Auctions have been the focus of much research in economics, mathematics and computer science, and have received attention in the AI and multi-agent communities as a significant tool for resource and task allocation. Beyond explicit auctions, as performed on the web (e.g., eBay) and in auction houses, auctions also model various real-life situations in which people (and machines) interact and compete for some valuable item. For example, companies advertising during the U.S. Superbowl are, in effect, bidding to be one of the few remembered by the viewer, and are thus putting in tremendous amounts of money in order to create a memorable and unique event for the viewer, overshadowing the other advertisers.
A particularly suitable auction for modeling various scenarios in the real world is the all-pay auction. In this type of auction, all participants announce bids, and all of them pay those bids, while only the highest bid wins the product. Candidates applying for a job are, in a sense, participating in such a bidding process, as they put in time and effort preparing for the job interview, while only one of them is selected for the job. This is a max-profit auction, as the auctioneer (employer, in this case), receives only the top bid. In comparison, a workplace with an “employee of the month” competition is a sum-profit auctioneer, as it enjoys the fruits of all employees’ labour, regardless of who won the competition.
The explosion in mass usage of the web has enabled many more all-pay auction-like interactions, including some involving an extremely large number of participants. For example, various crowdsourcing contests, such as the Netflix challenge, involve many participants putting in effort, with only the best performing one winning a prize. Similar efforts can be seen throughout the web, in TopCoder.com, Amazon Mechanical Turk, Bitcoin mining and other frameworks.
However, despite the research done on all-pay auctions in the past few years (DiPalantino and Vojnović, 2009; Chawla et al., 2012; Lev et al., 2013), some basic questions about all-pay auctions remain — in a full information setting, any equilibrium has bidders’ expected profit at , raising, naturally, the question of why bidders would participate.
Several extensions to the all-pay auction model have been suggested in order to answer this question. For example, Lev et al. (2013) showed that allowing bidders to collude enables the cooperating bidders to have a positive expected profit, at the expense of others bidders or the auctioneer. This paper addresses this question by suggesting a model in which the bidders have a positive expected profit.
Furthermore, in all-pay auctions, the number of bidders is a crucial information in order to bid according to the equilibrium (Baye et al., 1996). Hence, the number of participants must be known to the bidder. We suggest a relaxation of this assumption by allowing the possibility of bidders’ failure, that is, there is a probability that a bidder will not be able to participate in the auction. Therefore, we assume that the number of potential bidders and the failure probability of every bidder are common knowledge, but not the exact number of participants.
As most large-scale all-pay auction mechanisms have variable participation, we believe this helps capture a large family of scenarios, particularly for online, web-based, situations and the uncertainty they contain. We propose a symmetric equilibrium for this situation, we show when it is unique and prove its various properties. Somewhat surprisingly, allowing failures makes the expected profit for bidders positive, justifying their participation.
We start by reviewing related work in Section 2. We then introduce the model with and without failures. In Section 4, we first examine the case where each bidder has a different failure probability. Next, in Section 5, we study the potential manipulations possible in this model, such as announcing a false probability (e.g., saying that you will put all your time into a TopCoder.com project) and changing the probability of others (e.g., sabotaging their car). Finally, in Section 6, as calculations in this general case are complex, we examine situations where bidders have the same failure probability (as is possible when weather or web server failure, for example, are the main determinant of participation), enabling us to detail more information about the equilibrium in this state. In those situations we examine the effectiveness of changing the participation probability for all the bidders (e.g., convincing a deity to make it snow or attacking the server).
2 Related Work
Initial research on all-pay auction was in the political sciences, modeling lobbying activities (Hillman and Riley, 1989), but since then, much analysis (especially that dealing with the Revenue Equivalence Theorem) has been done on game-theoretic auction theory. When bidders have the same value distribution for the item, Maskin and Riley (2003) showed that there is a symmetric equilibrium in auctions where the winner is the bidder with the highest bid. A significant analysis of all-pay auctions in full information settings was Baye et al. (1996), showing (aided by Hillman and Riley (1989)) the equilibrium states in various cases of all-pay auctions, and noting that most valuations (apart from the top two), are not relevant to the winner’s strategies.
More recent work has extended the basic model. Lev et al. (2013) addressed issues of mergers and collusions, while several others directly addressed crowdsourcing models. DiPalantino and Vojnović (2009) detailed the issues stemming from needing to choose one auction from several, and Chawla et al. (2012) dealt with optimal mechanisms for crowdsourcing.
