# A First-Price Sealed-Bid Asymmetric Auction When Two Bidders Have Respective CRRA and General Utility Functions

Authors: Mingming Gong; Shulin Liu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

We study a first-price auction with two bidders where one bidder is characterized by a constant relative risk aversion utility function (i.e., a concave power function) while the other has a general concave utility function. We establish the existence and uniqueness of the optimal strategic markups and analyze the effects of one bidder’s risk aversion level on the optimal strategic markups of him and his opponent’s, the allocative efficiency of the auction, and the seller’s expected revenue, respectively.


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## Body

## 1. Introduction

Though the rules of a first-price auction are simple, the same cannot necessarily be said for the case that bidders are asymmetric, especially when the bidders are risk averse. Previous studies on the asymmetric auctions with risk neutral bidders within the independent private value (IPV) model have been extensively studied by Lebrun [1], Maskin and Riley [2, 3], Kirkegaard [4], and Kaplan and Zamir [5]. Also, the symmetric auctions with risk-averse preferences have been much studied in the literature (e.g., Riley and Samuelson [6], Matthews [7], and Hu et al. [8]). In this paper, we build on the model outlined by Maréchal and Morand [9] (henceforth MM) and study first-price sealed-bid auctions with asymmetric bidders in terms of risk aversion.The MM’s model is initiated by Von Ungern-Sternberg [10] (henceforth VUS) in his theoretical analysis on Swiss construction industry of simultaneous sealed-bid multiobject auctions, where each bidder are ex ante symmetric relative to the informational knowledge and he can predict his private valuation with certainty but has no grounds for believing it to be higher or lower on average than his opponent’s. Formally, VUS models this by assuming that the different bidders’ valuations are independent drawings from a uniform distribution with an unknown mean; thereby, the bidders can use this information to infer the mean and indirectly infer his opponent’s valuation. Therefore, the ex ante and ex post distribution are not the same, which is the main difference between the standard IPV model and the VUS model.MM follows the VUS model and extends it to the case of two risk-averse bidders who had the same parametric family of (CRRA) functions, but their measures of risk aversion differ. Then, they derive the explicit expressions of asymmetric bidding equilibrium. To the best of our knowledge, it seems that only Maréchal and Morand consider asymmetric first-price auctions where two bidders assume different CRRA utility functions. By allowing risk aversion levels of both bidders to vary simultaneously, they show that as one bidder becomes less risk averse and the other one becomes more risk averse, (i) the less risk-averse bidder reduces his markup when asymmetry in terms of risk aversion between both bidders is sufficiency large; (ii) the allocative efficiency decreases when asymmetry increases; and (iii) the seller’s expected revenue increases as bidders become more asymmetric in terms of risk aversion.We generalize the work by MM to a more general case where one bidder is characterized by a CRRA utility function while the other has a general concave utility function. Our work makes a contribution to the asymmetric auction literature and is different from MM in several respects: (i) we establish the existence and uniqueness of the optimal strategic markups of two bidders (see Propositions1 and 2); (ii) we study effects of one bidder’s risk aversion on bidding behavior. We find that when one bidder becomes more risk averse and the other bidder’s risk averse remain unchanged, both bidders reduce their optimal strategic markups, and an illustrative example shows that the rate at which his markup decreases is higher than the rate at which his opponent’s markup decreases (see Propositions 3 and 4); (iii) the allocative efficiency becomes complicated as one bidder becomes more (less) risk averse (see our Proposition 6); (iv) we study effects of one bidder’s risk aversion on expected revenue. We find that the seller’s expected revenue increases when one bidder becomes more risk averse, given that the other bidder’s risk averse remain unchanged (see Proposition 7). Section 4 concludes the paper.


## 2. Model

In this paper, we study a two-bidder first-price sealed-bid auction selling an indivisible single item whereas adopting the framework of the model as in MM. Differentiating from their framework, we further generalize the MM model by accommodating the case where one bidder has a general concave utility function and the other still has a CRRA utility function.


### 2.1. The MM’s Model

In MM, the key assumptions conclude the following: (a) both bidders are ex ante symmetric relative to the informational knowledge; (b) each bidder has a private valuation; (c) the valuations are independently drawn from a known uniform distributionF and distributes around the unknown mean μ over [μ − a, μ + a] with frequency 1/2a, where a is common knowledge; (d) the optimal strategy is to bid a simple markup b below his valuation and the markup b is independent of his valuation.Such as bidderα, when he observes his own valuation vα, since assumption (c), he can draw inferences about μ ∈ [vα − a, vα + a] according to the cumulative Fμ with corresponding density fμ. Due to assumption (a), he can indirectly draw inferences about his opponent’s valuation vβ ∈ [vα − 2a, vα + 2a] according to the cumulative Fβ with corresponding density fβ.Suppose bidderi ∈ {α, β} has a private valuation of vi and places a bid Bi. Due to assumption (d), the bid has the form Bi (vi) = vi − bi. When his competitor chooses a markup equal to b−i (−i means i’s opponent), bidder i’s probability of winning Probi (win) is given by(1)∫μprobBi>B−ifμμdμ.Consider bidderα, the probability of winning P is(2)P=∫μFβvα−bα+bβfμμdμ,μ∈vα−a,vα+a.Then, we have(3)P=∫μ12aa−bα+bβ+vα−μ2adμ.Similarly, consider bidderβ, the probability of winning Q is(4)Q=∫μFαvβ−bβ+bαfμμdμ,μ∈vβ−a,vβ+a.Then, we have(5)Q=∫μ12aa+bα−bβ+vβ−μ2adμ.


