Informed entry in auctions

Abstract

We examine entry decisions in first-price and English clock auctions with participation costs. Potential bidders observe their value and report maximum willingness to pay (WTP) to participate. Entry occurs if revealed WTP (weakly) exceeds the randomly drawn participation cost. We find no difference in WTP between auction formats, although males have a higher WTP for first-price auctions. WTP is decreasing in the number of potential bidders, but this reduction is less than predicted and small in magnitude.

Appendix A: Bidding behavior

Potential bidders enter more often than predicted. If bidders were reporting a WTP in keeping with their beliefs regarding their expected earnings, the observed over-entry would imply that they believe that bids are, on average, below the Nash prediction, thus increasing expected payoffs of entering the auction.²⁷ Since in English clock auctions bidders have a weakly dominant strategy to bid their value, potential bidding holding such beliefs seems unlikely. In first-price auctions equilibrium bids depend on both the observed number of bidders, as well as beliefs about the entry threshold employed by bidders. In particular, the Nash bidding function in first-price auctions is linear in the bidder’s value. The slope of this function, $(m-1)/m$, is increasing in the number of bidders. The intercept, $v_c/m$ is decreasing in the number of bidders and increasing in the cutoff entry value.²⁸ The entry cost has already been incurred at the bidding stage. In an English clock auction, it is a sunk cost that does not affect the weakly dominant bidding strategy. In a first price auction, it is still a sunk cost, but in equilibrium, it provides information on the minimum value of entrants. As such, it affects the minimum bid independent of value.

Fig. 4

Kernel densities of bid deviations from Nash predictions by auction format and number of potential bidders

Bidding behavior relative to Nash predictions is illustrated in Fig. 4. Summary statistics are contained in Table 9. Notice that bidding in first-price auctions is bimodal, one of which represents overbidding, the other of which represents underbidding. For English clock auctions we see a substantial amount of bidding in accordance with theory, as well as some underbidding. Significantly, note that in all treatments except FP5, bidding is, on average, below Nash predictions. This result is puzzling, since there is a large literature which finds that bidders in English clock auctions learn quickly to bid their value (Harstad 1990) and that bidders in first-price auctions tend to overbid (Kagel and Levin 1993). Many possible explanations for overbidding in first-price auctions. Explanations related to the preferences of bidders include risk averse bidders (Cox et al. 1983, 1988), a joy of winning (Cox et al. 1992; Holt and Sherman 1994), and regret averse bidders (Engelbrecht-Wiggans and Katok 2007; Filiz-Ozbay and Ozbay 2007; Engelbrecht-Wiggans and Katok 2009). Explanations which relax the assumptions of Nash equilibrium include quantal response equilibrium (Goeree et al. 2002) and level-k models of bidders (Crawford and Iriberri 2007).

The purpose of this paper is not to determine which, if any, of these models best explains our data. However, it is worth noting that risk aversion, a joy of winning, or regret aversion are not able to explain both the over-entry and under-entry we observe.

Our results on bidding are surprising, and suggest a need for further research. While we are not able to conclusively offer an explanation for this behavior, We conjecture that a portion of bidders are not treating the cost of entry as a sunk cost when formulating their bids. Bidding relative to a naive model of bidding, in which a bidders behaves as though his value were $v_{it}-c_{it}$ is illustrated in Fig. 5 which contains kernel densities of bids relative to this model by group size and auction format. Notice that the densities are bimodal, and that in English clock auctions, one of these modes corresponds with this naive model of bidding. Note that such a model is not able to fully explain the underbidding we observe in first-price auctions.

Fig. 5

Kernel densities of bid deviations from naive predictions by auction format and number of potential bidders

Another factor that may contribute to the observed underbidding in first-price auctions is the relatively high level of feedback utilized in our design. In particular, subjects are informed of all observed bids at the end of each period. Neugebauer and Selten (2006) and Isaac and Walker (1985) both find that providing feedback which includes the losing bids tends to reduce bids. Further research is needed to determine the extent to which feedback drives bidding behavior in our experiment.

Table 9 Summary statistics for bidding conditional on observed entry behavior

To further analyze bidding behavior, we estimate bidding functions for each auction format via GLS and include random effects to control for individual subject variation, and cluster standard errors at the session level. The dependent variable is the observed bid.²⁹ To determine the effect of value on bids, we include $v_{it}$. We also include the observed number of bidders ($m_{it}$ and the realized entry cost ($c_{it}$) of bidder i in period t. Additionally, we include $G_{it}^5$, and interact this dummy with $v_{it}$ $m_{it}$ and $c_{it}$. In some specifications we also include additional controls. In particular we control for $Male_i$, $Age_i$, $ln(t+1)$), and $GroupOrder_i$.

Table 10 reports results for English clock and first-auctions in which there in more than one bidder.³⁰ Notice that in English clock auctions, the coefficient on $v_{it}$ is predicted to be one, and all other coefficients are predicted to be zero. However, while the coefficient on $v_{it}$ is positive and highly significant it is statistically lower than one in all specifications.³¹ Thus, bidders in English clock auctions respond positively to value, but despite it being a weakly dominated bidding strategy, they bid less than their value. Note that bidders do seem to be learning, as evidenced by an increase of the coefficient for $v_{it}$ during the second half of the experiment. However, the coefficient is still less than one. Note that he conjecture that bidders are falling victim to the sunk cost fallacy is consistent with the negative and statistically significant coefficient on $c_{it}$. This coefficient decreases during the second half, but it is still negative and marginally significant.

Table 10 Random effects estimates of the determinants of bids in auctions with more than one bidder

Table 11 Random effects estimates of the responsiveness of bids in first-price auctions to equilibrium predictions

Also counter to theory, we find that under some specifications bids are increasing in $m_{it}$ when group size is three, but that this effect is negative for a group size of five. The magnitude of these effects are small and may reflect (anti) social preferences.³²

For first price auctions, we find a positive and statistically significant effect of value, and a negative and statistically significant effect of participation cost. Both are relevant variables for Nash bidding. However, the predicated coefficients depend on the number of bidders. To facilitate the comparison on bidding behavior with Nash predictions we report additional specifications which include $\nu _{it}=v_{i}\cdot (m-1)/m$ (i.e. the slope of the Nash bidding function) in place of $v_{it}$, and $\kappa _{it}= (c_{i} / 100)^{1/n}\cdot 100/m$ (i.e. the intercept of the Nash bidding function) in place of $c_{it}$.

Table 11 contain the results of these specifications. We find that the coefficient on $\nu _{it}$ is not only positive and significant, but we are unable to reject that it is equal to one in any specification. In the first half of the experiment, the interaction of $\nu _{it}$ and groups size is negative and significant, indicating that responsiveness is less than predicted by theory when group size is five.³³ However, if we restrict attention to the second half of the experiment, the coefficient on $\nu _{it} \cdot G_{it}^5$ is not longer significant: we cannot reject that bids respond to changes in value as predicted by theory, regardless of group size. Thus, the slope of our estimated bid functions are in line with theoretical predictions.

The coefficient on $\kappa _{it}$ is positive and statistically significant when we consider all periods, but we reject the null that it is equal to one. Furthermore, when we restrict attention to the second half of the experiment, the coefficient is no longer significant. That is, the intercept of the bid function seems to be lower than what theory predicts. As such, we conclude that deviations of bidding from theory are largely driven by bidders not accounting for the information that the entry cost conveys regarding the interval of bids from which the values of other bidders are drawn.