2.2.1 The Vickrey–Clark–Groves mechanism

An idealized benchmark to which auction formats are often compared is the Vickrey–Clark–Groves (VCG) mechanism (Vickrey, Reference Vickrey1961).Footnote ⁹ The VCG mechanism always allocates the items efficiently and its equilibrium outcome is always in the core in the environments we consider. Given its many practical drawbacks, see Ausubel and Milgrom (Reference Ausubel and Milgrom2006), this format is rarely used in practice and we likewise do not test it in our experiment. Nevertheless, it provides a useful theoretical comparison as a minimally-competitive mechanism.

In the VCG mechanism, bidders report their values to the seller and, based on these reports, the seller chooses the allocation that maximizes total surplus (i.e. the efficient allocation). Payments are designed such that it is a dominant strategy for bidders to report their true values to the seller. See Online Appendix A.1 for the formal description of the mechanism.

2.2.2 The first price auction

In the first price auction, bidders submit one bid for every possible package (i.e. subset) of items, which they pay if and only if they win the package. Since the X type bidders only value a pair of items, we need only consider their bids for two items; i.e. bids on one or three items are zero. Similarly, since the Y type bidders only value a package of three, we need only consider her bids for three items. An equilibrium in the first price auction in a particular environment will consist of a bidding function for each of the types present in that environment. The seller determines the feasible combination of bids that maximize revenue. Since each bidder only bids on a single package, these revenue-maximizing allocations are relatively simple to describe. In the $XXX$ and $XXY$ environments, at most two bids can be fulfilled; the seller allocates the requested number of items each to the top two bidders and nothing to the lowest bidder. In the $XYY$ environment, the seller can fulfill at most one bid from a Y type (for three items) and one bid from the X type (for two items); the seller allocates three items to the highest bidding Y type and two items to the X regardless of her bid. In the $YYY$ environment, at most one bid (for three items) can be fulfilled; the seller allocates three items to the highest bidder. With these revenue-maximizing allocations, equilibrium bid functions can be found using standard differential analysis. These calculations are described in Online Appendix A.2.

2.2.3 The simultaneous multiple-round auction

The simultaneous multi-round auction (SMRA) is modelled using five price clocks (one for each item), each of which ticks upward from zero whenever two or more bidders demand (i.e. bid on) the associated item. Bidders can only decrease the number of items they bid on after the auction starts. Given bidders preferences, each bidder will either bid on her entire demand (i.e. two items for type X, three items for type Y) or on no items; in the latter case we say the bidder is inactive or has dropped out. If only one bidder demands a particular item, its price clock is paused and this bidder is declared the provisional winner. If other bidders later demand this item, the price clock restarts and the item becomes provisionally unassigned. When demand on all items is at most one, the auction ends, items are assigned to their provisional winners and the winners pay the prices on the clocks for the items they won.

An equilibrium in the SMRA consists of a bidding function for each of the types present in the environment conditional on which types remain in the auction and at which prices others have dropped out. The equilibrium calculations are described in Online Appendix A.3. In environments $XXX$ and $XXY$ , as soon as any bidder drops out, the auctions ends. In the $XYY$ environment, the auction ends after a type Y bidder drops out but continues after the X type drops out. In the $YYY$ environment, the auction ends only after two Y type bidders drop out. In equilibrium, one bidder is randomly chosen to abstain from the auction while the remaining bidders compete.

2.3.1 Efficiency

Efficiency is calculated as

$\begin{array}{c}{\text{Efficiency}}_a=\frac{V_a-V_{\mathrm{random}}}{V_{\mathrm{opt}}-V_{\mathrm{random}}}\times 100\%\end{array}$

where $V_a$ denote the total surplus generated by mechanism $a\in \left\{\text{SMRA, First Price}\right\}$ , $V_{\mathrm{opt}}$ the total maximum surplus (generated by the VCG mechanism), and $V_{\mathrm{random}}$ the value of randomly assigning all the items to the bidders. This definition has the advantage that it is invariant when bidders’ values are multiplied by a common number (i.e. when they are measured in cents rather than dollars) or when a common number is added to all of them. Subtracting surplus generated by randomly assigning all items helps to isolate the added value of mechanisms being studied; it reflects the fact that the relevant alternative to the auction is not the withdrawal of the items from the market but random assignment of all items.Footnote ¹⁰

The first price auction is perfectly efficient in all but the $XXY$ environment, where it is at least 98.6% for $\alpha \le 2$ . Meanwhile, the efficiency of the SMRA varies widely between environments, with a low of 66.7% in the $YYY$ environment to a high of 100% in the $XXX$ environment. Both auctions approach perfect efficiency in the $XXY$ environment as $\alpha$ tends to infinity; for $\alpha \le 2$ , the first price auction is more efficient than the SMRA.

