Spying and imperfect commitment in first-price auctions: a case of tacit collusion

Fan, Cuihong; Jun, Byoung Heon; Wolfstetter, Elmar G. Article — Published Version Spying and imperfect commitment in first-price auctions: a case of tacit collusion Economic Theory Bulletin Provided in Cooperation with: Springer Nature Suggested Citation: Fan, Cuihong; Jun, Byoung Heon; Wolfstetter, Elmar G. (2023) : Spying and imperfect commitment in first-price auctions: a case of tacit collusion, Economic Theory Bulletin, ISSN 2196-1093, Springer International Publishing, Cham, Vol. 11, Iss. 2, pp. 255-275, https://doi.org/10.1007/s40505-023-00257-3 This Version is available at: https://hdl.handle.net/10419/313195 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Economic Theory Bulletin (2023) 11:255–275 https://doi.org/10.1007/s40505-023-00257-3 RESEARCH ARTICLE Spying and imperfect commitment in first-price auctions: a case of tacit collusion Cuihong Fan1·Byoung Heon Jun2·Elmar G. Wolfstetter3 Received: 25 May 2023 / Accepted: 31 July 2023 / Published online: 28 August 2023 © The Author(s) 2023 Abstract We analyze Stackelberg leadership in a first-price auction. Leadership is induced by an information system, represented by a spy, that leaks one bidder’s bid before others choose their bids. However, the leader may secretly revise his bid with some probability; therefore, the leaked bid is only an imperfect signal. Whereas leadership with perfect commitment exclusively benefits the follower, imperfect commitment yields a collusive outcome, even if the likelihood that the leader may revise his bid is arbitrarily small. This collusive impact shows up in all equilibria and is strongest in the unique pooling equilibrium which is also payoff dominant. Keywords Auctions ·Tacit collusion ·Espionage ·Second-mover advantage · Signaling ·Incomplete information JEL Classification L12 ·L13 ·L41 ·D43 ·D44 ·D82 Research support by the National Natural Science Foundation of China (Grant: 72171140), the Humanities and Social Sciences Research Foundation of the Ministry of Education of China (Grant:19YJA790009), and Korea University (Grant: K1919021) is gratefully acknowledged. We thank seminar participants at the Universidad de Los Andes (Buenos Aires) and the Academia Sinica (Taiwan) and in particular an anonymous referee whose comments and suggestions helped us significantly to improve our paper. BElmar G. Wolfstetter elmar[email protected] Cuihong Fan [email protected] Byoung Heon Jun [email protected] 1School of Economics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China 2Department of Economics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 136-701, Korea 3School of Business and Economics, Humboldt-University at Berlin, Spandauer Str. 1, Berlin 10178, Germany 123 256 C. Fan et al. 1 Introduction The present paper studies Stackelberg leadership in a first-price auction subject to incomplete information and imperfect commitment. There, sequential bidding is induced by an information system, represented by a spy, who observes one bidder’s bid, albeit imperfectly, and reports it to a rival bidder before the latter submits his own bid. The presence of a spy is common knowledge. Therefore, bidders know that the spied-at bidder is Stackelberg leader and the bidder served by the spy is follower. However, the spied-at bidder is able to secretly revise his bid with some commonly known probability. Unlike in a usual Stackelberg game, leadership and the leader’s commitment to a particular choice of action is not by that bidder’s intent. Instead, it is an inescapable consequence of the presence of the spy, and the thus induced commitment is imperfect because the leader may be able to secretly revise his bid. Our analysis begins with two special cases which serve as benchmarks: the case when the spy’s report is perfectly informative and the case when the spy’s signal is uninformative. Not surprisingly, if the spy’s report is a perfect signal, spying benefits only the follower; whereas, if the signal is uninformative, the presence of the spy is inconsequential and one obtains the equilibrium of the first-price auction without spying. As the leader is able to secretly revise his bid with some probability, the follower cannot be sure that the leaked bid is the leader’s true bid. This ambiguity gives rise to a peculiar signaling game where both sender and receiver of messages have private information, beliefs are multi-dimensional, and the sender has a chance to secretly revise his action with some probability. That game admits a unique pooling equilibrium and multiple partially separating equilibria. There, spying benefits both the leader and the follower, even if the likelihood that the leader is able to secretly revise his bid is arbitrarily small. This collusive impact shows up in all equilibria and is strongest