Incentive and Sampling Effects in Procurement Auctions with Endogenous Number of Bidders Youngwoo Koh∗ November 8, 2016
Abstract We study an auction contest for a procurement of innovation. Firms exert effort and the resulting quality of innovation is ex ante uncertain. Given this uncertainty, there is a trade-off regarding the number of participating firms in the contest: increasing the number of firms reduces each firm’s chance of winning the auction, leading the firms to make little efforts; meanwhile, the chance of obtaining a high quality of innovation increases with the number of firms due to the randomness of the quality. Thus, the procurer faces a nontrivial problem of how many firms to invite. We show that in the high level of randomness, it is optimal for the procurer to invites many firms. As the randomness vanishes, however, inviting only two firms is optimal. We also show that a fixed-prize tournament may outperform the auction when the randomness is large. Keywords: Procurement, Contest, Auction, Innovation, Quality
1
Introduction
Firms and governments increasingly rely on procuring goods and services from outside sources. In particular, they often seek to procure innovations, and the innovative activities require firms to undertake investments. For instance, when the Department of Defense procures a weapon system, defense contractors often make R&D investments to produce ∗
College of Economics and Finance, Hanyang University,
[email protected]. I wish to thank Yeon-Koo Che, Chang-Koo Chi, Booil Philip Jeon, Navin Kartik, Jinwoo Kim, Kyungmin Kim, Eiichi Miyagawa, Keeyoung Rhee, Mike Riordan and seminar participants at Columbia University, Yonsei University, Hanyang-Kobe-Nanyang Conference, Korean Economic Association Conference, and KIET. I also thank two anonymous referees for their insightful and constructive comments. Financial support from Hanyang University (HY-2014-G) is gratefully acknowledged.
prototypes and then participate in the procurement process.1 Although the quality of innovation depends on firms’ investment level, it is often the case that the quality is not solely determined by the exerted investment, making it ex ante uncertain. Moreover, if a firm’s investment cost cannot be recovered unless it wins the contract, then firms may refrain from exerting investments.2 A typical way of the procurer to maintain firms’ investment incentives is limiting the number of participants. For example, in procurement auctions funded by the World Bank, the procurer puts a limited number of contractors on a “short list” (see Fan and Wolfstetter, 2008). In this paper, we study a procurement of innovation in that the buyer can decide how many firms to invite. We consider a first-score auction in which the quality of a firm’s innovation stochastically depends on its investment, and the buyer procures an innovation from the firm who offers him the most favorable price-quality combination, called “score.” Such a scoring auction has served as a prominent contest mechanism. For instance, the US government procures highway constructions, as well as weapon systems, via scoring auctions, and the European Union mandates the use of scoring auctions for public procurements (see Asker and Cantillon, 2008). In such procurement auctions, the procurer is often restricted to specify the number of offers to elicit.3 That is, the buyer faces a nontrivial problem of how many firms to invite. If there are many competitors, participants may be discouraged from making any substantial investment for fear of losing the entire investment in case they do not win. We call this an incentive effect. On the other hand, due to the randomness on the quality realization, the more firms the buyer invites, the higher the chance he has of having a high quality innovation. We call this a sampling effect. Intuitively, if the randomness is negligible, limiting the number of participants is beneficial for the buyer. Otherwise the firms’ investment level would be low. For a large randomness, however, the sampling effect may prevail over the incentive effect, so it can be the case that inviting many firms is optimal. We will investigate how the optimal number of contestants varies depending on the degree of randomness. To isolate the trade-off between the two effects, we restrict our attention to an environment in which (i) firms invest nonmonetary efforts, (ii) the level of investment and the 1
Piccione and Tan (1996) and Jeitschko and Wolfstetter (2000) study procurement auctions in that bidders invest in quality before they place their bids. 2 This is true if the outcome of the investment per se does not have a market value. The investment costs can be understood as nonmonetary efforts or opportunity costs. 3 For instance, the Word Bank is bounded by its procurement rule to shortlist a limited number of firms (typically up to six) and only those firms are invited to bid. Similarly, the European Investment Bank also sends invitations to tender for a procurement only to shortlisted firms. The European Investment Bank states in its guideline for public-private partnerships that “The purpose of shortlisting is to reduce the number of bidders to generally between three to five. [...] Just as the presence of too few bidders results in poor competition, the presence of too many bidders on the shortlist may reduce the interest in participating and cause good bidders to drop out.” (European Investment Bank, 2011)
2
