Perfect bidder collusion through bribe and request
PPerfect bidder collusion through bribe and request
Jingfeng Lu ∗ Zongwei Lu † Christian Riis ‡ December 10, 2019
Abstract
We study collusion in a second price auction with two bidders in a dy-namic environment. One bidder can make a take-it-or-leave-it collusion pro-posal, which consists of both an offer and a request of bribes, to the opponent.We show there always exists a robust equilibrium in which the collusion suc-cess probability is one. In the equilibrium, the interim expected payoff ofthe collusion initiator Pareto dominates the counterpart in any robust equi-libria of the single-option model (Es¨o and Schummer (2004)) and any otherseparating equilibria in our model.
Keywords : second-price auction, collusion, multidimensional signaling,bribe
JEL codes : D44, D82 ∗ Department of Economics, National University of Singapore. Email: [email protected]. † School of Economics, Shandong University. Email: [email protected]. ‡ Department of Economics, BI Norwegian Business School. Email: [email protected]. a r X i v : . [ ec on . T H ] D ec Introduction
Under standard assumptions auctions are a simple yet effective economic insti-tution that allows the owner of a scarce resource to extract economic rents frombuyers. Collusion among bidders, on the other hand, can ruin the rents and henceshould be one of the top issues that are always kept in the minds of the auctiondesigners with a revenue-maximizing objective. In the literature, collusion in auc-tions are typically modelled as a static game in which bidders already agree to forma cartel and the mission of the cartel is to select a representative bidder (to win theobject at a low price in the auction) and decide the side-payments (e.g., Grahamand Marshall (1987), Mailath and Zemsky (1991), Marshall and Marx (2007), andMcAfee and McMillan (1992)). Although collusion can be guaranteed througha revelation game, the working of such a cartel typically requires the aid of anincentiveless third party. Furthermore, the static model misses a realistic and im-portant scenario when a particular bidder has strong bargaining power and thus isinterested in monetizing it through some bargaining process with other bidders.In a pioneering work, Es¨o and Schummer (2004) (hereafter, ES) consider a dy-namic model of bidder collusion in a second price auction before which a bidderhas the opportunity to make a take-it-or-leave-it offer of bribe in exchange for theopponent’s absence from the auction (or bidding zero). Since their work, alter-native models in the same vein have been proposed. For example, Rachmilevitch(2013) considers the same bargaining protocol in first price auctions and Rachmile-vitch (2015) assumes equal bargaining power between the bidders who can makean offer sequentially. Troyan (2017) extends ES’s model to interdependent valuesand affiliated signals.The papers cited above share a common feature, i.e., the collusion proposalspecifies a positive transfer from the initiator to the opponent. However, it is naturalthat if offering a bribe to the opponent is feasible then requesting a bribe from theopponent should also be available in the initiator’s toolbox.The collusion initiator may consider a combination of both a bribe and a re-quest for the following two reasons. First, with one more option available for theopponent to choose to accept, it seems to improve the probability of successfulcollusion (if the initiator cares about it), as compared to the single-option scheme.Second, and most importantly, a double-option scheme seems to be more lucrativefor the initiator, provided it can be successfully implemented.With the additional option, a multi-dimensional signaling game emerges and2he problem of adverse selection may be more severe, so that the existence and ro-bustness of equilibria are at stake. To examine the implementability, we consider asecond price auction with two bidders, as in ES. Specifically, we assume that a bid-der has an opportunity to make a take-it-or-leave-it collusion proposal, consistingof both a bribe and a request, to the opponent before a second price auction starts.The opponent can accept at most one option in the proposal, in which case bothbidders follow the proposal and bid cooperatively in the auction. If the opponentrejects the proposal, then both bidders compete non-cooperatively in the auction.The solution concept we use is weak perfect Bayesian equilibrium (hereafter,equilibrium). If an equilibrium survives the D1 criterion (Cho and Sobel (1990)),the same standard refinement used in ES, it is said to be robust. To illustrate ourmain points, we focus on separating equilibria. Although there may exist a multi-plicity of separating equilibria, we show that there always exists a robust separatingequilibrium in which the initiator’s request is equal to his valuation. In the equilib-rium, the opponent always accepts the collusion proposal (accepting one of the twooptions), i.e., for each type of the initiator, the probability of a successful collusionis one. The significance of this result is threefold. First, a separating equilibriumdoes not always exist in ES’s model. Secondly, in ES, the initiator typically can notbe guaranteed a successful collusion because the best response of the opponent is toaccept the bribe if her type is low and reject it otherwise. Hence, in our model, sucha double-option scheme generally improves the probability of successful collusionupon the single-option one. Thirdly, perhaps more importantly, the result showsthat second price auctions in this dynamic environment are as vulnerable as in thestatic model, even though now the bidders do not agree on collusion beforehandand both acts strategically. It is interesting to compare the expected payoff of the initiator in our modelwith the counterpart in ES. On one hand, it seems that the additional option itselfalone represents an additional channel through which the initiator can extract moresurplus from the opponent. On the other hand, the additional option tends to in-tensify the adverse selection problem. In particular, an interesting feature of theseparating equilibrium is that the bribe may exceed the initiator’s valuation, which Milgrom and Roberts (1986) apply the Intuitive Criterion (Cho and Kreps (1987)), which isweaker than the D1 criterion, for their multidimensional signaling model of price and advertisingexpenditure decisions by a firm trying to convey information about its product quality to consumers. In Rachmilevitch (2015), under some conditions, there exists an efficient equilibrium in which asuccessful collusion is also guaranteed. However, in that model the two bidders engage in two roundsof alternating offers of bribes and thus neither of them has full bargaining power.