The early major work on failures in auctions was McAfee and McMillan (1987), followed soon after by Matthews (1987), which introduced bidders who are not certain of how many bidders there will actually be at the auction. Their analysis showed that in first-price auctions (like our all-pay auction), risk averse bidders prefer to know the numbers, while it is the auctioneer’s interest to hide that number. In the case of neutral bidders (such as ours), their model claimed that bidders were unaffected by the numerical knowledge. Dyer et al. (1989) claimed that experiments that allowed “contingent” bids (i.e., one submits several bids, depending on the number of actual participants) supported these results. Menezes and Monteiro (2000) presented a model where auction participants know the maximal number of bidders, but not how many will ultimately participate. However, the decision in their case was endogenous to the bidder, and therefore a reserve price has a significant effect in their model (though ultimately without change in expected revenue, in comparison to full-knowledge models). In contrast to that, our model, which assumes a little more information is available to the bidders (they know the maximal number of bidders, and the probability of failure), finds that in such a scenario, bidders are better off not having everyone show up, rather than knowing the real number of contestants appearing. Empirical work done on actual auctions (Lu and Yang, 2003) seems to support some of our theoretical findings (though not specifically in all-pay auction settings).
In our settings, the failure probabilities are public information and the failures are independent. Such failures have also been studied in other game-theoretic fields. Meir et al. (2012) studied the effects of failures in congestion games, and showed that in some cases, the failures could be beneficial to the social welfare. Some earlier work focused on agent redundancy and agent failures in cooperative games, studying various solution concepts in such games (see, e.g., (Bachrach et al., 2011, 2014)).
We consider an all-pay auction with a single auctioned item that is commonly valued by all the participants. This is a restricted case of the model in Baye et al. (1996), where players’ item valuation could be different.
Formally, we assume that each of the bidders issues a bid of , , and all bidders value the item at . The highest bidders win the item and divide it among themselves, while the rest lose their bid. Thus, bidder ’s utility from a combination of bids is given by:
We are interested in a symmetric equilibrium, which in this case, without possibility of failure, is unique (Baye et al., 1996; Maskin and Riley, 2003). It is a mixed equilibrium with full support of , so that each bidder’s bid is distributed in
according to the same cumulative distribution function, with the density function (since it is non-atomic, tie-breaking is not an issue). As we compare this case to that of no-failures, this is a case similar to that presented in Baye et al. (1996), where various results on the behavior of non-cooperative bidders have been provided. We briefly give an overview of the results without failures in Subsection 3.1.
When we allow bidders to fail, we assume that each of them has a probability of participating — . As a matter of convenience, we shall order the bidders according to their probabilities, so . If a bidder fails to participate, its utility is .
3.1 Auctions without Failures
The expected utility of any participant with a bid is:
where and are the probabilities of winning or losing the item when bidding , respectively. In a symmetric equilibrium with
players, each of the bidders chooses their bid from a single bid distribution with a probability density functionand a cumulative distribution function . A player who bids can only win if all the other players bid at most , which occurs with probability . Thus, the expected utility of a player bidding is given by:
The unique symmetric equilibrium is defined by the CDF (Baye et al., 1996). This equilibrium has full support, and all points in the support yield the same expected utility to a player, for all . Since , this means that for all bids, . The various properties of an auction without failures can be found in Table 1 (Lev et al., 2013).
|Sum-profit principal utility|
|Max-profit principal utility|
4 Every Bidder with Own Failure Probability
In this section, we assume that each bidder has its own probability for participating in the auction, with . We can assume without loss of generality, that each bidder has a positive participating probability, that is, . If this is not the case, we can remove from the auction the bidders with zero probability of participating.
4.1 Equilibrium Properties
Before we present a symmetric Nash equilibrium, we will characterize any Nash equilibrium.
In common values all-pay auction when the item value is , if then there is a unique Nash equilibrium, in which the expected profit of every participating bidder is . Furthermore, there exists a continuous function , such that when a bidder , has a positive density over an interval, they bid according to over that interval, and if then .
In common values all-pay auction when the item value is , if then in every Nash equilibrium the expected profit of every participating bidder is . At least two bidders with randomize over with each other player randomizing continuously over , , and having an atomic point at of . There exists a continuous function , such that when a bidder, , has a positive density over an interval, they bid according to over that interval. For every , the atomic point at is equals to .
4.2 Symmetric Equilibrium
We are now ready to present a symmetric Nash equilibrium, we assume that . If , from Theorem 1 it follows that the equilibrium is unique. If the equilibrium is not unique, except for two bidders with , every bidder can place an arbitrary atomic point at . In the equilibrium that we present, every bidder has an atomic point at of , and thus the equilibrium is symmetric.
In order to simplify the calculations, we add a “dummy” bidder, with index 0, and , adding a bidder that surely will not participate in the auction, does not effect the other bidders and therefore does not influence the equilibrium.