in the pooling equilibrium which is also the payoff dominant equilibrium. Therefore, our analysis makes a strong case for tacit collusion. Spying out rivals’ bids or prices is abundant in competitive environments. The incentive for spying is particularly strong in a first-price auction where the winnertakes-all and the winner has to pay his bid, which drives bidders to strategically shade their bids. Generally such bid shading leads to ex post regret, either because the winner could have lowered his bid and yet won or some loser could have raised his bid and won while making a positive profit, which could have been avoided if he had known rivals’ bids. Although spying is intrinsically a secret operation, evidence of spying has surfaced on numerous occasions. For example, bidding for the construction of a new metropolitan airport in Berlin was reopened after investigators found out that Hochtief AG,the winner of the auction, had illegally acquired the application documents of the rival bidder IVG. Similarly, in 1996 Siemens AG was excluded from all public procurements in Singapore for a period of five years after the authorities determined that Siemens had acquired information about rival bids for a major power station construction project. 123 Spying and imperfect commitment in first-price auctions… 257 Circumstantial evidence of bid leakage abounds. Andreyanov et al. (2017) observed that in first-price procurement auctions a bidder is likely to have observed leaked bids if that bidder bid last, close to the deadline, and if, conditional on winning, his bid was close to the runner-up. Based on this pattern they suspect widespread bid leakage in at least 10% of a large sample of 4.3 million procurement auctions that took place in Russia between 2011 and 2016. Using weaker indicators and more sophisticated techniques, Ivanov and Nesterov (2020) confirmed these findings based on another data set of 600,000 Russian procurement auctions that took place between 2014 and 2018 and estimated that the outcome was influenced by leaked bids in around 9% of these auctions. Typically, the spy who leaks a rival’s bid is a “mole” or trusted insider who is driven by financial motives, or takes revenge for unfair treatment as an employee, or because he has been blackmailed into handing over sensitive information. Sometimes a gullible staff member is lured into passing over inconsequential information and, after having committed a minor offense, is blackmailed into leaking sensitive information. However, the spy may also be a corrupt agent auctioneer who has access to early bids or, for that matter, a “malware tool” that exploits vulnerabilities in computer software to transmit prospective or actual bids prepared on a PC or submitted online. The techniques used by spies range from low-key activities such as searching through wastebaskets, gaining access to unattended PCs and laptops, planting sophisticated malware that is able to secretly switch on cameras or recording devices of computers and mobile phones,1to participation in the bid preparation by a mole. Spying may also trigger counter-spying. When the identity of a spy has been exposed, the spy may find himself between “Scylla and Charybdis”, faced with the agonizing choice between either punishment or being “doubled”, and, after being doubled, serves the spied-at party by leaking distorted information.2 Our analysis relates to various strands of the literature. Xu and Ligett (2018) analyze the impact of commitment in first-price auctions, assuming a Stackelberg game where, in the first stage, one player publicly commits to a mixed bidding strategy, and, in the second stage, after having observed that strategy, all others submit their bids simultaneously. Focusing on bidding games under complete information, they show that in the subgame perfect equilibrium both first- and secondmovers benefit in expectation.3Interestingly, all bidders benefit equally so that they do not care who is selected as first-mover. Their approach differs from ours in several regards: whereas they assume that bidders observe the mixed bidding strategy of the first-mover, but not the bid drawn from that strategy, we assume that bidders observe the first-mover’s bid, and whereas they assume perfect commitment, we assume that the first-mover may secretly revise his bid with some probability. Moreover, whereas they assume that one player proactively commits to a mixed strategy, our analysis assumes that commitment is simply an inescapable consequence of the activity of a spy who reveals the bid, albeit imperfectly. 1For a detailed account of the many ways in which smartphone may be misused for spying see Farrow (2022). 2For a collection of case studies of the different kinds and techniques of economic espionage and counterespionage see Nasheri (2005) and Andrew (2019). 