We developed a novel platform for testing competitive decision making in a simulated Dutch auction. The platform allows manipulation of three fundamental design features—start price, rate of price change and size of price change [see Cox et al. (1982)] and records the winning-bid price and winning-bid time step across the various conditions. We briefly recap the main findings and then present a novel adaptation of Kahneman and Tversky’s prospect theory (1979, 1992) to account for qualitative patterns in the data, including the continuous and discrete step outcomes.

In both Experiments 1 and 2, there was no significant difference in the price or step (i.e. time at which participants bid) of the winning bid between the discrete price change and continuous price change conditions. These outcomes were supported by Bayes factor analysis which allowed the assessment of null effects, thereby overcoming limitations of frequentist tests. We also found no significant difference in either the Price or Step of the winning bid across testing blocks in either experiment.

### Effect of price change patterns on bidding behaviours

The findings of the current study suggest there was no significant difference in the Price or Step of the winning bid between discrete and continuous price-change conditions. This supports the findings of Katok and Kwasnica (2008) where the overall duration of the auction, rather than differences in patterns of price changes, influenced bidding behaviour. Katok and Kwasnica (2008) found that Dutch auctions with slow price changes (i.e. longer time intervals between price decreases) resulted in higher priced winning bids when compared to Dutch auctions with fast price changes (i.e. small time intervals between price decreases). However, their manipulations to the speed of price change resulted in changes in the overall duration of the auctions. The slow price change auction ran for a maximum of 10 min, while the fast price change auction ran for a maximum of 20 s. It is also worth noting that Katok and Kwasnica (2008) used financial incentives for both winning auctions and for finishing the task. They acknowledged that participants may have bid earlier, thus raising the price of bids, to end the auction and receive their payout earlier. In the current study, we maintained overall duration across the discrete and continuous price change conditions, so were able to focus on the effect of different patterns of price changes while controlling for the overall duration of the auction. With no difference observed in bidding behaviours between the discrete and continuous condition, it is possible that the overall duration of the auction affects bidding behaviours rather than the pattern of price changes.

Our findings support a different aspect of Katok and Kwasnica (2008) theory—that difference in bidding behaviours is caused by bidders considering time as a valuable resource, resulting in a trade-off between time saved and price. In the current study, the overall duration of the auctions was held constant across price conditions (fast, slow), so there was no need for bidders to trade-off between time saved and price, and without financial incentive for task completion there was no direct gain for ending auctions prematurely. This resulted in similar Price and Step of winning bids across the discrete and continuous price-change conditions.

This outcome may have been affected by the short duration of individual auction trials used in the current study. Each individual auction-trial in both the discrete and continuous price-change conditions ran for a maximum of 5 s. This short duration of individual auction [relative to Katok and Kwasnica (2008)] may have not allowed for the perceptual differences in the different patterns of price changes to visibly affect bidding behaviour. Future research may benefit from utilisation of our platform to examine the effect of different patterns of price changes over longer-duration auctions, where the perceptual difference is more apparent to the bidders.

### Consideration of hypothetical bidding in competitive decision-making

Next, we consider the potential effect of hypothetical funds on bidding behaviour. Without financial incentives to motivate participants to engage in real-world behaviours, it is possible our non-significant results may be an outcome of this design feature. However, in an extensive review of incentivised versus non-incentivised experiments in economics and psychology, Camerer and Hogarth (1999) concluded that incentives are less likely to affect mean performance in games, auctions, and risky choice tasks; however, incentives can reduce response variance. From a psychological perspective on participant effort, Erkal et al. (2018) found that non-monetary incentives like task enjoyment, desire to perform well and, importantly, competitiveness motivated participants to exert an equally high effort when compared to the same incentivised two-player task. Nonetheless, there are also studies suggesting different behaviour in incentivised versus non-incentivised tasks, and this could be tested in future investigations of Dutch auction bidding behaviour.