2.3.2 Seller revenue and bidder profit

The seller’s revenue is the sum of the winning bidders’ payments while bidder profit is the difference between the value of what bidders won and the payments they made.

As with efficiency, payments in the first price auction closely track those of the VCG; in all but the $XXY$ environment, expected seller revenue and bidders payoffs in the first price auction are equal to those in the VCG auction. In the $XXY$ environment, the seller’s expected revenue is higher in the first price auction while bidders’ profits are lower; the bidders thus absorb the loss of efficiency in this environment. Since the VCG mechanism is minimally competitive (i.e. generates the lowest competitive equilibrium revenue for the seller and highest competitive equilibrium payoffs for the bidders), the first price auction can be said to be reasonably competitive in all our environments. The seller’s revenue in the SMRA fluctuates around her first price/ VCG revenue between environments, being relatively low in the $YYY$ environment and high in the $XXY$ and $XYY$ environments; the opposite pattern holds for bidders’ profits. Thus, who bears the cost of the inefficiency in the SMRA depends on the environment; it is the seller in the $YYY$ environment and the buyers in the $XXY$ and $XYY$ environments.

Table 1 The table displays efficiency, revenue and bidder profit figures for the SMRA, first price and VCG auctions for $\alpha \in \{1,\frac{3}{2},2\}$

	Efficiency			Revenue			Profits
	SMRA (%)	First price (%)	VCG (%)	SMRA	First price	VCG	SMRA	First price	VCG
$\alpha =1$	86.4	100	100	0.440	0.458	0.458	0.631	0.646	0.646
$\alpha =\frac{3}{2}$	86.3	99.8	100	0.559	0.579	0.577	0.731	0.790	0.761
$\alpha =2$	86.3	99.7	100	0.660	0.694	0.688	0.852	0.880	0.886

Data are averaged across type environments

2.3.3 Price discovery

The use of a multiple-round auction is often justified by appealing to its ability to discover or reveal prices; that is, the process of competitive bidding is expected to determine a set of prices for items and packages of items that are “correct”, in the sense that they are close to what would prevail in a perfectly competitive environment with no uncertainty.Footnote ¹¹ The intuition is that bidders will reduce their demands gradually over the rounds of the auction as prices rise until supply equals demand, as in the classical Walrasian tâtonnement process. As shown in Milgrom (Reference Milgrom2000), this process leads to competitive prices if bidders bid truthfully (i.e. myopically) and all items are substitutes for all bidders.

Items are complements for our bidders by design. In the $XXX$ and $YYY$ environments, competitive prices are easy to determine: the price per item should be between the value (per item) of the highest losing bidding and the lowest winning bidder. When types are mixed, however, a competitive price may not exist. For example, consider the valuations in Table 2 for a $XXY$ type environment where the numbers in bold indicate the best allocation for a total surplus of 1.7.

Table 2 Example of bidders’ valuations in the $XXY$ environment

	Bidder 1 (Type X)	Bidder 2 (Type X)	Bidder 3 (Type Y)
2 items	0.7	0.8	0
3 items	0.7	0.8	0.9

To allocate two items to bidder 2 and none to bidder 1, the price of a single item p must satisfy $0.35\le p\le 0.4$ . On the other hand, to allocate three items to bidder 3, p must satisfy $p\le 0.3$ .

Because competitive equilibrium prices do not always exist in the presence of complementarities, attention has turned to the core as a proper benchmark for “reasonably” competitive outcomes.Footnote ¹² The core is defined by combinations of seller and buyers’ payoffs that satisfy certain stability constraints. The intuition is that auction payoffs are in the core when no coalition of bidders and seller can all do better than their auction payoffs. If we index the seller by $i=0$ and the three bidders by $i=1,2,3$ then the possible coalitions are the non-empty elements of the powerset of $\{0,1,2,3\}$ .Footnote ¹³ A vector of payoffs $\{{\pi }_0,{\pi }_1,{\pi }_2,{\pi }_3\}$ is in the core if

$\begin{array}{c}\sum _{i\in S}{\pi }_i\ge v\left(S\right)\end{array}$

for all $S\subseteq \{0,1,2,3\}$ where ${\pi }_i$ is the auction profit for coalition member $i\in S$ , and v(S) is the maximum surplus that coalition S can generate. Competitive equilibrium prices, when they exist, always produce core payoffs. But while competitive equilibrium prices may not exist, the core is always non-empty in auction applications. As such it seems the right benchmark for competitive outcomes in settings with complementarities.

How good is price discovery in the SMRA in the absence of these assumptions? Figure 1 suggests that it is poor, relative to the first price auction. For each environment we drew 1000 valuations and calculated core payoffs for each draw using the Bayes–Nash bidding functions in Online Appendix A. We then ran each auction and calculated the distance from the auction payoff to the set of core payoffs for each player. Figure 1 shows that mean distance over these 1000 draws, staggered over types to show the distance of each type to their set of core payoffs. Aside from the X!X!Y environment, the SMRA format generates payoffs that are further from core than does the FPSB format. In the X!X!Y environment, while FPSB payoffs are further from core payoffs than SMRA, neither format deviates very far.

