Equilibrium and Welfare Analysis in Second-Price Auctions with Resale and Costly Entry

Tis study investigates the efects of resale allowance on entry strategies, seller’s expected revenue, and social welfare in a second-price auction with two-dimensional private information on values and participation costs. We characterize the perfect Bayesian equilibrium in cutof strategies and identify sufcient conditions under which the equilibrium is unique. Our analysis suggests that resale allowance leads the low-value bidder to become more aggressive on entry, while high-value bidder has a lower incentive to enter. Furthermore, the allowance of resale can increase the social welfare under a sufcient condition, and its efect on expected revenue is ambiguous.


Introduction
When bidders participate in an auction, they often incur participation costs. For instance, sellers may charge an entry fee or require registration or pre-qualifcation for an auction. It may be costly for bidders to prepare bids, travel to the auction site, or acquire information about the auction rules and the values of the object to be auctioned. In the presence of such participation costs, not all potential bidders are willing to participate in an auction. In this case, to participate or not, as well as how to bid, should be modeled collectively. If the bidder with the highest value is excluded from the auction, the auction outcome is necessarily inefcient ex post. Tis sort of inefciency may create a motive for postauction resale and such resale opportunity afects signifcantly, as we shall show, bidders' entry strategies, social welfare, and expected revenue.
In this paper, we study the efects of a resale opportunity in second-price auctions when bidders are independently and privately informed about their participation costs and their valuations. Introducing post-auction resale into models with costly entry enriches the analysis, yet it complicates matters so that the existence and uniqueness of entry equilibrium are no longer obvious. Te additional difculty emerges since, with resale, the option value for staying out is also positive and varies with types. To make our analysis tractable, we apply a simple setting where the values, following a binomial distribution, can only be either high (v h ) or low (v l ). Te resale opportunity in the auction can have important implications for policy design and empirical studies on auctions. We believe that our research can bring insights on the study of second-price auctions with participation costs and resale under more general distributions on valuations.
First, we characterize the perfect Bayesian equilibrium in cutof strategies. For each case, we show that there is always an "intuitive" equilibrium in which the higher-value bidder participates in the auction more frequently than the lowervalue bidder does. While the symmetric entry equilibrium is unique in the no-resale benchmark, the uniqueness of the symmetric equilibrium cannot be established when resale is allowed. We also identify sufcient conditions that assure the uniqueness of the symmetric entry equilibrium in the resale case. Ten, we compare the symmetric entry equilibrium when resale is allowed to the equilibrium when resale is banned. Our frst fnding is that, with resale, the entry cutof is higher for low-value bidders and lower for high-value bidders. Tis suggests that when resale is allowed, low-value bidders become more aggressive on entry and high-value bidders are less likely to enter. Finally, we investigate the efects of a resale opportunity on seller's expected revenue and social welfare. Our fndings suggest that the opportunity of resale can increase the social welfare under a sufcient condition, and its efect on expected revenue is ambiguous, which has been well-documented in other auctions.
Few studies incorporate resale with costly entry. Che et al. [1] investigate the efect of resale allowance on entry strategies in a second-price auction with two bidders whose entries are sequential and costly. Tey show that there exists a unique threshold such that if the reseller's bargaining power is greater (less) than the threshold, resale allowance causes the leading bidder (the following bidder) to have a higher (lower) incentive on entry. Celik and Yilankaya [2] consider a setting in which valuations are private, but each bidder's cost is identical and commonly known. Tere may be resale due to asymmetric cutof equilibria, and the equilibria under resale are more asymmetric. Akyol [3] shows that resale induces more symmetric equilibria, higher revenues for the seller, and higher social welfare when valuations are commonly known, but costs are private. Akyol [4] considers second-price auctions with participation costs and investigates the revenue efects of a resale possibility. When values are drawn from a uniform distribution, resale increases (decreases) entry of the lower-cost (higher-cost) bidder and decreases the original seller' s expected revenue. Tese research studies on the equilibrium of second-price auctions apply single-dimensional framework, where either the valuations or the participation costs are common knowledge. Te closest paper to ours is by Xu et al. [5] who study the efect of resale allowance on entry strategies in a second-price auctions with two-dimensional private information on values and participation costs. Tey show that a symmetric equilibrium exists and is unique under some conditions. However, owing to the existence of incomplete multidimensional information, only some numerical examples can be derived for the efect of resale on revenue and efciency.
Te study proceeds as follows. Section 2 presents the model, after which Section 3 describes the equilibrium for the resale and no-resale cases. Tis section also compares the equilibria between the two cases. Sections 4 and 5 present the results for the seller's revenue and social welfare, respectively. Lastly, we conclude the paper. Appendix contains all proofs omitted from the main body of the study.

