Design competition through multidimensional auctions

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4 Assumption 2. The trade always takes place (even with the highest cost type a). CHE By the revelation principle (Myerson, 1982), an optimal outcome can be identified as a direct revelation mechanism. The following proposition is due to Laffont and Tirole ( 1987), McAfee and McMillan ( 1987), and Riordan and Sappington ( 1987).5 Proposition 1. In the optimal revelation mechanism, the firm with the lowest 8 is selected; the winning firm is induced to choose quality go, which for each 19 maximizes V(q) - J(q, O), where J(q, 8) = c(q, 8) + -F(B)cs(q, 8). f(8) A standard interpretation applies: in the optimal mechanism, quality is distorted downward to limit the information rents accruing to relatively efficient firms, while competition further curtails the absolute magnitude of the rents. In this article I take this optimal outcome as a benchmark and focus on how it can be implemented through two-dimensional auctions that capture the salient features of the DoD source selection. I consider the following three auction rules: first-score auction, second-score auction, and second-preferred-offer auction. In all three auction schemes, the buyer selects the winning firm based on a scoring rule; more specifically, a contract is awarded to a firm whose offer achieves the highest score.6 Let S = S(q, p) denote a scoring rule for an offer (q, p). The rule is assumed to be publicly known to the firms at the start of bidding, and initially it is assumed that the buyer can design and commit to a rule in his best interest. The assumption of commitment is subsequently relaxed. Lack of commitment power on the part of the buyer will imply that the true preference of the buyer must be reflected in the scoring rule: i.e., S(q, p) = U(q, p).' In general, I restrict attention to quasi-linear scoring rules: Assumption 3. S(q, p) = s(q) - p, where s(q) - c(q, 8) has a unique interior maximum in q for all 19E [fl, 81 and s(.) is increasing at least for q I argmax s(q) - c(q, 8). Many scoring rules belong to this class, including the naive rule U(q, p) and the optimal rule discussed in Section 4. To understand the differences among the three auction schemes, it is convenient to distinguish between a winning firm's offer and a contract finally awarded to the firm. In a first-score auction, a winning firm's offer is finalized as a contract. Thus, this auction rule can be considered a two-dimensional analogue of the first-price auction. Second-score and second-preferred-offer auction rules represent two variants of the second-price auction. In a second-score auction, a winning firm is required (in the contract) to match the highest rejected score. In meeting this score, however, the firm is free to choose any quality (or quality-price combination). A second-preferred-offer auction rule, suggested by Desgagne ( 1988), is the same as the second-score auction except that a winner has to match not only the score but also the exact quality choice of the highest losing firm.' These three schemes are not just abstract mechanisms. To some extent, the actual source-selection procedure employed by DoD resembles aspects of all three mechanisms. In fact, my model is a special. limiting case of Riordan and Sappington (1987)in which first- and secondperiod costs are perfectly correlated. It is also a special case of Laffont and Tirole ( 1987),which also incorporates a moral hazard problem. If there is more than one firm that achieves the highest score, then a random drawing determines the winner. As is typical with atomless distributions, a particular tie-breaking rule does not affect expected equilibrium payoffs. 'AS usual, using any monotone transfol-mation of U as a scoring rule will have the same effect. The following example illustrates the difference among the rules. Suppose two firms A and B offer (7, 5) and (5, 3). and the scoring rule is such that A gets 10 points and B gets 8 points (i.e., S(7,5) = 10,S(5, 3) = 8). Then, under all three rules A is declared the winner. However, the final contract awarded to A is: (7, 5) under the first-score auction; any (y. p) satisfying S(q,p) = 8 under the second-score auction: and (5, 3)under the secondpreferred-offer auction.

5 672 / THE RAND JOURNAL OF ECONOMICS Although the initial stage of sealed-bid competition is similar to the first-score auction, sealed-bid competition is often followed by a negotiation phase in which DoD approaches and asks each firm to beat the offers made by the other firms. The second-score auction can be implemented by a two-stage scheme where a winner is selected in the first period through a sealed-bid auction and engages in a negotiation in the second period making a take-it-orleave-it offer to the buyer. The second-preferred-offer auction can be implemented by a similar two-stage scheme, but with an added restriction on the negotiation that the winning firm should dominate other firms in both price and quality.9 Therefore, studying these alternative auction schemes will essentially amount to examining alternative aspects of current DoD practice. Comparisons between the auction rules will have policy implications for whether a negotiation phase is desirable and whether the negotiation, if desirable, should involve the added restriction. 3. Equilibria under alternative multidimensional auction rules In this section, the equilibrium under each alternative auction scheme is characterized for a general scoring rule that satisfies Assumption 3. Each auction rule can be viewed as inducing a Bayesian game where each firm picks a quality-price combination (q, p) as a function of its cost parameter. Without any loss of generality, the strategy of each firm can be equivalently described as picking a score and quality, (S, q). The following lemma establishes that the equilibrium quality bid can be determined separately from the choice of score in the case of first- and second-score auctions. Lemma I. With first- and second-score auctions, quality is chosen at q,(o) for all 0 E [O, 8), where q,(b) = argmax s(q) - c(q, 8). Proof See the Appendix. The simple intuition behind the result is presented in Figure 1. Suppose firm i wants to offer any arbitrary score S, represented by the buyer's iso-score curve (depicted as the concave curve in Figure 1). As long as the firm moves along the curve, its probability of FIGURE 1 Winning firm's Buyer's isoscore curve Such a restriction may result from the political pressure of Congress, which has often demanded that a procured system should be the best not only in quality but also in price. Such behavior of Congress may be an optimal response to a buying agent that may misrepresent its preference.

