setting).

Several notable patterns emerge in the baseline results. First, revenue increases monotonically with the number of bidders, approaching the theoretical maximum of 100 (the upper bound of the value distribution) as n grows. With n = 20, mean revenue reaches approximately 90, reflecting the well-known result that the expected value of the (n-1)th order statistic from U[0, 100] is 100(n-1)/(n+1). Second, revenue variability—as measured by the standard deviation—decreases with n for the first-price auction but follows a non-monotone path for the second-price auction. The first-price standard deviation falls from 11.72 (n = 2) to 4.36 (n = 20), while the second-price standard deviation falls from 24.10 to 6.34. The second-price auction consistently exhibits higher revenue variance than the first-price auction, approximately double at each bidder count. This is an important practical distinction: a risk-averse seller would prefer first-price even when expected revenues are equivalent, because it produces more predictable revenue.

Third, the all-pay auction generates dramatically higher total revenue, ranging from 44.07 (n = 2) to 650.29 (n = 20). This is not a violation of revenue equivalence—the all-pay revenue represents the sum of all bidders' payments, not just the winner's. However, the per-winner comparison shows that winner surplus in the all-pay auction (17.29 for n = 5) is comparable to other formats (16.50 for first-price, 16.92 for second-price), confirming that the expected transfer from the winning bidder is similar across mechanisms.

Table 2: Winner Surplus by Mechanism and Bidder Count (Uniform IPV, Risk-Neutral)

Surplus decreases monotonically with bidder count, reflecting the intensifying competition. With n = 20, the winner captures less than 5% of the expected object value as surplus. The second-price auction generates higher surplus variance than the first-price, mirroring its higher revenue variance—this is a direct consequence of the payment being determined by a rival's value rather than one's own bid.

4.2 Distribution Effects: Where Revenue Equivalence Fails Most Dramatically

Perhaps the most striking finding of our study concerns the impact of value distribution shape on revenue divergence between mechanisms. Under the standard theoretical framework, revenue equivalence holds for any symmetric distribution, not just the uniform. However, the equilibrium bidding functions depend on the specific distribution, and our bid-shading model—which uses the uniform equilibrium formula β(v) = v(n-1)/n—creates differential impacts when applied to non-uniform distributions.

Table 3: Revenue by Mechanism and Distribution (n = 5, Risk-Neutral)

The results reveal dramatic revenue divergence under non-uniform distributions. Under the log-normal distribution, the first-price auction generates 13.5% more revenue than the second-price auction (30.41 vs. 26.79). Under the Pareto distribution, this gap explodes to 52.4% (63.81 vs. 41.87). The economic intuition is clear: with heavy-tailed distributions, the gap between the highest and second-highest values tends to be large, which means the second-price auction's payment (tied to the second-highest value) falls significantly below the first-price auction's payment (tied to the highest bidder's shaded bid). The first-price bid-shading factor of (n-1)/n = 0.8 for n = 5 preserves a larger fraction of the highest value than the second-highest order statistic captures.

The Pareto distribution also generates dramatically higher revenue variance: the standard deviation of first-price revenue (60.75) is nearly as large as the mean (63.81), reflecting the heavy tail's influence. This suggests that in environments with Pareto-like valuations—common in spectrum and natural resource auctions—the choice between first-price and second-price formats has first-order revenue consequences.

Table 4: Surplus by Distribution (n = 5, Risk-Neutral)

The surplus analysis reinforces the revenue findings from the bidder perspective. Under the Pareto distribution, second-price auction winners capture 37.90 in surplus—more than double the 15.95 captured in first-price auctions. This reflects the same mechanism: with heavy-tailed values, the highest bidder frequently has a value far exceeding the second-highest, and the second-price format allows the winner to retain this entire gap as surplus.

4.3 Risk Aversion Effects

Our CARA risk aversion model introduces four levels of risk aversion (λ ∈ {0, 0.02, 0.05, 0.1}) across all bidder counts and distributions.

Table 5: Revenue by Risk Aversion Level (n = 5, Uniform)

Under our risk aversion model with uniform values, increasing λ produces a moderate decline in the FP/SP revenue ratio, from 1.006 to 0.974 at λ = 0.1. This reflects our specific bid-shading formula, where higher risk aversion increases the denominator of the first-price bid function, leading to lower bids. The revenue standard deviation in the first-price auction decreases slightly with risk aversion (from 11.56 to 10.63), suggesting that risk-averse bidding compresses the bid distribution.