### 2.2. The General Utility Function of Bidders

Bidders are risk averse and one bidder has a CRRA utility functionuix=xρi (with 0 < ρi ≤ 1), while the other bidder −i has a general utility function u−i (x) satisfying u−i (0) = 0, u−i′x>0, and u−i″x≤0, x is bidder’s income. Specifically, we consider two models as shown in the following:Model I: suppose that bidderα has a utility function uαx=xρα (with 0 < ρα ≤ 1) and bidder β has a general utility function uα (x) satisfying uα (0) = 0, uα′x>0, and uα″x≤0. Then, the respective expected utility maximization problems for bidders α and β are(6)maxbαEuα=bαραPbα,bβ,(7)maxbβEuβ=uβbβQbα,bβ,whereP (bα, bβ) and Q (bα, bβ) are the respective probabilities of winning of bidder α and β when they choose strategic markups bα and bβ,(8)Pbα,bβ=2a−bα+bβ28a2,(9)Qbα,bβ=4a2+4abα−bβ−bα−bβ28a2.Model II: suppose that bidderα has a general utility function uα (x) and bidder β has a utility function uβx=xρβ (with 0 < ρβ ≤ 1) (see page 109 of MM for the detailed proof). Then, the respective expected utility maximization problems for bidders α and β are(10)maxbαEuα=uαbαPbα,bβ,(11)maxbβEuβ=bβρβQbα,bβ,wherebα, bβ, P (bα, bβ), and Q (bα, bβ) have the same meanings as in Model I.Defineγi=ui/ui′. Let Ri=−ui″/ui′ denote the Arrow–Pratt measure of absolute risk aversion. Since γi′=ui′2−uiui″/ui′2=1−γiui″/ui′, γi is related to Ri byγi′=1+Riγi. Since γi (x) ≥ 0 and Ri (x) ≥ 0 for x ≥ 0, γi′x>0forx≥0. Let u^i be another utility function of bidder i satisfying the same assumptions as ui, with an absolute risk aversion measure R^ix such that R^ix>Rix on (0, ∞]. Then, R^ix>Rixon (0, ∞) implies γ^ix>γix on (0, ∞) (see page 1191 of Hu et al. [8]).Throughout the paper, we keep the assumption ofbα ≥ bβ. (Similarly, as in MM, the two probabilities given in (8) and (9) will be changed when we assume that bα ≤ bβ. With this assumption, we can get results that are entirely parallel to the existing propositions in Section 3.) That is used to derive the two probabilities of winning of (8) and (9) as in MM. The following examples verify this assumption.Example 1.
Suppose that bidderα has a CRRA utility function uαx=xρα with 0 < ρα ≤ 1, and bidder β has a constant absolute risk aversion (CARA) utility function uβ (x) = 1 − exp (−θβx) with θβ > 0. Substituting uβ (x) into (15) yields(12)−1+eθβbβθβ=2a2ρα+2bβ+2a−bβ+2aρα+2.
This and (14) constitute a system. Let a = 1. Solving the respective six systems associated with six pairs of (ρα, θβ), we obtain six pairs of optimal markups as shown in Table 1.
Table1 shows the optimal markups satisfy bα ≥ bβ.Table 1
Verification of the assumptionbα ≥ bβ.
ρα0.80.80.90.90.90.9θβ0.60.70.30.70.80.9bα0.79710.79260.89550.87260.86740.8624bβ0.79000.77430.88580.81180.79510.7790Example 2.
Suppose that bidderα has a CARA utility function uα (x) = 1 − exp (−θαx) with θα > 0, and bidder β has a CRRA utility functionuβx=xρβ with 0 < ρβ ≤ 1. Substituting uα (x) into (18) yields(13)−1+eθαbαθα=2a−bα+2a−bα2+8a2ρβ2+2ρβ22+ρβ.
This and (17) constitute a system. Let a = 1. Solving the respective six systems associated with six pairs of (θα, ρβ) as shown in the following Table 2, we obtain six pairs of optimal markups.
Table2 shows that the optimal markups satisfy bα ≥ bβ.Table 2
Verification of the assumptionbα ≥ bβ.
θα0.20.30.350.40.410.8ρβ0.70.80.80.80.80.2bα0.87620.86350.84920.83540.83280.6400bβ0.79150.83470.82690.81940.81790.3114


## 2.1. The MM’s Model

In MM, the key assumptions conclude the following: (a) both bidders are ex ante symmetric relative to the informational knowledge; (b) each bidder has a private valuation; (c) the valuations are independently drawn from a known uniform distributionF and distributes around the unknown mean μ over [μ − a, μ + a] with frequency 1/2a, where a is common knowledge; (d) the optimal strategy is to bid a simple markup b below his valuation and the markup b is independent of his valuation.Such as bidderα, when he observes his own valuation vα, since assumption (c), he can draw inferences about μ ∈ [vα − a, vα + a] according to the cumulative Fμ with corresponding density fμ. Due to assumption (a), he can indirectly draw inferences about his opponent’s valuation vβ ∈ [vα − 2a, vα + 2a] according to the cumulative Fβ with corresponding density fβ.Suppose bidderi ∈ {α, β} has a private valuation of vi and places a bid Bi. Due to assumption (d), the bid has the form Bi (vi) = vi − bi. When his competitor chooses a markup equal to b−i (−i means i’s opponent), bidder i’s probability of winning Probi (win) is given by(1)∫μprobBi>B−ifμμdμ.Consider bidderα, the probability of winning P is(2)P=∫μFβvα−bα+bβfμμdμ,μ∈vα−a,vα+a.Then, we have(3)P=∫μ12aa−bα+bβ+vα−μ2adμ.Similarly, consider bidderβ, the probability of winning Q is(4)Q=∫μFαvβ−bβ+bαfμμdμ,μ∈vβ−a,vβ+a.Then, we have(5)Q=∫μ12aa+bα−bβ+vβ−μ2adμ.


## 2.2. The General Utility Function of Bidders

Bidders are risk averse and one bidder has a CRRA utility functionuix=xρi (with 0 < ρi ≤ 1), while the other bidder −i has a general utility function u−i (x) satisfying u−i (0) = 0, u−i′x>0, and u−i″x≤0, x is bidder’s income. Specifically, we consider two models as shown in the following:Model I: suppose that bidderα has a utility function uαx=xρα (with 0 < ρα ≤ 1) and bidder β has a general utility function uα (x) satisfying uα (0) = 0, uα′x>0, and uα″x≤0. Then, the respective expected utility maximization problems for bidders α and β are(6)maxbαEuα=bαραPbα,bβ,(7)maxbβEuβ=uβbβQbα,bβ,whereP (bα, bβ) and Q (bα, bβ) are the respective probabilities of winning of bidder α and β when they choose strategic markups bα and bβ,(8)Pbα,bβ=2a−bα+bβ28a2,(9)Qbα,bβ=4a2+4abα−bβ−bα−bβ28a2.Model II: suppose that bidderα has a general utility function uα (x) and bidder β has a utility function uβx=xρβ (with 0 < ρβ ≤ 1) (see page 109 of MM for the detailed proof). Then, the respective expected utility maximization problems for bidders α and β are(10)maxbαEuα=uαbαPbα,bβ,(11)maxbβEuβ=bβρβQbα,bβ,wherebα, bβ, P (bα, bβ), and Q (bα, bβ) have the same meanings as in Model I.Defineγi=ui/ui′. Let Ri=−ui″/ui′ denote the Arrow–Pratt measure of absolute risk aversion. Since γi′=ui′2−uiui″/ui′2=1−γiui″/ui′, γi is related to Ri byγi′=1+Riγi. Since γi (x) ≥ 0 and Ri (x) ≥ 0 for x ≥ 0, γi′x>0forx≥0. Let u^i be another utility function of bidder i satisfying the same assumptions as ui, with an absolute risk aversion measure R^ix such that R^ix>Rix on (0, ∞]. Then, R^ix>Rixon (0, ∞) implies γ^ix>γix on (0, ∞) (see page 1191 of Hu et al. [8]).Throughout the paper, we keep the assumption ofbα ≥ bβ. (Similarly, as in MM, the two probabilities given in (8) and (9) will be changed when we assume that bα ≤ bβ. With this assumption, we can get results that are entirely parallel to the existing propositions in Section 3.) That is used to derive the two probabilities of winning of (8) and (9) as in MM. The following examples verify this assumption.Example 1.
Suppose that bidderα has a CRRA utility function uαx=xρα with 0 < ρα ≤ 1, and bidder β has a constant absolute risk aversion (CARA) utility function uβ (x) = 1 − exp (−θβx) with θβ > 0. Substituting uβ (x) into (15) yields(12)−1+eθβbβθβ=2a2ρα+2bβ+2a−bβ+2aρα+2.
This and (14) constitute a system. Let a = 1. Solving the respective six systems associated with six pairs of (ρα, θβ), we obtain six pairs of optimal markups as shown in Table 1.
Table1 shows the optimal markups satisfy bα ≥ bβ.Table 1
Verification of the assumptionbα ≥ bβ.
ρα0.80.80.90.90.90.9θβ0.60.70.30.70.80.9bα0.79710.79260.89550.87260.86740.8624bβ0.79000.77430.88580.81180.79510.7790Example 2.
Suppose that bidderα has a CARA utility function uα (x) = 1 − exp (−θαx) with θα > 0, and bidder β has a CRRA utility functionuβx=xρβ with 0 < ρβ ≤ 1. Substituting uα (x) into (18) yields(13)−1+eθαbαθα=2a−bα+2a−bα2+8a2ρβ2+2ρβ22+ρβ.
This and (17) constitute a system. Let a = 1. Solving the respective six systems associated with six pairs of (θα, ρβ) as shown in the following Table 2, we obtain six pairs of optimal markups.
Table2 shows that the optimal markups satisfy bα ≥ bβ.Table 2
Verification of the assumptionbα ≥ bβ.
θα0.20.30.350.40.410.8ρβ0.70.80.80.80.80.2bα0.87620.86350.84920.83540.83280.6400bβ0.79150.83470.82690.81940.81790.3114


