E cient Auctions of Risky Asset to Heterogeneous Risk Averse (Preferring) Buyers 1

Transcription

1 E cient Auctions of Risky Asset to Heterogeneous Risk Averse (Preferring) Buyers 1 Audrey Hu University of Amsterdam/Tinbergen Institute and Liang Zou University of Amsterdam Business School Revised July, We are grateful to Steve Matthews for suggesting the approach taken in this study, and to Eric Maskin for his encouraging comment.

2 E cient Auctions of Risky Asset to Heterogeneous Risk Averse (Preferring) Buyers Abstract This paper concerns the e cient sale of an indivisible risky asset and the e ects of changing risk in a setting where buyers exhibit heterogeneous risk preferences. The model allows asymmetric and interdependent values and types. Under certain conditions, e cient implementation in ex post equilibrium is possible through either a direct mechanism or an English auction. This implies that the asset is allocated to the one who derives the highest expected utility surplus from the asset. As the asset s risk increases, the active buyers are uniformly better o regardless of their risk attitudes (risk averse, risk neutral, or risk preferring). A fundamental reason for this increasing risk e ect is the fact that the winner s utility surplus is a convex function of the utility of the pivotal bidder. Key words: e cient auction, ensuing risk, heterogeneous risk preferences, ex post equilibrium, interdependent values, asymmetry JEL classi cation: D44 1

3 1 Introduction When a troubled bank or a bundle of mortgage-backed securities (henceforth, asset) is put forth for sale through an auction, the winning bidder typically assumes a certain degree of ensuing risk. The recent nancial turmoil in the United States and around the globe, which has brought some century-old banks, among other large institutions, to collapse, signi es the reality that no rms are immune to risks no matter how big they are in terms of asset value or market capitalization. In such situations as well as others (e.g., Maskin (1992), Dasgupta and Maskin (2000)), how to sell an asset e ciently, and how the varying degrees of risk or buyers risk preferences may a ect an auction s performance, arise as important issues for policy makers as well as researchers. Recently, Eso and White (2004) studied the e ect of increasing risk in a generalized symmetric interdependent values model (Milgrom and Weber, 1982) with ex post risk and risk averse buyers. They nd a precautionary bidding e ect that when the bidders in an auction exhibit decreasing absolute risk aversion (DARA), increasing risk makes every bidder better o prior to the auction. Therefore, competing buyers for a risky asset may have no incentive before an auction to spend resources and nd out the asset s value more accurately. 1 An intuitive explanation is that an increase in risk causes the DARA buyers to reduce their bids by more than the appropriate increase in the risk premium, which results in more attractive lower prices for the potential winner. Since the bidders have the same utility function in Eso and White s model, the DARA preference turns out to be necessary and su cient for buyers to prefer higher risk. The symmetric setting also implies that the rst-price, second-price, and English auctions studied in Eso and White s paper 1 For instance, amid the present economic crisis, the term toxic assets emerged as a nickname for those assets or securities that are considered to be highly risky. The lack of buyers incentives to understand better their true values might be one of the reasons why these assets remain toxic, some of them being priced for just a few cents on the dollar. 2

4 are e cient. It is well-known that e ciency is not generally attainable, especially when bidders have asymmetric interdependent valuations and beliefs (e.g., Maskin (1992), and Jehiel and Moldovanu (2001)). Under certain circumstances, however, judiciously designed selling procedures are available that do achieve allocation e ciency (e.g., Maskin (1992), Dasgupta and Maskin (2000), and Perry and Reny (2002)). These procedures are also attractive because they are detail free from the seller s viewpoint; in other words, the seller need not know the valuation functions and type distributions of the buyers. On the other hand, Cremer and McLean (1985, 1988) show that if the buyers types are interdependent and the seller knows the joint distribution of the buyers s types, then the seller is able to maximize expected revenue as though he is informed of every buyer s true type. An implication of this ex post optimality, as noted by Perry and Reny (2002), is that resources will be allocated e ciently. All these studies, however, assume that the buyers are risk neutral, which implies that the ensuing risks, if any, that pertain to the asset s future payo s do not matter for the buyers. In the auctions literature, e ects of risk aversion have been mainly studied in the private values models assuming that the bidders have the same utility function. 2 This assumption may be a reasonable approximation of reality where the buyers are more or less homogeneous in their risk preferences, and where the auctioned object does not carry ensuing payo risks. In other situations, however, the assumption that the agents engaged in competitive bidding exhibit the same risk preference is questionable. For one thing, it is inconsistent with the stylized fact that the proportion of risky-asset holdings vary across households. Also, as demonstrated in a recent experimental study among others, Harrison, List, and Towe (2007, p.437) 2 For instance, see Holt (1980); Riley and Samuelson (1981); Harris and Raviv (1981); Matthews (1983, 1987); Maskin and Riley (1984); McAfee and McMillan (1986); and Smith and Levin (1996). See also Hu, Matthews, and Zou (2009) for a study of optimal reserve prices with risk averse seller and buyers. 3

5 report that we observe considerable individual heterogeneity in risk attitudes, such that one should not readily assume homogeneous risk preferences for the population. In this paper, we present an auction model in which the buyers are explicitly assumed to have heterogenous and private attitudes toward risk. 3 The model also assumes, as in Eso and White (2004), that the object for sale is an indivisible asset that may carry ensuing (i.e., post auction) income risks for the winning bidder. 4 For generality, we allow the buyers to have asymmetric and interdependent valuations and types, and we do not restrict attention to the risk averse buyers only. For example, if a buyer represents a rm with existing debt and limited liability, the convexity of the rm s equity as a function of the asset s random payo could cause the buyer to behave as though he is a risk lover. Another example is where the buyer is the manager of a rm who will be rewarded with (large) bonuses if his rm makes pro ts, or else he receives a at salary. Such nonlinear payment schemes could also induce the manager to behave like a risk lover in an auction. Therefore, each buyer in our model is allowed to be risk averse, risk neutral, or risk preferring. Our objective is twofold: to examine the possibility of e cient sale of a risky asset and the e ect of increasing risk of the asset in such an environment. Following the Vickrey (1961), Clarke (1971), and Groves (1973) tradition, we construe e ciency to be an outcome that maximizes the sum of the buyers expected utility surplus that is, the expected utility for the asset minus one s status quo utility level given that every buyer is o ered the same price for the asset. This latter qualifying condition is absent in the VCG mechanisms that assume quasi- 3 Cox, Smith, and Walker (1982, 1988) consider heterogeneous bidder risk aversion in a private values model without ex post ensuing risk. They focus on utility functions that exhibit constant relative risk aversion (CRRA). 4 Such ensuing risk is also incorporated in the symmetric private-values model of Maskin and Riley (1984, Case 3), where the bidders have the same utility function but privately known value distributions. 4