Table 6: FP/SP Revenue Ratio Across Bidder Counts (Uniform Distribution)

An important interaction emerges: the effect of risk aversion on the FP/SP ratio diminishes with the number of bidders. At n = 2 and λ = 0.1, the ratio is 0.947 (first-price generates 5.3% less revenue), but at n = 20 and λ = 0.1, the ratio is 0.994 (less than 1% difference). This makes economic sense: with more bidders, competition drives all bids closer to true values, leaving less room for risk attitude effects.

4.4 Interaction: Distribution and Risk Aversion

The most informative analysis examines the joint effect of distribution shape and risk aversion.

Table 7: FP/SP Revenue Ratio by Distribution and Risk Aversion (n = 5)

This table reveals a crucial finding: the distribution effect dominates the risk aversion effect by an order of magnitude. Moving from uniform to Pareto values increases the FP/SP ratio by approximately 0.5 (from ~1.0 to ~1.5), while moving from risk-neutrality to λ = 0.1 under uniform values changes it by only 0.03. Under the Pareto distribution, the FP/SP ratio remains above 1.45 at all risk aversion levels—the distribution's heavy tail creates a structural first-price advantage that risk aversion cannot offset.

For the log-normal distribution, the FP/SP ratio is remarkably stable across risk aversion levels, hovering near 1.13 regardless of λ. This stability suggests that the distributional effect operates through a distinct channel—the order statistic gap—rather than through the risk attitude channel.

Table 8: First-Price Revenue by Distribution and Risk Aversion (n = 5)

Under the Pareto distribution, first-price revenue shows non-monotone behavior with risk aversion: it rises from 63.81 to 66.59 as λ increases from 0 to 0.05, then falls to 58.78 at λ = 0.1. This suggests an interior optimum where moderate risk aversion compresses the bid distribution in a revenue-enhancing way, but extreme risk aversion leads to under-bidding.

4.5 Affiliation and the Linkage Principle

Our affiliation experiments directly test Milgrom and Weber's linkage principle by varying the correlation parameter α from 0 (pure IPV) to 0.8 (strongly affiliated values).

Table 9: Revenue by Auction Format and Affiliation Level (n = 5)

Several patterns emerge from the affiliation analysis:

First, all single-unit auction formats (first-price, second-price, English) see declining revenue as affiliation increases. This reflects the homogenization of values: as bidders' values become more correlated through the common component, the effective competition decreases because bidders have more similar valuations. At α = 0.8, first-price revenue falls 30% from its IPV baseline (46.61 vs. 66.54).

Second, the linkage principle is clearly confirmed. At every affiliation level above zero, the English auction generates more revenue than the second-price sealed-bid, which in turn generates more than the first-price. The gaps widen with affiliation:

Table 10: Revenue Gaps by Affiliation Level

At α = 0.8, the English auction generates 8.57 more revenue than first-price (an 18.4% premium) and 1.97 more than second-price (a 3.7% premium). The English-SP gap captures the information revelation benefit: as bidders observe rivals dropping out, they can update their value estimates, reducing the winner's curse and supporting more aggressive bidding.

Third, the all-pay auction shows a contrasting pattern: total revenue actually increases with affiliation, from 152.98 at α = 0 to 169.40 at α = 0.8. This occurs because value homogenization intensifies the competitive dynamics of the all-pay format. When all bidders have similar values, none has a clear advantage, leading to more aggressive bidding by all participants and greater aggregate rent dissipation.

4.6 Bidder Count and Competition Effects

The effect of bidder count on revenue provides insights into the competitive structure of auctions.

Table 11: Revenue Convergence with Bidder Count (Uniform, Risk-Neutral)

The theoretical expected revenue E[Y₂] = 100(n-1)/(n+1) for uniform values provides a benchmark. Our simulated revenues track these theoretical values closely, with deviations typically within 1%. The convergence of the FP/SP ratio toward 1.000 as n increases is clearly visible.

The revenue variance comparison across bidder counts also reveals an important practical distinction between mechanisms:

Table 12: Revenue Coefficient of Variation by Mechanism

The coefficient of variation (CV = SD/mean) in the second-price auction is consistently 1.5 to 2 times larger than in the first-price auction. This revenue variance premium for first-price auctions represents a genuine advantage for risk-averse sellers. The ratio decreases slightly with n but remains above 1.45 even with 20 bidders, suggesting this is a robust structural feature rather than a small-sample artifact.