Related literature
Our paper contributes to the growing literature of collusion in auctions. As men-tioned in the introduction, the traditional approach is to consider a static gamebefore which some bidders have already agreed on participation of collusion. Onthe other hand, ES-like dynamic models typically assume the single-option col-lusion proposal, i.e., an offer of a bribe, from the bidder with the move to theopponent, in exchange for the latter’s cooperation. For example, apart from the pa-pers mentioned in the introduction, Chen and Tauman (2006) address the potentialcollusion problem of the opponent cheating by using shill bidders in second priceauctions. Kivetz and Tauman (2010) consider a complete information model of a4rst price auction. Balzer (2019) considers a more general set of mechanisms forfirst price and second price auctions. The main difference is that we consider asimple and natural double-option proposal in a second price auction and focus onits implementability and profitability for the initiator while his main objective is toexamine the difference between a first price and a second price auction in terms ofthe possibility of efficient collusion.More broadly, our paper is related to the strand of mechanism-design literaturethat studies the informed-principal problem, due to the seminal works of Myerson(1983), Maskin and Tirole (1990) and Maskin and Tirole (1992). In particular,Maskin and Tirole (1990) consider a private value setting in which the principalproposes a contract that itself is a game to be played when it is accepted by theagent and the whole game ends if it is rejected. They show that the principalis generally better off as compared to the scenario when his type is of completeinformation.Our paper is also related to the literature of multidimensional signaling gamespioneered by Milgrom and Roberts (1986) and Wilson (1985). In particular, Mil-grom and Roberts (1986) study the problem of a firm that can use both price andadvertising expenditure to signal its product quality to consumers. In their model,the firm has only two types and the advertising expenditure is dissipative. In con-trast, in our model the initiator has a continuum of types and neither of the signalsis dissipative.
Two risk-neutral bidders are about to attend a second price auction in which thereis a single indivisible object and no reserve price. Before the auction starts, bidder1 (he) has an opportunity to make a take-it-or-leave-it proposal to bidder 2 (she).The proposal consists of a nonnegative bribe and a nonnegative request, denotedby ( b , r ) with b , r ≥
0. If bidder 2 accepts the bribe, then bidder 1 pays b to her andshe bids zero in the auction so that bidder 1 wins the object at a price of zero. Ifbidder 2 accepts the request, then she pays r to bidder 1 who then bids zero in theauction and she wins the object at a price of zero. If bidder 2 rejects the proposal,then both bidders bid non-cooperatively in the auction.We assume bidders’ valuation (or type ) distributions are independent, but allowfor asymmetry. For i = ,
2, bidder i ’s valuation v i is independently distributed A discussion of a reserve price is given in section 5.1. F i ( v i ) on [ v i , ¯ v i ] . Each distribution function F i admits a continuousdensity function f i ( v i ) ∈ ( , ∞ ) for all v i . We assume v i ≥ v i < ∞ .We focus on pure strategy separating equilibria. So both bidders use purestrategies whenever they are supposed to take the move. We assume that if bidder2 rejects the proposal then both bidders bid truthfully in the second-price auctionsince it is a weakly dominant strategy.We let b ( v ) and r ( v ) be the bribing and requesting functions in an equilib-rium. One immediate observation of our model is that if v ≥ ¯ v , then there al-ways exists a separating equilibrium in which b ( v ) = r ( v ) = v . In theequilibrium, all types of bidder 2 accepts the zero bribe and bidder 1 realizes hisvaluation. Hence, below we focus on the non-trivial case of v < ¯ v . Suppose there exists a separating equilibrium in which bidder 1’s request is notgreater than his own valuation. We assume that bidder 2 is willing to accept aproposal if she is indifferent between accepting and rejecting it. In the equi-librium, upon receiving a separating proposal ( b , r ) from type v , for bidder 2,accepting b gives a payoff of b , accepting r gives a payoff of v − r and reject-ing the proposal gives a payoff of v − v . Thus, if r ≤ v , it is optimal for bid-der 2 to accept b if v ≤ b + r and accept r if v > b + r . The expected pay-off of bidder 1 with valuation v from a separating proposal ( b , r ) with r ≤ v is π ( v , b , r ) = F ( b + r )( v − b ) + ( − F ( b + r )) r , which can be rewritten as π ( v , b , r ) = F ( b + r )( v − ( b + r )) + r . For convenience of exposition, let v be the smallest value such that, if it exists, b ( v ) + v = ¯ v , (1)and thus F ( b ( v ) + v ) =
