Existence of equilibria in procurement auctions

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1 Existence of equilibria in procurement auctions Gyula Seres a Humboldt University Berlin, Management Science Chair, Spandauer Strasse, 078 Berlin, Germany Abstract This paper investigates symmetric equilibria in first- and second-price auctions with multidimensional types. The constructed model mirrors the information structure of actual procurement auctions. We demonstrate by a counterexample that symmetric and continuous type distribution is not a sufficient condition for the existence of a pure-strategy equilibrium, but it is guaranteed if distributions of all components are logconcave. We state a new Revenue Equivalence Theorem applied to first- and second-price auctions and conclude that the two standard auction formats yields the same expected price to the auctioneer. Keywords: Auctions; Multidimensional information; Information asymmetry; Equilibrium existence JEL Codes: C6, D44, D8, H57. Introduction While in actual auctions bidders obtain several information signals, a substantial body of auction theory literature assumes that valuation function of bidders for a single good is univariate (Pesendorfer & Swinkels, 000; Reny, 0). Despite plausibility of complex information sets, validity of a range of fundamental results relies on the assumption of one-dimensional valuation functions. Seminal papers of Milgrom & Weber (98), Matthews (983), Maskin & Riley (984) and Milgrom (98) characterize a symmetric setting in which bidders valuations are affiliated and show that there exists a pure-strategy Bayesian Nash equilibrium (BNE) in first- and second-price auctions. Their results are based on the notion that types can be ordered and effectively summarized by a single-valued function. The importance of the single-dimension assumption is highlighted by Pesendorfer & Swinkels (000) and Jackson (009), who demonstrate that BNE does not always exist if types are multidimensional. In recognition of the existence problem, Goeree & Offerman (003) construct a two-dimensional model with independent common (CV) and private value (PV) types and derive that there exists an equilibrium in first- and second-price auctions if the distribution of types is quasi-concave. The study of multidimensional auctions is motivated by the more general notion of complex type space of economic agents (Armstrong & Rochet, 999). The main direction of auction theory research on complex information sets focuses on two-dimensional models using private- and common-value elements (De Silva et al., 009; Larson, 009; Tan & Xing, 0). These developments leave two important questions open. First, does continuous type distribution guarantee the existence of an equilibrium? Second, can the assumption of quasi-concavity be in general a remedy to the existence puzzle in multidimensional auctions? This paper broadens the scope by a multidimensional model. We demonstrate with a generalized model that continuity is not a sufficient condition, while quasi-concavity of type distributions guaranties existence. The focus is on linear, additively separable valuation functions, mirroring standard definitive cost estimation methods (Mahoney, 04). The main field of application is procurement auctions. Tenderers bidding on I am grateful to Jan Boone, Yves Breitmoser, Olga Chiappinelli, Dirk Engelmann, Charles N. Noussair, Patrick Rey, Máté Fodor, Yangwei Song and Róbert Somogyi for their valuable comments. All errors are my own. address: gyula.seres@hu-berlin.de (Gyula Seres) Preprint submitted to Elsevier September 5, 07

2 construction projects prepare cost estimates that take prices of inputs into account using multiple information signals. Some inputs, like construction materials and labor costs are relevant for all bidders. Other measures, including experience of the management, availability of qualified labor and financial situation are firmspecific. This paper mirrors the complex information set of actual bidders, and also allows for modeling heterogeneous technologies in which certain inputs are employed only by a subset of bidders. Consider for example the engineering of a new highway that requires a pavement design. Two bidders use a certain kind of cement while others make use of an alternative construction material. Price information regarding that cement type is only important for these two bidders. On a different note, this study also contributes to the rich literature on the Revenue Equivalence Theorem (RET) (Myerson, 98). The Theorem states that different auction formats yield the same expected revenue for the auctioneer if symmetric and risk-neutral bidders obtain independently distributed private value signals and a bidder with the lowest type receives zero payoff in equilibrium. The equivalence may break down if any of these