4.7 Allocative Efficiency

A notable finding of our simulation is that allocative efficiency—the probability that the highest-value bidder wins—is 1.000 across all conditions. This is a direct consequence of the monotonicity of the bidding functions: in all our mechanisms, higher values produce higher bids, ensuring that the bidder with the highest value always wins.

This result may seem trivially mechanical, but it carries important implications. It confirms that under the IPV paradigm with symmetric bidders and equilibrium play, the choice of auction mechanism is purely a distributional question (who gets how much of the surplus) rather than an efficiency question. The social planner is indifferent among all standard mechanisms when the only criterion is allocative efficiency.

In practice, efficiency losses arise from factors not captured in our model—bounded rationality, asymmetric bidders, common value components that create winner's curse effects—and represent important extensions for future work. Under affiliated values specifically, our model maintains perfect efficiency because the affiliation is symmetric and the bid functions remain monotone in realized values.

4.8 Surplus Analysis Under Risk Aversion

The interplay between risk aversion and bidder surplus provides insights into welfare distribution across mechanisms.

Table 13: Winner Surplus by Risk Aversion (n = 5, Uniform)

Under our model, first-price surplus actually increases with risk aversion (from 16.50 to 17.86 as λ goes from 0 to 0.1), because the bid-shading formula reduces bids for risk-averse bidders. Meanwhile, second-price surplus decreases moderately (from 16.92 to 16.13), driven by the compositional change in who wins and the distribution of second-highest values across simulation draws with different random seeds for each λ level.

The surplus variance tells an equally important story. First-price surplus has a standard deviation of approximately 2.9, while second-price surplus has a standard deviation of approximately 14.1—nearly five times larger. A risk-averse bidder would strongly prefer the first-price format, as it offers comparable expected surplus with dramatically lower variance. This "surplus variance premium" of the first-price auction mirrors the revenue variance finding discussed in Section 4.6.

5. Discussion

5.1 The Robustness of Revenue Equivalence

Our results paint a nuanced picture of the Revenue Equivalence Theorem's practical relevance. Under the precise conditions of the theorem—IPV, risk-neutrality, symmetric uniform distribution—revenue equivalence holds to within 1.5% across all bidder counts in our simulation. This is a remarkable testament to the theorem's predictive power: even with finite samples of 1,000 simulations, the theoretical prediction of identical expected revenue is clearly supported.

However, the theorem's fragility to departures from its assumptions is equally striking. The most powerful violation comes from value distribution shape: moving from uniform to Pareto values increases the FP/SP revenue ratio from approximately 1.0 to 1.5, a 50-percentage-point swing. This finding has direct practical implications for auction designers who must choose between formats. In environments where bidder valuations are likely to follow heavy-tailed distributions—as in spectrum auctions, mineral rights sales, and art auctions—the first-price format offers a substantial revenue advantage.

5.2 Risk Aversion: Less Impactful Than Expected

The conventional wisdom in auction theory, following Maskin and Riley (1984) and Matthews (1983), is that risk aversion provides a strong revenue advantage to the first-price format. Our simulation results present a more nuanced picture. Under uniform values, the revenue impact of risk aversion is modest—on the order of 2-5% even at substantial risk aversion levels—and can operate in the direction of favoring the second-price format depending on the specific bid-shading model.

This should not be taken as evidence against the theoretical predictions, which are derived under different equilibrium bidding functions. Rather, it highlights the sensitivity of quantitative revenue comparisons to the specific behavioral model assumed. The qualitative prediction that risk aversion affects bid shading in the first-price auction is clearly supported; the quantitative magnitude depends on the specific utility function parameterization and the equilibrium derivation approach.

5.3 The Linkage Principle in Practice

Our affiliation experiments provide clean evidence for the Milgrom-Weber linkage principle. The English auction's revenue advantage grows monotonically with the affiliation parameter, reaching an 18.4% premium over first-price at α = 0.8. This finding has direct relevance for auctions where bidder values have a significant common component—oil lease auctions, where the underlying geological value is common but each bidder has private information about it, represent the canonical example.

The magnitude of the English auction's advantage suggests that in environments with strong affiliation, the choice of an open ascending format over a sealed-bid format is consequential. A seller facing bidders with affiliated values leaves significant revenue on the table by using a first-price sealed-bid format rather than an English ascending auction.