1. To save notation, we let π ( v ) ≡ π ( v , b ( v ) , r ( v )) .The following proposition shows that for any distribution F and F , there al-ways exists a separating equilibrium in which bidder 1 requests his own valuation. The single-option model in ES shares the same equilibrium in this case, i.e., b ( v ) = v . Accepting the proposal in this case is a best response (i.e., there is no profitable deviation). Although type v = b + r is indifferent between accepting b and accepting r , for simplicity weassume she accepts b when she accepts the proposal. roposition 1. There exists a separating equilibrium in which • for any type v ≤ v , b ( v ) is given byb (cid:48) ( v ) = f ( b ( v ) + v ) b ( v ) + F ( b ( v ) + v ) − , (2) with an initial condition b ( v ) = , which always admits a unique solution.For any type v > v , b ( v ) = b ( v ) . • for any type v , r ( v ) = v . Bidder 2 always accepts the equilibrium proposal ( b ( v ) , v ) , i.e., accepts b ( v ) ifv ≤ b ( v ) + v and accepts the request v if v > b ( v ) + v .Proof. For any type v < v , the on-path incentive compatibility (IC) condition foran equilibrium in which r ( v ) ≤ v is v ∈ arg max t π ( v , b ( t ) , r ( t )) = F ( b ( t ) + r ( t ))( v − ( b ( t ) + r ( t ))) + r ( t ) . (3)which implies [ f ( b ( v )+ r ( v ))( v − ( b ( v )+ r ( v ))) − F ( b ( v )+ r ( v ))]( b (cid:48) ( v )+ r (cid:48) ( v ))+ r (cid:48) ( v ) = . (4)In the equilibrium with r ( v ) = v , the bribing function b ( v ) satisfies [ f ( b ( v ) + v ) b ( v ) + F ( b ( v ) + v )]( b (cid:48) ( v ) + ) = , which in turn can be rewritten as (2).With the initial condition b ( v ) =
0, which implies b (cid:48) ( v ) >
0, the differentialequation in (2) always has a unique solution. It also follows that b (cid:48) ( v ) >
0. Fur-thermore, whenever b ( v ) = v , b (cid:48) ( v ) >
0. Thus b ( v ) ≥ v and is an admissible bribing function.Observe that (2) implies b ( v ) + v is strictly increasing for all v and thus theequilibrium is separating. So F ( b ( v ) + v ) = v ≥ v . It follows that By (2), b (cid:48) ( v ) = v > v . Technically, for any type v > v , r ( v ) can be any value such that b ( v ) + r ( v ) ≥ ¯ v and r ( v ) takes different values for different v . Alternatively, in an equilibrium, for any type v > v , r ( v ) canbe equal to v so that there is a pooling segment on the top. However, the equilibria are essentiallyequivalent. v > v , b ( v ) = b ( v ) , since for these types the equilibrium propos-als must be such that b ( v ) are always accepted by all types of bidder 2 and theequilibrium payoff of such types is v − b ( v ) .We now show that there is no profitable on-path deviation for any type v , i.e.,given that all other types of bidder 1 are following the equilibrium proposals, it isnot profitable for type v to send any on-path proposal ( b ( t ) , t ) , t (cid:54) = v . Considerthe expected payoff of type v from deviating to ( b ( t ) , t ) , i.e., π ( v , b ( t ) , t ) . While π (cid:48) v ( v , b ( t ) , t ) = F ( b ( t ) + t ) for any given t , the envelope theorem implies that π (cid:48) ( v ) = F ( b ( v ) + v ) . Because b ( v ) + v is strictly increasing, it implies that forany given t , π (cid:48) ( v ) − π (cid:48) v ( v , b ( t ) , t ) ≷ v ≷ t . Hence π ( v ) > π ( v , b ( t ) , t ) for any v (cid:54) = t .There are two types of off-path deviations in the equilibrium. First, there aremany unsent off-path proposals by bidder 1. Second, because in the equilibrium,given an on-path proposal, bidder 2 on the path is supposed to accept the proposalwith probability one, rejection becomes an off-path deviation as well. However, forthe latter type of off-path deviations, it is straightforward to see that for any belief ofbidder 1 it is not profitable for any type of bidder 2 to reject the received equilibriumproposal, since it is a weakly dominant strategy for bidder 1 to submit a bid of hisvaluation in the auction. So the only non-trivial type of off-path deviations is theformer, which is two dimensional and thus of a large set.Below, we show that there exists a system of off-path beliefs of bidder 2 suchthat given the resulting best-response of bidder 2, it is not profitable for bidder 1 todeviate to any off-path proposals. In particular, the off-path beliefs of bidder 2 arereasonable in the sense of the D1 criterion.Cho and Sobel’s D1 criterion says that if, upon observing an off-path action,any best responses of the receiver (based on some beliefs about the sender’s type)that imply a profitable deviation for a sender type imply also a profitable devia-tion for another sender type and the converse is not true, then, upon observing thatoff-path action the receiver’s posterior beliefs should place zero probability on theformer type. Roughly speaking in an alternative way, if a type of the sender is dom-inated by another type in terms of the expected payoff of the sender for possiblebest responses of the receiver, then the former type should be excluded from the setof reasonable beliefs of the receiver. Hence, if for an off-path action there existssome type of the sender that is not dominated by any other types, then a reasonablebelief of the receiver is that it is the sender’s type. If for each off-path action, there8xists such a reasonable belief that the best response of the receiver based on thebelief implies that it is not profitable for any type of the sender to deviate to theoff-path action, then the equilibrium survives the D1 criterion.We first describe the set of best responses of bidder 2 upon receiving an off-pathproposal ( b , r ) . Lemma 1.
Upon receiving an off-path proposal ( b , r ) , the best response of bidder2 can be described as a pair of critical values ( v b , v r ) such that she accepts b ifv ≤ v b , rejects the proposal if v b < v < v r and accepts r if v ≥ v r . In particular,v b ≤ min { b + r , ¯ v } ≤ v r ≤ ¯ v . (5) Proof.