3The authors also address games of incomplete information which however yield weaker results. 123 258 C. Fan et al. In another related contribution, Fischer et al. (2021) compare the effects of spying in first- and second-price auctions, using the framework of Arozamena and Weinschelbaum (2009), where a corrupt auctioneer leaks a rival’s bid to a favored bidder. However, while Arozamena and Weinschelbaum (2009) focus on first-price auctions, and show that the impact of leaked information on the behavior of the spied-at bidder is driven by the curvature of the probability distribution of valuations,4the authors assume a uniform distribution which implies that the spied-at bidder’s bidding is not affected by spying, and otherwise focus on the role of behavioral assumptions in first- and second-price auctions and their testing in controlled lab experiments. One limitation of their approach is the implicit assumption that the spy faithfully reports the bid of the spied-at bidder. This is where the present paper steps in. The distinct feature of our analysis is that we take into account that the spied-at bidder may be able to secretly revise his bid which in turn induces him to convey strategically distorted information. In the literature, spying on rivals’ bids has also been analyzed in the context of corruption, where a dishonest agent auctioneer either allows a predetermined favored bidder to adjust his bid after reporting rivals’ bids to him (as, for example, in Burguet and Perry 2007; Arozamena and Weinschelbaum 2009) or flexibly seeks a deal with the bidder who gains the most by either lowering or increasing his bid (as in Lengwiler and Wolfstetter 2010).5 Spying out a rival’s choice of action has also been analyzed in the context of Bertrand market games with differentiated products and incomplete information about firms’ unit cost (Fan et al. 2022,2023a), and in entry deterrence games (Solan and Yariv 2004; Barrachina et al. 2014,2021). Spying out a rival’s type is analyzed in Wang (2020). Altogether, spying on a rival’s type tends to increase competition and benefits consumers, whereas spying on a rival’s actions tends to support higher prices. The plan of the paper is as follows: In Sect.2we state the binary base model. Section3summarizes two benchmark games: the game with perfectly informative and with uninformative signals. In Sect.4we solve the game with imperfect commitment that yields imperfectly informative signals and fully characterize the unique pooling equilibrium and the family of partially separating equilibria. Section5summarizes the collusive impact of spying under imperfect commitment on players’ payoffs and offers an intuitive interpretation. In Sect.6we explain why both kinds of equilibria are not driven by unreasonable off-equilibrium beliefs and show that the unique pooling equilibrium is payoff dominant. In Sect. 7.1 we characterize alternative partially separating equilibria and in Sect.7.2 generalize our parsimonious binary model to more than two states and, in both extensions, show that essential results are preserved. The paper closes in Sect.8. Several long proofs are relegated to the downloadable Online- Appendix Fan et al. (2023b) (which can also been obtained upon request from the authors). 4Specifically, they find that the spied-at bidder’s behavior is not affected if the inverse of the reverse hazard rate function is linear (which occurs for the uniform distribution), whereas that bidder bids more/less aggressively if the inverse of the reverse hazard rate function is strictly convex/concave. 5If the spread between the two highest bids is sufficiently large, it is most profitable to let the highest bidder match the second highest bid, whereas if that spread is sufficiently small, due to bid shading it is most profitable to let the second highest bidder match the highest bid. 123 Spying and imperfect commitment in first-price auctions… 259 2Model Consider a first-price auction with two risk neutral bidders, 1 and 2, who compete for buying one unit of a valuable good. Each bidder knows his own valuation for that good but not that of the other. Bidder 2 has access to an information system, represented by a spy, who observes the bid of bidder 1, albeit imperfectly, before bidder 2 submits his bid, and this fact is common knowledge. The presence of a spy induces a sequential game where the spied-at bidder 1 is Stackelberg leader and bidder 2 is follower. However, bidder 1 is able to secretly revise his bid with some commonly known probability, and in that event makes two bids: the first-round bid observed and reported by the spy and the secretly revised (true) bid. Therefore, when bidder 2 receives the spy’s message he does not know whether he has learned the true bid or just a decoy. To keep the initial analysis as simple as possible, bidders’ values, Vi∈{0,v},v>0, are