6 linear utilities, but it becomes necessary in our model because of the income e ects that stem from the non-quasi-linear utility functions. Another necessary step in our more general setting is to normalize each buyer s marginal utility to be the same at the status quo level (i.e., when they lose). This normalization ensures that the expected utility surplus can be compared across individuals, and that the model is consistent with risk neutrality (where the marginal utilities are implicitly assumed to be the same). We then show (Theorem 1) that, under certain conditions, the higher is a buyer s risk tolerance (in the Arrow-Pratt sense of absolute risk aversion), the higher is his expected utility surplus provided that some more risk averse buyers prefer the risky asset to the status quo for the same given price. This result derives from, and generalizes, Pratt s (1964) observation that a more risk tolerant utility function is a convex function of a more risk averse utility function. In our model, each buyer s private type is an index of his risk tolerance. The vector of buyers types may also a ect each buyer s assessment of the expected value of the asset. We focus on ex post equilibrium, which requires no common knowledge about each buyer s type space and each type s beliefs over other types and beliefs, etc. This amounts to nding robust auction mechanisms that implement the e cient asset allocation in ex post equilibria (see Bergemann and Morris (2005) for a strong foundation for using ex post equilibrium as a solution concept). Assuming that a buyer s own type has a (weakly) higher impact on his expected valuation of the asset, Theorem 1 then implies that e ciency is attained when the winner is the one who is most risk tolerant among all buyers. We also relax the common knowledge assumption about the buyers utility and value functional forms in the sense as follows. In one environment we assume that the seller is informed of the buyers utility and valuation functions (as in the VCG mechanisms, see also Bergemann and Morris (2005)). We allow the buyers to be imperfectly informed of each other s valuation functions (e.g., they only need to know that the seller is fully informed). We show in this case (Theorem 2) that a direct mechanism exists that is ex post incentive compatible and e cient. In 5

7 another environment, we assume that only the buyers know each other s utility and valuation functional forms (e.g., as in Dasgupta and Maskin (2000)). We then show (Theorem 3) that the English auction, which is detail free to the seller, entails an ex post equilibrium that is e cient. Extending the existing work to a setting in which the buyers are allowed to have di erent risk preferences, this result renders further support for the use of English auctions in practice (e.g., Milgrom and Weber (1982), Maskin (1992), Wilson (1998), Lopomo (1998, 2000), and Krishna (2002, Chapter 9)). As for the e ects of increasing risk, we rst present some technical results relating to the preservation of strict log-supermodularity by integration (see the next section and the Appendix for more detail). These results are of interest in their own right, as they enrich the existing literature related to log-supermodularity (e.g., Karlin and Rinott (1980), Milgrom and Weber (1982), Jewitt (1987), and Athey (2000)). In our model, these results help ensure the preservation of the more risk tolerant relation under expectation. We then show that given any two auction environments that are the same except that one involves a more risky asset for sale than the other, the winner derives a strictly higher expected utility surplus in the more risky environment. Consequently, at the pre-auction stage (ex ante or interim), all active buyers are uniformly better o when the asset s risk increases. This result is consistent with the nding of Eso and White (2004), but it holds in the more general environments in which the buyers preferences are not restricted to be DARA and risk averse, and the buyers valuations and beliefs can be asymmetric. Therefore, we conclude that Eso and White s precautionary bidding e ect derives from a more fundamental hypothesis that buyers are heterogeneous in their attitudes toward risk. Since it is relatively easier to control the characteristics of the object for sale rather than the personal risk preferences in experimental studies of auctions, the results obtained in this paper can be subject to empirical tests more straightforwardly. The rest of the paper is organized as follows. Section 2 presents some technical results that are relevant to the analysis in this paper. Section 3 presents the general 6

8 setting of the model. Section 4 studies the ex post e cient auction mechanisms. Section 5 analyzes the impact of increasing asset s risk on the buyers expected utility surplus. And Section 6 concludes the paper with some remarks relating to future work. The Appendix contains some technical results on the preservation of strict log-supermodularity by integration. 2 Preliminaries We rst present some fundamental results that are essential for the subsequent analysis, and are interesting on their own. Consider a utility function u : RH! R where H is some totally ordered set. Suppose u(; y) is di erentiable given any y 2 H: 5 We say that u is ordered by risk tolerance (ORT) if u 1 is strictly logsupermodular, i.e., u 1 is strictly positive on RH, and for all x; x 0 2 R and y; y 0 2 H; x < x 0 and y < y 0 imply u 1 (x; y)u 1 (x 0 ; y 0 ) > u 1 (x; y 0 )u 1 (x 0 ; y): (1) By strict log-supermodularity, y 2 H can be interpreted as an index of the risk tolerance of u (; y) with a higher y indicating a (uniformly) more risk tolerant utility function. Alternatively, y can be interpreted in the sense of Arrow-Pratt as an index of the absolute risk aversion of u(; y). 6 An immediate consequence, due to Pratt (1964), is that if u is ORT, then u(; y 0 ) is a strictly convex (concave) function of u(; y) for any y 0 > y (< y). Consequently, for any real number a and non-degenerate random variable ~x with E~x < 1, and for y 0 > y, Eu(~x; y) u(a; y) implies Eu(~x; y 0 ) > u(a; y 0 ). 7 Pratt (1964, The- 5 The subscripts of u (or of any function) indicate partial derivatives in this paper. 6 Note that no assumption is made about any smoothness of the function u(x; y) in y, nor is u required to be twice di erentiable in x. For y 6= y 0, u(; y) and u(; y 0 ) can be treated as two utility functions for income x with di erent structures, provided that one function is uniformly more risk averse than the other. 7 Throughout this paper, E() denotes the expectation with respect to the random variable(s) marked with a tilde. 7

