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

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.

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 Propositions 1 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 distribution F 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 bidder α, when he observes his own valuation , since assumption (c), he can draw inferences about μ ∈ [ − a,  + a] according to the cumulative F_μ with corresponding density f_μ. Due to assumption (a), he can indirectly draw inferences about his opponent’s valuation  ∈ [ − 2a,  + 2a] according to the cumulative F_β with corresponding density f_β.

Suppose bidder i ∈ {α, β} has a private valuation of and places a bid B_i. Due to assumption (d), the bid has the form B_i () =  − b_i. When his competitor chooses a markup equal to b_−i (−i means i’s opponent), bidder i’s probability of winning Prob_i (win) is given by

Consider bidder α, the probability of winning P is

Then, we have

Similarly, consider bidder β, the probability of winning Q is

Then, we have

2.2. The General Utility Function of Bidders

Bidders are risk averse and one bidder has a CRRA utility function (with 0 < ρ_i ≤ 1), while the other bidder −i has a general utility function u_−i (x) satisfying u_−i (0) = 0, , and , x is bidder’s income. Specifically, we consider two models as shown in the following: Model I: suppose that bidder α has a utility function (with 0 < ρ_α ≤ 1) and bidder β has a general utility function u_α (x) satisfying u_α (0) = 0, , and . Then, the respective expected utility maximization problems for bidders α and β are where P (b_α, b_β) and Q (b_α, b_β) are the respective probabilities of winning of bidder α and β when they choose strategic markups b_α and b_β, Model II: suppose that bidder α has a general utility function u_α (x) and bidder β has a utility function (with 0 < ρ_β ≤ 1) (see page 109 of MM for the detailed proof). Then, the respective expected utility maximization problems for bidders α and β are where b_α, b_β, P (b_α, b_β), and Q (b_α, b_β) have the same meanings as in Model I.

Define . Let denote the Arrow–Pratt measure of absolute risk aversion. Since , γ_i is related to R_i by. Since γ_i (x) ≥ 0 and R_i (x) ≥ 0 for x ≥ 0, for. Let be another utility function of bidder i satisfying the same assumptions as u_i, with an absolute risk aversion measure such that on (0, ∞]. Then, on (0, ∞) implies on (0, ∞) (see page 1191 of Hu et al. [8]).

Throughout the paper, we keep the assumption of b_α ≥ b_β. (Similarly, as in MM, the two probabilities given in (8) and (9) will be changed when we assume that b_α ≤ b_β. 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 bidder α has a CRRA utility function with 0 < ρ_α ≤ 1, and bidder β has a constant absolute risk aversion (CARA) utility function u_β (x) = 1 − exp (−θ_βx) with θ_β > 0. Substituting u_β (x) into (15) yieldsThis and (14) constitute a system. Let a = 1. Solving the respective six systems associated with six pairs of (ρ_α, θ_β), we obtain six pairs of optimal markups as shown in Table 1.
Table 1 shows the optimal markups satisfy b_α ≥ b_β.

Table 1 

Verification of the assumption b_α ≥ b_β.

Example 2. Suppose that bidder α has a CARA utility function u_α (x) = 1 − exp (−θ_αx) with θ_α > 0, and bidder β has a CRRA utility function with 0 < ρ_β ≤ 1. Substituting u_α (x) into (18) yieldsThis and (17) constitute a system. Let a = 1. Solving the respective six systems associated with six pairs of (θ_α, ρ_β) as shown in the following Table 2, we obtain six pairs of optimal markups.
Table 2 shows that the optimal markups satisfy b_α ≥ b_β.

Table 2 

Verification of the assumption b_α ≥ b_β.

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 (b_α, b_β) are characterized by (when , we get the same optimal strategic markups as in MM)where(ii)There exists a unique optimal strategic markup (b_α, b_β) with b_α, b_β ∈ (0, aρ_α].(iii)b_α and b_β increase with the uncertainty parameter a.