6 CHE / 673 winning remains unchanged. Clearly, the firm's profit is maximized at the tangency point between its iso-profit curve and the buyer's iso-score curve, since that point allows the firm to maximize its profit without lowering its winning probability. Since the scoring rule is additively separable, quality choice is set independently of the score. As a result of Lemma 1, there is no loss of generality in restricting attention to qs(.) when searching for equilibria. (Essentially, Lemma 1 reduces a two-dimensional auction to a single-dimensional problem.) Let So(19) = max s(q) - c(q, 8) for all B E [O, 81. Then, by the envelope theorem, So(.) is strictly decreasing; therefore, its inverse exists. Now, consider the following change of variables: The problem can then be reinterpreted as one in which each firm, indexed according to its productive potential v with cumulative distribution H(-), proposes to meet the level of score b.in particular, letting b(.) denote an equilibrium bid function of v that is symmetric and increasing, the objective function facing each firm in the first-score auction can be described as n(qs(8), ~10) = [P- c(qs(19), 011 Prob {win/ S(qs(8), P I ) The following proposition is immediate: Proposition 2. (i) A unique symmetric equilibrium of a first-score auction is one in which each firm offers (ii) The second-score auction game has a dominant strategy equilibrium, where each firm with type 8 offers Proof: (i) follows from the above change of variables and the usual equilibrium result in first-price auctions (for example, Riley and Samuelson, 198 1); (ii) is immediate from Vickrey (1961 ) after applying the change of variables. Q.E.D. Several observations can be made. First, the equilibrium in the first-score auction is reduced to the equilibrium in the first-price auction if c(q, 19) = I9 (i.e., quality is fixed). Second, the Vickrey auction intuition applies for the second-score auction: Given the score So(8), if a firm with type 6' bids a higher score, it would risk winning at negative profits without increasing its profit conditional on winning; if it bids a lower score, it would forgo some opportunity of winning at positive profits. Finally, implementation of these auctions relies on the invertibility of So(.), which is always satisfied; this is in contrast to implementation via per-unit bid auctions, which requires costs to exhibit increasing returns to scale (Dasgupta and Spulber, 1990). 'O If a technology exhibits decreasing returns to scale, a more efficient firm may have a higher average cost (since it is induced to produce more quantity) and thus may offer a higher per-unit bid than a less efficient firm. Thus, a bid function may not be monotonic. This problem does not arise in my auction implementation, since the score bid function is always monotone decreasing in B.

