Strategic behavior and learning in all-pay auctions: an empirical study using crowdsourced data

Abstract

We analyze human behavior in crowdsourcing contests using an all-pay auction model where all participants exert effort, but only the highest bidder receives the reward. We let workers sourced from Amazon Mechanical Turk participate in an all-pay auction, and contrast the game theoretic equilibrium with the choices of the humans participants. We examine how people competing in the contest learn and adapt their bids, comparing their behavior to well-established online learning algorithms in a novel approach to quantifying the performance of humans as learners. For the crowdsourcing contest designer, our results show that a bimodal distribution of effort should be expected, with some very high effort and some very low effort, and that humans have a tendency to overbid. Our results suggest that humans are weak learners in this setting, so it may be important to educate participants about the strategic implications of crowdsourcing contests.

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Notes

  1. 1.

    www.topcoder.com, www.codechef.com.

  2. 2.

    Empirical results do not always exhibit overbidding. Some studies examining all-pay auctions between two players find no overbidding [16, 32], while studies examining auctions between more than two players find significant overbidding [11, 22] (recent work on this contains a more complete discussion [20]). Further, the degree of overbidding depends on the specifics of the contest and domain [7, 12, 19, 25,26,27].

  3. 3.

    We also allowed players to play against a specific friend, but such games were not used for the analysis in this paper.

  4. 4.

    We note that players who follow the mixed-strategy of the Nash equilibrium are extremely unlikely to be labeled as spammers. The symmetric mixed-strategy under the Nash equilibrium is choosing a bid uniformly at random over the range, making each possible strategy have a very small probability. In particular, for a player who selects bids uniformly at random, the probability of selecting the specific precluded bids in over 25% of the games is very low. This means that our spammer-detection rule is very unlikely to have a “false-positive”, and mistakenly labeling a player who is using the Nash mixed-strategy as a spammer.

  5. 5.

    Notably, such collusive strategies are predicted by a different game theoretic analysis of this as a repeated game setting, showing another way human behavior differs from idealized mathematical models in crowdsourcing contests.

  6. 6.

    Note that the uniform distribution over the bids gives an expected number of distinct bids of 70. Shortly, we provide two more precise comparisons with the uniform distribution.

  7. 7.

    In particular, such settings include auctions, especially when the winner is simply the highest bidder, rather than ones where the probability of winning is higher for the highest bidder [11], and Blotto games [10, 23].

  8. 8.

    A player who lost the auction can also lower their bid in an attempt to improve the negative utility they incur, but achieving a positive utility requires increasing the bids so as to win the auction.

  9. 9.

    We have also tried using only the bid difference \(d_{t-1} = x^{bid}_{t-1} - y^{bid}_{t-1}\) as a feature (i.e. using the two features \(W_{t-1}, d_{t-1}\), which achieves \(R^2=0.34\), still lower than the baseline model.

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Correspondence to Joel Oren.

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Joel Oren: The project was completed while all authors were employed by Microsoft Research, Cambridge, UK.

Appendix: Including spammers: revised results

Appendix: Including spammers: revised results

In this appendix section, we revisit our findings without the exclusion of the so-called Spammers (the 85 players whose at least 25% of their bids were taken from the set \(\{0, 1, 1000, 9999\}\)). Recall that in total, there are 340 players who played a total of 11, 327 games. The average revenue over all games, was 11, 832 (previously 13, 730).

Figure 7 shows the average player bid distributions, which is quite close to that in the original analysis (though of course including more “spam” bids).

Fig. 7
figure7

Average player distribution of bids (with spamming users)

Figure 8 shows the average bid per time period (and again, the results are similar to those occuring with spammers, although unsuprisingly there are more players who do not adjust their bid when including spammers, as these are by definition players who use the same bids frequently).

Fig. 8
figure8

Average bid per time period

Figures 9 and 10 show key clusters when including spammers in the clustering analysis. Again, the results are qualitatively similar to the original analysis.

Fig. 9
figure9

Cluster 1 (64 players) player bid distributions

Fig. 10
figure10

Cluster 2 (43 players) player bid distributions

Fig. 11
figure11

Average utility against the empirical bid distribution

Figure 11 shows the average utility of bids against the empirical bid distribution. The figure is very similar to that in the original analysis. In other words, although best-responses against “spam” bids do well against the bids taken only from spammers, the overall best responses (when reacting to the general population of all players, including both spammers and non-spammers) are very similar to what we found in the original analysis. Figure 12 shows the utility distribution of players, which is again very similar to the distribution found in our original analysis.

Fig. 12
figure12

Utility distribution of players

To conclude, repeating the analysis carried in the main paper which not filtering our players who frequently use the same “focal-bids” (but do filtering our uses who used fake profiles) does not yield significantly different results. This indicates that our results are relatively robust to our choice of mechanism for eliminating spammers.

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Bachrach, Y., Kash, I.A., Key, P. et al. Strategic behavior and learning in all-pay auctions: an empirical study using crowdsourced data. Auton Agent Multi-Agent Syst 33, 192–215 (2019). https://doi.org/10.1007/s10458-019-09402-4

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Keywords

  • All-pay auctions
  • Crowdsourcing contests
  • Learning