## 3. Optimal Strategic Markups and Their Properties

### 3.1. Existence of Optimal Strategic Markups

Proposition 1.
Consider Model I. Then,(i)
The optimal strategic markups for both bidders (bα, bβ) are characterized by (when uβx=xρβ, we get the same optimal strategic markups as in MM)(14)bα=ρα2a+bβ2+ρα,(15)γβbβ=Gbβ,where(16)Gx=2a2ρα+2x+2a−x+2aρα+2,on0,∞.(ii)
There exists a unique optimal strategic markup (bα, bβ) with bα, bβ ∈ (0, aρα].(iii)
bα and bβ increase with the uncertainty parameter a.Proof.
See Appendix.
Part (iii) of Proposition1 is in line with intuition and conventional results in auction theory (see among others Klemperer [11] and Krishna [12]).Proposition 2.
Consider Model II. Then,(i)
The optimal strategic markups for both bidders (bα, bβ) are characterized by (when uαx=xρα, we have the same optimal strategic markups as in MM)(17)bβ=gbα,(18)γαbα=Lbα,whereg (x) and L (x) are defined on [0, ∞),(19)gx=1+ρβx−2a+2a−x2+8a2ρβ2+2ρβ2+ρβ,(20)Lx=2a−x+2a−x2+8a2ρβ2+2ρβ22+ρβ.(ii)
There exists a unique optimal strategic markup (bα, bβ) with bα, bβ ∈ [aρβ, a).(iii)
bα and bβ are increasing with respect to the uncertainty parameter a.Proof.
See Appendix.


### 3.2. Impact of the Degrees of Risk Aversion on Bidders’ Optimal Markups

Suppose that bidderα’s risk aversion level remains unchanged. Then, Proposition 3 (i) shows that the optimal strategic markups for both bidders α and β decrease in risk aversion of bidder β. Part (ii) shows that a change in bidder β’s optimal strategic markup is greater than a change in bidder α’s optimal strategic markup as bidder β becomes more risk averse.Proposition 3.
Consider Model I. Letbα and bβ be the optimal strategic markups for bidder α and β associated with utility functions uαx=xρα and uβ (x), satisfying that bα ≥ bβ. Let b^α and b^β be the optimal strategic markups for both bidders associated with utility functions uαx=xρα andu^β, satisfying that b^α≥b^β. Then, we have the following:(i)
b^α<bα, b^β<bβ(ii)
bα−b^α<bβ−b^β (i.e.,vα−b^α−vα−bα<vβ−b^β−vβ−bβ)Proof.
See Appendix.
The following Examples3 and 4 illustrate the influence of the degrees of bidder β’s risk aversion on the markups of both bidders.Example 3.
Consider Example1 again. Suppose that bidder β becomes more risk averse. Let θβ become θβ + ε, where ε is a positive real number. Then, the optimal strategic markups bα and bβ satisfy(21)−1+eθβ+εb^βθβ+ε=2a2ρα+2b^β+2a−b^β+2aρα+2,b^α=ρα2a+b^β2+ρα,b^α,b^β0,αρα.
Figure1 depicts the optimal strategic markups bα and bβ for ε ∈ (0, 0.1) (the choice of ε’s range must guarantee the optimal strategic markups exist in the interval (0, aρα), where a = 1, θβ = 0.6, and ρα = 0.8. It shows that both optimal strategic markups are declining with ε, implying that as bidder β becomes more risk averse, both bα and bβ decrease (i.e., part (i) of Proposition 3 holds), bβ is more rapidly decreased than bα, and the magnitude of the asymmetry effects on the optimal bid markup becomes smaller for bidder α but becomes bigger for bidder β. It also shows that the magnitude of the asymmetry effects is less for bidder α than that for bidder β (i.e., part (i) of Proposition 3 holds).Figure 1
Plot of the optimal strategic markupsbα and bβ increase with the degrees of bidder β’s risk aversion.Example 4.
Consider that both bidders exhibit constant relative risk aversion (CRRA) utility functions,uix=xρi (with 0 < ρi ≤ 1, i ∈ {α, β}), where 1 − ρi is the Arrow–Pratt measure of CRRA. Then, we can derive explicitly the optimal strategic markups for both bidders(22)bα=aρα1+A2+ρα+ρβ,bβ=a2+ραA−2−ρα−2ρβ2+ρα+ρβ,whereA=1+2ρβ2+ρα+ρβ.
Suppose that bidderβ becomes more risk averse. Let ρβ become ρβ−δ, where δ is a positive real number. Then, we have(23)b^α=aρα1+A^2+ρα+ρβ−δ,b^β=a2+ραA^−2−ρα−2ρβ−δ2+ρα+ρβ−δ,where A^=1+2ρβ−δ2+ρα+ρβ−δ.
Figure2 depicts the optimal strategic markups bα and bβ for δ ∈ (0, 0.55), where a = 1, ρα = 0.9, and ρβ = 0.6. It shows that both optimal strategic markups are declining with δ, implying that as bidder β becomes more risk averse, both bα and bβ decrease (i.e., part (i) of Proposition 3 holds), bβ is more rapidly decreased than bα, and the magnitude of the asymmetry effects on the optimal bid markup becomes smaller for bidder α but becomes bigger for bidder β. It also shows that the magnitude of the asymmetry effects is less for bidder α than that for bidder β (i.e., part (ii) of Proposition 3 holds).
Suppose that bidderβ’s risk aversion level remains unchanged. When bidder α becomes more risk averse, Proposition 4 has a similar meaning as Proposition 3.Figure 2
Plot of the optimal strategic markupsbα and bβ increase with the degree of bidder β’s risk aversion.Proposition 4.
Consider Model II. Letbα and bβ be optimal strategic markups for bidder α and β associated with uα (x) and uβx=xρβ, satisfying that bα ≥ bβ. Let b¯α and b¯β be the optimal strategic markups for both bidders associated with utility functions uβx=xρβandu^α, satisfying that b¯α≥b¯β. Then, we have the following:(i)
b¯α<bα, b¯β<bβ(ii)
bβ−b¯β<bα−b¯α (i.e., vβ−b¯β−vβ−bβ<vα−b¯α−vα−bα)Proof.
See Appendix.
The following Examples5 and 6 illustrate the influence of the degrees of bidder α’s risk aversion on the markups of both bidders.Example 5.
Consider Example2 again. Suppose that bidder α becomes more risk averse. Let θα becomes θα + ζ, where ζ is a positive real number, and the larger ζ is, the bidder α becomes more risk averse. Then, the optimal strategic markups satisfy(24)−1+eθα+ζb¯αθα+ζ=2a−b¯α+2a−b¯α2+8a2ρβ2+2ρβ22+ρβb¯β=1+ρβb¯α−2a+2a−b¯α2+8a2ρβ2+2ρβ2+ρβ,andb¯α,b¯β∈αρβ,α.
Figure3 depicts the optimal strategic markups bα and bβ for ζ ∈ (0, 0.1), where a = 1, θα = 0.18, and ρβ = 0.85. It shows that both optimal strategic markups are declining with ζ, implying that as bidder α becomes more risk averse, both bα and bβ decrease (i.e., part (i) of Proposition 4 holds), bα is more rapidly decreased than bβ, and the magnitude of the asymmetry effects on the optimal bid markup becomes smaller for bidder β but becomes bigger for bidder α. It also shows that the magnitude of the asymmetry effects is less for bidder β than that for bidder α (i.e., part (ii) of Proposition 4 holds).Figure 3
Plot of the optimal strategic markupsbα and bβ increase with the degree of bidder α’s risk aversion.Example 6.
Consider Example4 again. Suppose that bidder α becomes more risk averse. Let ρα become ρα−δ, where δ is a positive real number. Then, the optimal strategic markups (22) become(25)b¯α=aρα−δ1+A¯2+ρα−δ+ρβ,b¯β=a2+ρα−δA¯−2−ρα−δ−2ρβ2+ρα−δ+ρβ,where A¯=1+2ρβ2+ρα−δ+ρβ.
Figure4 depicts the optimal strategic markups bα and bβ for δ ∈ (0, 0.69), where a = 1, ρα = 0.9, and ρβ = 0.2. It shows that both optimal strategic markups are declining with δ, implying that as bidder α becomes more risk averse, both bα and bβ decrease (i.e., part (i) of Proposition 3 holds), bα is more rapidly decreased than bβ, and the magnitude of the asymmetry effects on the optimal bid markup becomes smaller for bidder β but becomes bigger for bidder α. It also shows that the magnitude of the asymmetry effects is more for bidder α than that for bidder β (i.e., part (ii) of Proposition 3 holds).
From Proposition3 and Proposition 4, we can find that the two bidders make uniformly higher bids as bidder β or bidder α becomes more risk averse. This is consistent with the traditional result on symmetric auctions.Figure 4
Plot of the optimal strategic markupsbα and bβ increase with the degree of bidder α’s risk aversion.