### Effect of learning across blocks on bidding behaviours

Garvin and Kagel (1994) found that inexperienced bidders initially bid earlier at high prices, resulting in the winner’s curse (the tendency to bid higher than the value of the auctioned item). However, with experience these participants could adjust their bidding behaviours, reducing the magnitude of the winner’s curse.

We examined whether exposure to the auction format, and interaction with other competitors, would affect participant bids by assessing the group’s mean price and step of the winning bids across the five blocks (12 trials per block) of a testing session. We found that the mean price and step of winning bids in Dutch auctions with either a fixed number of units for sale (Experiment 1) or varying units for sale (Experiment 2) was not significantly different. These results may arise from participants beginning the experiment with a near optimal or good estimation of item value based on experimental design features. For example, in each block participants were asked to use a fixed allocation of funds to purchase stock to fill their fixed size virtual warehouse. By providing each participant with $250 to spend and a warehouse capacity of 500 units, a participant had to win a total of 5 trials (i.e. auctions) in Experiment 1 or an average of 5 trials in Experiment 2 at an average of $50 per bid to successfully fill their warehouse. Our results suggest that participant triads may have already determined a bidding price strategy, averaging a winning bid of around $55 across all blocks in both conditions. Alternatively, the optimal use of funds would be through five $50 bids. While the competitive environment may drive the final bids up from this optimal price, Turocy et al. (2015) have also found that some participants will not update their bidding strategy. The authors found that only some participants changed bid price across an auction while others exhibited winner’s curse bidding but did not utilise information to update their bidding strategy to reach an optimal bid-price. Whether all group members identified an equivalent bidding strategy from the auction design or some individuals adapted their bidding strategies and others did not, our results indicate that the price of winning bids within a group competitive bidding environment are not affected by exposure to other competitors or the auction format.

### Consideration of certainty and utility on bidding behaviour: a prospect theory account

A fundamental feature of the Dutch auction is the certainty of winning (and losing)—the bidder who is first to bid wins the available item with certainty (Turocy et al. 2007). However, this is only true to the extent that no other player had yet placed a bid at that point in time. With passing time, there is an increasing likelihood that other stakeholders will place a bid, reducing the chances the item is still available. Our experiments required participants to trade off between the certainty of winning the bid, which decreases over the time course of the auction trial, and the price they are willing to pay for the available items (which also goes down).

Balancing risk, certainty, and value (alternatively, utility) is a standard feature in theories of economic decision-making. One of the most influential theories is prospect theory (Tversky and Kahneman 1992), which is commonly applied to single-player scenarios with standard choices about gambles such as “would you rather take a certain gain of $100, or a 50–50 chance of winning $200?”. We developed a new adaptation of prospect theory to account for multiple players’ bidding behaviour in Dutch auctions.

Prospect theory has been widely considered in ascending price auction formats; however, it has not been applied to Dutch auctions in a quantitative manner. For example, Kuruzovich (2012) discussed processes by which bidders increase their valuation of items through interaction with an online auction mechanism (but not necessarily *Dutch* Auctions). They argued from a prospect theory perspective that Dutch auctions present the individual with a different decisional frame compared to other auction formats, as Dutch auctions begin at a high price point and decreases, rather than start at a small value that increases. By commencing auctions at a higher value, the auctioneer changes the external framing of the choice, which should theoretically result in a higher bid and overall revenue from the auction. Fu et al. (2018) used regret theory to claim that participants will bid earlier in Dutch auctions to avoid feelings of regret as they are *loss averse*, a central prediction of prospect theory. They hypothesised that higher starting prices in a Dutch auction would increase the perceived valuation of items, resulting in higher bids as individuals become more loss averse. Similarly, Dodonova and Khoroshilov (2009) argue that the endowment effect, another example of loss aversion where individuals place higher value on items they already own, should be seen in the reserve prices set in a Dutch auction by sellers. While prospect theory has provided a sound theoretical framework to develop hypotheses and interpret results, we are not aware of any quantitative adaptation of prospect theory to Dutch auction decision making.