binary random variables that are independently drawn with positive probabilities. This assumption is generalized in Sect.7.2. We refer to the bidder 1 who is committed to stick to his bid and cannot revise it as type c(mnemonic for “committed”) and the bidder 1 who can secretly revise his bid as type n(mnemonic for “not committed”), and to bidders with value vas type hand with value zero as type (mnemonic for “high” and “low”). Therefore, the type set of bidder 1 is T1={n, nh,c, ch}and that of bidder 2 is T2={, h}. The time-line is as follows: 1. Nature independently draws bidders’ types, (t1,t2)∈T1×T2, and each bidder privately observes his own type. 2. Bidder 1 chooses his first-round bid, b1, which the spy reports to bidder 2. That bid is the true submitted bid if bidder 1 is type cand a decoy if he is type n. 3. Bidder 2 submits his bid, b2, and bidder 1 type nrevises and submits his (true) second-round bid, br 1(mnemonic for “revised bid”). 4. The auctioneer selects the winner based on true bids: (b1,b2)if bidder 1 is type c, and (br 1,b2)if bidder 1 is type n, and collects payments according to the rules of the first-price auction. Note that it does not matter when bidder 1 type nliterally revises his bid; what matters is that bidder 2 and bidder 1 type nchoose their (true) bids “simultaneously”. Bidders’ prior probability of drawing the high value, v, is equal to θ∈(0,1)and that of bidder 1 being type nis μ∈(0,1). Values V1and types of bidder 1 are stochastically independent and the prior probabilities of T1are (Pr{nh},Pr{n})=μ(θ,1−θ), (Pr{ch},Pr{c})=(1−μ) (θ,1−θ). We denote bids (actions) by the letters b,br, and pure bidding strategies that map bidders’ values into bids by functions β, and mixed bidding strategies that prescribe ac.d.f. of bids by functions Fand G. As in other discontinuous games we need to invoke particular tie-breaking or sharing rules to assure existence of equilibrium. Simon and Zame (1990, p. 861) argue convincingly that “…payoffs should be viewed as only partially determined, and that whenever the economic nature of the problem leads to indeterminacies, the sharing 123 260 C. Fan et al. rule should be determined endogenously, i.e., as part of the solution to the model rather than as part of the description of the model.” We follow this advice and carefully design tie-breaking rules that assure existence of equilibrium and consider them as part of the equilibrium. Unless stated otherwise, we apply the following type-dependent tie-breaking rule6: Tie-breaking rule (T): In the event of a tie the item is awarded to the bidder with the higher value and, if this fails to break the tie, the item is awarded to bidder 2. We stress that in order to assure existence of equilibrium we need a tie-breaking rule that thus favors bidder 2 (unless one invokes cumbersome discrete bids with a smallest monetary unit). Without it, there would be no best-reply of bidder 2. Note, however, that thus favoring bidder 2 is least favorable for explaining a collusive outcome where even the spied-at bidder benefits from spying. It is not surprising that the spying bidder 2 benefits from spying; the more challenging part is to explain that the spied-at bidder 1 benefits as well. 3 Benchmark games We first consider two benchmark games: the standard first-price auction without spy, followed by the game with spy and a perfectly committed bidder 1. These models can be viewed as special cases of our model when the spy’s message is either uninformative because bidder 1 can revise his bid with probability one (μ=1), or perfectly informative because bidder 1 cannot revise his bid with probability one (μ=0). 3.1 Benchmark without spy (A) The benchmark bidding game without spy is a symmetric simultaneous moves game with the reduced type sets T1=T2={, h}; it is equivalent to a game with spy whose message is uninformative. It has a unique symmetric equilibrium: bidders type bid zero and bidders type hplay the mixed bidding strategy F(b)(see Maskin and Riley 1985, Sect. I): β(0)=0,F(b)=(1−θ)b θ(v −b),b∈[0,b∗],b∗:=θv, F(b∗)=1.(1) Randomization is essential because if a bidder type hcould be predicted to submit a particular bid less than θv with certainty, the other bidder type hwould match this bid if he is bidder 2 or outbid him if he is bidder 1 and win for sure. Of course, bidding more than θv is dominated by bidding 0. If bidder 1 type hbids θv, then the best response of bidder 2 type his either 0 or θv to neither of which the bid of bidder 1 type his a best response. 