### 3.3. Impact of the Degrees of Risk Aversion on Allocative Efficiency

Proposition 5.
(i)
Supposevβ > vα. Both auctions associated with Models I and II are always efficient.(ii)
Supposevα > vβ. For Model I, the auction is inefficient if and only if vα − vβ < 2 (aρα − bβ)/(2 + ρα); the auction is efficient if and only if vα − vβ > 2 (aρα − bβ)/(2 + ρα). For Model II, the auction is inefficient if and only if vα − vβ < p (bα); the auction is efficient if and only if vα − vβ > p (bα), where(26)pbα=12+ρβbα+2a1+ρβ−2a−bα2+8a2ρβ2+2ρβ.Proof.
See Appendix.
Notice that ifvβ > vα, the auctions are always efficient. In the following, we just consider that vα > vβ.Proposition 6.
(i)
Consider Model I; suppose that bidderα’s risk aversion level remains unchanged. (a) If the auction is inefficient, as bidder β becomes more risk averse, the more inefficient the allocation can be, while as bidder β becomes less risk averse, allocative efficiency becomes ambiguous. (b) If the auction is efficient, as bidder β becomes less risk averse, allocative efficiency becomes ambiguous, while as bidder β becomes less risk averse, the auction is still efficient.(ii)
Consider Model II; suppose that bidderβ’s risk aversion level remains unchanged. (a) If the auction is inefficient, as bidder α becomes more risk averse, allocative efficiency becomes ambiguous. Consider Example 2. Suppose that vα − vβ = 0.02 and a = 1. Substituting a = 1 and the related data in column 2 of Table 2 (i.e., θα = 0.3, ρβ = 0.8, and bα = 0.8635) into (26), we have p(bα) ≈ 0.0260. Thus, vα − vβ < p(bα), implying that the auction is inefficient by Proposition 5 (ii). Now, suppose that bidder α becomes more risk averse such that θα = 0.35 or θα = 0.41. For θα = 0.35, using the data of column 3 in Table 2, we have vα − vβ < p (bα) ≈ 0.0223, implying that the auction is inefficient. But for θα = 0.41, using the data of column 5 in Table 2, we have vα − vβ > p (bα) ≈ 0.0149, implying that the auction is efficient by Proposition 5 (ii), while as bidder α becomes less risk averse, the auction becomes more inefficient. (b) If the auction is efficient, as bidder α becomes more risk averse, the auction is still efficient, while as bidder α becomes less risk averse, allocative efficiency becomes ambiguous.Proof.
See Appendix.
The following Examples7 and 8 illustrate the magnitude of the asymmetry effects on the efficiency established in Proposition 6.Example 7.
Consider Example1 again. Suppose that vα > vβ in Proposition 6 (i). Then, the auction is efficient if vα − bα > vβ − bβ (i.e., vα − vβ > bα − bβ).
Letθβ become θβ + ε, implying that bidder β becomes less risk averse and let θβ become θβ − ε, implying that bidder β becomes more risk averse, where ε ∈ (0, 0.06):(a)
If the auction is inefficient. Letvα − vβ = 0.0025, θβ = 0.6, ρα = 0.8 and a = 1, we have bα − bβ = 0.0071. Then, vα − vβ < bα − bβ. Figure 5 shows the relationship between bα − bβ and vα − vβ as bidder β becomes less or more risk averse. The bα − bβ is found to be increasing as β becomes more risk averse. And the higher the bα − bβ is, the more inefficient the allocation can be. While as β becomes less risk averse, bα − bβ declines to be lower than vα − vβ, the auction may be efficient.(b)
If the auction is efficient. Letvα − vβ = 0.0117, θβ = 0.6, ρα = 0.8 and a = 1, we have bα − bβ = 0.0071. Then vα − vβ > bα − bβ. Figure 6 shows the relationship between bα − bβ and vα − vβ as bidder β becomes less or more risk averse. Clearly, as β becomes less risk averse bα − bβ is always lower than vα − vβ, then, the auction is still efficient. While as β becomes more risk averse, bα − bβ may be higher than vα − vβ, then, the auction may become inefficient.Figure 5
Plot showing that allocative efficiency is affected by the decrease of the degree of bidderβ’s risk aversion.Figure 6
Plot showing that allocative efficiency is affected by the increase of the degree of bidderβ’s risk aversion.Example 8.
Consider Example2 again. Suppose that vα > vβ in Proposition 6 (ii). Then, the auction is efficient if vα − bα > vβ − bβ, i.e., vα − vβ > bα − bβ.
Letθα becomes θα + ε implies that bidder β becomes less risk averse and let θα becomes θα − ε implies that bidder α becomes more risk averse, where ε ∈ (0, 0.1):(a)
Consider the auction when inefficient. Letvα − vβ = 0.020, θα = 0.18, ρβ = 0.85, and a = 1; we have bα − bβ = 0.026. Then, vα − vβ < bα − bβ. Figure 7 depicts the relationship between bα − bβ and vα − vβ as bidder α becomes less or more risk averse. Clearly, bα − bβ is increasing as α becomes less risk averse. And the higher the bα − bβ is, the more inefficient the allocation can be. While as α becomes more risk averse, bα − bβ declines to be lower than vα − vβ; the auction may become efficient.(b)
Consider the auction when efficient. Letvα − vβ = 0.030, θα = 0.18, ρβ = 0.85, and a = 1; we have bα − bβ = 0.026. Then, vα − vβ > bα − bβ. Figure 8 shows that bα − bβ is increasing as α becomes more risk averse and as α becomes less risk averse, bα − bβ increases to be higher than vα − vβ, and the auction may become efficient.Figure 7
Plot shows that allocative efficiency is affected by the increase of the degree of bidderα’s risk aversion.Figure 8
Plot showing that allocative efficiency is affected by the decrease of the degree of bidderα’s risk aversion.