Our approach was to extend prospect theory in the time domain, by assuming that each player makes a *sequence* of choices during the auction. These repeated choices are all binary decisions: each time, the player must decide whether to bid immediately, or to wait just a few moments longer. Bidding immediately—given the auction is still running—is a prospect comparable to the “certain $100” above, in that the player will be guaranteed to win the auction, so they know both the gain (the product for sale) and the loss (the current price) associated with the choice. Waiting is a prospect comparable to the risky option above: the player must estimate the risk associated with waiting longer, the probability that another player will bid in the next few moments. Prospect theory (Tversky and Kahneman 1992) provides a well-established way to predict the choices of people faced with decisions between these options. In prospect theory, the choice relies on the perceived value of the item for sale (its “utility” to the bidder), the perceived value of the loss of money (the utility loss associated with paying the price), and the uncertainty (the probability of being beaten by another player in the next few moments).

We operationalise the prospect theory model as follows. Suppose *t* represents the time in the auction, and *C*(*t*) represents the selling price, or cost, at time *t* (in our experiments, *C* is a linear function). Suppose also that the perceived value of the product to the bidder is *P* and that the chance of another player bidding (and ending the auction) in the next few moments between time *t* and time \(t+dt\) is *r*. Prospect theory converts the costs to subjective utilities via a simple power function, *U*, and converts the probability to a subjective weighted probability via a sigmoid function, \(\pi\). The details of those functions are standardised in prospect theory and are reproduced below as well. The choice to buy now has no risk, and so has a net utility of \(U(P)+U(-C(t))\). The net utility of buying the product in a few moments is better, because the price will be reduced, \(U(P)+U(-C(t+dt))\), but this outcome must be weighed against the probability of being beaten by another player, which has zero gain and loss. This makes the overall weighted utility associated with waiting \(\pi (1-r)(U(P)+U(-C(t)))\).

We assume standard forms for the utility and probability weighting functions. Utility (*U*) is a power function of price (*x*), with different behaviour on losses (negative prices) than gains:

$$\begin{aligned} U(x,\alpha ,\beta ) \ = \ x^{\alpha } \ \text {if} \ x>0, \\ -\lambda (-x)^{\beta } \ \text {if} \ x<0 \end{aligned}$$

The probability weighting function (\(\pi\)) is defined below by two parameters, to allow separate weights for gains and losses, where we replace \(\gamma\) with \(\delta\) for losses:

$$\begin{aligned} \pi (p,\gamma ) = (p^{\gamma }/(p^{\gamma } + -p^{\gamma }))^{1/\gamma } \end{aligned}$$

The above defines a weighted utility for each of the two choices— bidding now, at time *t* versus waiting to bid at time \(t+dt\). Prospect theory converts these utilities into a probability of choice using a “softmax” rule. Suppose we let \(U_t\) represent the weighted utility of bidding at time *t* and \(U_{t+dt}\) represent the weighted utility of bidding at time \(t+dt\). Then:

$$\begin{aligned} \mathrm {Pr}(\mathrm {bid} \, \mathrm {now}) = \frac{e^{-cU_t}}{e^{-cU_t} + e^{-cU_{t+dt}}} \end{aligned}$$

Parameter *c* represents a player’s sensitivity to differences in value. When *c* is large, the player will almost certainly choose the option with the larger weighted utility, even if the differences between options are very small. When *c* is small, the player sometimes chooses randomly, selecting the lower-utility option on some occasions.

The model relies on the player having some estimate of the probability that one of the other players will bid in the next few moments, *r*. We operationalise this by assuming that the player maintains some representation of the bidding-time distributions of the other players. If this distribution is assumed to be identical for the two other players, with density *f*(*t*) and cumulative distribution *F*(*t*), then it is simple to show that \(r= 1-(1-(F(t+dt)-F(t)))^2\).^{Footnote 2} For the simulations below, we made the simple assumption that each player estimated the other players’ bidding times as normally distributed, \(N(\mu ,\sigma )\).