Observe that bidder 2 never accepts r if her type v ≤ b + r and never accepts b otherwise. That is, for any type v ≤ b + r , her best response is either to accept b or reject the proposal (thus compete with bidder 1 in the auction). Similarly, forany type v > b + r , her best response is either to accept r or reject the proposal.As shown in ES, when bidder 2’s decision making problem is to determinewhether to accept or reject a bribe b (and thus compete non-cooperatively withbidder 1 in the auction), the decision rule requires that for any type v and v b , iftype v b accepts b , then any type v < v b accepts b .We show below that when bidder 2’s decision making problem is to determinewhether to accept or reject a request r (and thus compete non-cooperatively withbidder 1 in the auction), the decision rule requires that for any type v and v (cid:48) , iftype v (cid:48) accepts r , then any type v > v (cid:48) accepts r .First we suppose b + r < ¯ v . Let the set of types of bidder 1 sending proposal ( b , r ) be Q b , r . Suppose v , v (cid:48) ≥ b + r and v > v (cid:48) . For bidder 2 with type v ,the difference between the expected payoffs from accepting r and rejecting theproposal is ∆ v ≡ v − r − E [( v − v ) v < v | v ∈ Q b , r ] , where X is the indicatorfunction for event X . Thus, ∆ v − ∆ v (cid:48) = v − v (cid:48) − ( E [( v − v ) v < v | v ∈ Q b , r ] − E [( v (cid:48) − v ) v < v (cid:48) | v ∈ Q b , r ]) ≥ v − v (cid:48) − ( E [( v − v ) v < v | v ∈ Q b , r ] − E [( v (cid:48) − v ) v < v | v ∈ Q b , r ])= v − v (cid:48) − E [( v − v (cid:48) ) v < v | v ∈ Q b , r ]= ( v − v (cid:48) )( − E [ v < v | v ∈ Q b , r ]) ≥ . Therefore, there exists a v r ≥ b + r such that any type v ≥ v r accepts r and any9ype v ∈ ( b + r , v r ) rejects the proposal.If also follows from above that if b + r ≥ ¯ v , then the statement in the lemma isautomatically true. This completes the proof.For convenience, we record the following fact which follows directly from theenvelope theorem and (2). Fact 1. π (cid:48) ( v ) = F ( b ( v ) + v ) , π ( v ) = v − F ( b ( v ) + v ) b ( v ) and b ( v ) + v is strictly increasing.For an off-path proposal ( b , r ) , given a best response ( v b , v r ) of bidder 2 asdescribed in Lemma 1, the expected payoff of type v , denoted by π ( v , b , r , v b , v r ) ,is given by π ( v , b , r , v b , v r ) ≡ F ( v b )( v − b ) + (cid:90) min { v r , v } min { v b , v } ( v − v ) f ( v ) dv + ( − F ( v r )) r . (6)The derivative of π ( v , b , r , v b , v r ) w.r.t. v is π (cid:48) v ( v , b , r , v b , v r ) = F ( v b ) if v ≤ v b , F ( v ) if v ∈ ( v b , v r ] , F ( v r ) if v > v r . (7)Recall that π (cid:48) ( v ) = F ( b ( v ) + v ) . Because b ( v ) > v > v , (7)implies that given any v b and v r , π (cid:48) v ( v , b , r , v b , v r ) < π (cid:48) ( v ) for any type v ≥ v b .Thus we have the following fact. Fact 2.
For an off-path proposal ( b , r ) and a best response ( v b , v r ) of bidder 2, if π ( v b , b , r , v b , v r ) ≤ π ( v b ) , then for any type v > v b , π ( v , b , r , v b , v r ) < π ( v ) . The next two facts identify some off-path proposals that would never be madeby any type of bidder 1.
Fact 3.
Consider an off-path proposal ( b , r ) . If r ≤ v , then it is not profitable forany type v to deviate to ( b , r ) for any belief of bidder 2. Proof of Fact
3: Recall that when bidder 2 receives a proposal ( b , r ) , accepting r gives a payoff v − r while rejecting the proposal gives a payoff v − v . Thus, if r ≤ v , then the proposal is never rejected.10uppose b + r < ¯ v . Then for type v < b + r , it is more profitable to accept b than r . Hence, if r ≤ v , bidder 2’s best response is to accept b if v ≤ b + r andaccept r if v > b + r , i.e., v b = b + r = v r . So the expected payoff of type v is π ( v , b , r , b + r , b + r ) = F ( b + r )( v − ( b + r )) + r . Then π ( v ) − π ( v , b , r , b + r , b + r ) = v − F ( b ( v ) + v ) b ( v ) − [ F ( b + r )( v − ( b + r )) + r ]= ( − F ( b + r ))( v − r ) + F ( b + r ) b − F ( b ( v ) + v ) b ( v ) . If b ( v ) < b and b ( v ) + v < b + r , then obviously π ( v ) − π ( v , b , r , b + r , b + r ) >
0. Because r ≤ v , if b ( v ) < b and b ( v ) + v ≥ b + r , or b ( v ) ≥ b , then π (cid:48) v ( v , b , r , b + r , b + r ) = F ( b + r ) ≤ F ( b ( v ) + v ) = π (cid:48) ( v ) . Since r ≤ v , wehave that π ( v , b , r , b + r , b + r ) = F ( b + r )( v − b ) + ( − F ( b + r )) r ≤ F ( b + r ) v + ( − F ( b + r )) r ≤ v = π ( v ) . Hence, π ( v ) ≥ π ( v , b , r , b + r , b + r ) for any v if r ≤ v .Suppose b + r ≥ ¯ v . Then v b = v r = ¯ v . Then π ( v , b , r , ¯ v , ¯ v ) = v − b . So π ( v ) − π ( v , b , r , ¯ v , ¯ v ) = b − F ( b ( v ) + v ) b ( v ) If b ≥ b ( v ) , then obviously π ( v ) − π ( v , b , r , ¯ v , ¯ v ) ≥
0. If b < b ( v ) , then since r ≤ v , we have b ( v ) + v ≥ b + r ≥ ¯ v , which implies π (cid:48) ( v ) = F ( b ( v ) + v ) = = π (cid:48) v ( v , b , r , ¯ v , ¯ v ) . Again, since π ( v ) ≥ π ( v , b , r , ¯ v , ¯ v ) , we have π ( v ) ≥ π ( v , b , r , ¯ v , ¯ v ) for any v .Intuitively, since bidder 1’s value for bidder 2 is at least v , bidder 1 does nothave an incentive to request an amount lower than v . Fact 3 implies we can restrictattention to proposals with r > v .The following fact further excludes proposals ( b , r ) with r > v and b ≥ ¯ v − v . Fact 4.