Fig. 1 The figure shows the mean distance to the set of core payoffs for the first-price auction and the SMRA for each environment with $\alpha =2$ . The bar graphs are staggered over types to show the distance of each type to their core payoff

From the first bars of left panel of Fig. 2 it is clear that the one-stage FPSB delivers substantially higher efficiency than the corresponding SMRA. In fact, efficiency in FPSB comes very close to the theoretical maximum achieved by the VCG and remained on average above 90% across all type combinations. In SMRA efficiency is substantially lower and remains on average below two thirds of the theoretical maximum achieved by the VCG mechanism. In the other two graphs one sees that the FPSB also achieves, on average, higher seller revenue and lower bidder profits than the SMRA. This can be attributed to demand reduction on the part of bidders in the SMRA. Note however that in our experiment bidders in this mechanism frequently make losses. This is a consequence of the inability to protect themselves from the exposure problem. In the SMRA, bidders competing aggressively for a package of two or three items may end up winning only one. In addition, there may be fragmentation, i.e. a bidder winning non-contiguous items.

The top panels of Fig. 4 illuminate the shortcomings of the SMRA: this format often leads to allocations where one or two bidders get a single unit. The bottom panel of Fig. 4 displays the degree to which items are sold in non-consecutive packages or remain unsold in the standard SMRA; evidently, bidders had difficulty coordinating their bids effectively to form packages of consecutive items in the single-stage SMRA.

Fig. 4 Observed outcomes in the different mechanisms (top panel) and fragmentation in the SMRA (bottom panel)

Note that FPSB yields higher revenue on average than VCG despite its efficiency being less than VCG’s 100%. This comes at a cost to the bidders who make less than under VCG. Importantly, bidders’ profits are always positive under FPSB. The reason is that FPSB fully protects bidders from the exposure problem: they can specify a separate bid for each of the (combination of) items they might win and, by submitting bids that are less than values, never risk a loss.

Result 1

Compared to the SMRA the FPSB is more efficient, yields more revenue and lower bidder profit. Bidders frequently incur losses in the SMRA due to exposure problems.

Support. The result is supported by the data shown in the graphs in Fig. 2. Further support is given in Fig. 3, where the top row graphs display the distributions for efficiency, revenue, and bidder profits for the SMRA and FPSB. This is confirmed by non-parametric tests. For instance, the Wilcoxon Signed-Rank test comparing mean efficiency in FPSB and SMRA gives a p value of below 0.008. This is further supported by regressions controlling for the groups’ value type combinations. While the simple Wilcoxon Signed-Rank tests comparing revenues and bidder profits in the two treatments do not give significant differences (a Mann–Whitney U test rejects equality for revenues), the regressions controlling for value type combinations indicate that the treatment effect is indeed significant for at least some value type combinations. In fact, this is the case for revenues for all type combinations, once the data from all one-stage treatments is pooled (see Sect. 4.4). For bidder profits, the treatment effect is significant for all but one value type combination (XYY). For all significance tests and the regressions controlling for value type combinations, see Tables 5 to 13 in Online Appendix B.

The two-stage SMRA is used in practice to help bidders overcome some of the problems with its single-stage counterpart. First, it avoids fragmentation, as items won are contiguous by design. It also requires bidders to focus on a specific attribute in each stage: first the number of items, then on their location. Unfortunately, it has a double exposure problem: bidders who compete aggressively for a package may end up winning only a subset (as in the single-stage SMRA) and when competing for the number of items, bidders do not know whether item A will be included or not. In fact, this second exposure problem is independent of the underlying mechanism and is inherent to the two-stage process. It can therefore also affect results in the two-stage FPSB.

The bars grouped on the right of each panel in Fig. 2 allow us to compare efficiency, revenue and bidder profits between FPSB-2 and SMRA-2. In both cases we observe a substantial efficiency loss compared to their two-stage counterparts, but on average efficiency is higher in the FPSB-2. The comparison of revenues and bidder profits seems to indicate that SMRA-2 suffers from a larger exposure problem. Bidders make high losses that translate to high seller revenues. In FPSB-2, while bidder profits are significantly lower than the ones in the VCG theoretical benchmark, losses are not very common. In fact, seller revenues are not significantly different than what they are in the VCG benchmark.

Result 2

Compared to the SMRA-2 the FPSB-2 is more efficient. But SMRA-2 yields higher revenue and lower bidder profit mainly because bidders incur losses due to exposure problems.

