## Abstract

In this article we experimentally investigate reverse multi-unit Dutch auctions in which bidders compete to sell their single unit to a buyer who wants to purchase several objects. Our study yields three insights: (i) bids are substantially higher than Nash equilibrium bids predicted by standard economic theory; (ii) these higher-than-predicted prices gradually decline in later periods; and (iii) bid pooling (or simultaneous bidding) is frequently observed—the majority of bidders submit their bids immediately after the first bidder has sold his unit. A model that distinguishes between myopic and sophisticated bidding strategies helps to organize these patterns both on the aggregate and on the individual level.

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## Notes

In these auction-like settings, all buyers pay the same price, but they may incur different opportunity costs depending on the point in time they decide to purchase their tickets. Another application is the problem of how to deal with stranded passengers when flights are overbooked. The use of auctions as a solution to the airline-overbooking problem was suggested by Simon (1968) (see also Simon 1994).

Sherstyuk (1999, 2002) analyzes multi-unit English auctions and shows that collusion (without bid pooling) occurs if bidders can match their offers. This is in accordance with standard theoretical predictions, as she allows for bid matching, which means that competitors may immediately match any deviating bid, rendering deviation unprofitable. In contrast, our observations in a multi-unit Dutch auction cannot be explained by standard theory.

Moreover, in a recent article, Cox and James (2012) compare Dutch auctions and centipede games with private information about payoffs and highlight that the mode of presentation (clock or tree structure) has a decisive impact on behavior.

This feature is similar to some treatments of the oral-auction studies by Sherstyuk (1999, 2002) in which, after observing the bid of a competitor at a given price, a bidder could decide to match it. The possibility of matching bids strongly facilitated collusion. In our design, a subject can realize only after a given bid price that a competitor has entered the auction, thereby running the risk that all items have already been sold.

At the same time, we acknowledge that repeated game effects may arise in our setting. Therefore, an interesting extension of our experiment would be to investigate the bidding patterns and market prices that emerge if subjects are matched with new competitors in every period.

Instructions translated from German can be found in Appendix A.

Given that the expected payoff resulting from equilibrium play accounted for only \(3/4 \times 0.2\)€ \(\times \) \(20 = 3.00\) €, the high show-up fee was chosen to ensure that the average payoff in our experiment would not be significantly below the typical level.

We choose a start price of 20 ECU to get a unique equilibrium under standard assumptions.

The exception is the final time interval (periods 16–20). There, the share of auctions in which the highest prices stay constant rises by about 20 % points, which is due to the fact that many auctions have already reached the minimum price of 20 by then.

A somewhat related argument is made by Suetens and Potters (2007) to explain dynamic behavioral patterns in Bertrand settings. When information about previous behavior is provided, subjects might imitate the best performer in the last round.

We formalize this idea in Appendix B.

For example, if the share of sophisticated bids is 50 %, then it is optimal for a bidder following the sophisticated bidding strategy to enter exactly one price step below \({\bar{b}}_{t-1}\) for any \({\bar{b}}_{t-1}\in \{20,\ldots ,100\}\).

For higher values of

*x*, there may be unraveling to the minimum price.This assumption seems plausible, because bidders need to perform elaborate computations to follow the sophisticated strategy. On the contrary, when following the myopic strategy, a bidder simply copies a price in the next period.

The term “realized price” here refers to the initial price at which one or more bidders enter the auction. Note that whether the prices at which subsequent bids are submitted are equal to or lower than the highest price achieved in the previous round depends on the initial bid in the present round.

Many studies (see, for example, Duffy and Nagel 1997; Ho et al. 1998; Bosch-Domènech et al. 2002; Kocher and Sutter 2005; Costa-Gomes and Crawford 2006 and the references cited therein) have provided empirical support for level-

*k*models. A robust phenomenon is that the majority of choices is in line with one or two steps of iterated reasoning; the share of more sophisticated players is typically small. For recent discussions on the general validity of the level-*k*approach, see, e.g., Penczynski (2011) and Georganas et al. (2015). Investigations on the empirical relevance of level-*k*thinking in static auction settings yielded mixed results (see Crawford and Iriberri 2007; Ivanov et al. 2010; Georganas 2011; Kirchkamp and Reiss 2011). More generally, other studies showed the importance of boundedly rational bidding strategies in auctions (see, for example, Cooper and Fang 2008; Shachat and Wei 2012; Kirchkamp and Reiss 2014).The exception are auctions in which the initial bid is placed one price step below the highest price achieved in the previous period. Bidders who do not enter at this price are sure to follow the myopic strategy according to our definition.