The Setup
We consider an environment with one seller and two potential bidders. Te seller is risk neutral and has an indivisible object to be auctioned. Te seller values the object at zero. Each bidder i's value is v i ∈ v l , v h with 0 < v l < v h ≤ 1, and they are risk neutral and independent. It is assumed that In order to participate in the auction, bidder i must incur a nonrefundable participation cost c i which is a private information and drawn from a distribution function G(·) with support [0, 1]. Te corresponding density for participation costs is g(c). When a bidder is indiferent between participating in the seller's auction or not, we assume that he participates for illustration convenience. When a bidder submits a bid, he knows who else will submit a bid. Bidders do not know the participation decisions of other bidders when they make their own decisions.
Te auction format is the usual second-price sealed-bid auction. A bidder with the highest bid wins the auction and pays the second highest bid. We investigate two cases. In the benchmark case, which we call the no-resale case, the game ends after the auction ends. In the second case, which we call the resale case, after the auction takes place, the winner of the object, if any, has the opportunity to sell the object to the other bidder by a standard Nash bargaining game. Te bargaining power parameters of the reseller and the buyer are λ and 1 − λ, respectively, where λ ∈ (0, 1). Furthermore, we assume that the resale stage is costless.
Since bidders are risk neutral, they participate in the auction if and only if the expected payof they can get from participating is greater than or equal to his expected revenue from staying out of the initial auction; we naturally restrict ourselves to equilibria in which each bidder uses a cutof entry strategy. Tat is, if a bidder's participation cost is below a certain threshold level, he participates in the auction. Otherwise, he chooses not to participate in the auction. In both cases, we look for (Perfect Bayesian Nash) equilibria where a bidder's strategy consists of an entry decision and bidding behavior. More precisely, the individually rational action set for any type of bidder is No where c i ′ denotes bidder i's entry cutof and m denotes the number of bidders participating in the auction. For the game described above, each bidder's action is to choose a cutof and decides how to bid when he participates. Tus, an (Bayesian-Nash) equilibrium of the sealed-bid second-price auction with participation costs is composed of bidders' cutof strategies as well as the corresponding bidding strategies.

Equilibrium Characterization
In this section, we characterize the equilibrium in the noresale benchmark case and in the case when resale is allowed. We will focus on the symmetric entry equilibrium characterized by a pair of entry thresholds, (c * l , c * h ) in the case when resale is banned and (c l , c h ) in the case when resale is allowed, so that a bidder with a type (c, v k ) enters the auction if and only if c ≤ c * k in the no-resale benchmark and c ≤ c k in the case with resale, k ∈ l, h { }; in other words, bidders with the same value will follow the same entry cutof in a symmetric entry equilibrium.

Defnition 1. When two bidders have the same distribution on participation costs, an equilibrium
; otherwise, it is an asymmetric equilibrium.