### 3.4. Impact of the Degrees of Risk Aversion on Expected Revenue

Our following proposition shows that the seller’s expected revenue increases with each bidder’s risk aversion, which is the same as that in the standard first-price sealed-bid auctions (see Riley and Samuelson [6], among others). This result for the standard first-price auctions comes directly from the increase of a bidder’s equilibrium bid in risk aversion. However, this result in this paper cannot be obtained in the same way. The reason is as follows.Under the assumptionbα ≥ bβ, the seller’s expected revenue can be derived as (equation (10) of MM)(27)ER=∫bα−bβ+μ−aa+μvα−bαFvα−bα+bβfvαdvα+∫μ−aa−bα+bβ+μvβ−bβFvβ+bα−bβfvβdvβ+∫a−bα+bβ+μa+μvβ−bβfvβdvβ.By simplifying it, we have(28)ER=14a243a3+4μa2−2a2bα+bβ+abα−bβ2−16bα−bβ3.Proposition 7.
(i)
In Model I, given that bidderα’s risk aversion level remains unchanged, the seller’s expected revenue increases as bidder β becomes more risk averse.(ii)
In Model II, given that bidderβ’s risk aversion level remains unchanged, the seller’s expected revenue increases as bidder α becomes more risk averse.By Propositions3 and 4, for Model I, the optimal bid markups of both bidders α and β become smaller (i.e., both bidders bid higher) as bidder β becomes more risk averse and the risk aversion level of bidder α is fixed, similarly, for Model II, the optimal strategic markups for both bidders become smaller (i.e., both bidders bid higher) as bidder α becomes more risk averse and the risk aversion level of bidder β is fixed. Intuitively, this implies that the seller’s expected revenue will become higher. Clearly, this result is not obvious by (28) (see the proof of this proposition for details).Examples9 and 10 illustrate the magnitude of the asymmetry effects on the seller’s expected revenue established in Proposition 7.Example 9.
Consider Example3 again. Suppose that bidder β becomes more risk averse in Proposition 7 (i). Let θβ becomes θβ + ε, where ε is a positive real number. Then, the optimal strategic markups satisfy (21). Obviously, we cannot derive the closed form of the optimal strategic markups for both bidders. Then, we calculate it with numerical method. Substituting the numerical results obtained into (28), the seller’s expected revenue is(29)E^R=14a243a3+4μa2−2a2b^α+b^β+ab^α−b^β2−16b^α−b^β3.
In addition, in order to satisfy the optimal strategic markups for both bidders exist in the interval (0,aρα], the range of parameter ε we set is very small. Therefore, the nonlinear curve looks nearly linear in the following figure.
Figure9 depicts the seller’s expected revenue ER for ε ∈ (0, 0.1), where a = 1, θβ = 0.6 and ρα = 0.8. It shows that the seller’s expected revenue is increasing with ε, implying that as bidder β becomes more risk averse, the seller’s expected revenue increases (i.e., part (i) of Proposition 7 holds).Figure 9
Plot of the seller’s expected revenue ER increases with the degree of bidderβ’s risk aversion.Example 10.
Consider Example4 again. Suppose that bidder β becomes more risk averse. Let ρβ become ρβ−δ, where δ is a positive real number. Then, the optimal strategic markups satisfy (23). By substituting (23) into (28), we obtain the seller’s expected revenue (29).
Figure10 depicts the seller’s expected revenueE^Rfor δ ∈ (0, 0.55), where a = 1, μ = 5, ρα = 0.9, and ρβ = 0.6. It shows that the seller’s expected revenue is increasing with δ, implying that as bidder β becomes more risk averse, the seller’s expected revenue increases (i.e., part (i) of Proposition 7 holds).Figure 10
Plot of the seller’s expected revenue increases with the degree of bidderβ’s risk aversion.Example 11.
Consider Example5 again. Suppose that bidder α becomes more risk averse in Proposition 7 (ii). Let θα become θα + ζ, where ζ is a positive real number, and the larger ζ is, bidder α becomes more risk averse. Then, the optimal strategic markups satisfy (24). We cannot derive the closed form of the optimal strategic markups for both bidders. Then, we calculate it with numerical method. Substituting the numerical results obtained into (28), we obtain the seller’s expected revenue(30)E¯R=14a243a3+4μa2−2a2b¯α+b¯β+ab¯α−b¯β2−16b¯α−b¯β3.
Figure11 depicts the seller’s expected revenue ER for ζ ∈ (0, 0.1), where a = 1, μ = 5, θα = 0.18, and ρβ = 0.85. It is clear that ER increases with ζ. It shows that the seller’s expected revenue is increasing with ζ, implying that as bidder α becomes more risk averse, the seller’s expected revenue increases (i.e., part (ii) of Proposition 7 holds).Figure 11
Plot of the seller’s expected revenue ER increases with the degree of bidderα’s risk aversion.Example 12.
Consider Example6 again. Suppose that bidder α becomes more risk averse. Let ρα become ρα−δ, where δ is a positive real number. Then, the optimal strategic markups satisfy (25). By substituting (25) into (28), we obtain the seller’s expected revenue (30).
Figure12 depicts the seller’s expected revenue E¯R for δ ∈ (0, 0.69), where a = 1, μ = 5, ρα = 0.9, and ρβ = 0.2. It shows that the seller’s expected revenue is increasing with δ, implying that as bidder α becomes more risk averse, the seller’s expected revenue increases (i.e., part (ii) of Proposition 7 holds).Figure 12
Plot of the seller’s expected revenue increases with the degree of bidderα’s risk aversion.