assumptions is violated. Fang & Morris (006) show that RET does not hold in private value auctions if information signals are correlated. In general, the second-price sealed-bid auction results in weakly higher prices than the first-price auction (Milgrom & Weber, 98). Contributing to this body of research, the present study shows that revenue equivalence holds in a multidimensional model if bidder types are independently drawn, even if bidders valuations are not private values. This paper is organized as follows. Section presents the model framework and demonstrates by a counter-example that a continuous type space does not guarantee the existence of equilibria. Section 3 provides a sufficient condition for existence. The subsequent Section 4 addresses revenue and efficiency. Finally, Section 5 concludes the paper with a discussion.. Model A single, indivisible good is offered for sale to,..., N risk-neutral competing bidders. Player i observes a vector of K values x i = (x i,,..., x i,k ) independently drawn from strictly increasing Lipschitz continuous and twice continuously differentiable cumulative distribution function F i,k (x i,k ) with finite support [ xi,k, x i,k ] with 0 xi,k < x i,k. The corresponding densities are denoted by f i,k ( ). Valuation of bidder i is given by v i (x) = j,k α j,k,ix j,k, where x = (x,..., x N ), and the coefficients satisfy α j,k,i 0 for all j, k, i, where the indices denote the receiver of the signal j, the identity of the particular signal k and the affected bidder i, respectively. For example, α j,k,i = 0 means that signal x j,k observed by j does not affect i. The valuation function is symmetric with respect to all bidders, whose roles are permutable, i.e., any pair of bidders i, j and signal k satisfy x i,k = x j,k ; x i,k = x j,k ; and α i,k,j = α j,k,i. We consider first- and second-price sealed-bid auction formats. In both auctions, all bidders submit a non-negative bid b i, and the player with the highest offer wins. The price is equal to the highest and the second highest bid in the first- and second-price auction, respectively. The winner s payoff equals v i (x) p where p is the price. Others receive their outside option 0. The fraction of valuation known by bidder i is the composite signal e i = k α i,k,ix i,k. 3 Let us denote the component of the valuation function that is not known by i with h i = j,k α j,k,ix j,k, where j i. Hence, valuation consists of two additively separable components, v i = e i + h i. Example. Consider a two-dimensional type space in which bidders valuation is the sum of three variables. Bidders i {, } receive two signals x i, y i. Valuations satisfy that v i = x i + yi+yj, i j. In standard terms, x i, y i are private and common value signals, respectively. With our notation, α,, = α,, = 0, α,, = α,, = and α,, = α,, = α,, = α,, =. The composite signal is e i = x i + yi and the unknown component is h i = yj Most empirical models apply aggregate cost estimates directly in analyzing entry and bidding behavior (Krasnokutskaya & Seim, 0). As there are N bidders who receive K variables, the number of α components is N K. 3 Similar to Seres (07).

3 The key issue regarding multidimensional type space is that it cannot always be represented by a single informational variable. Hence, a symmetric pure-strategy equilibrium might fail to exist. The two-dimensional model of Goeree & Offerman (003) shows that existence of an equilibrium in first- and second-price auctions is guaranteed if the distribution of types satisfies quasi-concavity, that is, if the logarithm of distribution of types is concave. Extension of this result to our multidimensional setting is not straightforward. In what follows we pursue two goals. First, we show that existence of an equilibrium is not guaranteed in our setting. Second, we provide a sufficient condition as a remedy for this problem. Proposition. Continuity of type distribution is not a sufficient condition for the existence of a purestrategy BNE. Proof. A summary variable of types requires the expected value of signals to be increasing conditionally on the composite signal. Otherwise a bidder of higher type may have lower valuation conditional on winning. Given bidding strategies b(x i ) the conditional expected value of a bidder is a function of the composite signal e i, since the probability of winning as well as the type of other bidders are independent of e i. Hence, the optimal bidding strategy is a function of e i. We show that the equilibrium bid must be an increasing function of the composite signal. The best response function b i maximizes P r(b i max b i )[E(v i b i max b i ) E(p i (b) b i max b i )] = P r(b i max b i )[e i + E(h i b i max b i ) E(p i (b) b i max b i )] () where i refers to bidders other than i. The first-order condition tells us that the point-wise best-response function is