We cannot use data from auctions in the first round, because it is not possible here to separate sophisticated and myopic bids from each other. Due to the assumption that without a bidding history, the myopic strategy consists of randomizing over the strategy space, every observed bid in the first auction could in principle be sophisticated or myopic. Moreover, initial bids exceeding the highest previous level in any auction in periods \(t\ge 2\) or initial bids at \({\bar{b}}_{t-1}\) by less than four bidders are not captured by our approach and, therefore, excluded (a total of 43 auctions). Finally, we have to exclude 50 auctions from the analysis, because goods are sold for prices near or at the Nash equilibrium under standard assumptions. In these cases, a distinction between sophisticated and myopic bids and equilibrium play is no longer possible.

The exact approach, functions, and derivation can be found in Appendix D.

We acknowledge that with our estimation strategy, we may understate the propensity of making mistakes. As prices successively approach the Nash equilibrium bid for rational players, bidders have less and less possibility to underbid by many price steps. Yet, our results do not change substantially when we restrict our analysis to auctions in which the highest previous bid was 40 ECU or higher. For these auctions, sophisticated bidders still can erroneously underbid by three steps. Repeating our estimation for the restricted sample, 52.1 % (47.9 %) of the bids are classified as sophisticated (myopic); of the sophisticated bids, 75.7 % are estimated to be exactly one price step below the highest bid from the previous round.

In classic level-

*k*models, subjects are assumed to stick to the strategy associated with their types. In a recent approach to modeling level-*k*-thinking, Ho and Su (2013) assume that players may change their strategy after observing unexpected behavior of others. In their model, players are in principal capable of playing according to any level-*k*-thinking type, but choose their behavior as to maximize payoffs given their beliefs about the distribution of the thinking types of the other players.We provide a formal proof concerning the validity of this procedure below.

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## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

We would like to thank the Editor, two anonymous referees, Wieland Müller, Hans-Theo Normann, Axel Ockenfels, Henry S. Schneider, Michael Visser, Christian Waibel, as well as seminar and conference audiences in Cologne, Munich, Murcia (JEI), Paris, and Washington (IIOC) for very helpful comments and suggestions. Financial support of the German Research Foundation (DFG) through the Gottfried Wilhelm Leibniz Prize awarded to Axel Ockenfels and through the Research Unit “Design & Behavior” (FOR 1371) is gratefully acknowledged.

## Appendices

### Appendix A: Instructions

*Below we include a translation from German of the instructions that we used in the experiment.*

Welcome to our experiment! In this experiment you can earn money. How much you will get depends on your own decisions and on the decisions of the other participants. **From now on please do not communicate with other participants. If there are any questions, please raise your hand!** We will come to you and answer your question. If you break this rule, we will have to exclude you from this experiment and all payments.

In this experiment, ECU is used as the currency. **At the end of the experiment, your ECU-payoffs will be converted to Euro and paid out in cash.** The conversion rate is **100 ECU = 1 Euro**.

### 1.1 Procedures

The experiment consists of **20 auctions in total**. In each auction, you have the opportunity **to sell a fictitious good**.

At the beginning of the experiment, **four sellers** are randomly chosen and randomly assigned to each other. These sellers **will interact in each of the 20 auctions**.

At most **three goods** are sold in each auction. Thus, not every seller will be able to sell her good. An auction proceeds as follows:

**The price starts at 20 ECU**. The sellers then have **5 s to decide whether they want to sell their goods at this price** by clicking on the **button “Sell the good at this price”.** The remaining time is displayed in the upper right corner of the screen.