Consider an off-path proposal ( b , r ) . If r > v and b ≥ ¯ v − v , then it isnot profitable for any type v to deviate to ( b , r ) for any belief of bidder 2. Proof of Fact
4: If r > v and b ≥ ¯ v − v , then the bribe is accepted by all typesof bidder 2, since b ≥ v − v implies that for any type v , accepting b is moreprofitable than competing with bidder 1 in the auction and b ≥ ¯ v − v > v − r b is more profitable than accepting r , i.e., the best response ofbidder 2 is described by v b = ¯ v = v r . So the expected payoff of bidder 1 withtype v is π ( v , b , r , ¯ v , ¯ v ) = v − b . If ¯ v < ¯ v , then π ( ¯ v , b , r , ¯ v , ¯ v ) = ¯ v − b < ¯ v − b ≤ v = π ( v ) . Since π ( v , b , r , ¯ v , ¯ v ) is non-decreasing in v and π ( v ) isstrictly increasing, π ( v , b , r , ¯ v , ¯ v ) < π ( v ) for any type v ∈ [ v , ¯ v ] . If ¯ v ≥ ¯ v ,then for v = ¯ v , π ( ¯ v , b , r , ¯ v , ¯ v ) = ¯ v − b ≤ v = π ( v ) . By the monotonicity of π ( v , b , r , ¯ v , ¯ v ) and π ( v ) , we have π ( v , b , r , ¯ v , ¯ v ) ≤ π ( v ) for any type v ≤ ¯ v .Then by Fact 2, π ( v , b , r , ¯ v , ¯ v ) ≤ π ( v ) for any type v ∈ [ v , ¯ v ] .Fact 3 and Fact 4 together imply we can restrict attention to proposals with r > v and b < ¯ v − v . Fact 5.
Consider an off-path proposal ( b , r ) with r > v and b < ¯ v − v . If b > v >
0, then π ( v , b , r , b + v , ¯ v ) < π ( v ) for any type v . Proof of Fact
5: From (6), π ( v , b , r , b + v , ¯ v ) = F ( b + v )( v − b ) + (cid:90) v min { b + v , v } ( v − v ) f ( v ) dv If b >
0, then b + v > v . Since π ( v , b , r , b + v , ¯ v ) is non-decreasing and π ( v ) is strictly increasing, if ¯ v ≤ b + v , then π ( ¯ v , b , r , b + v , ¯ v ) = F ( b + v )( ¯ v − b ) ≤ v = π ( v ) which in turn implies that π ( v , b , r , b + v , ¯ v ) < π ( v ) for any type v ∈ [ v , ¯ v ] . If ¯ v > b + v , then π ( b + v , b , r , b + v , ¯ v ) = F ( b + v ) v ≤ v = π ( v ) . By the monotonicity of π ( v , b , r , b + v , ¯ v ) and π ( v ) , we have π ( v , b , r , b + v , ¯ v ) < π ( v ) for any type v ≤ b + v . Then by Fact 2, π ( v , b , r , b + v , ¯ v ) < π ( v ) for any type v ∈ [ v , ¯ v ] .If b = v >
0, then π ( b + v , b , r , b + v , ¯ v ) = F ( v ) v < v = π ( v ) since we assume v < ¯ v . By the similar arguments in the case of b >
0, we alsohave π ( v , b , r , b + v , ¯ v ) < π ( v ) for any type v ∈ [ v , ¯ v ] . This completes theproof.We are now ready to show that the equilibrium identified above survives theD1 criterion. Proposition 2.
The equilibrium identified above survives the D1 criterion.Proof.
By Fact 3 and Fact 4, we can focus on off-path proposals ( b , r ) with r > v and b < ¯ v − v . So b + v < b + r and b + v < ¯ v . For convenience of exposition,12elow we consider only the case b + r ≤ ¯ v . The analysis of the case of b + r > ¯ v is identical and omitted. The lowest belief of bidder 2 is v = v . If bidder 2 believes v = v , accepting b gives b , rejecting the proposal gives v − v . So any type v ≤ b + v < b + r always accepts b , i.e., v b ≥ b + v . Hence, below we can focus on the beliefs suchthat v b ∈ [ b + v , b + r ] and v r ∈ [ b + r , ¯ v ] . Suppose b = v =
0. When v b = v =
0, and v r = ¯ v , i.e., the proposal isrejected by all types of bidder 2 and both bidders compete in the auction, only type v is indifferent between the equilibrium payoff and the payoff from the deviationwhile all other types of bidder 1 are strictly worse off. To see this, the expectedpayoff from the deviation is the one from a standard second price auction, whichhas a slope F ( v ) , which in turn is strictly smaller than π (cid:48) ( v ) = F ( b ( v ) + v ) .Hence, type v = v = v =
0. Since b =
0, the bestresponse of bidder 2 with this belief is then to reject the proposal and compete withbidder 1 in the auction. Then the above confirms it is not profitable for any typeof bidder 1 to deviate to the off-path ( b = , r ) . Hence, below we only need toexamine the event b > v > ( b , r ) , let M ( v b , v r ) = max v π ( v ) − π ( v , b , r , v b , v r ) . (8)Clearly for any given v b ∈ [ b + v , b + r ] and v r ∈ [ b + r , ¯ v ] , π ( v ) − π ( v , b , r , v b , v r ) is well defined on [ v , ¯ v ] and continuous in v . By the extreme value theorem, itadmits a maximum for any v b , v r in the relevant intervals. The solution to maxi-mization problem in (8) is continuous and thus M ( v b , v r ) is continuous.Suppose for a given proposal ( b , r ) , there exists no v b ∈ [ b + v , b + r ] and no v r ∈ [ b + r , ¯ v ] such that M ( v b , v r ) >
0. Then π ( v , b , r , v b , v r ) ≤ π ( v ) for any type v , i.e., it is not profitable for any type of bidder 1 to deviate to ( b , r ) for any beliefof bidder 2 and we are done.Suppose for a given proposal ( b , r ) , for some v b = v b (cid:48) ∈ [ b + v , b + r ] and v r = v r (cid:48) ∈ [ b + r , ¯ v ] , M ( v b (cid:48) , v r (cid:48) ) >