Proof. See Appendix.
Part (iii) of Proposition 1 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 (b_α, b_β) are characterized by (when , we have the same optimal strategic markups as in MM) where (x) and L (x) are defined on [0, ∞),(ii)There exists a unique optimal strategic markup (b_α, b_β) with b_α, b_β ∈ [aρ_β, a).(iii)b_α and b_β 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 bidder α’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. Let b_α and b_β be the optimal strategic markups for bidder α and β associated with utility functions and u_β (x), satisfying that b_α ≥ b_β. Let and be the optimal strategic markups for both bidders associated with utility functions and, satisfying that . Then, we have the following:(i), (ii) (i.e.,)

Proof. See Appendix.
The following Examples 3 and 4 illustrate the influence of the degrees of bidder β’s risk aversion on the markups of both bidders.

Example 3. Consider Example 1 again. Suppose that bidder β becomes more risk averse. Let θ_β become θ_β + ε, where ε is a positive real number. Then, the optimal strategic markups b_α and b_β satisfyFigure 1 depicts the optimal strategic markups b_α and b_β for ε ∈ (0, 0.1) (the choice of ε’s range must guarantee the optimal strategic markups exist in the interval (0, aρ_α), where a = 1, θ_β = 0.6, and ρ_α = 0.8. It shows that both optimal strategic markups are declining with ε, implying that as bidder β becomes more risk averse, both b_α and b_β decrease (i.e., part (i) of Proposition 3 holds), b_β is more rapidly decreased than b_α, 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 markups b_α and b_β increase with the degrees of bidder β’s risk aversion.

Example 4. Consider that both bidders exhibit constant relative risk aversion (CRRA) utility functions, (with 0 < ρ_i ≤ 1, i ∈ {α, β}), where 1 − ρ_i is the Arrow–Pratt measure of CRRA. Then, we can derive explicitly the optimal strategic markups for both bidderswhere.
Suppose that bidder β becomes more risk averse. Let ρ_β become ρ_β−δ, where δ is a positive real number. Then, we havewhere .
Figure 2 depicts the optimal strategic markups b_α and b_β for δ ∈ (0, 0.55), where a = 1, ρ_α = 0.9, and ρ_β = 0.6. It shows that both optimal strategic markups are declining with δ, implying that as bidder β becomes more risk averse, both b_α and b_β decrease (i.e., part (i) of Proposition 3 holds), b_β is more rapidly decreased than b_α, 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 bidder β’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 markups b_α and b_β increase with the degree of bidder β’s risk aversion.

Proposition 4. Consider Model II. Let b_α and b_β be optimal strategic markups for bidder α and β associated with u_α (x) and , satisfying that b_α ≥ b_β. Let and be the optimal strategic markups for both bidders associated with utility functions and, satisfying that . Then, we have the following:(i), (ii) (i.e., )

Proof. See Appendix.
The following Examples 5 and 6 illustrate the influence of the degrees of bidder α’s risk aversion on the markups of both bidders.

Example 5. Consider Example 2 again. Suppose that bidder α becomes more risk averse. Let θ_α becomes θ_α + 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 satisfyFigure 3 depicts the optimal strategic markups b_α and b_β for ζ ∈ (0, 0.1), where a = 1, θ_α = 0.18, and ρ_β = 0.85. It shows that both optimal strategic markups are declining with ζ, implying that as bidder α becomes more risk averse, both b_α and b_β decrease (i.e., part (i) of Proposition 4 holds), b_α is more rapidly decreased than b_β, 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 markups b_α and b_β increase with the degree of bidder α’s risk aversion.