Support. The result is supported by the data shown in the graphs in the bottom row of Fig. 2. Further support is given in Fig. 3, where the bottom row graphs display the distributions for efficiency, revenue, and bidder profits for the SMRA-2 and FPSB-2. Overall, the differences between the two two-stage formats are not statistically significant: the p values are 0.25 for efficiency, 0.148 for revenue and 0.055 for bidder profits. The regressions controlling for value type combinations reveal that the treatment effect is significant for efficiency when the type combination is XYY, while for revenue and bidder profits it is the case with type combinations XXY and YYY.

Looking at Fig. 2 it becomes clear that the two-stage process did not help bidders in our experiment. For the SMRA efficiency did not improve significantly moving from one stage to two. At the same time, it seems that the exposure problem intensified, leading to much higher seller revenues and losses for bidders in the two-stage mechanism compared to the one-stage SMRA. As an illustration, consider a case where in the first stage, a type Y bidder might compete fiercely to win three items but finally give in (at high prices) and settle for two items. In the second stage, the value of what was won may depreciate further if the type Y bidder places the lowest bid. As a result, bidder losses in the two-stage SMRA are common and substantial. In terms of protecting bidders’ from exposure risk, this format is least desirable. This is an important finding as regulators have started using such a two-stage format in spectrum applications. For example, Ofcom in the UK used two stages in the 2.3 GHz and 3.4–3.6 GHz auction in 2018.Footnote ¹⁷

For the FPSB, efficiency decreased with the addition of the second stage. Apparently, bidders are able to deal well with the relatively high number of bids required in the one-stage FPSB (six bids in total). Breaking down the process in two stages does not bring additional benefits, but introduces a new exposure problem. Unlike in SMRA-2, bidders in FPSB-2 do appear to protect themselves against this problem, as their profits are not significantly different than those in FPSB. Still, seller revenues are lower.

Result 3

The two-stage mechanisms result in lower efficiency (FPSB) or an exacerbated exposure problem (SMRA).

Support. The result is supported by the data shown in the graphs of Fig. 2. For the Wilcoxon Signed-Rank test comparing mean efficiency in SMRA and SMRA-2 we get p value 0.383 and for mean efficiency in FPSB versus FPSB-2 we get p value below 0.023. For the same test comparing revenues in SMRA versus SMRA-2 we get p value below 0.078 and for FPSB versus FPSB-2 we get p value 0.195. For the test comparing bidder profits in the SMRA versus SMRA-2 we get p value 0.195 and for FPSB versus FPSB-2 we get p value 0.742. Using the Mann–Whitney U test yields very similar results.

In the baseline SMRA and FPSB treatments subjects knew not only their own type and valuations, but also the type of the other bidders in their group. In real applications it is not unreasonable to think that telecom companies may have some information about their competitors preferences and the degree of complementarity they face. Nevertheless, it is a valid concern that the problematic performance of the SMRA in our experiment, as stated in Result 1, may be driven by this design feature combined with the particular choice of valuation distributions used. To test the robustness of our main result with respect to the informational environment, we conducted additional treatments of the one-stage mechanisms in which bidders know their valuations but are entirely uninformed about the distribution of other bidders’ valuations. These treatments are dubbed FPSB-U and SMRA-U respectively.

The second row in Fig. 2 presents the results for these additional treatments. We find that the comparison between FPSB-U and SMRA-U yields very similar results as that between FPSB and SMRA. The signs of the differences remain unchanged for all three measures: efficiency, revenue and bidders’ profit. In terms of magnitudes, the difference in efficiency is reduced but remains substantial: the FPSB-U is approximately 20% more efficient than the SMRA-U. There is also a reduction in the difference in revenue between the two mechanisms, but it remains significant. Bidders’ profit is again higher in the SMRA, although now the difference in not statistically significant.

Result 4

The FPSB’s better performance compared to the SMRA is robust to changes in the information available to bidders about others’ distribution of valuations.

Support. The result is supported by the data shown in the corresponding group of bars in the graphs of Fig. 2. Further support is given in Fig. 3, where the middle row graphs display the distributions for efficiency, revenue, and bidder profits for the SMRA-U and FPSB-U. For the Wilcoxon Signed-Rank test comparing mean efficiency in SMRA-U and FPSB-U we get p value below 0.008. For the same test comparing revenues in SMRA-U and FPSB-U we get p value below 0.023. For the test comparing bidder profits in the SMRA-U and FPSB-U we get p value 0.383. Using the Mann-Whitney U test yields very similar results. The regressions controlling for value type combination provide further support.

Overall, the change in the information available to bidders does not seem to have any effect on the average outcomes of the FPSB or the SMRA mechanism.Footnote ¹⁸ Based on this, and to facilitate the presentation of results regarding price discovery, in the analysis in the following section we pool the data for each mechanism across the two informational environments. If anything, this would work in favour of the SMRA.