6This rule can be implemented without knowing bidders’ valuations. One way is to endow bidders with a voucher of small value with the proviso that: (1) the voucher can either be used as a bid (in which case it is paid to the seller regardless of winning or losing) or traded-in to the seller for money, and (2) a voucher bid wins against a bid equal to zero, and a positive bid wins against a voucher bid. 123 Spying and imperfect commitment in first-price auctions… 261 The equilibrium outcome is efficient and bidders’ interim equilibrium expected payoffs,7πA(t), t∈{, h}, their ex ante equilibrium expected payoff, A, and the seller’s expected revenue, A 0,are: πA() =0,π A(h)=(1−θ)v, A=θ(1−θ)v (2) A 0=v1−(1−θ)2−2A=θ2v. (3) 3.2 Benchmark game with perfect commitment (St) The benchmark game with spy but with perfect commitment is a Stackelberg game with reduced type sets T1={c, ch},T2={, h}where the spy’s message is perfectly informative. The perfect equilibrium of that bidding game is: bidder 1 bids zero, regardless of his type and bidder 2 matches the bid of bidder 1 unless it exceeds his own value: β1(V1)=0,β 2(V2,b1)=min {V2,b1},V1,V2∈{0,v}.(4) In equilibrium bidder 1 wins if and only if V1=vand V2=0; otherwise, bidder 2 wins. In either case, the highest bid is equal to zero and the equilibrium outcome is efficient. Therefore, Proposition 1 The equilibrium outcome of game St exhibits a strong second-mover advantage; the spying bidder 2 fully extracts the seller’s payoff while the payoff of the spied-at bidder 1 is not affected by spying: πSt 1(c) =πSt 2() =πA(), πSt 1(ch)=πA(h), πSt 2(h)=v>π A(h)(5) St 2=θ(1−θ) +θ2v=A+A 0, St 0=0, St 1=θ(1−θ)v =A. (6) Only the spying bidder 2 gains. Spying does not yield a collusive outcome for which all bidders would have to gain, at the expense of the seller. The intuition is straightforward: It is optimal for bidder 2 to match the bid of bidder 1 (unless it exceeds his value); therefore, bidder 1 type hhas only a chance to win if he faces bidder 2 type who must bid zero, and in that event, due to tie-rule T, bidding zero makes bidder 1 type hwin and earn the greatest possible payoff equal to v. Therefore, bidding zero is a best response to the strategy of bidder 2 and vice versa. 4 The game with imperfect commitment We now assume that bidder 1 is type n(able to secretly revise his bid) with probability μ∈(0,1). 7Throughout this paper interim equilibrium expected payoffs are defined as function of bidders’ type t. 123 262 C. Fan et al. After observing the first-round bid, b1, reported by the spy, bidder 2 updates his beliefs about the type of bidder 1. His posterior beliefs are denoted by Pr{t1|b1}, t1∈T1={n, nh,c, ch}. The fact that bidder 1 may revise his bid with some probability gives rise to a non-standard signaling game where both sender and receiver of messages have private information and the sender has a chance to make an unobserved move. It admits pooling and partially separating perfect equilibria. 4.1 Pooling equilibrium In a pooling equilibrium all types of bidder 1 submit the same first-round bid; consequently, the first-round bid is uninformative. Because bidder 1 type must bid zero, it follows naturally that the equilibrium first-round bid of bidder 1 is equal to zero, regardless of his type. After observing the equilibrium first-round bid, in the second round the players are bidder 1 type nand bidder 2. For bidders type hrandomization is essential for the same reason that explains randomization in benchmark game A. The main difference is that the mixed strategy of bidder 2 now has a mass point at zero because bidder 2 can thus take advantage of the fact that bidder 1 is bound by his zero bid if he is type c, together with the fact that in this event bidder 2 type hwins with a zero bid because of the way in which the tie-rule favors bidder 2. An immediate implication is that the mixed strategy of bidder 1 type nh cannot also have a mass-point at zero, because if it did, bidder 1 type nh could increase his expected payoff by moving that probability mass upwards to a bid slightly greater than zero. The equilibrium is supported by a non-restrictive variety of off-equilibrium beliefs and, given these beliefs, the pooling equilibrium turns out to be unique. Applying the concept of a perfect equilibrium we obtain the following unique pooling equilibrium: Proposition 2 (Pooling equilibrium) First-round bids: Bidder 1 bids zero regardless of his type. Second-round bids: Bidder 1 type nbids zero; type nh plays the mixed strategy: F1(b)=(1−θμ)b θμ(v −b),b∈[0,¯ b],¯ b=μθv, F1(¯ b)=1.(7) Bidder 2 type bids zero; type h plays the mixed strategy, F2,ifb 1=0has been observed: F2(b)=F2(0)+(1−θμ)b θ(v −b),b∈[0,¯ b],F2(0)=1−μ, F2(¯ b)=1,(8) and the equilibrium strategy of benchmark game A, F(b),ifb 1>0has been observed. Posterior beliefs: If the spy reported the first-round bid b1=0prior beliefs are confirmed: Pr{t1|b1}=Pr{t1}for all t1∈T1; if he reported an off-equilibrium bid, 123 Spying and imperfect commitment in first-price auctions… 269 Fig. 2 FSD relationships: pooling (solid), partially separating (dashed) bidder, imperfectly informative spying does yield a collusive outcome even though the assumed tie-rule favors the spying bidder 2. That collusive impact shows up in all equilibria of the game with imperfect commitment. However, the pooling equilibrium has the strongest collusive impact. Among all partially separating equilibria (including the alternative equilibria characterized in Sect.7.1), the collusive impact is diminishing in qand, as qis diminished and approaches zero, the equilibrium expected payoffs approach those of the unique pooling equilibrium: ∂qS i<0,i∈{1,2},∂ qS 0>0,and lim q→0S i=P i,i∈{0,1,2}.