Example 6. Consider Example 4 again. Suppose that bidder α becomes more risk averse. Let ρ_α become ρ_α−δ, where δ is a positive real number. Then, the optimal strategic markups (22) becomewhere .
Figure 4 depicts the optimal strategic markups b_α and b_β for δ ∈ (0, 0.69), where a = 1, ρ_α = 0.9, and ρ_β = 0.2. It shows that both optimal strategic markups are declining with δ, implying that as bidder α becomes more risk averse, both b_α and b_β decrease (i.e., part (i) of Proposition 3 holds), b_α is more rapidly decreased than b_β, 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 Proposition 3 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 markups b_α and b_β 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)Suppose  > . Both auctions associated with Models I and II are always efficient.(ii)Suppose  > . For Model I, the auction is inefficient if and only if  −  < 2 (aρ_α − b_β)/(2 + ρ_α); the auction is efficient if and only if  −  > 2 (aρ_α − b_β)/(2 + ρ_α). For Model II, the auction is inefficient if and only if  −  < p (b_α); the auction is efficient if and only if  −  > p (b_α), where

Proof. See Appendix.
Notice that if  > , the auctions are always efficient. In the following, we just consider that  > .

Proposition 6. (i)Consider Model I; suppose that bidder α’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 bidder β’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  −  = 0.02 and a = 1. Substituting a = 1 and the related data in column 2 of Table 2 (i.e., θ_α = 0.3, ρ_β = 0.8, and b_α = 0.8635) into (26), we have p(b_α) ≈ 0.0260. Thus,  −  < p(b_α), implying that the auction is inefficient by Proposition 5 (ii). Now, suppose that bidder α becomes more risk averse such that θ_α = 0.35 or θ_α = 0.41. For θ_α = 0.35, using the data of column 3 in Table 2, we have  −  < p (b_α) ≈ 0.0223, implying that the auction is inefficient. But for θ_α = 0.41, using the data of column 5 in Table 2, we have  −  > p (b_α) ≈ 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 Examples 7 and 8 illustrate the magnitude of the asymmetry effects on the efficiency established in Proposition 6.

Example 7. Consider Example 1 again. Suppose that  >  in Proposition 6 (i). Then, the auction is efficient if  − b_α >  − b_β (i.e.,  −  > b_α − b_β).
Let θ_β become θ_β + ε, implying that bidder β becomes less risk averse and let θ_β become θ_β − ε, implying that bidder β becomes more risk averse, where ε ∈ (0, 0.06):(a)If the auction is inefficient. Let  −  = 0.0025, θ_β = 0.6, ρ_α = 0.8 and a = 1, we have b_α − b_β = 0.0071. Then,  −  < b_α − b_β. Figure 5 shows the relationship between b_α − b_β and  −  as bidder β becomes less or more risk averse. The b_α − b_β is found to be increasing as β becomes more risk averse. And the higher the b_α − b_β is, the more inefficient the allocation can be. While as β becomes less risk averse, b_α − b_β declines to be lower than  − , the auction may be efficient.(b)If the auction is efficient. Let  −  = 0.0117, θ_β = 0.6, ρ_α = 0.8 and a = 1, we have b_α − b_β = 0.0071. Then  −  > b_α − b_β. Figure 6 shows the relationship between b_α − b_β and  −  as bidder _β becomes less or more risk averse. Clearly, as β becomes less risk averse b_α − b_β is always lower than  − , then, the auction is still efficient. While as β becomes more risk averse, b_α − b_β may be higher than  − , then, the auction may become inefficient.

Figure 5 

Plot showing that allocative efficiency is affected by the decrease of the degree of bidder β’s risk aversion.

Figure 6 

Plot showing that allocative efficiency is affected by the increase of the degree of bidder β’s risk aversion.

Example 8. Consider Example 2 again. Suppose that  >  in Proposition 6 (ii). Then, the auction is efficient if  − b_α >  − b_β, i.e.,  −  > b_α − b_β.
Let θ_α becomes θ_α + ε implies that bidder β becomes less risk averse and let θ_α becomes θ_α − ε implies that bidder α becomes more risk averse, where ε ∈ (0, 0.1):(a)Consider the auction when inefficient. Let  −  = 0.020, θ_α = 0.18, ρ_β = 0.85, and a = 1; we have b_α − b_β = 0.026. Then,  −  < b_α − b_β. Figure 7 depicts the relationship between b_α − b_β and  −  as bidder α becomes less or more risk averse. Clearly, b_α − b_β is increasing as α becomes less risk averse. And the higher the b_α − b_β is, the more inefficient the allocation can be. While as α becomes more risk averse, b_α − b_β declines to be lower than  − ; the auction may become efficient.(b)Consider the auction when efficient. Let  −  = 0.030, θ_α = 0.18, ρ_β = 0.85, and a = 1; we have b_α − b_β = 0.026. Then,  −  > b_α − b_β. Figure 8 shows that b_α − b_β is increasing as α becomes more risk averse and as α becomes less risk averse, b_α − b_β increases to be higher than  − , and the auction may become efficient.