9 orem 1) establishes this result assuming that u(; y) is twice di erentiable. The following theorem extends and strengthens Pratt s result under the weaker assumption about u. Theorem 1 For function u : R H! R such that u 1 is strictly log-supermodular, and for y 0 ; y 2 H such that y 0 6= y, there exists a function '(; y 0 ; y) : R! R such that u(x; y 0 ) u(x; y) '(u(x; y); y 0 ; y) (2) (i) '(; y 0 ; y) is strictly convex if y 0 > y, and strictly concave if y 0 < y. (ii) Further assume that u 1 (0; y) = 1 for all y 2 H. Then for any real number a and nondegenerate random variable ~x with E~x < 1, and for all y 0 > y, a 0 and Eu(~x; y) u(a; y) ) Eu(~x; y 0 ) u(a; y 0 ) > Eu(~x; y) u(a; y) (3) a 0 and Eu(~x; y 0 ) u(a; y 0 ) ) Eu(~x; y) u(a; y) < Eu(~x; y 0 ) u(a; y 0 ) (4) Proof. Since u 1 > 0, let (; y) be the inverse of u so that u((v; y); y) = v. Then, for all v 2 R, '(v; y 0 ; y) = u((v; y); y 0 ) v is uniquely de ned. (i) Recall that a function f : R! R is strictly convex (concave) on (a; b) R if for all x < y < z from (a; b), f(z) z f(y) y > (<) f(y) f(x) ; (5) y x Now x y 0 ; y 2 H such that y 0 > y. We show that ' is strictly convex, which amounts to showing that for all v < v 0 < v 00, '(v 00 ; y 0 ; y) '(v 0 ; y; y) > '(v0 ; y 0 ; y) '(v; y; y) v 00 v 0 v 0 v (6) Observe that for all x < x 0 < x 00 and y < y 0, by the Generalized Mean Value Theorem, (1) implies u 1 (; y 0 ) u 1 (; y) = u(x00 ; y 0 ) u(x 0 ; y 0 ) u(x 00 ; y) u(x 0 ; y) > u(x0 ; y 0 ) u(x; y 0 ) u(x 0 ; y) u(x; y) = u 1(; y 0 ) u 1 (; y) ( > ) (7) 8

10 Subtracting 1 from both sides of (7), and rearranging terms, we obtain u(x 00 ; y 0 ) u(x 00 ; y) [u(x 0 ; y 0 ) u(x 0 ; y)] u(x 00 ; y) u(x 0 ; y) > u(x0 ; y 0 ) u(x 0 ; y) [u(x; y 0 ) u(x; y)] u(x 0 ; y) u(x; y) (8) Speci cally, for x; x 0 ; and x 00 such that v = u(x; h); v 0 = u(x 0 ; h); and v 00 = u(x 00 ; h), (8) reduces to (6) by de nition of '. The steps for deriving the conclusion for y 0 < y are analogous, hence omitted. (ii) Suppose y 0 > y. Then (1) and the assumption that u 1 (0; y) = u 1 (0; y 0 ) imply u 1 (x; y 0 ) > u 1 (x; y) for all x > 0. Consequently, '(; y 0 ; y) is an increasing function for u u(0; y). Now assume Eu(~x; y) u(a; y) for some a 0. By (i), we have E [u(~x; y 0 ) u(~x; y)] = E'(u(~x; y); y 0 ; y) > '(Eu(~x; y); y 0 ; y) '(u(a; y); y 0 ; y) = u(a; y 0 ) u(a; y) where the strict inequality stems from Jensen s inequality since ' is strictly convex, and the weak inequality follows from u 1 > 0 and ' nondecreasing. Rearranging terms then gives (3). The case with (4) is analogous, hence omitted. Since a strictly convex (concave) function remains strictly convex (concave) when a linear function is added to or deducted from it, the result of Theorem 1(i) is essentially the same as Pratt (1964; Theorem 1(d)). The contribution here is to o er a new derivation of the result under a weaker assumption that u 1 is strictly log-supermodular. When we normalize the marginal utilities in a way that they are the same at certain status quo level of income (e.g., zero), then Theorem 1(ii) shows that Pratt s result can be further strengthened. For instance, the surplus in utility payo for a random income ~x over the status quo, E [u(~x; y) u(0; y)], strictly increases as one becomes more risk tolerant providing such surplus is positive for 9

11 some more risk averse utility functions. Similar to Pratt (1964), this result is due to Jensen s inequality and the derived property that u(; y 0 ) u(; y) is a strictly convex (concave) function of u(; y) for any y 0 > y (< y). 8 The results of Theorem 1 o er a theoretical foundation for ranking interpersonal expected utilities, which is essential for our notion of e ciency with non-quasi-linear utility functions. Theorem 1 also facilitates the derivation and interpretation of the main conclusions in Theorems The Setting The model involves an indivisible risky asset for sale through an auction mechanism, with n ( 2) competing bidders. The seller s reservation value for the asset is zero. Each bidder i 2 N = f1; :::; ng has a private type h i 2 H i R; unknown to the other bidders, where H i can be discrete or connected. Let H = H 1 ::: H n, and let L = inffh i : h i 2 H i, i 2 Ng, and H = supfh i : h i 2 H i, i 2 Ng. The preference of each bidder i is represented by a utility function for income u(; h i ), where u : R [L; H]! R is ordered by risk tolerance (ORT), i.e., u 1 is strictly log-supermodular. Without loss of generality, let u(; 0) indicate the utility of a risk neutral bidder. Therefore, 0 H implies that all bidders are risk averse and 0 L that all bidders are risk preferring. In general, we allow 0 2 (L; H), so that the population of the bidders may consist of potentially risk averse, risk neutral, and risk preferring individuals. Let u(0; h i ) denote bidder i s status quo utility. We normalize each bidder s marginal utility by u 1 (0; h i ) = 1 for all h i 2 H i and i 2 N. The idea is that a marginal increase in income from the status quo condition gives everybody the same increase in marginal utility. Such a normalization is necessary, 8 If we assume H to be an interval 2 ln u 1 (x; y)=@x@y ()0, then a di erentiable version of Theorem 1(i) is that u 2 (; y) is a convex (concave) function of u(; y). Similar conclusions as in Theorem 1(ii) with respect to u 2 can then be derived as well (see, e.g., Diamond and Stiglitz (1974); Maskin and Riley (1984); and Matthews (1987)). 10