No-Resale
Case. Note that, once a bidder enters the auction, he can observe who has also entered the auction and thus update his belief about others' valuation distributions. If we observe that bidder i participates in the auction, it can be inferred that bidder i's cost is lower than or equal to c * i . Ten, by Bayes' rule, for bidders who enter the auction, . We focus on the symmetric equilibrium in which all bidders use the same cutof strategy We also know 0 < c * i < 1, since bidder i will not enter if his cost is higher than his valuation (c * i ≤ v i < 1). In the frst bidding stage, upon entry, it is a weakly dominant strategy for each bidder to bid his value, which is shown by Tan and Yilankaya [6], Xu et al. [5], Cao and Tian [7], Celik and Yilankaya [2], and Cao et al. [8].
In the entrance decision making stage, a bidder participates if and only if , the expected revenue from participating is equal to his participation cost. For any bidder i, the expected revenues from participating are given by the following equations for For bidder i with value v l , he has positive surplus only when he is the only bidder submitting a bid. In this case, he can bid 0 and pay nothing. Tus, low-value bidder, with cost c * l , has an expected payof of from participating in the auction. If he does not enter, his payof is 0. For bidder i with value v h , the expected payof can be divided into two parts. Te frst part is the payof when he is the only bidder submitting a bid. In this case, he can bid 0 and obtain the object being auctioned. Te second part is the expected payof when there is another bidder submitting bid. Tus, high-value bidder, with cost c * h , has an expected payof of from participating in the auction. If he does not enter, his payof is 0. In equilibrium, bidder i with c * i must be indiferent between entering and not entering. Terefore, the equilibrium (c * l , c * h ) is a solution of the following equation system: We then have the following result.

Proposition 1. Suppose that each bidder's participation cost is independently drawn from a distribution function G(·)
with support [0, 1]. In the no-resale benchmark case, there always exists a unique symmetric equilibrium (c * l , c * h ) where each bidder uses the same cutof strategy determined by (P1).

Resale Case.
We now augment the model just analyzed by allowing a resale stage where the auction winner may resell the item. Similar to the benchmark case without resale, we will focus on symmetric entry equilibria characterized by entry thresholds (c l , c h ). In reality, resale can be conducted in diferent formats, such as bargaining, optimal auction, and monopoly pricing. In this study, we follow Gupta and Lebrun [9], Pagnozzi [10], Cheng [11], and Zhang and Wang [12] to assume that the resale stage is conducted in a standard Nash bargaining game. Te bargaining power parameters of the reseller and the buyer are λ and 1 − λ, respectively, where λ ∈ (0, 1).
To characterize symmetric entry equilibria, we start our analysis from the last stage. Given the special features in our model, it is easily seen that, in equilibrium, the initial auction winner can only possibly beneft from resale when his value is v l . So, resale can only be initiated when low-value bidder wins the initial auction, and the potential buyer participating in resale must be a high-value bidder who stays out of the initial auction.
Next, we consider the auction stage. Note that, if bidder i enters the auction while the other does not enter, bidder i bids zero. If both bidders participate in the auction, upon entry, the auction format is the standard second-price auction. In that case, there is an equilibrium in which both bidders bid their values. Absent costs are shown by Hafalir and Krishna [13] and Virág [14]. Since participation costs are sunk costs, they do not afect the analysis. Tus, "bid-yourvalue" constitutes a robust equilibrium independent of the value distributions. When there are more than two potential bidders, "bid-your-value" may not be an equilibrium. In this case, the low-value bidder would bid the "adjusted value," that is, the expected value from a possible resale to the highvalue bidder, which is larger than his true value. Tis is shown by Xu et al. [5] and Celik and Yilankaya [2].
Finally, we look at entry behavior in the initial auction. When there is a resale opportunity after the auction ends, the lower-value bidder will still get the object only if no other bidder enters the auction, and if possible, he can sell it in the resale stage at a price λv l + (1 − λ)v h . Tus, the low-value bidder with cost c l has an expected payof of from participating. If he does not enter, he will get 0 with certainty, regardless of the other bidder's entry decision.
For bidder i with value v h , he will get the object with certainty if he enters the auction; the expected payof can be divided into two parts. Te frst part is the payof when he is the only bidder submitting a bid. In this case, he can bid 0 Discrete Dynamics in Nature and Society 3 and obtain the object being auctioned. Te second part is the expected payof when there is other bidder submitting bids. In this case, bidder will bid truth value. Tus, the high-value bidder with cost c h has an expected payof of from participating. Note that if he does not enter, he will get the object in the resale stage if low-value bidder enters the auction in the frst stage, but will have to pay λv l In equilibrium, bidder i with c i must be indiferent between entering and not entering. Terefore, the equilibrium (c l , c h ) is a solution of the following equation system:

Proposition . Assume that each bidder's participation cost is independently drawn from a distribution function G(·) with positive density g(c i ) over (0,1). In the resale case, (i) there always exists a symmetric equilibrium where each bidder uses the same cutof strategy (c l , c h ), and (ii) if
, then the symmetric equilibrium is unique.
Te condition in Proposition 2 can be easily satisfed, for instance, when participation costs are more dispersed. As an illustrative example when G(.) follows a uniform distribution on [0, 1], in this case, independent of the distribution of the valuations, the equilibrium is unique.
In general, it is very difcult to obtain closed-form solutions for the entry equilibrium in both the no-resale and resale cases; nonetheless, we can establish the following comparison results.