resulting quality of innovation are unverifiable, (iii) firms are liquidity-constrained, and (iv) there is no outside market for the firms except selling the innovation to the buyer.4 Under such an environment, we develop a simple model. There are identical firms, and the buyer decides how many of them to invite in the contest. Any participating firm exerts investment, draws quality from a distribution, and then bids a price to be paid if it wins. We assume that the quality of a firm’s innovation is additively separable in its investment level and a random variable, which is independent on the firm’s investment. We first show that if the randomness is larger than a certain threshold value, then there is a symmetric pure-strategy equilibrium. In such equilibrium, we identify the incentive and sampling effects and show that it is optimal for the buyer to invite as many firms as possible. The reason is that even though each firm’s investment level decreases in the number of contestants, the buyer gains more by inviting many firms due to the large randomness on the quality. That is, the sampling effect dominates the incentive effect. We then consider the case that the randomness is small. In this case, we show that any (mixed-strategy) equilibrium converges to an equilibrium as the randomness vanishes. Based on the characterization of the equilibrium at the limit, we show that the optimal number of contestants is two. This is because the small randomness results in a minimal sampling effect, and inviting many firms merely reduces the firms’ incentive to invest. In such a case, the buyer prefers to solicit only two firms. We also discuss implications of the innovation technology and the optimality of the auction mechanism. Because of the additive formulation of innovation technology, there is a concern if the results still hold when the randomness depends on firms’ investments. We consider an alternative innovation technology incorporating such a possibility and show that the results with large randomness still carry over, but those with small randomness do not follow. Next, we consider a fixed-prize tournament, in which the best innovator receives a fixed prize, and show that when the randomness is large, the buyer invites many firms as in the auction and benefits by using the tournament rather than the auction. Lastly, we discuss the difficulties of characterizing equilibrium and comparing the buyer’s payoffs under different numbers of firms for an intermediate level of randomness.5 There are several works studying the optimal number of contestants. Che and Gale (2003) 4
When the quality is verifiable, the terms of contracts can be made contingent on the realized quality of innovation. Any optimal mechanism selects the firm who delivers the highest quality at a minimal cost (Laffont and Tirole, 1986; McAfee and McMillan, 1987; Riordan and Sappington, 1987). If the cost associated with innovation activities is observable, one could implement various reimbursement schemes (Rogerson, 2003; Chu and Sappington, 2007). Even with unverifiable quality, the buyer may use a simple “option contract” that requires suppliers to pay an up-front fee (Taylor, 1993) or use an entry auction (Fullerton and McAfee, 1999; Giebe, 2014) if firms do not have liquidity constraints. 5 But, we provide an example of a symmetric mixed-strategy equilibrium with two firms in Appendix A.3 for a various degree of randomness.
3
show that when the innovation technology is deterministic, the first-score auction with two firms is optimal among various contest mechanisms. A crucial difference between their model and ours is the randomness on the quality. We show that with a large randomness, the buyer benefits by inviting many firms in the auction, but the auction is not necessarily optimal. Nevertheless, we also show that their results still hold when the randomness is sufficiently small. In this sense, our work complements theirs. Taylor (1995) and Fullerton and McAfee (1999) consider a fixed-prize tournament with random quality and show that the buyer restricts the number of contestants. As opposed to our model, each firm decides the number of draws of quality on the same distribution in the former and has to pay a fixed cost to make an investment in the latter. Terwiesch and Xu (2008) study a trade-off regarding the number of contestants, similar to the incentive and sampling effects, in fixed-prize tournaments and provide conditions under which it is optimal for the buyer to invite many firms.6 For the comparison of auctions and tournaments, Fullerton et al. (2002) combine theoretical and experimental comparison based on Taylor (1995) and show desirability of auctions. Sch¨ottner (2008) provides conditions under which the auction outperforms the tournament and vice versa when the innovation technology involves randomness. Giebe (2014) also compares those two mechanisms with an entry auction and shows that both mechanisms implement the same level of investment conditional on the same number of participants in both mechanisms. However, both Sch¨ottner (2008) and Giebe (2014) assume that the number of participants is fixed and do not consider how the optimal number of firms is determined endogenously.7 Our model shares similarity with Piccione and Tan (1996) and Jeitschko and Wolfstetter (2000) which study the impact of investments in quality before procurement takes place. Our work is also related to the literature studying bidders’ endogenous entry decisions. Li and Zheng (2009) study the effects of the number of potential bidders on the buyer’s revenue in procurement auctions.8 Menezes and Monteiro (2000) and De Silva et al. (2009) show that the seller’s (buyer’s) revenue may decrease when the number of potential bidders increases in an auction (procurement). The rest of the paper is laid out as follows. Section 2 introduces the model. Section 2.1 and Section 2.2 consider the cases of large and small randomness, respectively. Section 3 6
See also Coleff and Garc´ıa (2013) which show that the buyer attracts only two bidders when the buyer can send a signal to the firms and Lauermann and Wolinsky (2013) which study the case that the invited bidders do not know how many other bidders have been invited. 7 See also Letina and Schmutzler (2016) which show that when the quality of innovation depends on the “research approach” rather than effort of innovation, a variant of fixed-prize tournament is optimal and the buyer may invite many firms. 8 They identify the “entry effect”—meaning that as the number of potential bidders increases, each bidder’s probability of entry falls, which leads to a reduced auction price. See also Marmer et al. (2013) that develops a “selective entry” model.