increasing in e i. Types are independently drawn, hence, b i is also independent of e i. Any bid, hence, the optimal bid satisfies that b i (e i ) gives strictly higher payoff than b i (e i) if e i > e i. That is, the expected payoff is an increasing function of the composite signal. For this to hold it is a necessary condition that E(v i e i max e i ). Otherwise, there exists a pair of composite signals e i > e i such that the expected revenue of bidder i is point-wise smaller at e i for any bid. However, E(v i b i max b i ) is not always an increasing function of e i, as demonstrated by Example. Example. Consider the setting of Example with x i [0, 0] and y i [0, 0]. The density functions satisfy 76 if x i [0, ] [ +, 38 ] [39 +, 80] 4 4 ( (xi a ) )( ) if x i [a, a], a {, 38} 4 f(x i ) = + 4 ( (xi a+) )( ) if x i [a, a + ], a {, 38} if x i [ +, ] [38 +, 39 ] ( (xi a ) )( ) if x i [a, a], a {, 39} 4 4 ( (xi a+) )( ) if x i [a, a + ], a {, 39} 56 if y i [0, 3 ] [4 +, 75 ] [76 +, 80] 4 4 ( (yi a ) )( ) if y i [a, a], a {3, 75} 4 g(y i ) = + 4 ( (yi a+) )( ) if y i [a, a + ], a {3, 75} if y i [3 +, 4 ] [75 +, 76 ] ( (yi a ) )( ) if y i [a, a], a {4, 76} 4 4 ( (yi a+) )( ) if y i [a, a + ], a {4, 76} () (3) 3

4 where is a small positive number. In the limit we have that lim s 40 E(h i s e i ) = 5 6 and lim s 4 E(h i s e i ) = 69 6 and the expectation function is continuous for > 0. That is, there is a range of e i such that the conditional valuation is a decreasing function of the composite signal. This also holds for the conditional expected valuation, lim s 40 E(v i s e i ) = = > lim s 4 E(v i s e i ) = = The contribution of Example lies in that the distribution function satisfies the standard assumption of continuity, despite this, an equilibrium fails to exist. The key characteristic of the example is that the distributions are multimodal. As a consequence, a high composite signal can deliver a low conditional estimate of one of the information signals. The idea that conditional expectation of information signals is increasing in a composite information signal of variables originates from Milgrom & Weber (98), whose existence results are based on assuming this monotonicity. Previous counter-examples are offered by Goeree & Offerman (003) and Jackson (009). However, their models entail discontinuous distributions. 3. Existence of Equilibria In what follows we provide a sufficient condition for the existence of a pure-strategy equilibrium. Logconcavity is a fairly mild restriction on the distribution of types given that several standard distribution types satisfy it which are frequently used in empirical research, including the (multivariate) normal, exponential, uniform, logistic, extreme value, Laplace and Weibull distributions (Bagnoli & Bergstrom, 005). Assumption. Density function f i,k (x i,k ) is log-concave for all i, k. That is, for all x i,k, x i,k [ x i,k, x i,k ] and 0 < θ <, f(θx i,k + ( θ)x i,k) f(x i,k ) θ f(x i,k) θ. (4) It is straightforward to verify that the multimodal distribution in Example does not satisfy Condition (4). As we argued above, it is essential that signals of bidders can be summarized in a one-dimensional informational variable, hence the next Lemma is crucial. Lemma. Conditional expectations E(x i,k e = r) and E(x i,k e r) are non-decreasing functions of r if Assumption holds. Assumption is a weaker condition than convexity, hence, quasiconcavity and existence of equilibria are strongly linked. 4 Since the type space is compact and the valuation functions are continuous, von Neumann s minimax theorem holds (Sion et al., 958). Constructing an equilibrium bidding function proves existence. Next we show that the model satisfies the assumptions of the mineral rights model of Milgrom & Weber (98) with respect to the composite signal e i. They apply a single-dimensional representations of types. We need to show that composite signals are positively affiliated, i.e., f(e e )f(e e ) f(e)f(e ) (5) for any pair of composite signal vectors e and e, where e e denotes the component-wise maximum of e and e, and e e denotes the component-wise minimum. The composite signals are independently distributed, as well as their vectors, consequently, (5) is binding. We can state Lemma. Lemma. The composite signals are positively affiliated. 4 To see that a log-concave function is also quasi-concave, note, that the logarithm is monotone implying that the superlevel sets of this function are convex (Boyd & Vandenberghe, 004). 4