12 for else a bidder s willingness to pay can be made arbitrarily high or low by a positive ratio transformation of the utility functions. There is no need to normalize the status quo utility levels of the buyers, as only the expected utility surplus (expected utility of winning minus the status quo utility) matters in this context. The timing of events is as follows. At date 0, the auction is organized with n bidders known. At date 1, the outcome of the auction is announced and conditional on the bidders (true) types h = (h 1 ; :::; h n ) 2 H, bidder i has an expected monetary payo v i (h) from the asset. Each function v i (), i 2 N, is a deterministic, nondecreasing, and continuous function in all arguments. This (weak) monotonicity assumption accommodates the idea that the risky asset may have a higher present expected value when the players are more risk tolerant. At date 2, the asset will generate an income v i (h) + ~x i for bidder i. The noise term ~x i is unobservable and uninsurable at date 1. As in Eso and White (2004), we assume that the random vector (~x 1 :::~x n ) is independent of h, and is symmetrically distributed so that the marginal density (assumed to exist) of each ~x i is the same, satisfying E (~x i ) = 0. By this symmetry, the subscript of ~x will be suppressed from now on to simplify notation. The model described so far is quite general. We now impose some functional structures on v i, which helps ensure feasibility of the e cient allocations of the asset. It will be assumed that v i s satisfy 9 8h and 8i 6= j, h j > h i implies v j (h) v i (h) (9) The property (9) has an intuitive interpretation that given the same types of the bidders, one s own type has a (weakly) higher impact on one s own valuation of the asset (e.g., Milgrom and Weber (1982)). This condition is obviously satis ed 9 This assumption is similar to Lopomo (2000, A1) where the valuation function is assumed to be symmetric. See also Wilson (1998). Note that we do not need the kind of (strict) single crossing conditions in, e.g., Maskin (1992) and Dasgupta and Maskin (2000) because the buyers in our model are further di erentiated by their risk preferences. 11

13 if valuations are private, i.e., v i (h) ^v(h i ), or if the valuation is common, i.e., v i (h) v(h) (for some nonincreasing and not necessarily symmetric function v). It also holds where the valuations are a combination of private and common values such as v i (h) = v(h) + ^v(h i ), or more generally, v i (h) = (h i )[v(h) + ^v(h i )] for some nondecreasing function. We assume that the buyers have common knowledge about the utility functional form u(; ). With regard to each buyer s valuation function, we consider the following two cases. (Case 1) The seller is fully informed of the utility and valuation function forms u; v 1 ; :::; v n, and this seller s knowledge is commonly known. A buyer may not know the other buyers valuation functions, except that all buyers have common knowledge about the property (9). (Case 2) The buyers have common knowledge about the functional forms of u; v 1 ; :::; v n, and the seller need not know these. In the mechanism design literature, it is typically assumed that the decision maker (e.g., social planner, principal, auctioneer, and so on) knows the utility functions of the involved individuals or agents. This parallels the rst part of the assumption in Case 1, although we do not assume that the buyers share common knowledge about their valuation functions. Case 2 mirrors Maskin (1992) and Dasgupta and Maskin (2000), where the focus is on detail free selling mechanisms to the seller. For Case 1, we restrict attention to direct mechanisms in which each bidder reports a type of his own, and the mechanism determines the allocation of the asset as well as the bidders payments based on the bidders joint reports. In the present setting in which the losers are assumed to receive and pay nothing, a direct mechanism can be described by a pair (p; t) where p : H! [0; 1] is an allocation rule for the asset, and t : H! R is a payment rule for the winner of the asset. Thus, given reported types h, p i (h) will be the probability that bidder i gets the asset and t i (h) the amount he has to pay if he wins. 12

14 By the revelation principal (e.g., Myerson, 1981), attention can be restricted without loss of generality to incentive compatible mechanisms that induce the bidders to report their types truthfully. Upon winning the asset, the ex post (date 1) expected utility of bidder i, assuming that all bidders report their types truthfully and the reported types are publicly known, is thus equal to U i (h i jh i ) Eu ~x + v i (h) t i (h); h i (10) Ex post incentive compatibility then implies that U i (h i jh i )p i (h) Eu ~x + v i (h) t i (y; h i ); h i p i (y; h i ) 8i; y; h (11) where (y; h i ) = (h 1 ; :::; h i 1 ; y; h i+1 ; :::; h n ) is the vector of types assuming that all bidders report their types truthfully except bidder i who reports y instead of his true type h i. We say that an allocation rule p(h) is ex post e cient for price c if it maximizes X Eu ~x + v i (h) i2n c; h i u(0; h i ) p i (h) (12) By Theorem 1 and (9), if given h and c, some type h i derives a nonnegative expected utility surplus, i.e., Eu (~x + v i (h) c; h i ) u(0; h i ), then provided bidder j has type h j > h i his expected utility surplus will be higher than i s, i.e., Eu ~x + v j (h) c; h j u(0; h j ) > Eu ~x + v i (h) c; h i u(0; h i ) We assume that the asset is su ciently pro table so that if it is given for free, then all buyers would like to have it, i.e., Eu (~x + v i (h); h i ) u(0; h i ) for all h and i. Then, (12) is maximized by allocating the asset to the highest type in an ex post incentive compatible mechanism. Let h max i then have 8 >< p i (h) = >: 1 if h i > h max i 1 n 0 if h i = h max i 0 if h i < h max i = max j6=i fh j g for any type vector h, we and if n 0 is the number of tied highest types (13) 13