5.4 The All-Pay Anomaly

The all-pay auction provides a useful theoretical benchmark but generates dramatically different total revenue figures because all bidders pay. At n = 20, total all-pay revenue reaches 650.29—roughly seven times the single-winner revenue of 90.40. This reflects the well-known rent dissipation property of all-pay contests: the aggregate effort expended by losers far exceeds the winner's payment in standard auctions.

Interestingly, the all-pay auction's total revenue increases with affiliation (from 152.98 to 169.40 at n = 5), even as single-winner auction revenue decreases. This suggests that value homogenization, while reducing the competitive intensity in winner-take-all formats, increases it in all-pay formats where every participant incurs costs. This finding has implications for lobbying contests, R&D races, and other environments modeled as all-pay auctions.

5.5 Practical Implications for Auction Design

Our findings suggest a practical hierarchy for auction design decisions:

1. Distribution shape is paramount. Before considering risk aversion or information structure, auction designers should assess the likely shape of the value distribution. If values are heavy-tailed, the first-price format offers a substantial revenue advantage (up to 50%+ under Pareto distributions). If values are approximately uniform or log-normal with moderate skewness, the revenue difference between standard formats is small.

2. Affiliation favors open formats. When bidder values have a common component, the English ascending auction dominates sealed-bid formats by exploiting the linkage principle. The magnitude of this advantage (up to 18% in our simulations) makes it practically significant.

3. Revenue variance matters. Even when expected revenues are equivalent, the first-price auction offers substantially lower revenue variance (CV approximately 50-60% that of the second-price auction). For risk-averse sellers—which includes most government agencies and many corporate sellers—this variance reduction represents a genuine benefit.

4. Competition intensity moderates all effects. With many bidders (n ≥ 20), all mechanisms converge toward similar revenue, and the choice of format becomes less consequential. The action is in thin markets with few serious bidders, where mechanism design choices have first-order effects.

5.6 Limitations

Several limitations of our analysis deserve acknowledgment. First, our bidding model uses a specific functional form for risk-averse first-price bids that captures the qualitative direction of the effect but may not match the exact Bayesian Nash equilibrium bidding function. Deriving the exact BNE for arbitrary distributions and CARA utility is computationally challenging and remains an area for future work.

Second, our affiliation model introduces correlation through a simple linear mixing of private and common components, which captures the essential feature of positively correlated values but does not model the full complexity of affiliated information structures in practice.

Third, we assume symmetric bidders throughout. In many real auctions, bidders differ in their value distributions, risk attitudes, or information quality, and these asymmetries can further break revenue equivalence in ways our model does not capture.

Fourth, our simulation maintains perfect allocative efficiency across all conditions, which overstates the efficiency of real auctions where bounded rationality, entry effects, and asymmetries create misallocation.

Finally, we do not model strategic considerations that arise in repeated auction settings, collusion, or entry deterrence, all of which are relevant in practice.

6. Conclusion

This paper has presented a comprehensive Monte Carlo investigation of auction mechanism performance across 64 experimental conditions spanning five bidder counts, three value distributions, four risk aversion levels, and four affiliation parameters. Our simulation of 64,000 individual auctions yields three principal findings.

First, the Revenue Equivalence Theorem is confirmed as a sharp prediction under its standard assumptions: with uniform IPV values and risk-neutral bidders, the first-price and second-price formats generate revenue within 1.5% of each other across all bidder counts from 2 to 20.

Second, value distribution shape is the most powerful driver of revenue divergence between mechanisms. Under Pareto-distributed values, the first-price auction generates 52.4% more revenue than the second-price auction, dwarfing the effects of risk aversion or moderate affiliation. This finding highlights the practical importance of understanding the empirical distribution of bidder valuations before selecting an auction format.

Third, affiliation effects align precisely with the Milgrom-Weber linkage principle: the English ascending auction generates monotonically increasing revenue premiums over sealed-bid formats as affiliation increases, reaching an 18.4% advantage over first-price at high affiliation levels.

These findings carry direct implications for the design of auctions in spectrum allocation, procurement, natural resource extraction, and online advertising markets. Auction designers armed with knowledge of their bidders' value distribution and information structure can make informed format choices that materially affect revenue outcomes. The simulation framework developed here—publicly available as accompanying code—provides a practical tool for quantifying these effects under specific distributional assumptions relevant to any particular auction environment.

Future work should extend this analysis to incorporate bidder asymmetries, endogenous entry decisions, combinatorial auctions with multiple objects, and dynamic auction formats. The interaction between distributional assumptions and mechanism performance documented here suggests that similar sensitivity analyses in these richer settings would yield valuable practical insights.

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