Figure 7 

Plot shows that allocative efficiency is affected by the increase of the degree of bidder α’s risk aversion.

Figure 8 

Plot showing that allocative efficiency is affected by the decrease of the degree of bidder α’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 assumption b_α ≥ b_β, the seller’s expected revenue can be derived as (equation (10) of MM)

By simplifying it, we have

Proposition 7. (i)In Model I, given that bidder α’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 bidder β’s risk aversion level remains unchanged, the seller’s expected revenue increases as bidder α becomes more risk averse.

By Propositions 3 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).

Examples 9 and 10 illustrate the magnitude of the asymmetry effects on the seller’s expected revenue established in Proposition 7.

Example 9. Consider Example 3 again. Suppose that bidder β becomes more risk averse in Proposition 7 (i). Let θ_β becomes θ_β + 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 isIn addition, in order to satisfy the optimal strategic markups for both bidders exist in the interval (0, aρ_α], the range of parameter ε we set is very small. Therefore, the nonlinear curve looks nearly linear in the following figure.
Figure 9 depicts the seller’s expected revenue ER for ε ∈ (0, 0.1), where a = 1, θ_β = 0.6 and ρ_α = 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 bidder β’s risk aversion.

Example 10. Consider Example 4 again. Suppose that bidder β becomes more risk averse. Let ρ_β become ρ_β−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).
Figure 10 depicts the seller’s expected revenuefor δ ∈ (0, 0.55), where a = 1, μ = 5, ρ_α = 0.9, and ρ_β = 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 bidder β’s risk aversion.

Example 11. Consider Example 5 again. Suppose that bidder α becomes more risk averse in Proposition 7 (ii). Let θ_α become θ_α + 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 revenueFigure 11 depicts the seller’s expected revenue ER for ζ ∈ (0, 0.1), where a = 1, μ = 5, θ_α = 0.18, and ρ_β = 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 bidder α’s risk aversion.

Example 12. Consider Example 6 again. Suppose that bidder α becomes more risk averse. Let ρ_α become ρ_α−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).
Figure 12 depicts the seller’s expected revenue for δ ∈ (0, 0.69), where a = 1, μ = 5, ρ_α = 0.9, and ρ_β = 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 bidder α’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.