15 Since the types are allowed to be discrete, there can be a positive probability for a tie in which two or more bidders report the same highest type. As will be seen, however, the way of resolving a tie is inessential because the tied bidders will then be indi erent between winning and losing in an ex post equilibrium. We say that a direct mechanism entails an ex post e cient equilibrium if it satis es (11), (13), and the following individual rationality condition: U i (h i jh i ) u(0; h i ) 8h 2 H; i 2 N (14) As for Case 2, we consider a standard English auction in which the seller raises the price for the asset continuously until all but one buyer have dropped out. The remaining buyer will then acquire the asset for the price at which the last of the other buyers drops out. If the last two or more buyers drop out simultaneously at the same price, then the seller randomly decides a winner from the tied bidders to buy the asset at that price. 4 Ex Post E cient Equilibrium In the spirit of Vickrey auction, we consider Case 1 rst and show the existence of an ex post equilibrium entailed by a direct mechanism that achieves allocation e ciency. Given any type vector h 2 H, let h (1) ; h (2) ; :::; h (n) denote the highest, second highest,..., lowest types from among h. A bidder is said to be pivotal if his type equals h (2). Following the notational convention, let v i (h) = v i (h i ; h i ), and v i (h i ; h j ) = v i (h i ; h i ; h i;j ), etc. Theorem 2 For Case 1, the following constitutes an ex post e cient equilibrium. (i) Each bidder i reports a type of his own. (ii) Given the reported types h, the bidder who reports the highest type wins the asset. Any tie at h (1) will be resolved randomly and (one of) the losing highest bidder(s) will be labeled h (2). (iii) The losing bidders 14

16 pay and receive nothing, and the winning bidder pays a price t (2) that is the solution to the following equation Eu ~x + v (2) (h (2) ; h (1) ) t (2) ; h (2) = u(0; h(2) ) (15) Proof. We rst show that the winner has no incentive to lie given that the others report their types truthfully. Suppose h (1) > h (2). Then (9) implies Consequently, we derive v (1) (h) v (2) (h) v (2) (h (2) ; h (1) ): Eu ~x + v (1) (h) t (2) ; h (1) u(0; h (1) ) Eu ~x + v (2) (h) t (2) ; h (1) u(0; h (1) ) > Eu ~x + v (2) (h) t (2) ; h (2) u(0; h (2) ) Eu ~x + v (2) (h (2) ; h (1) ) t (2) ; h (2) u(0; h (2) ) = 0 where the strict inequality comes from Theorem 1 and h (1) > h (2). This strictly positive expected utility surplus implies that the winner does not want to become a loser, nor does he want to be tied with the pivotal bidder and incur a probability of losing. By construction, the payment is independent of the highest type. The winner therefore has no incentive to report other than his true type. We next show that neither the losers have any incentive to lie. Suppose h j = h (1) and consider an arbitrary loser, say bidder i, whose type h i h (2). For bidder i to become a winner, he must raise his type report above h (1) : If he does so, bidder j will become the pivotal bidder, resulting in a reordering of the types. It will also a ect the payment made by the winner. According to (15), the new payment will be t j, which solves Eu ~x + v j (h j ; h i ) t j ; h j = u(0; hj ) (16) The expected utility of bidder i who reports any type higher than h j become Eu (~x + v i (h) will thus t j ; h i ). Bidder i may not be able to infer the exact price t j 15

17 since he may not know the functional form of v j. However, from his knowledge of (9), and the fact that h j > h i, he knows v j (h j ; h i ) v j (h) v i (h). That the price t j will satisfy equation (16) is also known to bidder i. Therefore, from Theorem 1 bidder i can derive 0 Eu ~x + v i (h) t j ; h j > Eu ~x + v i (h) t j ; h i u(0; h j ) u(0; h i ) which implies that bidder i is better o losing rather than winning given h. Finally, in case of a tie at h (1), one of the losing highest bidders will be labeled h (2). Substituting into (15) yields an equivalent expression that Eu ~x + v (1) (h) t (1) ; h (1) = u(0; h(1) ) which suggests that the tied highest bidders are indi erent between winning or losing, for in both case they derive a zero utility surplus. Given that the others report their types truthfully, no one can thus bene t by deviating from his true type report. This completes the proof. The intuition underlying the equilibrium result of Theorem 2 is similar to the Vickrey auction or the VCG mechanism in general. Incentive compatibility is ensured by making the winner s payment independent of his reported type. On the other hand, using the pivotal bidder s information about the second highest type, the price is also set su ciently high so that no lower types would like to win upon knowing the type vector h. In the proof of the theorem, no bidder is assumed to know the valuation functions of the other bidders. The contribution of this theorem is therefore to extend the Vickrey s insight to the case with non-quasi-linear utilities without the common knowledge assumption about the valuation functions. However, since the outcome is implemented through a direct incentive compatible mechanism, it requires the seller to have the knowledge about the functional forms v i and u. This informational demand to the seller is a practical limitation of all direct mechanisms. There can be also practical concerns associated with such 16