7 674 / THE RAND JOURNAL OF ECONOMICS Substituting equilibrium bids from ( 1) yields the buyer's expected utility under the first-score auction: where 8, is, with a slight abuse of notation, the lowest-order statistic (i.e., 0, = min { 0, )El).I ' Since in general revenue equivalence may not hold, the expected utility under the second-score auction is separately obtained. where ps(o1, 02) is the price a winning firm actually pays to the buyer; i.e., Applying the argument used for the second-score auction, one can show that, in the second-preferred-offer auction, each firm will offer a score that will earn the firm zero profit, should the bid be finalized as a contract. However, in the second-preferred-offer auction a winning firm has no control over the quality choice. Therefore, each firm's zero-profit score offer depends on the quality-price bids of other firms. Hence, a dominant strategy equilibrium does not exist. In fact, the following proposition shows that there are many symmetric Bayesian-Nash equilibria in which each firm offers a zero-profit score. Proposition 3. In the second-preferred-offer auction, any (q(.),p(.))is a symmetric Bayesian- Nash equilibrium strategy for each firm ifp(8) = c(q( O), O), and S(q(O), p(0)) is decreasing in 8. Proof Suppose all the firms except for the one under consideration follow the suggested equilibrium strategy, (q( 6 ),p(.)). Define the equilibrium score offer S,(O) = S(q(O), p(o)) for all O E [e, 81, and let S2denote the highest score of all the scores offered by the remaining N - 1 firms. Then, there exists a distribution of S2, G, induced by the equilibrium strategy such that G(S2) = F(s;'(s~))~-'. (The inverse function is well defined since S,(-) is decreasing.) The firm under consideration (with cost type c) faces the following problem: = max J-, S(4,p) [c(4(si1(s2)), s,i(s~)) - c(q(si1(s2)), O)I~G. 4,P Since S, is decreasing, the integrand is positive if and only if S,(O) < S2. Therefore, the optimal strategy is any (q, p) such that S(q, p) = S,(O). In particular, the firm can do no better than offer the suggested (q(o), p(0)). Q.E.D. From the set of equilibria identified by this proposition, consider the equilibrium in which each firm offers the highest score that does not incur losses, (i.e., (qs(0), fi(8))). This equilibrium is the most score-efficient in that the offered score is the highest (among all the equilibria) for a given price offer. From now on, I shall restrict attention to this particular equilibrium. In this equilibrium, each firm offers q,(o), just as for the two other auction rules. But since the winning firm is obligated to produce the quality offered by the highest losing firm, " Analogously, I use 6'2 for the second-lowest-order statistic.

9 ~ 676 / THE RAND JOURNAL OF ECONOMICS FIGURE 2 1 Buyer's indifference curve Buyer's isoscore curve power, the only feasible scoring rule is one that reflects the buyer's preference ordering. The following proposition shows that, in this noncommitment case, all three schemes yield the same expected utility to the buyer-a two-dimensional extension of the revenue equivalence theorem. Proposition 5. (equivalence). If the buyer is unable to commit to a scoring rule differing from U(.,.), all three auction schemes yield the same expected utility to the buyer, equal to where q*( 8,) = argmax V(q)- c( q, 8,),the first-best quality level. Proof See the Appendix. Several remarks are in order. First, in first- and second-score auctions, the winning firm produces the first-best level of quality, which the buyer finds excessive (relative to the optimal quality in Proposition 1 ). The reason that the lack of commitment power induces more-than-optimal quality must be clear from Proposition 4; i.e., the true utility function fails to internalize the informational costs associated with increasing quality. Second, the equivalence property has an interesting policy implication. Recalling that the second-preferred-offer auction can be implemented with a phase of restricted negotiation, the above result can be interpreted as suggesting that the added restriction on negotiation does not harm DoD. Third, even though all three auction rules are equivalent from the buyer's standpoint, the same equivalence may not hold for firms. Since in the second-preferredoffer auction a winning firm picks the highest losing firm's first-best level of quality, q*( 02) (which is lower than its own first-best level, q*(o1)), the total surplus under the secondpreferred-offer auction is strictly smaller than under the other two auction rules. Since the expected utility for the buyer is the same under all auction rules, we obtain the following corollary. Corollary. Expected profits of the winning firm are the same in the first- and second-score auctions but strictly smaller in the second-preferred-offer auction.

11 678 / THE RAND JOURNAL OF ECONOMICS There are several ways to extend the model. The multidimensional bidding model discussed in this article does not address distinctive features of quality: q in the model could well be interpreted as quantity. In the context of defense procurement, however, quality is distinguished from quantity by the sequential decision making: quality is determined first and quantity is determined afterwards as a result of the budgetary decisions of Congress. Another feature of quality is its imperfect verifiability. In practice, evaluation of bids is based on signals of quality rather than actual quality. An interesting extension will be to incorporate these features of quality in the framework of multidimensional bidding. Appendix Proofs of Lemma I and Propositions 4 and 5 follow ProofofLemma 1. Suppose to the contrary an equilibrium bid (q, p) has q # q, at least for one firm with fl < 8. A contradiction is derived by showing that the bid is strictly dominated by an alternative bid (q', p') where q' = q, and p' = p + s(q,) - s(q).notice that S(q, p) = S(qr, p'). Also, Prob {win 1 S(q, p)) > 0. This can be proved as follows: First let &' = inf {SI Prob {win1 S } > 0 ) and S,(B) = max, s(q)- c(q,8). Then, &' i So(\$).To prove this claim, suppose, to the contrary, the probability of winning for some type B < 8 is zero. Then, the choice of the score S must be that S 5 S. But since So is decreasing in c, there is an alternative score S' E (S,S,(B)),which allows positive profits for the type 0, contradicting the optimality of the score S. Now, as desired. Q.E.D. Proof of Proposition 4. First I show that the new scoring rule implements the optimal quality schedule go(.)under first- and second-score auctions. Under both schemes, quality is chosen to maximize V(q)- A(q)- c(q, 0). Notice To check the second-order condition, let y --= - F(B) cqs(q,b). By Assumption 1, y is increasing in 0. Then, for all f(b) B E [O,81 and all q E [q,(g), qo(b)l, The second equality follows from the definition of q,(.), and the last inequality uses Assumption I. Thus, I have proved that the optimal quality schedule is implemented by the modified scoring rule \$(q, p). Now, to show I use a method similar to that in Milgrom (1989).Consider an abstract mechanism M. The expected profit of a firm with cost B under M is given by

