1 Introduction

Auctions remain an important mechanism of government procurement, contracting, and natural resource allocation. Government auctions often produce new sales records: for instance, the 5G spectrum auctions in the United States generated a revenue of more than $50 billion. Auction theory and mechanism design are important tools for creating optimal and efficient mechanisms for public goods allocation. We now know that the optimal (and/or efficient) auction format heavily depends on many attributes of the allocation problem such as the goal of the auctioneer, specifics of the goods to be sold, value dependencies, and buyer characteristics.

One crucial consideration is the presence of budget constraints. If bidders have limited financial resources, many benchmark predictions such as revenue equivalence no longer hold (Che et al., Reference Che, Gale and Kim2012). While there is a lack of empirical research on the impact of financial constraints, they are common in business practice and are sometimes public knowledge. Consider this example from a radio spectrum auction (Wall Street Journal, 22 Jan. 2015): “AT&T Inc. said Thursday that it has entered into two credit pacts worth a total of more than $11 billion as the company beefs up its balance sheet to fund its bids for U.S. wireless spectrum.”

Financial constraints can lead to new strategic considerations that might have potentially undesired consequences for the auctioneer. One prominent example is the order of sale. In absence of buyers’ budget constraints, the order of sale is typically not important for the seller. However, Benoit and Krishna (Reference Benoit and Krishna2001) show that the seller’s revenue can vary drastically even in a basic setting of complete information with two items that are sold sequentially. Specifically, if the items have different values, the bidder with the highest budget might have an incentive to bid aggressively on the first item, letting the bidder with a smaller budget win that item, and deplete his budget. She can then pick up the higher-value item in the second auction against weaker opposition.

To illustrate this point, consider the following example. Three bidders (1, 2, and 3) have respective budgets of $70M, $40M, and $20M. They are competing for 2 items, Item A worth $60M and Item B worth $30M. The seller is using a simple English (increasing bid) auction and can choose whether to sell Item A first (Order AB) or Item B first (Order BA). All ties are resolved randomly. In Order AB, Bidder 1 gets Item A for $40 M, leaving a budget of $30M for the second round. Bidder 2 then wins Item B for $30M, with the seller earning $70M.

In Order BA, Bidder 1 could try to bid up to the value of Item B ($30M) and take a chance with a random allocation. She, however, is better off by only bidding up to $20M, letting Bidder 2 win Item B and reduce his remaining budget to $20M. Bidder 1 then wins Item A for just $20M in the second auction. The seller earns only $40M from selling the items.

One could argue that common knowledge of budgets is an unrealistic assumption. There are two potential considerations here. First, in some settings such as the FCC auctions mentioned above, bidders do have a public estimate of each other’s budget. Second, we use this setup as a starting point to study behavior in budget-constrained settings in order to understand behavioral mechanisms underlying bidding choices that respond to variations in other bidders’ budgets (which would be not the case for private budgets).

In this paper, we use a laboratory experiment to pursue two goals. First, we test the predictions of Benoit and Krishna (Reference Benoit and Krishna2001) using two popular auction formats: the English clock auction (EC, a version of an ascending bid auction) and a first-price sealed bid (FP) auction. In a between-subject design, we explore the optimality of these formats in several value/budget settings and two orders of sale (Order AB vs. Order BA). Second, we analyze the behavioral considerations and learning mechanisms underlying bidding strategies in both formats.

Empirical evidence on the effects of budget constraints in auctions is lacking, given that budget constraints are often unobservable to the researcher. A laboratory experiment thus has a few important advantages. We can easily control the values and budgets, as well as information available to the bidders, and compute specific predictions in a simplified setting. Additionally, we can estimate counterfactual predictions under different orders of sale and auction formats for the same value and budget settings.