Appendix

Proof. ofProposition 1.
Part (i) The first-order condition of bidder α’s expected utility maximization problem (6) isSubstituting (8) into (A.1) and simplifying, the authors obtain (14). The first-order condition of bidder β’s expected utility maximization problem (7) isSubstituting (9) into (A.2) and simplifying, the authors obtainBy substituting (14) into (A.3) and simplifying, the authors obtain (15). (ii) Differentiating (16) with respect to x, the authors obtainDefine F (x) = γ_β (x) − G (x) on [0, ∞). Then, F (0) = γ_β (0) − G(0) = − G(0) < 0 since . The concavity of u_β implies that γ_β (x) ≥ x. Thus,where . Clearly, H’ (x) > 0 for x ∈ [0, ∞). Solving H (x) > 0, or equivalently, (3 + ρ_α) x² + a (8 + 2ρ_α) x − a² (4 + 2 + 8ρ_α) > 0, we have x > x₁, where H (x₁) = 0,Let x₂ > x₁; then, H (x₂) > H (x₁) = 0 since H (x) is increasing. Thus, using (A.4), the authors have F (x₂) > 0. By the intermediate value theorem, there is a  ∈ (0, x₂) such that F() = 0. Combining the assumption of b_α ≥ b_β with (14), the authors have b_β ≤ aρ_α. Since x₂ > x₁ > aρ_α, there exists bidder β’s optimal markup with  ∈ (0, aρ_α]. It follows that bidder α’s optimal markup also exists because of (14). Since  > 0 and G′ (x) < 0 for x ≥ 0, F′ (x) > 0 for x ≥ 0. Thus, is unique. It follows from (14) that is also unique. Furthermore, the authors have 0 <  ≤ ρ_α (2a + aρ_α)/(2 + ρ_α) = aρ_α by (14). The authors need to verify  ≥ . In fact, using (14) and aρ_α ≥ , the authors have  ≥ (2 + ρ_α)/(2 + ρ_α) = .
Part (iii) Define F (a, ρ_α; b_β) = γ_β (b_β) − G (b_β). Then, by (15) and part (ii) of Proposition 1, the optimal markup over (0, aρ_α] satisfies F (a, ρ_α; b_β) = 0. Since and G′ (x) < 0 for, . Since b_β ∈ (0, aρ_α], the authors haveThus,Then, b_β is increasing with a. It follows by (14) that b_α is also increasing with a.

Proof. ofProposition 2.(i)The first-order condition of bidder β’s expected utility maximization problem (11) isSimilarly, the first-order condition of bidder α’s expected utility maximization problem (10) isSubstituting (8) into (A.10) and simplifying, the authors obtainSubstituting (9) into (A.9) and simplifying, the authors obtainAfter simplifying, the authors obtain a quadratic equation with the unknown variable of b_β:Solving it for b_β, the authors obtain two solutions:whereFurthermore, the concavity of u_α implies that γ_α (b_α) ≥ b_α for b_α ≥ 0. Thus, using (A.11), the authors have b_α ≤ [2a − (b_α − b_β)]/2. This and b_α ≥ b_β imply that b_α ≤ a. Since b_α ≤ a,  < 0. The authors reject on economic grounds and get (17) after omitting the subscript 1. By substituting (17) into (A.11) and simplifying, the authors obtain (18).
Part (ii) Combining b_α ≥ b_β with (17) and (19), the authors haveThus,Therefore, b_α ≥ aρ_β. Then, using b_α ≤ a from the proof of part (i), the authors have b_α ∈ [aρ_β, a].
Differentiating (20) with respect to x, the authors obtainDefine on (0, 1]. Then, L (a) = ah (ρ_β) by (20). SinceL (a) = ah (ρ_β) ≤ ah (1) = a.
Define Q (x) = γ_α (x) − L (x) on [0, ∞). Then, Q (0) = γ_α (0) − L (0) < 0 by L (0) > 0. The concavity of u_α implies that γ_α (x) ≥ x. Thus,By the intermediate value theorem, there is a  ∈ (0, a) such that Q () = 0. Since the authors have proved that b_α ∈ [aρ_β, a], there exists bidder α’s optimal markup with  ∈ [aρ_β, a). It follows that bidder β’s optimal markup also exists because of (17). Since  > 0 and L′ (x) < 0 for x ≥ 0, Q′ (x) > 0 for x ≥ 0. Thus, is unique. It follows from (17) that is also unique.
Differentiating (19) with respect to x, the authors obtainwhich implies that (x) is increasing on [aρ_β, a). It follows from  ∈ [aρ_β, a) and (17) that we haveBy (17), we need to verify  ≤ . In fact, using (17) and aρ_β ≤ , we havePart (iii) Define F (a, ρ_β; b_α) = γ_α (b_α) − L (b_α). Then, by (18) and part (ii) of Proposition 2, the optimal markup over [aρ_β, a) satisfies F (a, ρ_β; b_α) = 0. Since and L′ (x) < 0 for, . Since b_α ∈ [aρ_β, a), the authors haveThus,Then, b_α is increasing with a. It follows by (17) that b_β is also increasing with a.