(24) The collusive impact occurs even if the probability that the spy is able to secretly revise his bid is arbitrarily small, as long as that probability is positive. The collusive impact of spying has a simple intuitive interpretation. In the pooling equilibrium bidder 2 observes a zero bid with probability one. He knows that bidder 1 cannot revise his bid with positive probability; he also knows that, if he is type h, he will win against a zero bid by bidder 1 because the tie-rule favors him if he also bids zero. He takes advantage of this by bidding zero with sufficiently high probability. This benefits the spying bidder 2. Moreover, as μis decreased, starting from the benchmark case of no spying, μ=1, the entire probability distribution of bids by bidder 2 type hshifts upwards, because ∂μF2(b;μ) =−v/v−b<0. Therefore, there is a first-order stochastic dominance (FSD) relationship (with strict inequality if F2(b;μ)<1): μ>μ⇒F2(b;μ)≤F2(b;μ). (25) In other words, as μis decreased, bidder 2 type hbids less and less aggressively which benefits bidder 1 type nh and does not affect the other types of bidder 1. Similarly, the family of partially separating equilibria exhibits the same first-order stochastic dominance relationship, regardless of whether bidder 2 observes either b1= 123 270 C. Fan et al. 0orb1=b∗, because: ∂μF2(b;μ, q)=−v(1−θq) /(v−b)(1−θq(1−μ))2<0.This explains why bidder 1 benefits from spying also in all partially separating equilibria (including the alternative equilibria characterized in Sect.7.1). Moreover, F2is decreasing in q:∂qF2(b;μ, q)=−vθ(1−μ)μ /(v−b)(1−θq(1−μ))2<0,and as qgoes to zero, the F2 function converges to that of the pooling equilibrium. Therefore, for all q∈(0,¯q] bidder 2 type hbids more aggressively than in the pooling equilibrium. This also explains why the collusive impact of spying is the strongest in the unique pooling equilibrium. These results are illustrated in Fig.2. There the solid curves plot the F2functions that solve the unique pooling equilibrium and the dotted curves plot the F2functions that solve the corresponding partially separating equilibrium.19 As μis decreased, starting from the benchmark case without spy, μ=1, bidder 2 bids less and less aggressively, but this effect is significantly stronger in the pooling equilibrium. We also stress that if one picks a partially separating equilibrium with a lower value of q, the dashed curves move continuously closer towards the corresponding solid curves and coincide as qapproaches zero. Even though bidder 2 bids less and less aggressively as μis decreased, the equilibrium expected payoff of bidder 1 is however not monotone decreasing in μ.Asμis decreased bidder 1 benefits from the less aggressive bidding of bidder 2; however, he is then also less likely able to revise his bid. 1is an average of π1(ch)and π1(nh) and when μis decreased more weight is given to the smaller term π1(ch). Therefore, there is a trade-off and bidder 1’s net gain from spying reaches a maximum at some μ∈(0,1)(which is incidentally equal to 1 /2in the pooling equilibrium). 6 Equilibrium selection Like other signaling games, the spying game with imperfect commitment has multiple equilibria – a unique pooling equilibrium and a family of partially separating equilibria. In classical signaling games the multiplicity of equilibria is driven by “unreasonable” out-of-equilibrium beliefs and one can eliminate pooling equilibria by invoking standard equilibrium refinements such as the intuitive criterion by Cho and Kreps (1987). This does not apply to the present game in which, unlike in standard signaling games, both sender and receiver of messages have private information, the type space of bidder 1 is two-dimensional, and the sender, bidder 1, has a chance to make an unobserved move, his revised bid, independent of his first-round bid. The idea of the intuitive criterion is that a belief system is unreasonable if one can identify a type t∈{c, ch,n, nh}of bidder 1 who can convince bidder 2, by choosing a particular off-equilibrium action, that he should recognize him as the type who he is, because triggering this belief change is beneficial only to him. We now sketch briefly why, in the present model, this criterion has no bite. Consider the pooling equilibrium. Obviously, type ccannot benefit from an offequilibrium first-round bid b1>0 that triggers bidder 2 to recognize his type. If type ch makes an off-equilibrium bid b1>0 and triggers bidder 2 to recognize him, then 19 The plots are computed assuming (θ, v, q)=(1/2,1,¯q). 