We find robust empirical support for the theoretical predictions, with the revenue in Order AB clearly dominating the revenue in Order BA. Specifically, in Order AB, the seller earns 15% more on average than in Order BA. We find that individuals indeed recognize the strategic effects of budget depletion, with 94% of subjects at least once demonstrating strategic behavior in the role of the strongest bidder. However, the aggregate behavior deviates from theoretical predictions; only 27% of auction outcomes can be classified as fully consistent with theory. We find that the resulting revenues can be explained by a mixture of rational predictions and a naive bidding model motivated by limited foresight frameworks. We observe that the revenue gap between the two orders of sale increases as subjects gain more experience in the experiment.

We also demonstrate that the sealed-bid auction yields a higher average revenue compared to the English auction. We find that this effect is linked to differences in the speed of learning and the feedback induced by the two auction formats: in the English auction, subjects are significantly more likely to learn the benefits of playing the equilibrium strategy by the end of the experiment compared to the sealed-bid first price auction where opponents’ bids are often unobservable to the subjects.

To summarize, our results confirm the general comparative statics predictions of Benoit and Krishna’s model and additionally highlight the importance of behavioral effects related to the information environment induced by the auction format.

The rest of the paper follows the standard structure. Section 1 reviews related literature; Sect. 2 presents the theoretical model and hypotheses; Sect. 3 lays out the experimental design and methods; Sect. 4 presents the results.

2 Related literature

Theory. We directly test the predictions of the model introduced by Benoit and Krishna (Reference Benoit and Krishna2001). Following their paper, our main goal is to identify the auction design most beneficial to the seller. We use the standard English clock auction as a faster, simpler alternative to the ascending auction described in the original paper (Kagel & Levin, Reference Kagel and Levin2014).

Auction formats. Additionally, we explore another auction format that could potentially increase the revenue of the seller: the first-price sealed bid (FP) auction, which has been shown to yield revenue higher than the ascending (English) auction in many different settings. Overbidding in sealed-bid and first-price auctions compared to English auctions in the lab is a common result in the literature; a number of explanations have been suggested over the years: competitiveness (Kagel & Levin, Reference Kagel and Levin1993, Reference Kagel and Levin1986), risk aversion (Cox et al., Reference Cox, Smith and Walker1988), joy of winning (Goeree et al., Reference Goeree, Holt and Palfrey2002), fear of losing (Cramton et al., Reference Cramton, Filiz-Ozbay, Ozbay and Sujarittanonta2012a, Reference Cramton, Filiz-Ozbay, Ozbay and Sujarittanontab; Delgado et al., Reference Delgado, Schotter, Ozbay and Phelps2008), regret (Filiz-Ozbay & Ozbay, Reference Filiz-Ozbay and Ozbay2007), cognitive ability (Crawford & Iriberri, Reference Crawford and Iriberri2007), and simplicity of the mechanism (Li, Reference Li2017).Footnote ¹ Budget considerations add additional complexity to bidders’ strategies. Weaker informational feedback in the first-price auction can thus provide foundations for a new behavioral mechanism that leads to higher seller revenues.

It is worth noting that Benoit and Krishna (Reference Benoit and Krishna2001) also introduce a hybrid ascending auction format that is based on the standard sequential auction. In the hybrid auction, both objects are simultaneously on sale in multiple rounds of bidding, and in each round, a bidder can either outbid the highest bid on one of the objects (by posting/announcing their new bid) or pass. If there were no bid changes in the last two rounds, then the expensive item is sold for the current highest bid, and the auction for the cheaper item continues.

While this format might have interesting empirical implications, our pilot sessions revealed two unintended consequences that prevented us from running a full treatment with the hybrid design. First, many subjects found the rules of the hybrid auction confusing. Second, and most importantly, subjects often engaged in very long bid wars, where each subject would only increase their previous bid by 1 ECU, with the other subject following suit, repeating the same pattern over and over again. As this happened in almost every auction, this design resulted in unreasonably long session times. We thus decided to focus on a much faster English clock auction and not pursue the hybrid design. However, it still remains a potentially interesting subject for future studies (as well as other formats such as the Dutch auction or the all-pay auction).