We begin by defining a few helpful functions. First, we define , and we define the following expressions for all :
For the virtual “0” index, we use . Note that because the ’s are ordered, so are the ’s: . 111An equivalent definition of is , we alternate between those two definitions.
We are now ready to define the CDFs for our equilibrium, for every bidder :
, uniquely, while it is very similar to in its piecewise composition, has an atomic point at of , so:
Note that all CDFs are continuous and piecewise differentiable,222Note that when , and is undefined for some , then there is no range for which that is used. and when it follows that ; therefore, this is a symmetric equilibrium. The intuition behind this equilibrium is that bidders that participate rarely will usually bid high, while those that frequently participate in auctions with less competition would more commonly bid low.
In the course of proving this is indeed a equilibrium, we shall calculate the expected utility of the bidders when they participate.
When bidder bids according to this distribution, i.e., for :
If bidder bids outside their support, i.e., for , the same equation becomes:
and hence . Plugging it all together,
Finally, as , , hence , and therefore . ∎
When a bidder actually participates their expected utility, in the equilibrium, is , and therefore the overall expected utility for bidder is (which, naturally, decreases with ). Notice that, as is to be expected, a bidder’s profit rises the less reliable their fellow bidders are, or the fewer participants the auction has. However, the most reliable of the bidders does not affect the profits of the rest. If a bidder can set its own participation rate, if there is no bidder with , that is the best strategy; otherwise, the optimal probability should be , as that maximizes .
4.3.1 Expected Bid
In order to calculate the expected bid by each bidder, we need to calculate the bidders’ equilibrium PDF, for :
and . In the equilibrium, the expected bid of bidder , for is:
For every :
The expected bid decreases with , indicating, as in the no-failure model, that as more bidders participate, the chance of losing increases, causing bidders to lower their exposure. The proof of Theorem 4 can be found at the appendix.
4.3.2 Auctioneer - Sum-Profit Model
In the equilibrium, the expected profit of the auctioneer in the sum-profit model is given by:
When summing over all bidders, we receive a much simpler expression.
The sum-profit auctioneer’s equilibrium profits are:
In this case, growth with is monotonic increasing, and hence, any addition to is a net positive for the sum-profit auctioneer. The proof can be found at the appendix.
4.3.3 Auctioneer - Max-Profit Model
To calculate a max-profit auctioneer’s profits, we need to first define the max-profit auctioneer’s profits equilibrium CDF:
This is differentiable, and hence we can find and the max-profit auctioneer’s expected profit.
In the equilibrium, the max-profit auctioneer’s profits are:
From Theorem 6 we can see that the max-profit auctioneer would prefer to minimize , have two reliable bidders (), and the other bidders as unreliable as possible. The proof of Theorem 6 can be found at the appendix.
Consider how four bidders interact. Our bidders have participation probability of , , and . Let us look at each bidder’s equilibrium CDFs:
A graphical illustration of the bidders’ CDFs and PDFs can be found in the appendix. The expected utility for bidder 1 is , for expected bid of ; for bidder 2, for expected bid of ; for bidder 3, for expected bid of ; and for the last bidder, for expected bid of .
A sum-profit auctioneer will see an expected profit of , while a max-profit one will get, in expectation, .
As a comparison, in the case where we do not allow failures, the CDF of the bidders is with expected bid of and expected utility of . The expected profit of the sum-profit auctioneer is , while the expected profit of the max-profit auctioneer is .
5 False Identity and Sabotage
Now, suppose our bidder can influence others’ perceptions, and create a false sense of its participation probability. What would its best strategy be, and how should the participation probability be altered? Any bid beyond is sure to win, but as that would give profit of less than , which is less than the expected profit for non-manipulators, it is not worthwhile. Therefore, our bidder will bid in its support, with the expected profit being . However, our bidder may increase its expected profit by trying to portray its participation probability as being as low as possible, thus lulling the other bidders with a false sense of security. Of course, this reduces the payment to auctioneers of any type, and therefore, they would try to expose such manipulation.
More interesting is the possibility of a player’s changing another player’s participation probability by using sabotage; thus our bidder would be the only bidder knowing the real participation probability. Our bidder, , sabotages bidder , with a perceived participation probability of , changing its real participation probability to . Bidder ’s expected profit with bid is:
The values of this function change according to the relation between , and . To find the optimal strategy for a bidder, we must examine all the options.
Let be the announced participation probabilities, and let be bidder real participation probability. For every , Algorithm 1 finds the optimal bid for bidder .
To summarize, bidder best interest is to bid in the intersection of its support and bidder ’s support. Given the index of the saboteur bidder, the index of the sabotaged bidder, and the participation probability after the sabotage, Algorithm 1 finds the optimal bid. The full proof can be found at the appendix.