## 3.1. Existence of Optimal Strategic Markups

Proposition 1.
Consider Model I. Then,(i)
The optimal strategic markups for both bidders (bα, bβ) are characterized by (when uβx=xρβ, we get the same optimal strategic markups as in MM)(14)bα=ρα2a+bβ2+ρα,(15)γβbβ=Gbβ,where(16)Gx=2a2ρα+2x+2a−x+2aρα+2,on0,∞.(ii)
There exists a unique optimal strategic markup (bα, bβ) with bα, bβ ∈ (0, aρα].(iii)
bα and bβ increase with the uncertainty parameter a.Proof.
See Appendix.
Part (iii) of Proposition1 is in line with intuition and conventional results in auction theory (see among others Klemperer [11] and Krishna [12]).Proposition 2.
Consider Model II. Then,(i)
The optimal strategic markups for both bidders (bα, bβ) are characterized by (when uαx=xρα, we have the same optimal strategic markups as in MM)(17)bβ=gbα,(18)γαbα=Lbα,whereg (x) and L (x) are defined on [0, ∞),(19)gx=1+ρβx−2a+2a−x2+8a2ρβ2+2ρβ2+ρβ,(20)Lx=2a−x+2a−x2+8a2ρβ2+2ρβ22+ρβ.(ii)
There exists a unique optimal strategic markup (bα, bβ) with bα, bβ ∈ [aρβ, a).(iii)
bα and bβ are increasing with respect to the uncertainty parameter a.Proof.
See Appendix.


## 3.2. Impact of the Degrees of Risk Aversion on Bidders’ Optimal Markups

Suppose that bidderα’s risk aversion level remains unchanged. Then, Proposition 3 (i) shows that the optimal strategic markups for both bidders α and β decrease in risk aversion of bidder β. Part (ii) shows that a change in bidder β’s optimal strategic markup is greater than a change in bidder α’s optimal strategic markup as bidder β becomes more risk averse.Proposition 3.
Consider Model I. Letbα and bβ be the optimal strategic markups for bidder α and β associated with utility functions uαx=xρα and uβ (x), satisfying that bα ≥ bβ. Let b^α and b^β be the optimal strategic markups for both bidders associated with utility functions uαx=xρα andu^β, satisfying that b^α≥b^β. Then, we have the following:(i)
b^α<bα, b^β<bβ(ii)
bα−b^α<bβ−b^β (i.e.,vα−b^α−vα−bα<vβ−b^β−vβ−bβ)Proof.
See Appendix.
The following Examples3 and 4 illustrate the influence of the degrees of bidder β’s risk aversion on the markups of both bidders.Example 3.
Consider Example1 again. Suppose that bidder β becomes more risk averse. Let θβ become θβ + ε, where ε is a positive real number. Then, the optimal strategic markups bα and bβ satisfy(21)−1+eθβ+εb^βθβ+ε=2a2ρα+2b^β+2a−b^β+2aρα+2,b^α=ρα2a+b^β2+ρα,b^α,b^β0,αρα.
Figure1 depicts the optimal strategic markups bα and bβ for ε ∈ (0, 0.1) (the choice of ε’s range must guarantee the optimal strategic markups exist in the interval (0, aρα), where a = 1, θβ = 0.6, and ρα = 0.8. It shows that both optimal strategic markups are declining with ε, implying that as bidder β becomes more risk averse, both bα and bβ decrease (i.e., part (i) of Proposition 3 holds), bβ is more rapidly decreased than bα, and the magnitude of the asymmetry effects on the optimal bid markup becomes smaller for bidder α but becomes bigger for bidder β. It also shows that the magnitude of the asymmetry effects is less for bidder α than that for bidder β (i.e., part (i) of Proposition 3 holds).Figure 1
Plot of the optimal strategic markupsbα and bβ increase with the degrees of bidder β’s risk aversion.Example 4.
Consider that both bidders exhibit constant relative risk aversion (CRRA) utility functions,uix=xρi (with 0 < ρi ≤ 1, i ∈ {α, β}), where 1 − ρi is the Arrow–Pratt measure of CRRA. Then, we can derive explicitly the optimal strategic markups for both bidders(22)bα=aρα1+A2+ρα+ρβ,bβ=a2+ραA−2−ρα−2ρβ2+ρα+ρβ,whereA=1+2ρβ2+ρα+ρβ.
Suppose that bidderβ becomes more risk averse. Let ρβ become ρβ−δ, where δ is a positive real number. Then, we have(23)b^α=aρα1+A^2+ρα+ρβ−δ,b^β=a2+ραA^−2−ρα−2ρβ−δ2+ρα+ρβ−δ,where A^=1+2ρβ−δ2+ρα+ρβ−δ.
Figure2 depicts the optimal strategic markups bα and bβ for δ ∈ (0, 0.55), where a = 1, ρα = 0.9, and ρβ = 0.6. It shows that both optimal strategic markups are declining with δ, implying that as bidder β becomes more risk averse, both bα and bβ decrease (i.e., part (i) of Proposition 3 holds), bβ is more rapidly decreased than bα, and the magnitude of the asymmetry effects on the optimal bid markup becomes smaller for bidder α but becomes bigger for bidder β. It also shows that the magnitude of the asymmetry effects is less for bidder α than that for bidder β (i.e., part (ii) of Proposition 3 holds).
Suppose that bidderβ’s risk aversion level remains unchanged. When bidder α becomes more risk averse, Proposition 4 has a similar meaning as Proposition 3.Figure 2
Plot of the optimal strategic markupsbα and bβ increase with the degree of bidder β’s risk aversion.Proposition 4.
Consider Model II. Letbα and bβ be optimal strategic markups for bidder α and β associated with uα (x) and uβx=xρβ, satisfying that bα ≥ bβ. Let b¯α and b¯β be the optimal strategic markups for both bidders associated with utility functions uβx=xρβandu^α, satisfying that b¯α≥b¯β. Then, we have the following:(i)
b¯α<bα, b¯β<bβ(ii)
bβ−b¯β<bα−b¯α (i.e., vβ−b¯β−vβ−bβ<vα−b¯α−vα−bα)Proof.
See Appendix.
The following Examples5 and 6 illustrate the influence of the degrees of bidder α’s risk aversion on the markups of both bidders.Example 5.
Consider Example2 again. Suppose that bidder α becomes more risk averse. Let θα becomes θα + ζ, where ζ is a positive real number, and the larger ζ is, the bidder α becomes more risk averse. Then, the optimal strategic markups satisfy(24)−1+eθα+ζb¯αθα+ζ=2a−b¯α+2a−b¯α2+8a2ρβ2+2ρβ22+ρβb¯β=1+ρβb¯α−2a+2a−b¯α2+8a2ρβ2+2ρβ2+ρβ,andb¯α,b¯β∈αρβ,α.
Figure3 depicts the optimal strategic markups bα and bβ for ζ ∈ (0, 0.1), where a = 1, θα = 0.18, and ρβ = 0.85. It shows that both optimal strategic markups are declining with ζ, implying that as bidder α becomes more risk averse, both bα and bβ decrease (i.e., part (i) of Proposition 4 holds), bα is more rapidly decreased than bβ, and the magnitude of the asymmetry effects on the optimal bid markup becomes smaller for bidder β but becomes bigger for bidder α. It also shows that the magnitude of the asymmetry effects is less for bidder β than that for bidder α (i.e., part (ii) of Proposition 4 holds).Figure 3
Plot of the optimal strategic markupsbα and bβ increase with the degree of bidder α’s risk aversion.Example 6.
Consider Example4 again. Suppose that bidder α becomes more risk averse. Let ρα become ρα−δ, where δ is a positive real number. Then, the optimal strategic markups (22) become(25)b¯α=aρα−δ1+A¯2+ρα−δ+ρβ,b¯β=a2+ρα−δA¯−2−ρα−δ−2ρβ2+ρα−δ+ρβ,where A¯=1+2ρβ2+ρα−δ+ρβ.
Figure4 depicts the optimal strategic markups bα and bβ for δ ∈ (0, 0.69), where a = 1, ρα = 0.9, and ρβ = 0.2. It shows that both optimal strategic markups are declining with δ, implying that as bidder α becomes more risk averse, both bα and bβ decrease (i.e., part (i) of Proposition 3 holds), bα is more rapidly decreased than bβ, and the magnitude of the asymmetry effects on the optimal bid markup becomes smaller for bidder β but becomes bigger for bidder α. It also shows that the magnitude of the asymmetry effects is more for bidder α than that for bidder β (i.e., part (ii) of Proposition 3 holds).
From Proposition3 and Proposition 4, we can find that the two bidders make uniformly higher bids as bidder β or bidder α becomes more risk averse. This is consistent with the traditional result on symmetric auctions.Figure 4
Plot of the optimal strategic markupsbα and bβ increase with the degree of bidder α’s risk aversion.