Proof. ofProposition 3.
Part (i) By Proposition 1 (i), the authors haveThe authors prove by contradiction. Suppose that or . The authors first assume . Sinceimplies, the authors have . Since γ_β (x) strictly increases in x, . Therefore, the authors have. Since G (x) is strictly decreasing in x, . It follows from the first equality of (A.26), γ_β (b_β) < G (b_β), which contradicts with the second equality of (A.26). Next, the authors assume . Then, which contradicts with .
By Proposition 1 (i),Using (14), (A.27), and , the authors have. (ii) Using (14) and (A.27), . Thus, by ρ_α ∈ (0, 1], the authors have .

Proof. ofProposition 4.(i)By Proposition 2 (i), b_β =  (b_α) and γ_α (b_α) = L (b_α), , and. We can provein a similar way by which we provein Proposition 3. Thus, since (x) is increasing on [aρ_β, a).(ii)Let ,. Then, using (17), we obtainThe second equality holds by using ; the first inequality is true because a < 2a − b_α < 2a by b_α ∈ [aρ_β, a), 8a² > 4a² > (2a − b_α)², and then

Proof. ofProposition 5.(i)If  > , we have  − b_β >  − b_α, since b_α ≥ b_β that is assumed throughout the paper. Thus, the auction is always efficient.(ii)If  > . Substituting (14) into  − b_α < (>)  − b_β and rearranging, we have  −  < (>) 2 (aρ_α − b_β)/(2 + ρ_α). Substituting (17) into  − b_α < (>)  − b_β and rearranging, we have  −  < (>) p (b_α). This completes the proof.

Proof. ofProposition 6.(i)Since the auction in Model I is inefficient. Equivalently, we have  −  < 2 (aρ_α − b_β)/(2 + ρ_α) by Proposition 5 (ii). When bidder β becomes more (less) risk averse, we have since by Proposition 3 (i). Thus, the auction may become more inefficient (may be efficient or may be not). Suppose that the auction in Model I is efficient. Equivalently, we have  −  > 2 (aρ_α − b_β)/(2 + ρ_α) by Proposition 5 (ii). When bidder β becomes more (less) risk averse, we have since by Proposition 3 (i). Thus, the auction may be efficient or may be not (is still efficient).(ii)Since the auction in Model II is inefficient. Equivalently, we have  −  < p (b_α) by Proposition 5 (ii). Using (26), we haveWhen bidder α becomes more (less) risk averse, from Proposition 4 (i), the authors have , implying . Thus,  −  < p (b_α) <  and then the auction in Model II becomes more inefficient if bidder α becomes less risk averse. But when the bidder α becomes more risk averse the auction may be efficient or may be not since the authors cannot conclude the inequality  −  > .
Suppose that the auction in Model II is efficient. Equivalently, the authors have  −  > p (b_α) by Proposition 5 (ii). Since p′ (b_α) > 0, when bidder α becomes more (less) risk averse, from Proposition 4 (i), the authors have , implying . Thus,  −  > p (b_α) >  and then the auction in Model II is still efficient if bidder α becomes more risk averse. But when the bidder α becomes less risk averse, the auction may be efficient or may be not since the authors cannot conclude the inequality  −  > .

Proof. ofProposition 7.
Consider Model I. Substituting (14) into (28), the authors obtain the expected revenue of the seller E_R (b_β) = V (b_β), whereSimilarly, when the bidder β becomes more risk averse and has a utility function , the expected revenue of the seller is .where , , b_β ∈ (0, aρ_α] and ρ_α ∈ (0, 1].
The authors have 4b_β + 12a − 6aρ_α > 0, 12a − 6aρ_α > 0 for b_β ∈ (0, aρ_α], ρ_α ∈ (0, 1]. When, b_β = aρ_α, the authors obtain. Then, . Since . Thus, the authors have .
Consider Model II. When bidder α becomes more risk averse, the expected revenue of the seller isBy (28), the authors obtainBy simplifying, the authors havewhere the first inequality follows from and (Proposition 4) and the second one holds since (Proposition 4).

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 71571044.