The above functions describe the probability of a single player making a bid in the next few moments. This is a hazard function, which can be converted to a probability density function (say, *g*) and associated cumulative distribution function (say, *G*) by standard transformations. However, the empirical distributions collected in our experiment depict the price and time of the winning bids, across all three players (e.g. player 1 could have won the first auction, but another player could have won the next auction in the block). To link our theoretical predictions with the data, we must infer from the model the empirical distribution of bidding prices in all auctions, by marginalising over *all* players—not just a single player. This reflects the summary distributions shown in figures such as Fig. 14. Thus, the final step is to let the model define the behaviour of three concurrent players and to derive from this the distribution of the minimum bidding times. This is done by taking the minimum over the three players, which has cumulative distribution \(1-(1-G(t))^3\).

We now provide a sufficiency proof for the model, by demonstrating that it is capable of generating bidding patterns similar to those observed in empirical data (at least for some parameter combinations—a sufficiency proof). We simulated three-player group bidding data over 1000 fixed unit auctions. We set the behaviour of the auction price clock, *C*(*t*) to be randomly sampled from a uniform distribution between $80 and $120 and decrease to $0 linearly over 5 s.

We set the time step to \(dt=0.5\), matching the discrete condition or to \(dt=0.05\), matching the continuous condition. Figure 14 illustrates empirical distributions of Experiment 1 data (top, similar to the distributions plotted in Fig. 6) and the simulated bid data produced by the model predictions of the winning bid price. We used parameter values taken from the best fits determined by Tversky and Kahneman (1992): \(\alpha =0.88, \beta = 0.88, \lambda = 2.25, \gamma = 0.61, \Delta = 0.69\). For the other parameters, we investigated numerous combinations and plot here one example set. The mean (\(\mu = 2.01\)) and standard deviation (\(\sigma = 0.97\)) of the estimated bid time were derived from the empirical data as the mean and standard deviation of group bidding time and remained fixed for the continuous and discrete estimates but, given the difference in step size and associated price difference between steps, the perceived value (*V*) of goods and sensitivity (*c*) parameters differed between the continuous (\(V = 1.27, c = 2.1\)) and discrete (\(V = 1.43, c = 0.3\)). These parameter combinations, and others, can generate distributions of winning bid price similar to the empirical data, as can be seen in Fig. 14. To make the model accurately predict both the average bid and the detailed distribution of bids, we had to simulate the model using a range of starting prices ($80 to $120) smaller than the range used in the experiment ($50 to $150). This represents an interesting theoretical observation that participants were less sensitive to changes in the stated cost (i.e. the auction-clock price) than the most straightforward application of prospect theory would predict. This may suggest a reference point effect. We constrained our model to use a very standard account, in which all prospects were treated as changes from a zero-dollar reference point. However, our players may have treated the prospects differently, as gains and losses relative to their current circumstances. With nonzero reference points, the nonlinear effects of the weighting and utility functions are changed, leading to effects such range compression and expansion (e.g. the range of [0.5, 1] is the same as the range [10.5, 11], but their ranges are very different after log scaling, such as in a utility function). This represents an interesting insight that may be investigated in future research.

### Effect of starting price on winning bids

The model produces predicted distributions of bids that are similar to the empirical data, for some parameter values. We wanted to test more specific predictions of the model by examining predictions of auction starting price, for which prospect theory can be used to make predictions in Dutch auctions (see Kuruzovich 2012). There is a lack of consensus in the general auction literature (not necessarily Dutch auctions) concerning the effect of starting price. Some evidence shows lower starting prices lead to higher bids (e.g. Ku et al. 2006), while others find evidence to the contrary (lower start prices lead to lower bid, e.g. Ariely and Simonson 2003). Ku et al. (2006) analyzed how low starting prices attract an entry of new bidders and affect final prices. Based on eBay field data and survey experiments, the authors found low starting prices attracted more bidders and thereby result in higher final prices. Walley and Fortin (2005) confirmed in their controlled field experiment that lower starting prices increase the number of bidders and eventually final prices. Ariely and Simonson (2003) also confirm the results of Ku et al. (2006), stating that low starting prices attract more bidders, with data from a controlled field experiment on eBay. However, and in contrast to Ku et al. (2006), Ariely and Simonson (2003) find that these auctions yield on average fewer bids and lower final prices. The authors argue that this can be explained by an anchoring effect: starting prices may serve as a value signal for bidders, with higher starting prices indicating a higher product value. Although this effect has not been examined in Dutch auctions, Kuruzovich (2012) argued that because Dutch auctions start at a high price and decrease they should theoretically produce higher revenue compared to ascending price auctions; however, this was not empirically tested.