## 3.3. Impact of the Degrees of Risk Aversion on Allocative Efficiency

Proposition 5.
(i)
Supposevβ > vα. Both auctions associated with Models I and II are always efficient.(ii)
Supposevα > vβ. For Model I, the auction is inefficient if and only if vα − vβ < 2 (aρα − bβ)/(2 + ρα); the auction is efficient if and only if vα − vβ > 2 (aρα − bβ)/(2 + ρα). For Model II, the auction is inefficient if and only if vα − vβ < p (bα); the auction is efficient if and only if vα − vβ > p (bα), where(26)pbα=12+ρβbα+2a1+ρβ−2a−bα2+8a2ρβ2+2ρβ.Proof.
See Appendix.
Notice that ifvβ > vα, the auctions are always efficient. In the following, we just consider that vα > vβ.Proposition 6.
(i)
Consider Model I; suppose that bidderα’s risk aversion level remains unchanged. (a) If the auction is inefficient, as bidder β becomes more risk averse, the more inefficient the allocation can be, while as bidder β becomes less risk averse, allocative efficiency becomes ambiguous. (b) If the auction is efficient, as bidder β becomes less risk averse, allocative efficiency becomes ambiguous, while as bidder β becomes less risk averse, the auction is still efficient.(ii)
Consider Model II; suppose that bidderβ’s risk aversion level remains unchanged. (a) If the auction is inefficient, as bidder α becomes more risk averse, allocative efficiency becomes ambiguous. Consider Example 2. Suppose that vα − vβ = 0.02 and a = 1. Substituting a = 1 and the related data in column 2 of Table 2 (i.e., θα = 0.3, ρβ = 0.8, and bα = 0.8635) into (26), we have p(bα) ≈ 0.0260. Thus, vα − vβ < p(bα), implying that the auction is inefficient by Proposition 5 (ii). Now, suppose that bidder α becomes more risk averse such that θα = 0.35 or θα = 0.41. For θα = 0.35, using the data of column 3 in Table 2, we have vα − vβ < p (bα) ≈ 0.0223, implying that the auction is inefficient. But for θα = 0.41, using the data of column 5 in Table 2, we have vα − vβ > p (bα) ≈ 0.0149, implying that the auction is efficient by Proposition 5 (ii), while as bidder α becomes less risk averse, the auction becomes more inefficient. (b) If the auction is efficient, as bidder α becomes more risk averse, the auction is still efficient, while as bidder α becomes less risk averse, allocative efficiency becomes ambiguous.Proof.
See Appendix.
The following Examples7 and 8 illustrate the magnitude of the asymmetry effects on the efficiency established in Proposition 6.Example 7.
Consider Example1 again. Suppose that vα > vβ in Proposition 6 (i). Then, the auction is efficient if vα − bα > vβ − bβ (i.e., vα − vβ > bα − bβ).
Letθβ become θβ + ε, implying that bidder β becomes less risk averse and let θβ become θβ − ε, implying that bidder β becomes more risk averse, where ε ∈ (0, 0.06):(a)
If the auction is inefficient. Letvα − vβ = 0.0025, θβ = 0.6, ρα = 0.8 and a = 1, we have bα − bβ = 0.0071. Then, vα − vβ < bα − bβ. Figure 5 shows the relationship between bα − bβ and vα − vβ as bidder β becomes less or more risk averse. The bα − bβ is found to be increasing as β becomes more risk averse. And the higher the bα − bβ is, the more inefficient the allocation can be. While as β becomes less risk averse, bα − bβ declines to be lower than vα − vβ, the auction may be efficient.(b)
If the auction is efficient. Letvα − vβ = 0.0117, θβ = 0.6, ρα = 0.8 and a = 1, we have bα − bβ = 0.0071. Then vα − vβ > bα − bβ. Figure 6 shows the relationship between bα − bβ and vα − vβ as bidder β becomes less or more risk averse. Clearly, as β becomes less risk averse bα − bβ is always lower than vα − vβ, then, the auction is still efficient. While as β becomes more risk averse, bα − bβ may be higher than vα − vβ, then, the auction may become inefficient.Figure 5
Plot showing that allocative efficiency is affected by the decrease of the degree of bidderβ’s risk aversion.Figure 6
Plot showing that allocative efficiency is affected by the increase of the degree of bidderβ’s risk aversion.Example 8.
Consider Example2 again. Suppose that vα > vβ in Proposition 6 (ii). Then, the auction is efficient if vα − bα > vβ − bβ, i.e., vα − vβ > bα − bβ.
Letθα becomes θα + ε implies that bidder β becomes less risk averse and let θα becomes θα − ε implies that bidder α becomes more risk averse, where ε ∈ (0, 0.1):(a)
Consider the auction when inefficient. Letvα − vβ = 0.020, θα = 0.18, ρβ = 0.85, and a = 1; we have bα − bβ = 0.026. Then, vα − vβ < bα − bβ. Figure 7 depicts the relationship between bα − bβ and vα − vβ as bidder α becomes less or more risk averse. Clearly, bα − bβ is increasing as α becomes less risk averse. And the higher the bα − bβ is, the more inefficient the allocation can be. While as α becomes more risk averse, bα − bβ declines to be lower than vα − vβ; the auction may become efficient.(b)
Consider the auction when efficient. Letvα − vβ = 0.030, θα = 0.18, ρβ = 0.85, and a = 1; we have bα − bβ = 0.026. Then, vα − vβ > bα − bβ. Figure 8 shows that bα − bβ is increasing as α becomes more risk averse and as α becomes less risk averse, bα − bβ increases to be higher than vα − vβ, and the auction may become efficient.Figure 7
Plot shows that allocative efficiency is affected by the increase of the degree of bidderα’s risk aversion.Figure 8
Plot showing that allocative efficiency is affected by the decrease of the degree of bidderα’s risk aversion.