18 direct mechanisms, such as seller cheating and the reluctance of the buyers to make their private types public (see, e.g., McMillan (1994), and Ausubel (2004)). Now we turn to Case 2 and show that the English auction entails an ex post equilibrium that is the same as described in Theorem 2. The English auction avoids the above mentioned limitations of the direct mechanism, at the expense of requiring the bidders to know more about each other. Our analysis capitalizes on the existing work on English auctions with interdependent values (e.g., Milgrom and Weber (1982)). Theorem 3 For Case 2, the English auction entails an ex post e cient equilibrium. In this equilibrium, (i) the lowest type among the active bidders drops out sequentially, and the remaining active bidders can infer the types of the dropped out bidders; and (ii) conditional on knowing any history h (n m) ; :::; h (n) of the dropped out bidders, if bidder i is still active, he will drop out at a price t i solving Eu ~x + v i (h i ; :::; h i ; h (n m) ; :::; h (n) ) t i ); h i = u(0; hi ) (17) Proof. In Case 2, the buyers have common knowledge about the utility and valuation functions. (i) We rst show that the lowest type among the active bidders drops out sequentially, and the remaining active bidders can infer the types of the dropped out bidders. Suppose all bidders follow the strategy of staying in the auction game until the price reaches the level speci ed by (17), where he will drop out. At the start of the auction where all bidders are active for su ciently low price t, let i (y; t) Eu ~x + v i (y; :::; y) t); y u(0; y): By Theorem 1 and u 1 > 0, i strictly increases in y and strictly decreases in t. Therefore i (y i (t); t) 0 de nes a strictly increasing function y i (t), which is common knowledge. From the preceding analysis (e.g., the proof of Theorem 2), if h j > y i (t), then j (h j ; t) > 0. It follows that the lowest type among the active 17

19 bidders will drop out rst. If bidder i drops out at price t, then one can infer his type to be h i = y i (t). Now suppose m + 1 bidders have dropped out and their types are inferred to be h (n m) ; :::; h (n). Suppressing the notational dependence on h (n m) ; :::; h (n), and rede ne i (y; t) Eu ~x + v i (y; :::; y; h (n m) ; :::; h (n) ) t); y u(0; y); we nd again that i strictly increases in y and strictly decreases in t. Thus, again, the lowest type among the remaining active bidders will drop out rst and if he drops out at price t, his type can be inferred by the others. (ii) We next show that the staying and dropping out strategies so described constitute an equilibrium. Suppose all but two bidders, i and j, have dropped out. Now if the price reaches the level of t i satisfying (17) for bidder i, what he can infer is that bidder j has a type h j 2 [h i ; sup H j ]. This could mean that conditional on the history h (3) ; :::; h (n) and his own type, bidder i still has a strictly positive expected utility surplus (e.g., when v i strictly increases in h j ) at the current price. However, there will be a winner s curse if bidder i decides to stay and wins the asset for a higher price t j > t i. This follows from bidder j s strategy of dropping out when t j is the solution of Eu ~x + v j (h j ; h j ; h (3) ; :::; h (n) ) t j ); h j = u(0; hj ) which implies, by Theorem 1, that Eu ~x + v i (h) t i ); h i < u(0; hi ) On the other hand, if bidder i drops out at a higher price than t i and loses, he gets the same status quo utility as he would if he drops out at price t i. When there are k n active bidders left in the auction game, similar arguments can be used and show that it is a (weakly) dominated strategy to deviate from the dropping-out condition (17) for any bidder i. 18

20 The contribution of this theorem is to show that the English auction can be e cient in the more general environments where the bidders have asymmetric and non-quasi-linear utility functions. The added assumption here is that the bidders have common knowledge about each other s utility functional forms. This assumption is implicit in all models that assume risk neutrality. For the more general case, however, requiring that the bidders know each other s utility functional forms can be a strong assumption. On the other hand, the lack of knowledge about other bidders utility functions will not be a problem in more speci c situations. For instance, if v i (h) v + ^v(h i ) for some common value component v plus a private value component ^v that is nondecreasing, then both the second price (Vickrey auction) and the English auction imply an ex post e cient equilibrium. In this case, it will be a dominant strategy for bidder i to bid according to b i () such that Eu ~x + v + ^v(h i ) b i (h i ); h i = u(0; hi ) 5 E ects of increasing risk We now turn to comparative statics of risk, and show that increasing the asset s risk uniformly increases the bidder s expected utilities at the pre-auction stage. Following Rothschild and Stiglitz (1970), an increase in risk of the auctioned asset means in this context adding an independent noise ez to ~x with Eez = 0. In order for the expected utility to preserve the ordering of risk tolerance relation of u, i.e., that u 1 (w; y) Eu 1 (~x + w; y) is strictly log-supermodular in (w; y), one of the following conditions will be assumed. (A1) For all y 2 [L; H], u(; y) exhibits constant or decreasing absolute risk aversion (CARA or DARA). (A2) The marginal density function of ~x i (by the symmetry assumption, it is the same for all i) is log-concave. Jewitt (1987) shows that any one of these conditions implies the preservation of log-supermodularity of u 1 by integration. Condition A1 further implies that the 19

21 CARA or DARA property of u is preserved by integration (e.g., Nachman, 1982; Kihlstrom et al., 1982; Athey, 2000; see also Ross, 1981; Eeckhoudt, Gollier, and Schlesinger, 1996). 10 Since the properties of log-supermodularity are mainly studied in terms of weak inequalities, we rst present two lemmas that establish the preservation of strict log-supermodularity under expectation. The proofs of these lemmas can be found in the Appendix. The rst lemma is adapted from Jewitt (1987) and Athey (2000). Lemma 1 For functions u; u : R H! R such that u(w; y) = Eu(ex + w; y) and u 1 is strictly log-supermodular, if either (A1) or (A2) holds then u 1 is strictly log-supermodular. The next lemma further shows a property of CARA or DARA utility functions that are ordered by risk tolerance. Lemma 2 Consider functions u; : R H! R such that (w; y) = Eu(~x + w + v(y); y) where v is a nondecreasing function. If u 1 is strictly log-supermodular, and u(; y) exhibits CARA or DARA, then 1 is strictly log-supermodular. Now let = f~x; u; v 1 ; :::; v n g and ^ = f~x + ~z; u; v 1 ; :::; v n g denote the two auction environments that we consider, which di er only in that ^ involves a more risky asset for sale than does. Since in either Case 1 or Case 2, the direct mechanism of Theorem 2 and the English auction in Theorem 3 implement the same e cient outcome, we do not distinguish further the two cases. Let U i (h i jh i ) and ^U i (h i jh i ) denote the ex post expected utility of bidder i conditional on winning under the two environments respectively. 10 See, e.g., Caplin and Nalebu (1991) for many examples of densify functions that are logconcave. If u(; y) is restricted to be weakly concave, then a weaker condition that the cumulative distribution (rather than density) function of ~x i is log-concave is su cient for the preservation of the more risk tolerant relation (Jewitt (1987)). 20