123 Spying and imperfect commitment in first-price auctions… 271 bidder 2 type hwill match his bid, and he cannot be better off. Similarly, type n cannot benefit either. If type nh makes an off-equilibrium first-round bid, b1>0, and is thus recognized by bidder 2, bidder 2 will ignore the spy’s report, and bidder 2 type hplays the equilibrium mixed strategy of game A, while bidder 1 plays a mixed strategy with mass point at zero (see Maskin and Riley 1985, p. 153) which yields the payoff (1−θ)v, which is smaller than his equilibrium payoff, (1−μθ)v. Therefore, the equilibrium satisfies the intuitive criterion also in this case. While there is a continuum of partially separating equilibria, their equilibrium payoffs converge to those of the unique pooling equilibrium, as qapproaches zero. As one can see from (20)–(21) and (22), that pooling equilibrium is payoff dominant which suggests that it may be a plausible selection of equilibrium. 7 Extensions 7.1 Other partially separating equilibria A disturbing feature of the above family of partially separating equilibria is that bidder 1 type ch may bid zero with a considerably high probability even if the probability that bidder 2 is type is close to zero. Bidder 1 type ch cannot revise his bid and if he bids zero he can only win in the event when bidder 2 is type because of the way in which the tie-rule favors bidder 2. Nevertheless bidder 1 type ch bids zero with probability 1 −q,q≤¯qwhich can be as high as 1 (as qapproaches 0), even if the probability that bidder 2 is type is arbitrarily close to zero. For example, if μ=1 /2 bidder 1 type ch may bid zero with probability 1 even if he then stands practically no chance to win because 1 −θis arbitrarily close to zero and 1 −qcan be arbitrarily close to 1. One may thus wonder whether one can find other partially separating equilibria in which bidder 1 type ch never makes a first-round bid equal to zero and fully separates himself from bidder 1 type c. The answer to this question is in the negative: Proposition 4 There is no equilibrium where bidder 1 type ch fully separates himself from type cby bidding b1=b∗with probability 1. However, the above family of partially separating equilibria is not unique. All equilibria in this family have the property that b∗>¯ b. Allowing for b∗≤¯ b, we find alternative equilibria. These other equilibria share the same essential properties: Proposition 5 The game has also partially separating equilibria where b∗≤¯ b. Like the equilibria summarized in Proposition3(that exhibit b∗>¯ b) they are all payoff dominated by the unique pooling equilibrium, both the spying and the spied-at bidder benefit from spying, and bidder 1 type ch does not fully separate himself from type c (i.e., q <1). The elaborate proofs of Propositions4and 5are relegated to the Online-Appendix A. 123 272 C. Fan et al. Fig. 3 Equilibrium mixed strategies. Left: with b∗<¯ b; Right: with b∗=¯ b The equilibrium mixed strategies, F1,F2, of these alternative equilibria are illustrated in Fig.3; the strategies of bidder 2, F2, exhibit mass points at b=0and at b=b∗. 7.2 More than two states We have chosen a parsimonious model with binary valuations. This raises the question whether our main results hinge on the assumption of binary valuations or generalize to more than two valuations. In order to answer this question we have extended the analysis to three valuations: V1,V2∈0,v,v,v>v>0, with associated prior probabilities {θ0,θ,θ}, that are positive, less than one, and add up to one. This analysis also indicates how it can be adapted to cover arbitrarily many valuations. As a natural extension of our notation, we denote bidder 1 type nwith V1=vas type nH bidder 1 type cwith V1=vas type cH and bidder 2 with V2=vas type H. Therefore, type sets are T1={c, ch,cH,n, nh,nH}and T2={, h,H}. The extension is more involved than the above analysis of the binary model. However, the bulk of it employs essentially the same solution procedures. Therefore, in the following, we provide only a brief summary of results and elaborate only on issues that make the analysis different. The detailed analysis is relegated to the Online-Appendix B. We focus on equilibria that are monotone (efficient) in the sense that they exhibit a monotone allocation rule, i.e., a high value bidder wins against a low value bidder except in the event of a tie. The game has a unique monotone pooling equilibrium. Like in the binary case, the first-round bids of bidder 1 are equal to zero, regardless of his type, while, in the second round, bidder 1 type nh and nH now play distinct mixed strategies, F1,G1, and bidder 2 type hand Hplay distinct mixed strategies, F2,G2with neighboring supports, as illustrated in Fig.4.20 Again, both bidders benefit from spying in the strong sense that spying increases their interim equilibrium expected payoffs. The fact that the equilibrium strategy of bidder 2 type H,G2, has a mass point at ¯ b, G2(¯ b)=1−μ, implies that one can only assure existence of a monotone equilibrium 20 The plots assume (θ0,θ,θ,v,v,μ)=(1/3,1/3,1/3,1,2,1/2). 