Resale allowance leads the low-value bidders to become more aggressive on entry and high-value bidders have a lower incentive to enter.
Resale afects entry cutofs for both types. Our comparison in Proposition 3 suggests that when resale is allowed, the entry cutof for the high-value bidder becomes lower, implying that he has a lower incentive to enter; because the resale opportunity is available and he might be able to obtain the item from the post-auction resale, the bidder would prefer to directly attend the resale market to avoid the participation cost in the auction. On the contrary, the resale allowance would also encourage bidders to enter the auction, i.e., low-value bidders become more aggressive on entry, because the possibility of reselling the object to the other bidder may generate a higher expected payof.

Social Welfare
We next investigate the efect of resale on social welfare. Tere is a line of thought that resale will ensure full efciency if there is any inefciency in the allocation of an auction. However, Hafalir and Krishna [15] show that this may not be the case when there is uncertainty regarding values. Tat is, resale may not induce fully efcient outcomes. To examine the welfare efect of resale, we consider the social surplus as a function of two entry cutofs. Tis surplus function is constructed under the assumption that, once the bidders enter in or stay out of the initial auction according to these entry cutofs, they follow the equilibrium bidding and resale strategies described above.
First, we consider the no-resale case; the total surplus (TS) when bidders conform to cutof strategies (c * h , c * l ) is the sum of the payofs of the bidders and the seller, minus the expected participation costs of the bidders. Tat is, Te frst term refers to the expected surplus if the good is allocated to a low-value bidder with v l . Te second term is the expected surplus if the good is allocated to a high-value bidder with v h . Te last term measures the expected costs of participation. Simplifying it, we obtain 4 Discrete Dynamics in Nature and Society Next, we consider the case when resale is allowed. Similarly, for any (c h , c l ), we have Compared to TS(c * h , c * l ), the frst two terms refer to the expected surplus if the initial auction allocates the good to a low-cutof bidder with v l , which is calculated by taking the possibility of resale into account. Simplifying it, we obtain For fxed cutofs, the possibility of resale increases total welfare: Tis diference is simply the surplus gain of transferring the object from a low-value bidder to a high-value bidder in the resale phase. Consider an equilibrium in the benchmark case of no-resale with symmetric cutofs c * l and c * h . As we just observed, if bidders were to use the same cutofs when resale is allowed, then the surplus would be higher, i.e., TS R (c * l , c * h ) > TS(c * l , c * h ). However, the possibility of resale may also change equilibrium participation behavior of the bidders. Terefore, we need to know how the value of function TS R changes as we move from the no-resale equilibrium cutofs (c * l , c * h ) to resale equilibrium cutofs (c l , c h ). With our next result, we provide a sufcient condition for the social surplus to increase when resale is allowed.

Proposition 4. Assume that each bidder's participation cost is independently drawn from a distribution function G(·)
with support [0,1] and λ < (1/2). Te social welfare is higher in the resale case compared with that in the no-resale case.
Proposition 4 provides a sufcient condition on the bargaining power parameter λ only. Hence, it is independent of the distribution of participation costs and the magnitude of the valuations.

Remark 1.
Taking the derivative of the social surplus function with respect to λ, it is easy to know (zT S R /zλ) < 0. Tus, the social surplus is decreasing in λ.