0. Observe that Fact 5 implies M ( b + v , ¯ v ) < v b = v b ∗ ∈ [ b + v , v b (cid:48) ] and See footnote 10. When b + r > ¯ v , only the bribe will be considered by bidder 2 seriously. Thus, in this case, v b ∈ [ b + v , ¯ v ] and v r = ¯ v . So the analysis for this case is the direct translation from the currentone below, i.e., replacing b + r and v r by ¯ v . r = v r ∗ ∈ [ v r (cid:48) , ¯ v ] such that M ( v b ∗ , v r ∗ ) =
0. Let V be the set of v such that M ( v b ∗ , v r ∗ ) =
0. Then by the fact π (cid:48) ( v ) − π (cid:48) v ( v , b , r , v b , v r ) > v ≥ v b (from (7) and Fact 1), we can conclude that for any v = v ∗ ∈ V , v ∗ < v b ∗ ≤ b + r . (9)By the fact that π (cid:48) v ( v , b , r , v b , v r ) = v b for all v ≤ v b and π ( v ) is strictly convex(again from (7) and Fact 1), V is a singleton.Thus, if for some type v , there exists some v b ∈ [ b + v , b + r ] and v r ∈ [ b + r , ¯ v ] such that π ( v , b , r , v b , v r ) ≥ π ( v ) , then there must exist a pair of ( v b ∗ , v r ∗ ) such that π ( v , b , r , v b ∗ , v r ∗ ) is tangent to π ( v ) at some unique v = v ∗ , in the sense that forall v (cid:54) = v ∗ , π ( v , b , r , v b ∗ , v r ∗ ) < π ( v ) and π ( v ∗ , b , r , v b ∗ , v r ∗ ) = π ( v ∗ ) . Hence, type v ∗ is not excluded by the D1 criterion. So a reasonable belief of bidder 2 is thatbidder 1’s type is v = v ∗ and we adopt this belief for the analysis below.Suppose the tangency point is at v and thus a reasonable belief of bidder 2is v = v . Then for bidder 2, accepting b gives b , accepting r gives v − r andrejecting the proposal gives v − v . Since r > v and b < ¯ v − v , it is optimal forbidder 2 to accept b if v ≤ b + v and reject the proposal otherwise. Then Fact 5implies it is not profitable for any type of bidder 1 to deviate to the proposal.It follows then that below we can focus on interior tangency points, i.e, v ∗ ∈ ( v , v b ∗ ) . It in turn follows that π (cid:48) v ( v ∗ , b , r , v b , v r ) = π (cid:48) ( v ∗ ) , which implies v b ∗ = b ( v ∗ ) + v ∗ .Suppose v b ∗ = b + r = v r ∗ , which implies b ( v ∗ ) + v ∗ = b + r . If v ∗ ≥ r , thenthe best response of bidder 2 is to accept b if v ≤ b + r and accept r otherwise.Then the expected payoff of any type v from the best response is then exactly π ( v , b , r , v b ∗ , v r ∗ ) , which, by tangency, is no greater than π ( v ) and we are done.The sub-case v ∗ < r is analyzed below together with the case of v b ∗ < b + r or b + r < v r ∗ .Suppose v b ∗ < b + r or b + r < v r ∗ . Then we can always conclude v ∗ < r . Tosee this, observe that type v ∗ is indifferent between π ( v ∗ , b , r , v b ∗ , v r ∗ ) and π ( v ∗ ) ,i.e., F ( v b ∗ )( v ∗ − b ) + ( − F ( v r ∗ )) r = v ∗ − F ( b ( v ∗ ) + v ∗ ) b ( v ∗ ) , which, by the fact that v b ∗ = b ( v ∗ ) + v ∗ , can be rearranged into v ∗ − r = F ( v b ∗ )( b ( v ∗ ) + v ∗ − b ) − F ( v r ∗ ) r . v b ∗ < b + r or if b + r < v r ∗ , then v b ∗ < v r ∗ and v b ∗ = b ( v ∗ ) + v ∗ ≤ b + r ,which in turn implies v ∗ < r .So below we can focus on the belief v = v ∗ < r , with which the best responseof bidder 2 is to accept b if v ≤ b + v ∗ and reject the proposal otherwise. And theexpected payoff of bidder 1 from the proposal is then π ( v , b , r , b + v ∗ , ¯ v ) .By Fact 2, we can for now restrict our attention to type v ≤ b + v ∗ . For anytype v ≤ b + v ∗ , the expected payoff is π ( v , b , r , b + v ∗ , ¯ v ) = F ( b + v ∗ )( v − b ) . The tangency condition says, for any type v , π ( v , b , r , v b ∗ , v r ∗ ) = F ( b ( v ∗ ) + v ∗ )( v − b ) + ( − F ( v r ∗ )) r ≤ v − F ( b ( v ) + v ) b ( v )= π ( v ) . Suppose b ≤ b ( v ∗ ) . Then clearly π ( v , b , r , b + v ∗ , ¯ v ) ≤ π ( v , b , r , v b ∗ , v r ∗ ) ≤ π ( v ) for any type v .Suppose b > b ( v ∗ ) . Then π ( v ∗ ) − π ( v ∗ , b , r , b + v ∗ , ¯ v ) = v ∗ − F ( b ( v ∗ ) + v ∗ ) − F ( b + v ∗ )( v ∗ − b )= v ∗ ( − F ( b + v ∗ )) + F ( b + v ∗ ) b − F ( b ( v ∗ ) + v ∗ ) b ( v ∗ ) ≥ . Furthermore, recall that π ( v ) is strictly convex and observe that π ( v ∗ , b , r , b + v ∗ , ¯ v ) has a constant slope for v < v ∗ . So b + v ∗ > b ( v ∗ )+ v ∗ implies π (cid:48) v ( v , b , r , b + v ∗ , ¯ v ) = F ( b + v ∗ ) > F ( b ( v ) + v ) = π (cid:48) ( v ) for any type v < v ∗ . So for any type v < v ∗ , π ( v , b , r , b + v ∗ , ¯ v ) < π ( v ) . For any type v ∈ ( v ∗ , b + v ∗ ] , π ( v ) − π ( v , b , r , b + v ∗ , ¯ v ) = v ( − F ( b + v ∗ ))+ F ( b + v ∗ ) b − F ( b ( v )+ v ) b ( v ) . If b ( v ) < b and b ( v ) + v < b + v ∗ , then obviously π ( v ) − π ( v , b , r , b + v ∗ , ¯ v ) >
0. If b ( v ) < b and b ( v ) + v ≥ b + v ∗ , or if b ( v ) ≥ b , then π (cid:48) ( v ) = F ( b ( v ) + v ) > F ( b + v ∗ ) = π (cid:48) v ( v , b , r , b + v ∗ , ¯ v ) for any type v ∈ ( v ∗ , b + v ∗ ] . Since fromabove, π ( v ∗ ) > π ( v ∗ , b , r , b + v ∗ , ¯ v ) , we have that for any v ∈ ( v ∗ , b + v ∗ ] , π ( v ) > π ( v , b , r , b + v ∗ , ¯ v ) . Fact 2 then implies π ( v ) > π ( v , b , r , b + v ∗ , ¯ v ) for any v ∈ [ v , ¯ v ] . 15ggregating all above cases, the proof of the equilibrium surviving the D1criterion is completed.The following example sheds some light on the equilibrium. Example 1.