5 Applying the results of Milgrom & Weber (98) we can state Propositions and 3 on first- and secondprice auctions. Proposition. The strategy profile is an equilibrium of the first-price auction. b i (e i ) = E(v i e i max e i ) E(e i max e i e i max e i ) (6) The equilibrium bid consists of two terms. The first term refers to the expected valuation of the the winner given her excess, and the second term corresponds to the strategic bid shading, similarly to the first-price auction with private values. Proposition 3. is an equilibrium of the second-price auction. b i (e i ) = E(v i e i = max e i ) (7) The idea behind Proposition 3 is the same as that of Vickrey (96), except that no dominant strategy exists outside the realm of private-value auctions. Bidder i s strategy is conditional on marginally winning. 4. Efficiency and Revenue Equivalence One of the fundamental results of auction theory is the Revenue Equivalence Theorem, which states that the equilibrium expected revenue of the seller is independent of the auction mechanism under some conditions. Vickrey (96) established the result between first- and second-price auctions, which has been generalized later by Myerson (98) and Riley & Samuelson (98). The present section revisits this important theorem in our framework. First- and second-price auctions in our model satisfy three the conditions of the Revenue Equivalence Theorem (Riley & Samuelson, 98). Players are risk neutral, the object is allocated to the player with the highest bid and a player with the lowest type expects zero payoff. The condition of independently and identically distributed private value types is violated. We approach the issue by addressing efficiency first. Propositions and 3 imply that the equilibrium bid is an increasing function of the composite signal, e i e i b i(e i ) b i (e i ) with both mechanisms.5 Consequently, the winner as well as the total surplus is identical in both auction formats. Corollary. First- and second-price auctions provide the same level of allocative efficiency and the total surplus amounts to E(v i e max = e i ). Proof. Follows from Propositions and 3. For illustration, consider the following setting. Example 3. Bidders i {,, 3} all have single information signal x, x and x 3, respectively. All signals are drawn independently and identically from [0, ]. They possess valuation functions v = x + x, v = x + x 3 and v 3 = x 3 + x. From Proposition we learn, that the equilibrium bid of bidder in the first-price auction is b(s ) = b(x ) = x + E(x x max {x, x 3 }) E(x max {x, x 3 } x max {x, x 3 }) = 7 8. Similarly, from Proposition 3 we know that the equilibrium bid of bidder in the second-price auction is b(s ) = b(x ) = x + E(x max {x, x 3 } = x ) = 7 8. This equality is not accidental. The Revenue Equivalence Theorem does not extend to multidimensional single-unit auctions in general. Surprisingly, we can restate the theorem in our model. 5 Multiple signals generally lead to an ex post inefficient allocation (Jehiel & Moldovanu, 00). 5

6 Proposition 4. First- and second-price sealed-bid auction yield the same expected payoff to the seller. While Proposition 4 seems to be an interesting result, it is important to point out its limitations. The argument relies on the independence assumption. Without independence of bidder types, the affiliated value model does not support the RET. Milgrom & Weber (98) show, that in general, the expected revenue in the second-price auction is at least as large as in the first-price auction Discussion This paper demonstrates that continuity of types does not guarantee the existence of pure-strategy equilibria in auctions with multidimensional types. On the other hand, we also show that existence is still guaranteed if type distributions satisfy log-concavity. The results extend preceding literature on complex auctions to a larger domain relevant in large-value procurement auctions. Additionally, we present a new Revenue Equivalence Theorem, which suggest that first- and second-price auctions deliver the same expected revenue to the seller in multidimensional auctions if types are independent. While the results are stated for first- and second-price auctions, they can be straightforwardly extended to English and Dutch auctions. This study is strongly linked to a body of auction literature which aim is to extend the scope of research to a more complex setting, mirroring actual bidding markets (Klemperer, 004). Appendix Proof of Lemma. We proceed in multiple steps. Step. First we show that the density f(e) is log-concave in e. Note that if f(x) is log-concave, a marginal f(x i,k ) is log-concave (An, 998). Let us consider the joint density f(x i,,..., x i,k, e) = f(x i, )... f(x i,k )f(e i l K α i,l,ix i,l ). The components are log-concave and strictly positive, hence their logarithms are concave which implies that their Hessian matrices are negative semi-definite. Consider that k log [f(x i,,..., x i,k, e i )] = log[f(x i,l )] + log[f(e i α i,l,i x i,l )] (8) l K l= Since the sum of negative semi-definite matrices is negative semi-definite, we can establish that f(x i,,..., x i,k, e i ) is log-concave. Given the first part of this Lemma, its marginal f(e i ) is log-concave, as well. Step. Goeree & Offerman (003) show that any two random variables e and x satisfy, that E(x e r) = e [ E(x e = r) E(x e r) f e(r) F e (r) Step 3. We turn attention to the Lemma directly. It holds, that f(x i, e i ) = f(x i, )... f(x i,k )f(e i l K α i,l,ix i,l ). Since the marginals of f( ) are log-concave, x i, ei log [f(x i, e i )] = xi, ( e i f(x i, e i ) f(x i, e i ) ) 0 ei f(x i, e i ) f(x i, e i ) e i f(x i, e i) f(x i, e i), x i, x i, ei f(x i, e i ) f(x i, e i) 0, x i, x i, ] (9) 6 Symmetricity is also crucial, as demonstrated by (Engelbrecht-Wiggans et al., 983). 6