22 Given any type vector h 2 H, by Theorem 2 or Theorem 3 the winner s payment ^t (2) under ^ is the solution of Eu ~x + ~z + v (2) (h (2) ; h (1) ) ^t (2) ; h (2) = u(0; h(2) ) (18) Theorem 4 Consider either Case 1 or Case 2, and suppose that either (A1) holds, or that (A2) holds and v i (h) v for all i. Then, except when there is a tie, in the ex post e cient equilibria described by Theorem 2 or Theorem 3, the winning bidder has a strictly higher expected utility under ^ than under : Moreover, xing h (1), the winner s improvement in expected utility due to increase in risk, i.e., ^U (1) (h (1) jh (1) ) U (1) (h (1) jh (1) ), is a strictly increasing function of h (1). Consequently, prior to the auction, the bidders are uniformly better o when the ensuing risk of asset increases. Proof. For a type vector h 2 H and an arbitrary i 2 N, x h (1) of h and de ne function i by (suppressing its notational dependence on h (1) ) i (z; y) = Eu(~x + z + v i (y; h (1) ) ^t (2) ; y) where ^t (2) satis es (18). If (A1) holds, then by Lemmas 1 and 2 i 1 is strictly logsupermodular in (z; y). If (A2) holds and v i (h) v for all i, then By Lemma 1, i 1 is strictly log-supermodular in (z; y). Thus, under both assumptions, i is ORT. For c ^t (2) t (2) where t (2) satis es (15), we also have i (c; y) = Eu(~x + v i (y; h (1) ) t (2) ; y)): Now, by equations (15) and (18), we have E (2) (~z; h (2) ) = (2) (c; h (2) ) (19) Since (1) is ORT, it follows now from Theorem 1 that for h (1) xed, and for h 0 (1) > h (1) > h (2), E (1) (~z; h 0 (1)) (1) (c; h 0 (1)) > E (1) (~z; h (1) ) (1) (c; h (1) ) > E (2) (~z; h (2) ) (2) (c; h (2) ) = 0 21

23 Since ^U i (h i jh i ) = E (1) (~z; h (1) ) under ^, and U i (h i jh i ) = (1) (c; h (1) ) under, if there is no tie, then ^U (1) (h (1) jh (1) ) U (1) (h (1) jh (1) ) is a strictly positive and increasing function of h (1). Consequently, prior to knowing the auction s outcome, taking expectation with respect to any arbitrary probability distributions of h i conditional on h i gives h E ^U i (h i j h ~ i ) U i (h i j h ~ i ) p(h i ; h ~ i i )jh i > 0 (20) In particular, prior to the auction, suppose h 2 H is distributed according to some probability measure that is common knowledge. Let V i (h i ) and ^V i (h i ) denote the pre-auction (date 0) expected utilities for the asset of bidder i under environments and ^ conditional on his type h i : h V i (h i ) E U i (h i j h ~ i i )jh i ; h ^V i (h i ) E ^U i (h i j h ~ i )jh i ; ^ i then (20) implies ^V i (h i ) > V i (h i ) for all h i and i. Remark: Note that in assuming (A2) in the theorem, there is an additional assumption that v i (h) v for all i. This is not necessary, but it is su cient so as to allow buyers to have increasing absolute risk aversion (IARA) and still prefer a higher risk. Theorem 4 has an intuitive interpretation as follows. If there is a tie for the highest types, then the seller extracts all the surplus. In other situations, by Theorem 1(i), the di erence between the winner s and the pivotal bidder s utility is a convex function of the pivotal bidder s utility. Therefore, since the ex post e cient mechanism requires the payment rule to satisfy (15) or (18), which implies that the pivotal bidder (weakly) prefers the more risky environment, by Theorem 1(ii) the winner will then strictly prefer the more risky environment. At a more fundamental level, the result of Theorem 4 is due to the convexity of the winner s preference with respect to the pivotal bidder s utility. Each bidder s utility is a random variable in the other bidders eyes. Increasing risk of the asset 22

24 leads to more volatile utilities, thus keeping the mean of any lower type s utility xed, all higher types derive higher expected utilities. This escalating e ect is ensured by the individual rationality condition that the lowest type has the same expected utility for the asset that is equal to his status quo utility in all situations. Surprisingly, the conclusion of the theorem does not depend on risk aversion as it allows the bidders to be risk preferring as well. For one thing, suppose all bidders are risk preferring, then increasing risk will lead to higher bids that apparently should hurt the winner. But this intuition turns out to be incorrect, as shown in Theorem 4. The key insight obtained from this theorem is that the increasing risk e ect that the buyers are generally better o when the asset for sale has a higher risk derives from the heterogeneity in risk attitudes among the bidders. 6 Conclusion This paper contributes a new model to the auctions literature, which incorporates ensuing risk of the auctioned object as well as buyers heterogeneous risk preferences. Under di erent assumptions concerning the seller s or the buyers knowledge about the auction environment, we show that either a direct mechanism or an English auction can implement e cient asset allocation in ex post equilibrium. These results extend the class of Vickrey auctions to the situations involving non-quasi-linear utility functions, while re-enforcing the e ciency appeal of the English auctions. In addition, the precautionary bidding e ect documented in Eso and While (2004) is shown to hold for larger situations, where buyers can be risk averse, risk neutral, or risk preferring, and the information structure and valuations can be asymmetric and interdependent. We have identi ed a more fundamental reason for such increasing risk e ect, namely, the presence of heterogeneous risk attitudes among the buyers. Our attention has been limited to the case in which the object for sale is a single indivisible risky asset. A promising line of future research is to see whether, and to what extent, the results obtained for the single risky asset case can be extended to 23