Budget constraints in the laboratory. Although various forms of auctions with budget constraints have been extensively studied in the theoretical literature (Che et al., Reference Che, Gale and Kim2012; Che & Gale, Reference Che and Gale1999; Laffont & Robert, Reference Laffont and Robert1996; Malakhov & Vohra, Reference Malakhov and Vohra2008; Pai & Vohra, Reference Pai and Vohra2014; Pitchik, Reference Pitchik2009; Fang & Parreiras, Reference Fang and Parreiras2002; Dobzinski et al., Reference Dobzinski, Lavi and Nisan2012; Pai & Vohra, Reference Pai and Vohra2014; Kotowski, Reference Kotowski2020; Bobkova, Reference Bobkova2020), experimental evidence on behavior in such settings remains limited.

The closest study to our design is Pitchik and Schotter (Reference Pitchik and Schotter1988). The setup of their experiment is similar to Benoit and Krishna (Reference Benoit and Krishna2001). One of the treatments investigates the role of order of sale. Pitchik and Schotter (Reference Pitchik and Schotter1988) find that selling a more expensive item first to two bidders with equal budgets yields a higher revenue to the seller. We extend this approach substantially by analyzing multiple value and budget settings, and considering two separate auction formats (the English clock auction and first-price sealed bid auction).

Ausubel et al. (Reference Ausubel, Burkett and Filiz-Ozbay2017) study auctions with endogenous budget constraints, where a principal imposes a budget limit on a bidder in response to a principal-agent problem. The study shows that the empirical revenue difference between the first- and second-price auction seems to be persisting with and without budget constraints. Bae & Kagel (Reference Bae and Kagel2021) compare a proportional auction and a first-price auction for the case of budget-constrained bidders and a single item, reporting that the proportional auction is revenue-maximizing under strong budget constraints.

Order of sale. A number of studies analyze the “order-of-sale” effect and price trends in sequential auctions, both in the field (Deltas & Kosmopoulou, Reference Deltas and Kosmopoulou2004; Ashenfelter, Reference Ashenfelter1989; Raviv, Reference Raviv2006; Hong et al., Reference Hong, Kremer, Kubik, Mei and Moses2015) and in the lab (Pitchik & Schotter, Reference Pitchik and Schotter1988; Estenson, Reference Estenson1994; Keser & Olson, Reference Keser and Olson1996). It is typical for prices in sequential auctions for identical items to decrease, producing a so-called “declining price anomaly”, although the evidence is mixed (Raviv, Reference Raviv2006; Deltas & Kosmopoulou, Reference Deltas and Kosmopoulou2004). Note that we are considering the case of items of different values, however, the same (psychological) mechanisms might lead to a price decrease in this case.

Looking at bidders with budget constraints in fantasy football auctions, Boudreau and Shunda (Reference Boudreau and Shunda2016) show that individuals often tend to overbid early, irrespective of the underlying value of the item(s); they then switch to underbidding, and finish overbidding again. There are many behavioral explanations for initial overbidding suggesting that it is caused by behavioral effects in the first rounds of the auction (reflecting “warming up”, or competitive behavior, or familiarizing oneself with the rules of the auction) and is not caused by equilibrium strategic behavior (however, also see Huang et al. (Reference Huang, Devanur and Malec2012) for a theoretical treatment of the declining price phenomenon). Our results do not directly correspond to these observations. In each auction, we have only 2 items for sale, with known and observable values, and our subjects experience these two-stage auctions repeatedly.

3 Theoretical framework

3.3 Experimental settings

For the purposes of the experiment, we select several value and price settings inspired by Benoit and Krishna (Reference Benoit and Krishna2001). We use a fixed set of numbers (30, 40, 60, and 70 experimental currency units, or ECUs) and consider all their permutations as value and price amounts. Out of those permutations, we select those that satisfy $w_1>w_2>w_3$ . Bidder 2’s budget is chosen so that she is always financially constrained ( $w2<V_A+V_B$ ) in all settings (in addition, in Setting 3 $w_2<V_A$ ). Bidder 3’s budget is always set to 20 and thus was lower than the value of either item ( $w_3<V_B$ ). We use the resulting six combinations as the settings in the experiment (Table 1).