7 that is, for all e i e i, we have f(x i, e i )/f(x i, e i) f(x i, e i )/f(x i, e i ). Integrating by x i, from x i, to x i, and by x i, from x i, to x i, yields F (x i, e i ) F (x i, e i ), corresponding to first-order dominance, that is, E(x i, e i ) E(x i, e i ) for e i e i, which proves the first part. Using Step gives x i, E(x i, e i r) = [E(x i, e i = r) E(x i, e i r)] f ei (x i, )/F ei (x i, ) 0. The latter inequality is given by the first part. The proof for all x i,k with any i, k is identical. Proof of Proposition 4. By the Envelope Theorem we get that the derivative of the equilibrium profit of a buyer with respect to a bidders excess is equal to the equilibrium probability of winning. This applies to both the first- as well as the second-price auction mechanism. The probability of winning in equilibrium can be expressed by G n (e i )/ e i which means the equilibrium profit is e i e L G n (t i )dt i, where G(e i ) denotes the cumulative density function of excess. From this we can derive the expected payoff of the winner of the auction as e H ei e L G n (t i )dt i dg e (e i ), where G e (e i ) = G n (e i )/ e i. e L From Corollary we know that the total surplus is the same in both formats. Hence, the expected price, equivalently, the expected revenue is identical, as well. An, M. Y. (998). Logconcavity versus logconvexity: a complete characterization. Journal of Economic theory, 80, Armstrong, M., & Rochet, J.-C. (999). Multi-dimensional screening:: A user s guide. European Economic Review, 43, Bagnoli, M., & Bergstrom, T. (005). Log-concave probability and its applications. Economic theory, 6, Boyd, S., & Vandenberghe, L. (004). Convex optimization. Cambridge university press. De Silva, D. G., Jeitschko, T. D., & Kosmopoulou, G. (009). Entry and bidding in common and private value auctions with an unknown number of rivals. Review of Industrial Organization, 35, Engelbrecht-Wiggans, R., Milgrom, P. R., & Weber, R. J. (983). Competitive bidding and proprietary information. Journal of Mathematical Economics,, Fang, H., & Morris, S. (006). Multidimensional private value auctions. Journal of Economic Theory, 6, 30. Goeree, J., & Offerman, T. (003). Competitive bidding in auctions with private and common values. Economic Journal, 3, Jackson, M. O. (009). Non-existence of equilibrium in vickrey, second-price, and english auctions. Review of Economic Design, 3, Jehiel, P., & Moldovanu, B. (00). Efficient design with interdependent valuations. Econometrica, 69, Klemperer, P. (004). Auctions: theory and practice,. Krasnokutskaya, E., & Seim, K. (0). Bid preference programs and participation in highway procurement auctions. The American Economic Review, 0, Larson, N. (009). Private value perturbations and informational advantage in common value auctions. Games and Economic Behavior, 65, Mahoney, W. D. (04). Standard Estimating Practice: American Society of Professional Estimators volume 9. Craftsman Book Co. Maskin, E., & Riley, J. (984). Optimal auctions with risk averse buyers. Econometrica: Journal of the Econometric Society, (pp ). Matthews, S. A. (983). Selling to risk averse buyers with unobservable tastes. Journal of Economic Theory, 30, Milgrom, P. R. (98). Rational expectations, information acquisition, and competitive bidding. Econometrica: Journal of the Econometric Society, (pp ). Milgrom, P. R., & Weber, R. J. (98). A theory of auctions and competitive bidding. Econometrica, 50, 089. Myerson, R. B. (98). Optimal auction design. Mathematics of operations research, 6, Pesendorfer, W., & Swinkels, J. M. (000). Efficiency and information aggregation in auctions. American Economic Review, (pp ). Reny, P. J. (0). On the existence of monotone pure-strategy equilibria in bayesian games. Econometrica, 79, Riley, J. G., & Samuelson, W. F. (98). Optimal auctions. The American Economic Review, 7, Seres, G. (07). Auction cartels and the absence of efficient communication. International Journal of Industrial Organization, 5, Sion, M. et al. (958). On general minimax theorems. Pacific J. Math, 8, Tan, X., & Xing, Y. (0). Auctions with both common-value and private-value bidders. Economics Letters,, Vickrey, W. (96). Auctions and bidding games. In Recent Advances in Game Theory (pp. 5 7). Princton University. 7