Sellers who do not want to sell their good at this price do not have to do anything. After 5 s, the experiment proceeds automatically.

When all sellers have made their decisions **and not all three goods were sold, the price will be raised by 5 ECU to 25 ECU** and all remaining sellers in the auction have 5 s to decide whether they want to sell their goods at the new price.

When not all three goods are sold at this price, the price will again be raised by 5 ECU to 30 ECU and all remaining sellers in the auction decide again.

The price will be raised **by 5 ECU-steps** until **either the three goods are all sold**
**or the price reaches the upper limit of 100 ECU**, without three sellers having sold at this price.

If **more sellers want to sell their goods** at a certain price **than goods are demanded in the auction**, it will be **randomly determined** which seller is allowed to sell her good.

The sellers have no costs. **This means that sellers who sell their goods in the auction receive a payoff equally to their selling prices.** Sellers who did not sell their good in an auction do not receive a payoff from this auction.

After each auction, the sellers will be informed about all prices that were realized in this auction.

### 1.2 Concluding remarks

At the end of the experiment, the **sum of payoffs from all 20 auctions** will be converted into Euro and paid to you. **In addition**, you will receive an amount of **7.50 Euro for your participation** irrespective of the decisions in the experiment.

### Appendix B: A more elaborate approach to modeling first-round bidding

We consider the simplest case in which a bidder following the sophisticated bidding strategy expects the three other bidders to pursue the myopic strategy. In the first auction, as there is no information on bidding behavior from earlier periods available, a sophisticated bidder optimizes his bidding behavior under the assumption that myopic bids follow a discrete uniform distribution on the interval \(\{20, 25, \ldots ,100\}\).^{Footnote 24} In our case, this assumption reflects that in the very first auction, there is no anchor to which players can adjust their bids. To determine the optimal sophisticated bidding behavior at the current price *b*, we need to distinguish between three cases depending on the number of products that have already been sold during the auction.

Suppose none of the competing bidders has submitted a bid at price \(b-5\). Now let \(p{:}=(b-(b-5))/(100-(b-5))=5/(105-b)\) be the probability that a competing bidder bids at the current price \(b>20\). Then, the expected payoff of a player with the sophisticated strategy from submitting a bid at *b* is given by

Note that \(1-p^3\) is the probability that at most two other competitors bid at the same time, which means that the bidder sells his product with certainty. Similarly, \(p^3\) represents the probability that all other bidders simultaneously submit bids, which means that the bidder has a winning probability of only 3 / 4.

Let \({\tilde{p}} \mathrel {\mathop :}=(b+5-b)/(100-b)=5/(100-b)\) be the probability that the myopic bid is \(b+5\). Similar to the case in which the bidder submits a bid, the expected payoff from waiting at the current price *b* then amounts to

Solving \(\mathbb {E}\left[ \pi _s^{\mathrm{bid}}\right] =\mathbb {E}\left[ \pi _s^{\mathrm{wait}}\right] \) gives \(b\approx 85.07\) as the (relevant) solution. Hence, if no product has been sold, the sophisticated strategy consists of accepting a clock price of 90 (where \(\mathbb {E}\left[ \pi _s^{\mathrm{bid}}\right] >\mathbb {E}\left[ \pi _s^{\mathrm{wait}}\right] \)).

Suppose next that one of the competing bidders has already sold his product at a price lower than \(b-5\). Then, the expected payoff for a bidder following the sophisticated strategy and submitting a bid at *b* is given by

Analogously, waiting for another tick of the price clock yields an expected payoff of

Again, solving \(\mathbb {E}\left[ \pi _s^{\mathrm{bid}}\right] =\mathbb {E}\left[ \pi _s^{\mathrm{wait}}\right] \) gives \(b\approx 76.33\) as the (relevant) solution. Hence, if one product has been sold, the sophisticated bid is 80.