Suppose F ( x ) = x on [ , ] . Suppose v is distributed on [ , ] . Thenfor all v satisfying b ( v ) + v ≤
1, (2) becomes b (cid:48) ( v ) = b ( v ) + v − . The solution is b ( v ) = (cid:16) W (cid:16) − e − v − (cid:17) − v + (cid:17) , where W ( x ) solves x = W ( x ) e W ( x ) . With the initial condition b ( ) =
0, the equilibrium bribing function b ( v ) isplotted in Figure 1.Figure 1: The equilibrium bribing and requesting functions when F ( x ) = x and v = b (cid:48) ( v ) = / F ( v ) −
1, this ispossible if F ( v ) < . The fact that the equilibrium bribe may exceed bidder 1’svaluation suggests that it can be very costly for some types of bidder 1 to signalhis strength. In contrast, in ES’s single-option model, the separating bribes do notexceed bidder 1’s valuation. It is interesting to investigate whether it is moreprofitable for bidder 1 to implement the double-option scheme.
In this section we first compare the expected payoff of bidder 1 in the identifiedrobust equilibrium in our model to the counterpart in ES.In ES, bidder 1 commits to offering a variable take-it-or-leave-it bribe to bidder2. For separating bribes, let the bribing function be B es ( v ) in their model. Thenthe incentive compatibility condition for type v with a separating bribe is v ∈ arg max t Π ( v , t ) = F ( B es ( t ) + t )( v − B es ( t )) + (cid:90) v min { B es ( t )+ t , v } ( v − x ) f ( x ) dx , which implies B (cid:48) es ( v ) = f ( v + B es ( v ))( v − B es ( v )) F ( v + B es ( v )) − f ( v + B es ( v ))( v − B es ( v )) . (10)Under some regularity conditions, ES identify the set of equilibria that survive theD1 criterion. In such an equilibrium, for some ˆ v , the bribing function, denoted by B ( · ) , is B ( v ) = B es ( v ) if v < ˆ v , ˆ B ≡ ˆ v − F ( ˆ v + B es ( ˆ v ))( ˆ v − B es ( ˆ v )) otherwise. (11)where ˆ B ≥ ¯ v − E [ v | v ≥ ˆ v ] . The expected payoff function of bidder 1, denoted by In the robust equilibria in ES, there may exist a pooling bribe which for some types of bidder 1may exceed the valuations. But in that case, the bribe is accepted by all types of bidder 2 and thusbidder 1 benefits from the acceptance of high types of bidder 2. ( · ) , in such an equilibrium is Π ( v ) = F ( v + B ( v ))( v − B ( v )) if v ≤ ˆ v , v − ˆ B otherwise. (12) Proposition 3.
Suppose there exists a robust equilibrium in ES. Then π ( v ) ≥ Π ( v ) for any v .Proof. In our model the expected payoff of bidder 1 is π ( v ) = v − F ( b ( v ) + v ) b ( v ) and π (cid:48) ( v ) = F ( b ( v ) + v ) . In a robust equilibrium with a bribing func-tion B ( v ) in ES, the expected payoff of bidder 1 is Π ( v ) = F ( B ( v ) + v )( v − B ( v )) and the envelope theorem implies Π (cid:48) ( v ) = F ( B ( v ) + v ) for type v ≤ ˆ v ,which is also true for type v > ˆ v in fact. Thus π ( v ) − Π ( v ) = v ( − F ( B ( v )+ v ))+ F ( B ( v )+ v ) B ( v ) − F ( b ( v )+ v ) b ( v ) . So, whenever B ( v ) ≥ b ( v ) , π ( v ) − Π ( v ) ≥
0. On the other hand, whenever B ( v ) < b ( v ) , π (cid:48) ( v ) > Π (cid:48) ( v ) . Since π ( v ) = v ≥ Π ( v ) , π ( v ) ≥ Π ( v ) for anytype v .So far, we have focused on the separating equilibrium in which bidder 1’s re-quest is his valuation. Clearly any pair of ( b ( v ) , r ( v )) satisfying the incentivecompatibility condition in (4), with r ( v ) ≤ v , can sustain a separating equilib-rium if ( b ( v ) , r ( v )) is separating. Moreover, there may also exist some equilibriain which for some type v , r ( v ) > v so that the request is never accepted. A natu-ral question is whether the expected payoff of bidder 1 can be improved in some ofthose separating equilibria against the equilibrium identified above. We show be-low that in term of bidder 1’s expected payoff, any separating equilibrium in which r ( v ) (cid:54) = v is dominated by the one identified above with r ( v ) = v . Proposition 4.