Seller's Expected Revenue
In this section, we examine how the seller's expected revenue is afected by resale. Te seller obviously benefts when both bidders enter more frequently. However, in our setup, if one bidder enters more frequently, then the other enters less frequently, as shown in Proposition 3. Terefore, comparing the seller's revenue between the resale and no-resale case is not a trivial question, furthermore, we may face multiple equilibria in the resale case. Tus, a comparison becomes more difcult, and we need to consider all equilibria and the expected revenues they induce.
Here,SS(c * l , c * h ) denotes the original seller's expected revenue in the no-resale case; the seller's expected revenue is Te frst term is the expected payment when two lowvalue bidders participate in the auction; the second term is the expected payment when both low-value bidder and highvalue bidder participate in the auction; the third term is the expected payment when two high-value bidders participate in the auction.
In the resale case, the seller's expected surplus is the total surplus minus the bidders' surplus. Te bidders' surplus is BS R (c h , c l ) with resale, so we can utilize SS R (c h , c l ) � TS R (c h , c l ) − BS R (c h , c l ) to compute the seller's expected surplus in the resale case and Discrete Dynamics in Nature and Society 5 where is the highvalue bidder's expected payof, which is calculated by taking the possibility of resale into account. c l 0 (c l − c)dG(c) is the low-value bidder's expected payof. Simplifying it, we obtain Tus, Comparing (14) and (17), we can also utilize SS(c h , c l ) to denote the seller's expected surplus in the resale case, since the initial seller obtains revenue only if both bidders enter the auction. Tat is,the seller's expected surplus only depends on the bidders' cutofs when both bidders enter the auction. According to the derivatives of this seller's expected surplus function with respect to its two arguments, it is easy to know the seller's expected surplus is increasing in equilibrium cutofs for the set of points. We need to know how the value of function SS changes as we move from the no-resale equilibrium cutofs (c * l , c * h ) to resale equilibrium cutofs (c l , c h ). As implied in Propositions 3, c * l < c l , c * h > c h , resale induces the competition efect (low-value bidders bid more aggressively) and displacement efect (high-value bidders are replaced by low-value bidders). While the competition efect pushes up expected revenue, the displacement efect works in the other direction. Tus, the net impact of resale on expected revenue depends on which efect dominates, and either efect may dominate in our setting.

Proposition 5. Assume that each bidder's participation cost is independently drawn from a distribution function G(·)
with support [0,1] and 0 < v l < v h ≤ 1. Te efect of resale allowance on the original seller's expected revenue is ambiguous.
As an illustrative example when G(.) follows a uniform distribution on [0, 1], we show the unclear resale allowance. Tat resale has an ambiguous efect on expected revenue which has been well-documented in other auction settings with resale (e.g., [16,18]; it can be explained intuitively in our setting based on the competition efect and the displacement efect induced by resale opportunities. Tis contradicts earlier fndings that resale increases the seller's expected revenue, as demonstrated by Hafalir and Krishna [15], Akyol [3], and Garratt and Georganas [17].

Conclusion and Further Study
In this paper, we study the efects of resale allowance in second-price auctions with two-dimensional private information on values and participation costs. Te values can only be either high (v h ) or low (v l ) while the participation costs are drawn from any general distributions. We demonstrate that the symmetric entry equilibrium is characterized by entry cutofs, and we identify conditions under which such an equilibrium is unique in the resale case. Our comparison shows that high-value bidders have a lower incentive to enter, and they would prefer to directly attend the resale market to avoid the participation cost. However, low-value bidders become more aggressive on entry because the possibility of reselling the object to the other bidders may generate a higher expected payof.
We assume that there are no participation costs at the resale stage; our analysis suggests that the opportunity of resale can increase the social welfare under a sufcient condition, and its efect on expected revenue is ambiguous. Te implication is that a market regulator, whose objective is to maximize the social surplus or expected revenue, should exercise caution in suggesting whether or not resale should be permitted in auction settings similar to what is under our consideration.
Tis study gives a try to integrate and analyze both entry and resale in the second-price auction model with two-dimensional private information, but our analysis relies on several key assumptions; relaxation of those assumptions to 6 Discrete Dynamics in Nature and Society allow for a more general analysis on entry and resale is left for future research [18].
Proof of Equation 4. In the resale case, the seller's expected surplus is the total surplus minus the bidders' surplus; we have SS R c h , c l � TS R c h , c l − BS R c h , c l ,

□ Data Availability
Te data that support the fndings of this study can be obtained from the corresponding author upon reasonable request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.