Last, consider the case where only one more product can be sold to the buyer. Then, the expected payoffs amount to

Analogously, waiting for another tick of the price clock results in an expected payoff of

From \(\mathbb {E}\left[ \pi _s^{\mathrm{bid}}\right] =\mathbb {E}\left[ \pi _s^{\mathrm{bid}}\right] \Leftrightarrow b=48.75\), it follows that the sophisticated bid is 50.

The bidding behavior in the first round can thus be summarized as follows:

and

This bidding behavior implies that actual bids in the first period of the reverse multi-unit Dutch auction are significantly higher than the one predicted by standard economic theory, which equals 20 ECU.

Note that although we only covered the case where one player applies a sophisticated strategy, the above argument also holds for the case where two to four players follow the sophisticated bidding strategy. This is due to the fact that if the bidder applying the sophisticated bidding strategy expects three other players following a sophisticated strategy as well, he is going to enter the auction at a price of 20 for \(g\in \{1,2,3\}\). As the bidding strategy continuously depends on the distribution of bid types and given the intermediate value theorem, sophisticated bids higher than 20 can be supported for certain ranges of shares *x*.

### Appendix C: Trembling-hand equilibrium

Assume a share of *x* bidders follow the sophisticated bidding strategy of bidding \({\bar{b}}_{t-1}-5\) but might err in doing so. Assume the errors are distributed according to the Poisson distribution \(P(\lambda ,k)\), where *k* is the number of steps of deviation and \(\lambda \) is the variance/expected value of the distribution. The share \(1-x\) follows the myopic strategy. A best response to this setup might be to bid \({\bar{b}}_{t-1}-5\) depending on *x* and \(\lambda \). This is the case if the following holds:

The left-hand side is the reduction in profits if one were to bid \({\bar{b}}_{t-1}-10\) instead of \({\bar{b}}_{t-1}-5\). The first line of the right-hand side is the probability that there is at least one bidder following the myopic bidding heuristic plus the probability that all other bids are sophisticated and non-erring, which leads to a winning probability of 3 / 4. The second line is the probability that there is one trembling bidder who follows the sophisticated bidding strategy and submits an initial bid strictly smaller than one step below \({\bar{b}}_{t-1}\) times the resulting winning probability and profit when everyone enters at the next step. The third line follows the same logic given that there are two sophisticated trembling bids at the same bid step strictly smaller than one step below \({\bar{b}}_{t-1}\). This can be rearranged to

This is satisfied for large ranges of \(\lambda \) and *x*. In particular, it is fulfilled for all combinations of \(\lambda \), *x*, and \({\bar{b}}_{t-1}\) that we find empirically in Sect. 4. The only exception here are the periods 16–20 in which *x* and \(\lambda \) are such that bidding \({\bar{b}}_{t-1}-5 \) is not trembling-hand perfect for the entire support of \({\bar{b}}_{t-1}\). However, in these five periods, the observed values for \({\bar{b}}_{t-1}\) are low enough such that together with \(x\approx 0.65\) and \(\lambda \approx 0.208\), they form a trembling-hand perfect equilibrium again.

### Appendix D: Estimation procedure

In what follows, we first characterize the estimation procedure (subsection “Procedure and intuition”) and then show that it is a valid approach (subsections “Choice of norm”, “Existence of a minimum”, and “Uniqueness of a minimum”).

### 1.1 Procedure and intuition

We can derive the ex ante probabilities of observing between one and four bids (as four bidders interact) at a given number of steps below the highest previous bid in each auction. Table 4 lists all corresponding ex ante probabilities that *n* bids are of either myopic or sophisticated type. By comparing the ex ante probabilities with the observed number of auctions with a given number of initial bids, we can estimate the distribution of sophisticated and myopic strategies that fits our data best.

We capture the errors in the sophisticated bidding strategy by assuming that they are distributed according to a discrete probability mass function. High price undercuts are rare and our model shows that underbidding by exactly one price step is a trembling-hand perfect equilibrium for large parameter spaces. Transferring this idea to our formulation of the error term, we need a distribution function that allows for relatively high probabilities for small errors and vice versa. For this reason, we assume that the errors are Poisson distributed, \(P(y;\lambda )=\lambda ^y{\mathrm {e}}^{-\lambda }/y!\), where *y* is the number of steps below \({\bar{b}}_{t-1}\), and estimate \(\lambda \), the parameter that measures the expected probability of placing a bid based on erroneous beliefs in our case.