Suppose in our model there exists a different separating equilib-rium with a pair of bribing and requesting functions ( β ( v ) , γ ( v )) . Let the ex-pected payoff of bidder 1 in the different equilibrium be π ( v ; β , γ ) . Then π ( v ) ≥ π ( v ; β , γ ) .Proof. Suppose first that γ ( v ) ≤ v for any type v . Clearly, π ( v ; β , γ ) = F ( β ( v )+ γ ( v ))( v − ( β ( v )+ γ ( v )))+ γ ( v ) . The envelope theorem implies that π (cid:48) ( v ; β , γ ) = ( β ( v ) + γ ( v )) . Thus, π ( v ) − π ( v ; β , γ ) = v − F ( b ( v ) + v ) b ( v ) − [ F ( β ( v ) + γ ( v ))( v − ( β ( v ) + γ ( v ))) + γ ( v )]= ( v − γ ( v ))( − F ( β ( v ) + γ ( v )))+ F ( β ( v ) + γ ( v )) β ( v ) − F ( b ( v ) + v ) b ( v ) . If b ( v ) < β ( v ) and b ( v )+ v < β ( v )+ γ ( v ) , then obviously π ( v ) − π ( v ; β , γ ) >
0. If b ( v ) < β ( v ) and b ( v ) + v ≥ β ( v ) + γ ( v ) , or if b ( v ) ≥ β ( v ) , then π (cid:48) ( v ) = F ( b ( v ) + v ) ≥ F ( β ( v ) + γ ( v )) = π (cid:48) ( v ; β , γ ) . Since π ( v ) = v ≥ π ( v ; β , γ ) , π ( v ) ≥ π ( v ; β , γ ) for any type v .Suppose now that γ ( v ) > v for some type v . For any such a type v , therequest is never accepted and the bribing function β ( v ) must satisfy the incentivecompatibility condition in ES’s model. Consequently, the expected payoff of anysuch type v must be no greater than the one in the equilibrium with r ( v ) = v .This completes the proof. We note that the previous results remain essentially unchanged even if there is areserve price in the auction. Specifically, given a positive reserve price, there existsa robust separating equilibrium in which conditional on the object being sold, i.e.,at least one of the bidders’ valuations exceeds the reserve price, the the collusionsuccess probability is one. The intuition is the following. Let the reserve pricebe R . Observe that any type of bidder 1 below R has no value for bidder 2. Inthe separating equilibrium similar to the above, these types are correctly identifiedas below R . So for any type v ≤ R , the equilibrium bribe b ( v ) =
0, which isnever accepted by any v > R , and any request r ( v ) > = ) will be rejected(accepted) by any type of bidder 2. On the other hand, for any type v > R , theequilibrium request is r ( v ) = v − R , i.e., the valuable part for bidder 2 since ifbidder 2 accept the request, she still needs to pay R to the auctioneer.More formally, when a proposal ( b , r ) from type v is separating and r ≤ v − R , then bidder 2 accepts b if v ≤ b + r + R and accepts r if v > b + r + R . So theexpected payoff of bidder 1 with type v is π ( v ) = F ( b + r + R )( v − b − R ) + − F ( b + r + R )) r , which can be rewritten as π ( v ) = F ( b + r + R )( v − ( b + r + R )) + r . The IC condition for the separating equilibrium with r ( v ) ≤ v − R becomes [ f ( b ( v ) + r ( v ) + R )( v − ( b ( v ) + r ( v ) + R )) − F ( b ( v ) + r ( v ) + R )]( b (cid:48) ( v ) + r (cid:48) ( v )) + r (cid:48) ( v ) = . In the equilibrium with r ( v ) = v − R , it is the same as (2). The only differenceis that the initial condition is changed into b ( v ) = r ( v ) = v ≤ R .Hence, the equilibrium proposal of any type of bidder 1 is always accepted bybidder 2 (although a zero request has no value for her if her valuation is not higherthan the reserve price). We observe that with the double-option scheme the initiator may also be able tosecure a successful collusion even if the auction format is changed to a first-priceauction. Rachmilevitch (2013) considers the same single-option scheme as in Es¨oand Schummer (2004) but with first price auctions. The important result from hismodel is that existence of a separating equilibrium is generally impossible and theremay even be no pooling equilibrium. In his model, an important feature of the (purestrategy) equilibrium of the continuation games is that when a proposal reveals theinitiator’s type perfectly and is rejected, the initiator bids his valuation, v , andthe rejectors submit the “minimally winning” bid if they find winning worthwhile,i.e., a value of v + , which wins with certainty against any v (cid:48) ≤ v (but pays v )and loses against any v (cid:48) > v . The driving force for his results of nonexistenceof equilibrium is that a high-type initiator has the incentive to cheat the opponentby mimicking a low type v because on the path the bribe b ( v ) of the low type v can be rejected and once it is rejected the high type can bid a value marginallyhigher than v + and win the auction at a low price. We note that a double-optionmodel for a first price auction shares the same feature of truthful bidding of theinitiator when the auction format is changed to a first price auction. It then impliesthat in the double-option model with a first price auction, for any given separatingproposal, the best response of the opponent is the same as the one in the case ofa second price auction. Hence, the expected payoff of the initiator is the same as20ell. Since on the path the proposal is always accepted, the continuation gamesare never played and thus become off-path events. Hence, the driving force forthe nonexistence of a separating equilibrium in Rachmilevitch (2013) disappearin the double-option model. Hence, the same separating weak perfect Bayesianequilibrium as in the case of a second price auction exists. On the other hand,whether the equilibrium survives any reasonable off-path refinement is beyond thescope of the current paper. We have examined a collusion model for second price auctions in which a bidderhas the opportunity to propose a combination of an offer and a request of bribes tothe other bidder. The bidders are involved in a multidimensional signaling game.Even when asymmetry is allowed, we show the collusion initiator with full bar-gaining power can always secure a successful collusion in such a dynamic environ-ment, as in the previous literature with a static environment. This result confirmsthe vulnerability of second price auctions in dynamic environment and is in sharpcontrast to the previous dynamic models with single-option considered in Es¨o andSchummer (2004). Furthermore, we show that the initiator’s payoff from sucha double-option scheme is generally improved against the single-option scheme.Thus an initiator with full bargaining power has strong incentive to implementsuch a scheme.
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