6 Uniform Failure Probabilities
If we allow our bidders to have the same probability of failure (e.g., when failures stem from weather conditions), many of the calculations become more tractable, and we are able to further understand the scenario.
As this case is a particular instance of the general case presented above, we can calculate the expected equilibrium bid of every bidder and its variance.
The expected equilibrium bid of every bidder is:
and the variance of the bid is:
The expected bid and the variance are neither monotonic in nor in .
We are now ready to examine the profits of all the parties, the bidder and the auctioneer, both in the sum-profit model and the max-profit model.
From the general case we may deduce that expected equilibrium profit of every bidder is . Note the profit decreases as increases, and is maximized when . We can now compute the variance of bidder profit.
The variance of the bidder equilibrium profit is:
And the variance is monotonic increasing in .
6.2.2 Auctioneer - Sum-Profit Model
The expected bid of every bidder is , therefore, the expected profit of the sum-profit auctioneer, in the equilibrium, is:
which increases with and . Therefore, the auctioneer best interest is to have as many bidders as possible. Note that as grows, the auctioneer’s expected revenue approaches that of the no-failure case. From Theorem 9 we get the variance of the auctioneer equilibrium profit in the sum profit model:
6.2.3 Auctioneer - Max-Profit Model
For the max-profit auctioneer, the expected profit in equilibrium is:
which is, monotonically increasing in and (for ); for large enough it approaches the expected revenue in the no-failure case.
The variance of the auctioneer equilibrium profit in the max profit model is:
7 Conclusion and Discussion
Bidders failing to participate in auctions happen commonly, as people choose to apply to one job but not another, or to participate in the Netflix challenge but not a similar challenge offered by a competitor. Examining these scenarios enables us to understand certain fundamental issues in all-pay auctions. In the complete reliability, classic versions, each bidder has an expected revenue of . In contrast, in a limited reliability scenario, such as the one we dealt with, bidders have positive expected revenue, and are incentivized to participate in the auction. Auctioneers, on the other hand, mostly lose their strong control of the auction, and no longer pocket almost all revenues involved in the auction. However, by influencing participation probabilities, max-profit auctioneers can effectively increase their revenue in comparison to the no-failure model.
The idea of the equilibrium we explored was that frequent participants could allow themselves to bid lower, as there would be plenty of contests where they would be one of the few participants, and hence win with smaller bids. Infrequent bidders, on the other hand, would wish to maximize the few times they participate, and therefore bid fairly high bids. As exists in the no-failure case as well, as more and more participants join, there is a concentration of bids at lower price points, as bidders are more afraid of the fierce competition. Hence, it is fairly easy to see in all of our results that as approached larger numbers, the various variables were closer and closer to their no-failure brethren.
There is still much left to explore in these models — not only more techniques of manipulation by bidders and potential incentives by auctioneers, but also further enrichment of the model. Currently, participation rates are not influenced by other bidders’ probability of participation, but, obviously, many scenarios in real-life have, effectively, a feedback loop in this regard. We assumed that the item is commonly valued by all the bidders and the cost of effort is common, which is not always the case. A future research could examine a more realistic model with heterogeneous costs or valuations. In our model the failure happened before the bidder placed their bid, but in other models the failure could happen after the bidders place their bid and before the auctioneer collected the bid. Finding a suitable model for such interactions, while an ambitious goal, might help us gain even further insight into these types of interactions.
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Proof of Theorem 7
Let be the announced participation probabilities, and let be bidder real participation probability. For every , bidder ’s expected profit from bid is:
The values of this function change according to the relation between , and . The following lemmata offer some insights regarding the optimal strategy for a bidder.
Let be the announced participation probabilities, and let be bidder real participation probability. For every , the expected profit of bidder from bidding , for and , is:
The expected profit of bidder from bidding , for and , is:
In this case, this is larger than , and grows with the bid, though the maximal bid (due to the fact that ) is .
When either or , bidder can bid in the support of both bidders.
Let be the announced participation probabilities, and let be bidder real participation probability. For every , the expected profit of bidder from bidding , such that is:
The expected profit of bidder from bidding , such that is:
Since , this means that , hence ; again, this is an increase over , the position without sabotaging.
Let be the announced participation probabilities, and let be bidder real participation probability. For every and for every , there exists , such that .
When , bidder can bid in bidder ’s support and not in its own support, i.e., bidder can bid for and . We first note that bidder can bid in its support with expected utility of:
When is in bidder ’s support and not in bidder ’s support:
If then monotonic grows with , and maximized when ; While if , differentiating twice with respect to gives:
that is, for every : , hence, the extremum is a minimum point and the utility maximized when either or .
When bidding the expected utility is:
If then .
Otherwise, if , since it holds that , and therefore and
Similarly, . That is, for every : . Therefore, bidder