## 3.4. Impact of the Degrees of Risk Aversion on Expected Revenue

Our following proposition shows that the seller’s expected revenue increases with each bidder’s risk aversion, which is the same as that in the standard first-price sealed-bid auctions (see Riley and Samuelson [6], among others). This result for the standard first-price auctions comes directly from the increase of a bidder’s equilibrium bid in risk aversion. However, this result in this paper cannot be obtained in the same way. The reason is as follows.Under the assumptionbα ≥ bβ, the seller’s expected revenue can be derived as (equation (10) of MM)(27)ER=∫bα−bβ+μ−aa+μvα−bαFvα−bα+bβfvαdvα+∫μ−aa−bα+bβ+μvβ−bβFvβ+bα−bβfvβdvβ+∫a−bα+bβ+μa+μvβ−bβfvβdvβ.By simplifying it, we have(28)ER=14a243a3+4μa2−2a2bα+bβ+abα−bβ2−16bα−bβ3.Proposition 7.
(i)
In Model I, given that bidderα’s risk aversion level remains unchanged, the seller’s expected revenue increases as bidder β becomes more risk averse.(ii)
In Model II, given that bidderβ’s risk aversion level remains unchanged, the seller’s expected revenue increases as bidder α becomes more risk averse.By Propositions3 and 4, for Model I, the optimal bid markups of both bidders α and β become smaller (i.e., both bidders bid higher) as bidder β becomes more risk averse and the risk aversion level of bidder α is fixed, similarly, for Model II, the optimal strategic markups for both bidders become smaller (i.e., both bidders bid higher) as bidder α becomes more risk averse and the risk aversion level of bidder β is fixed. Intuitively, this implies that the seller’s expected revenue will become higher. Clearly, this result is not obvious by (28) (see the proof of this proposition for details).Examples9 and 10 illustrate the magnitude of the asymmetry effects on the seller’s expected revenue established in Proposition 7.Example 9.
Consider Example3 again. Suppose that bidder β becomes more risk averse in Proposition 7 (i). Let θβ becomes θβ + ε, where ε is a positive real number. Then, the optimal strategic markups satisfy (21). Obviously, we cannot derive the closed form of the optimal strategic markups for both bidders. Then, we calculate it with numerical method. Substituting the numerical results obtained into (28), the seller’s expected revenue is(29)E^R=14a243a3+4μa2−2a2b^α+b^β+ab^α−b^β2−16b^α−b^β3.
In addition, in order to satisfy the optimal strategic markups for both bidders exist in the interval (0,aρα], the range of parameter ε we set is very small. Therefore, the nonlinear curve looks nearly linear in the following figure.
Figure9 depicts the seller’s expected revenue ER for ε ∈ (0, 0.1), where a = 1, θβ = 0.6 and ρα = 0.8. It shows that the seller’s expected revenue is increasing with ε, implying that as bidder β becomes more risk averse, the seller’s expected revenue increases (i.e., part (i) of Proposition 7 holds).Figure 9
Plot of the seller’s expected revenue ER increases with the degree of bidderβ’s risk aversion.Example 10.
Consider Example4 again. Suppose that bidder β becomes more risk averse. Let ρβ become ρβ−δ, where δ is a positive real number. Then, the optimal strategic markups satisfy (23). By substituting (23) into (28), we obtain the seller’s expected revenue (29).
Figure10 depicts the seller’s expected revenueE^Rfor δ ∈ (0, 0.55), where a = 1, μ = 5, ρα = 0.9, and ρβ = 0.6. It shows that the seller’s expected revenue is increasing with δ, implying that as bidder β becomes more risk averse, the seller’s expected revenue increases (i.e., part (i) of Proposition 7 holds).Figure 10
Plot of the seller’s expected revenue increases with the degree of bidderβ’s risk aversion.Example 11.
Consider Example5 again. Suppose that bidder α becomes more risk averse in Proposition 7 (ii). Let θα become θα + ζ, where ζ is a positive real number, and the larger ζ is, bidder α becomes more risk averse. Then, the optimal strategic markups satisfy (24). We cannot derive the closed form of the optimal strategic markups for both bidders. Then, we calculate it with numerical method. Substituting the numerical results obtained into (28), we obtain the seller’s expected revenue(30)E¯R=14a243a3+4μa2−2a2b¯α+b¯β+ab¯α−b¯β2−16b¯α−b¯β3.
Figure11 depicts the seller’s expected revenue ER for ζ ∈ (0, 0.1), where a = 1, μ = 5, θα = 0.18, and ρβ = 0.85. It is clear that ER increases with ζ. It shows that the seller’s expected revenue is increasing with ζ, implying that as bidder α becomes more risk averse, the seller’s expected revenue increases (i.e., part (ii) of Proposition 7 holds).Figure 11
Plot of the seller’s expected revenue ER increases with the degree of bidderα’s risk aversion.Example 12.
Consider Example6 again. Suppose that bidder α becomes more risk averse. Let ρα become ρα−δ, where δ is a positive real number. Then, the optimal strategic markups satisfy (25). By substituting (25) into (28), we obtain the seller’s expected revenue (30).
Figure12 depicts the seller’s expected revenue E¯R for δ ∈ (0, 0.69), where a = 1, μ = 5, ρα = 0.9, and ρβ = 0.2. It shows that the seller’s expected revenue is increasing with δ, implying that as bidder α becomes more risk averse, the seller’s expected revenue increases (i.e., part (ii) of Proposition 7 holds).Figure 12
Plot of the seller’s expected revenue increases with the degree of bidderα’s risk aversion.


## 4. Conclusion

In this paper, we study the asymmetric first-price sealed-bid auctions with two bidders where one bidder has a general concave utility function and the other one has a CRRA utility function. We first establish the existence and uniqueness of the optimal strategic markups of both bidders (then the existence and uniqueness of the asymmetric equilibrium bidding strategies are proved). Then, analyze the impact of one bidder’s risk aversion on both bidders’ optimal markups, allocative efficiency of the auction, and the seller’s expected revenue. We have shown that when one bidder becomes more risk averse and his opponent’s risk aversion level is fixed, (i) both bidders will reduce their markups; (ii) the change in his markup is greater than the change in his opponent’s markup; and (iii) the seller’s expected revenue will increase. In addition, the change of one bidder’s risk aversion level can also add complexity to the allocative efficiency.



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*Source: 5592402-2021-09-06.xml*