123 Spying and imperfect commitment in first-price auctions… 273 Fig. 4 Equilibrium mixed strategies F1,F2and G1,G2 if one amends the auction rules as follows: Prior to bidding bidder 1 must pay an entry fee, f=(θ0+θ)R, and, after his final bid bhas been submitted to the auctioneer, bidder 1 receives a refund, R=θ(1−μ)(v −¯ b), conditional on b<¯ b. To explain the role of this amendment, note that, because G2has a mass point at ¯ b, bidder 1 type nh can discontinuously increase his probability of winning by bidding slightly more than ¯ b. However, by bidding more than ¯ bhe no longer qualifies to collect the refund Rwhich must be set in such a way that bidding more than ¯ bis not profitable. In turn, the refund may induce bidder 1 type nH to bid slightly lower than ¯ band thus qualify to collect the refund. Therefore, the refund has to be set in such a way that it also deters bidder 1 type nH from bidding slightly less than ¯ b. The prescribed R achieves both. One might think that a monotone equilibrium could also be established without amending the auction rules by shifting the support of G1,G2upwards to [ˆ b,¯ b] with ˆ b>¯ bto such an extent that bidder 1 type nh does not have an incentive to deviate to bid slightly more than ˆ b. However, then bidder 2 type Hwould gain by shifting the probability mass G2(ˆ b)downwards to ¯ band thus reduce his expected payment by (ˆ b−¯ b)G2(ˆ b)without reducing his probability of winning. Alternatively, one might think that a monotone equilibrium could be established by allowing for overlapping supports. However, then, for all bids in the intersection of the supports, the indifference requirements of mixed strategies would be violated, as we show in the Online-Appendix C. The monotone pooling equilibrium is unique and yields a collusive outcome. The game has also a family of monotone partially separating equilibria which are, however, payoff dominated by the pooling equilibrium. For some parameter values the game has also a non-monotone (inefficient) pooling equilibrium, where a low value bidder wins against a high value bidder with positive probability. This equilibrium does not require amending the auction rules because there G2has no mass point (see Fig.5). 123 274 C. Fan et al. Fig. 5 Non-monotone (inefficient) equilibrium 8 Conclusion In the present paper we examine the impact of spying in a first-price auction, assuming the spied-at bidder may secretly revise his bid with some known probability. Whereas perfectly informative spying exclusively benefits the spying bidder, the ambiguity about the spied-at bidder’s type gives rise to a collusive outcome where both bidders benefit from spying. The collusive impact suggests that once a bidder has procured the service of a spy, the spied-at bidder passively tolerates being spied-at by not taking measures to neutralize the spy and by not firing the spy if his identity has become known. This way tacit collusion is established and persists. While bidders benefit from spying, the seller is hurt. He may thus contemplate to replace the first-price auction by a Vickrey auction in which spying has no effect.21 However, Vickrey auctions pose their own problems and are susceptible to collusion (see, for example, Rothkopf et al. 1990; Ausubel and Milgrom 2006). Supplementary Information The online version contains supplementary material available at https://doi. org/10.1007/s40505-023-00257-3. Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 21 Unless bidders engage in spiteful bidding which has been observed in experiments by Fischer et al. (2021). However, one wonders whether spiteful bidding would also have occurred if they had assumed that the leaked information is subject to noise in which case spiteful bidders are at risk to suffer losses. 123 Spying and imperfect commitment in first-price auctions… 275 References Andrew, C.: The Secret World: A History of Intelligence. Yale University Press, New Haven (2019) Andreyanov, P., Davidson, A., Korovkin, V.: Detecting Auctioneer Corruption: Evidence from Russian Procurement Auctions. mimeo., UCLA, Stanford (2017) Arozamena, L., Weinschelbaum, F.: The effect of corruption on bidding behavior in first-price auctions. Eur. Econ. Rev. 53, 645–657 (2009) Ausubel, L., Milgrom, P.: The lovely but lonely Vickrey auction. In: Cramton, P., Shoham, Y., Steinberg, R. (eds.) 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