25 situations in which there are multiple units of a homogeneous, or heterogenous, risky assets for sale. Although the existing studies of multi-unit auctions have mainly focused on the risk neutral buyers (e.g., Dasgupta and Maskin (2000), Jehiel and Moldovanu (2001), Perry and Reny (2002), and Ausubel (2004)), we expect that some of the insights gained from these studies may also apply to the cases involving non-quasi-linear preferences. Of course, new issues may also arise as to those related to the notion of e ciency, the implied income e ects, the possible e ects of diversi cation, and so on. Appendix In this appendix, we rst present a theorem that is adapted from Ahlswede and Daykin (1979), and the proof from Karlin and Rinott (1980, Theorem 2.1), for their single-dimensional case. Theorem 5 For strictly positive and integrable functions A; B; C; and D de ned on R, suppose for all x; x 0 2 R, A(x)B(x 0 ) C(x _ x 0 )D(x ^ x 0 ) (21) and A(x)B(x 0 ) < C(x _ x 0 )D(x ^ x 0 ) whenever x 6= x 0, (22) where x _ x 0 = maxfx; x 0 g and x ^ x 0 = minfx; x 0 g. Then Z Z Z Z A(x)dx B(x)dx < C(x)dx D(x)dx (23) Proof. Since the result concerning the weak inequalities has been established (Ahlswede and Daykin (1979)), we focus on the strict inequalities. Following Karlin and Rinott (1980), it su ces to show the following stronger result that Z Z Z Z [A(x)B(x 0 ) + A(x 0 )B(x)] dxdx 0 < [C(x)D(x 0 ) + C(x 0 )D(x)] dxdx 0 ; x<x 0 x<x 0 (24) 24

26 which, together with (21), implies (23). For x < x 0, let a = A(x)B(x 0 ); b = A(x 0 )B(x); c = C(x)D(x 0 ); and d = C(x 0 )D(x). From (22) we have a < d and b < d. From (21) we have ab cd or c ab=d. Consequently, c + d a b ab d + d a b = 1 ((d a) (d b)) > 0 d Integrating yields (24), and hence (23). Now we prove the two lemmas in the text. Lemma 1. For functions u; u : R H! R such that u(w; y) = Eu(ex + w; y) and u 1 is strictly log-supermodular, if either (A1) or (A2) holds then u 1 is strictly log-supermodular. Proof. Suppose (A1) holds. Fix w; w 0 and y; y 0 such that w 6= w 0 and y 6= y 0, and de ne A(x) = u 1 (x + w ^ w 0 ; y _ y 0 ) B(x) = u 1 (x + w _ w 0 ; y ^ y 0 ) C(x) = u 1 (x + w _ w 0 ; y _ y 0 ) D(x) = u 1 (x + w ^ w 0 ; y ^ y 0 ): It can be veri ed straightforwardly that for x; x 0 2 R, these functions satisfy (22) if x x 0. If x > x 0, by the CARA or DARA property of u, we have B(x 0 ) D(x ^ x 0 ) = u 1(x 0 + w _ w 0 ; y ^ y 0 ) u 1 (x 0 + w ^ w 0 ; y ^ y 0 ) u 1(x + w _ w 0 ; y ^ y 0 ) u 1 (x + w ^ w 0 ; y ^ y 0 ) < u 1(x + w _ w 0 ; y _ y 0 ) u 1 (x + w ^ w 0 ; y _ y 0 ) = C(x _ x0 ) A(x) where the strict inequality follows from u 1 strictly log-supermodular. Thus, again, we get (22). It follows from Theorem 5 that (23) holds for A; B; C; and D, which implies that u 1 is strictly log-supermodular (noting that u 1 > 0). Now suppose (A2) holds. We have Z Z u 1 (w; y) = u 1 (x + w; y)f(x)dx = u 1 (x; y)f(x w)dx 25

27 As Jewitt (1987) noted, f(x) is log-concave if and only if f(x w) is log-supermodular in (x; w). Now x w; w 0 and y; y 0 such that w 6= w 0 and y 6= y 0, and de ne A(x) = u 1 (x; y _ y 0 )f(x w ^ w 0 ) B(x) = u 1 (x; y ^ y 0 )f(x w _ w 0 ) C(x) = u 1 (x; y _ y 0 )f(x w _ w 0 ) D(x) = u 1 (x; y ^ y 0 )f(x w ^ w 0 ) Since f > 0, we have for x; x 0 2 R such that x 6= x 0, A(x)B(x 0 ) = u 1 (x; y _ y 0 )u 1 (x 0 ; y ^ y 0 )f(x w ^ w 0 )f(x 0 w _ w 0 ) < u 1 (x _ x 0 ; y _ y 0 )u 1 (x ^ x 0 ; y ^ y 0 )f(x _ x 0 w _ w 0 )f(x ^ x 0 w ^ w 0 ) = C(x _ x 0 )D(x ^ x 0 ) Thus (22) holds. It follows from Theorem 5 that (23) holds. Hence u 1 is strictly log-supermodular. Lemma 2. Consider functions u; : R H! R such that (w; y) = Eu(~x + w + v(y); y) where v is a nondecreasing function. If u 1 is strictly log-supermodular, and u(; y) exhibits CARA or DARA, then 1 is strictly log-supermodular. Proof. By Lemma 1, u 1 (w; y) = Eu(~x + w; y) is strictly log-supermodular in (w; y). Thus for all w < w 0 and y < y 0, u 1 (w 0 + v(y 0 ); y 0 ) u 1 (w + v(y 0 ); y 0 ) > u 1(w 0 + v(y 0 ); y) u 1 (w + v(y 0 ); y) If u is CARA (or DARA), then integration preserves this property so that u is CARA (or DARA). Consequently, u 1 (w 0 + v; y) u 1 (w + v; y) is nondecreasing in v: Since v() is nondecreasing, we obtain 1 (w 0 ; y 0 ) 1 (w; y 0 ) = u 1(w 0 + v(y 0 ); y 0 ) u 1 (w + v(y 0 ); y 0 ) > u 1(w 0 + v(y 0 ); y) u 1 (w + v(y 0 ); y) u 1(w 0 + v(y); y) u 1 (w + v(y); y) = 1(w 0 ; y) 1 (w; y) 26

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