For each setting, we derive equilibrium predictions for the English and first price auctions (see Table 1 for the revenue predictions). While these equilibria can still be calculated by hand in some cases, in many others (specifically in the first price auction) they are too cumbersome to derive analytically (given the potential number of subgames and the size of normal form games within the subgames, under all value settings). We therefore apply an automated algorithm to derive equilibrium predictions computationally. For a detailed description of the equilibrium calculation procedure, see Appendix B. All data and code used to generate the equilibrium predictions are available through Open Science Framework.Footnote ²

Table 1 Value and budget settings used in the experiment

Setting	$V_A$	$V_B$	$w_1$	$w_2$	$w_3$	English auction		FP auction
						$E\left(R_{\mathrm{AB}}\right)$	$E\left(R_{\mathrm{BA}}\right)$	$E\left(R_{\mathrm{AB}}\right)$	$E\left(R_{\mathrm{BA}}\right)$
1	60	30	70	40	20	70	41	69–70	42
2	70	40	60	30	20	60	49	61	51
3	40	30	70	60	20	60	60	61	61
4	70	30	60	40	20	60	40	62	42
5	60	40	70	30	20	60	60	62	51/56
6	70	60	40	30	20	50	40	50–51	42

$V_j$ are the values of items A and B, $w_i$ are the budgets of bidders 1, 2, and 3. E(R) show the predicted theoretical revenue of the seller

4 Experimental design

4.3 Experimental treatments

Combinations of the sale order $\in \{AB,BA\}$ and auction format $\in \{EC,FP\}$ provided a $2\times 2$ treatment design on the session level (between subject design) resulting in 4 experimental treatments: EC-AB, EC-BA, FP-AB, and FP-BA (EC-AB: English clock auction, Order AB; EC-BA: English clock auction, Order BA; FP-AB: first-price sealed bid auction, Order AB; FP-BA: first-price sealed bid auction, Order BA).

4.3.1 Order of sale

The order of sale treatment determines the order in which the high-value Item A and low-value Item B are sold. In Order AB, the high-value Item A is sold first. In both treatments, subjects receive identical instructions.

4.3.2 Auction formats

Fig. 1 Experimental interface: English clock auction (left panel) and first-price sealed bid auction (right panel). The first item is displayed on the left, the second item is displayed on the right. The remaining budgets are shown underneath the currently auctioned item. If a bidder leaves the auction, their budget is removed from the screen. The subject’s own budget is displayed in bold red color. In the center of the screen, the subject can see their own remaining budget

English clock auction (EC) In these treatments, each item is sold via an ascending (English) clock auction. On the screen, subjects see a price clock for the current item. The clock starts at 0 and increases up to the item’s value at the rate of 1 ECU per second. Subjects can leave the auction at any time by clicking on an on-screen button (Fig. 1). If the price clock reaches a bidder’s budget, the bidder is forced to drop out of the bidding. If 2 out of 3 bidders leave the auction, the remaining bidder receives the item at the price at which the last bidder left.

First-price sealed-bid auction (FP) In these treatments, subjects submit a single sealed bid for each item by entering a number and clicking on a button. Bids are enforced to not exceed an item’s value or a subject’s budget. The bidder with the highest bid wins the item for a price equal to their bid.

In both formats, subjects can see the values of both items and the remaining budgets of all bidders at any time during the bidding period (see Fig. 1). All ties (bidders leaving the auction at the same price in EC or bidding the same amount in FP) are resolved randomly. The price of the first item is subtracted from the budget of a winner for the second auction. Budgets do not carry over across periods.