In both Experiments 1 and 2, there was a positive correlation between starting price and price of winning bids, in both fixed- and multi-unit Dutch auctions. We found that the price of winning bids was significantly different when separated into high and low starting price bins. Here, we implement the same analysis on the simulated data from our prospect theory based model and show the model predicts this qualitative pattern.

Figure 15 depicts the positive relationship between the auction starting price and the price of the winning bid for model-simulated data. The correlation was significant in both the continuous (\(r = 0.574, R^2 = 0.330, p < 0.001\)) and discrete (\(r = 0.417, R^2 = 0.174, p < 0.001\)) conditions.

We further assessed the model-simulated winning bid prices when bids were separated into auctions commencing at a low ($80–$100) or high ($100–$120) price (Fig. 15, right). We found a significant difference between the low and high starting price auctions with strong evidence in the continuous (\(t(998) = -17.1, p < 0.001, Log(BF_{10}) = 124.56\)) and discrete (\(t(998) = -11.47, p < 0.001 , Log(BF_{10}) = 58.5\)) conditions. We explored this relationship across different bin sizes in Appendix A and found it to be a robust effect in both data sets.

These results suggest starting price is related to the price of the winning bid in fixed unit Dutch auctions under continuous and discrete step-rates. This finding is important for both theoretical and practical reasons. From a theoretical perspective, it allows to compare the empirical pattern with model predictions. Practically, starting price of an auction is a design choice of the auctioneer or market designer. If they start the auction with too high a price, they might waste valuable time, which is especially critical when selling perishable items (as is the case in most Dutch auctions). If they start too low, they might miss out on additional revenue as predicted by both our data and model.

### Conclusion and future directions

The current study aimed to develop a computerised platform for Dutch auctions and test how different design parameters affect the decision-making processes involved in this competitive group context. Results from Experiments 1 (fixed item quantity) and 2 (variable item quantity) showed no significant effect for different patterns of price changes on the price or time-step of the winning bid. There was no difference in the price or step of the winning bid between testing blocks. This suggests that participants either (1) began with a good estimate of item value or (2) did not change their bidding behaviour through experience.

Empirical data were collected in-lab. This had limited the sample size, as the scheduling of multiple participants to concurrent testing is non-trivial. Future studies could employ online testing to obtain larger sample. However, in-lab testing rewarded the study with very real and vivid group context. In post testing interviews, some participants reported they were excited by the competitive nature of the task. Future research may look into physiological measures, to assess arousal and how it affects bidding behaviour (a-la Malhotra et al. 2008).

In conclusion, this paper offers theoretical and practical contributions. On the theoretical side, we developed an adaptation of prospect theory that can account for bidding behaviour in Dutch auction. Our results also have practical implications: to the extent that they can be generalised, they suggest that (1) for a given rate of price-change, the pattern of change (small decrements over many steps or large decrements over few steps) has little-to-no effect of bidders’ behaviour, and (2) that increasing the starting price of a fixed unit Dutch auction results in an increase in the price of bids. The model successfully captured both patterns. Our testing platform provides an exciting avenue for future research into biding behaviour. The Dutch auction allows to investigate the way multiple participants within a group balance risk (of missing the bid) and cost (bidding price) and serves as an ideal context for the study of competitive decision making.