Given the share *x* of sophisticated bids, the functions in Table 4 minus the observed shares define the distance between the ex ante and the observed probabilities of *n* initial bids of either type. Let us denote these functions \(f_j^{\{z\}}\), where *z* is the number of initial bids of either type and \(j\in \{m,s,se\}\) stands for the three strategies: myopic, sophisticated, or sophisticated with error. Take the example where exactly \(z=2\) initial bids are classified as type \(j=s\) in 55 of all 325 auctions. This means that according to the above definition, we have a function

From the table it becomes clear that we are looking at an over-determined system of equations. To estimate the probabilities given the number of observations per case (i.e., the values for *x* and \(\lambda \)), we transform the system into a minimization problem by defining a function \(f: \mathbb {R}^{2} \rightarrow \mathbb {R}^{9}\) with equations \(f_j^{\{z\}}\) as components.^{Footnote 25} For a classical solution \(\tilde{\mathbf {x}}=(\tilde{x},\tilde{\lambda })\), it holds that

In combination with the positive homogeneity of a norm, one can minimize the norm of *f* under the constraints

With \(\mathbf {x}=\left( x,\lambda \right) \), this problem can be written as

Thus, we look for those ex post probabilities \(\mathbf {x}\) in the model functions, such that the distance between *x* and the observed share is minimized.

We use our approach—rather than a maximum likelihood estimation conducted with individual data—because based on our model assumption concerning bidding behavior, each initial bid in an auction below the highest price in the previous auction is unambiguously assigned to the sophisticated bidding strategy, which means that the observed distribution of initial bids is deterministic. Hence, our method finds the best estimate for the underlying distribution of sophisticated and myopic individual bids. In addition, our approach provides us with an estimate of the distribution of errors among the share of sophisticated bids.

### 1.2 Choice of norm

In the minimization problem, we use the Euclidean (or 2-)norm. Since one could also minimize the \(L^1\) or even an \(L^p\) norm, this choice may not be obvious, but actually follows naturally from the problem. One can only minimize a function \(f: \mathbb {K} \mapsto \mathbb {R}\) with \(\mathbb {K}\subset \mathbb {R}^n\), but in our case, the function maps to \(\mathbb {R}^{9}\). Thus, we minimize

where \(\langle \cdot ,\cdot \rangle \) is the standard inner product in \(\mathbb {R}\). The norm induced by \(\langle \cdot ,\cdot \rangle \) is the 2-norm

and, therefore, the choice of the norm follows directly from the problem.

### 1.3 Existence of a minimum

The existence of a minimum (and not just an infimum) is guaranteed, because \(\left\| f(\mathbf {x})\right\| _2\) is continuous. The constraints require *x* to be in the compact space [0, 1]. \(\lambda \) represents the expected value of the Poisson distribution and is, therefore, bounded by \(\lambda _{\text {max}}=15\), because that is the maximum number of steps a bidder can erroneously deviate (given \({\bar{b}}_{t-1}=100\)). Continuous functions attain a minimum on compact spaces (Theorem of Weyerstraß).

### 1.4 Uniqueness of a minimum

Every norm is a convex function, because by the triangle inequality and the positive homogeneity, it holds that

Therefore, the worst-case scenario is that \(\left\| f(\mathbf {x)}\right\| \) is constant for a small space \(\mathbb {S}\subset [0,1]\) around a critical point. However, it can be easily check that for every critical point \(\mathbf {x}\) found, the Hessian matrix is strictly positive definite. It follows that the minimum is unique.

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Gillen, P., Rasch, A., Wambach, A. *et al.* Bid pooling in reverse multi-unit Dutch auctions: an experimental investigation.
*Theory Decis* **81**, 511–534 (2016). https://doi.org/10.1007/s11238-016-9546-z

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DOI: https://doi.org/10.1007/s11238-016-9546-z