12 CHE / 679 ~ X M PM,, 4~10) = XM(PM - c(~m,o)), where XM denotes the probability of winning. Define the optimized value of the expected profit Then, using the envelope theorem, r;(fl) dx;, P;, q;/o). From this it follows that T:(o) = lx;(t)ce(a;(t). t)dt + ~ 2 8 ). Now, notice that xfs = x& = [ I - FIN-.' and *Fs(8) = ~ : ~ ( = 8 ) 0. Since equilibrium quality is the same under both schemes equal to go, T& = T& = JS,8 c0(qo(f), f)[l - ~(f)]~-ldt and hence Since the expected total surplus is the same in the two schemes and equal to E{V(q,(O1))- c(q,(bl), B,))}, subtracting above-expected profits from the expected total surplus yields expected utilities under the two schemes: Finally, that the second-preferred-offer auction implements only a random quality schedule proves that it cannot implement the optimal mechanism. Q.E.D. Proof ofproposition 5. When S(.,.) = U(.,.). from Propositions 2 and 3, firms propose the first-best quality, and p,(bl, B2) = V(q*(O,))- V(q*(02))+ c(q*(02), 02). Substituting this into (3)yields that The rest of the proof follows, since EUss = E{V(q*(82))- c(q*(e2), 82)) = E~SPO. Euss = E ~SPO= E{V(q*(82))- c(q*(82), 82)) = E{v(q*(81))- J(q*(81),OI)} = EUFs. The third equality uses integration by parts and the envelope theorem. Q.E.D. References BARON,D.P. ANDMYERSON,R.B. "Regulating a Monopolist with Unknown Costs." Econometrics, Vol. 50 ( 1982), pp DASGUPTA,S. AND SPULBER,D.F. "Managing Procurement Auctions." Information Economics and Policy, Vol. 4 ( 1990),pp DESGAGNE, B.S. "Price Formation and Product Design Through Bidding." Mimeo, INSEAD, Fox, J.R. Arming America: How the US'. Buys Weapons. Cambridge, Mass.: Harvard University Press, HANSEN,R.G. "Auctions with Endogenous Quantity." RAND Journal of Economics, Vol. 19 ( 1988),pp LAFFONT,J.-J. AND TIROLE,J. "Using Cost Observations to Regulate Firms." Journal of Political Economy, Vol. 94 (1986), pp

13 680 / THE RAND JOURNAL OF ECONOMICS AND -. "Auctioning Incentive Contracts." Journal QfPolitical Economy, Vol. 95 ( 1987), pp AND -. "Auction Design and Favoritism." International Journal oflndzistrial Organization, Vol. S ( 1991), pp LICHTENBERG, F.R. "The Private R&D Investment Response to Federal Design and Technical Competitions.' American Economic Revievt: Vol. 78 ( 1988), pp MCAFEE, R.P. AND MCMILLAN, J. "Competition for Agency Contracts." RAND Journal of Economics, Vol. IE ( 1987), pp MILGROM,P.R. "Auctions and Bidding: A Primer." Jolirnal of Economic Perspectives, Vol. 3 ( 1989), pp MYERSON,R.B. "Optimal Auction Design." Mathematics ofoperations Research, Vol. 6 (1981), pp "Optimal Coordination Mechanisms in Generalized Principal-Agent Problems." Jotlrnal ofmathematica, Economics, Vol. 10 ( 1982), pp RILEY,J.G. AND SAMUELSON, W.P. "Optimal Auctions." American Economic Review, Vol. 71 (1981), pp RIORDAN,M.H. AND SAPPINGTON, D.E.M. "Awarding Monopoly Franchises." American Economic Review, Vol 77 (1987), pp AND ---. "Second Sourcing." RAND Journal ~feconomics, Vol. 20 (1989), pp ROGERSON,W.P. "Quality vs. Quantity in Military Procurement." American Economic Review, Vol. 80 ( 1990) pp VICKREY,W. "C~~nter~pec~lation, Auctions, and Competitive Sealed Tenders." Jotirnal of Finance, Vol. 16 ( 1961) pp

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