4
discusses the implications of firms’ innovation technology and the suboptimality of the auction. The difficulties of analysis in the intermediate level of randomness are also discussed. Section 4 concludes the paper. Proofs are provided in Appendix A.1 unless stated otherwise.
2
Model
A buyer wishes to procure an innovation from a set of identical firms, M = {1, . . . , N }, where N ≥ 2. In order for a firm to achieve an innovation, it must exert investment effort x ∈ [0, ∞) with cost of ψ(x), where ψ(·) satisfies that ψ(0) = 0, ψ 0 (·) > 0, ψ 00 (·) > 0 and ψ 000 (·) ≥ 0. The quality of an innovation of firm i is summarized by qi , which depends on the firm’s investment level as follow: qi = xi + δε, (1) where δ ≥ 0 and ε is the realization of a random variable following a continuous distribution H(·) with a smooth density function h(·) over the support E. We assume that ε has zero expected value and a finite variance.9 Let q(x) := x+δε and q(x) := x+δε, where ε := inf(E) and ε := sup(E) for notational simplicity. Note that the distribution of quality conditional on investment level x is given by F (q|x) = H
q−x δ
.
Denote by f (q|x) the conditional density. It is easy to see that Fx (q|x) ≤ 0, showing that for a given δ, if a firm invests a higher x, then it has a higher probability of getting a high quality. We assume that Fxx (q|x) ≥ 0, meaning that the investment is subject to diminishing returns to scale in the sense of the convexity of the distribution function condition. 10 Note also that for a given x, δ measures the degree of randomness in the sense that greater δ means greater dispersion of q.11 Since the randomness does not depend on the firms’ investment level, the innovation technology (1) clearly distinguishes the effects of the investment and the randomness on the quality, thereby helping understand the consequences of them in the buyer’s revenue. (In Section 3.1, we consider an alternative innovation technology in that the randomness depends on the investment level.) The timing of the game is as follows. At date 0, the buyer decides how many firms to 9
The assumption that E[ε] = 0 is made for simplicity but does not affect the results. See Jeitschko and Wolfstetter (2000) for the same assumption. This assumption is satisfied for large families of distributions (see Section 3.1), and we need to assume this to have a symmetric pure-strategy equilibrium in Section 2.1 as in the standard moral hazard problem. A sufficient condition to obtain this in our model is that the density of ε, h(·), is (weakly) increasing. 11 It is easy to see that the distributions F (q|x; δ) satisfy mean-preserving spread for a given x. That is, for δ > δ 0 , F (q|x; δ) ≥ F (q|x; δ 0 ) for all q < x and F (q|x; δ) ≤ F (q|x; δ 0 ) for all q > x. 10
5
invite in the auction. At date 1, all firms decide whether to participate in the contest. Each of the participating firms simultaneously chooses its investment level x, bearing cost ψ(x), and draws quality q according to the innovation technology (1). The quality is unverifiable but is observable by the buyer (but not by the other firms). At date 2, each firm submits a price p, and the buyer selects the firm that offers him the highest score, s = q − p. The winning firm is paid p.12 In what follows, we first consider the case that the randomness δ is large. We then consider the limit case when δ becomes small. Throughout this paper, we denote the firm’s payoff by π and the buyer’s payoff by R.
2.1
With Large Randomness
In this section, we focus on a symmetric pure-strategy equilibrium and investigate the propˆ so that such an equilibrium erties it satisfies. We then show that there is a threshold value δ, ˆ The restriction on symmetric equilibrium seems well justified, espeexists whenever δ ≥ δ. cially when all agents are symmetric and no pattern of asymmetry is known a priori as in our model. Suppose n firms have participated in the contest, and let N = {1, . . . , n} ⊆ M be the set of participants. We consider the bidding stage at first, assuming that all firms have chosen the same investment level x in the investment stage (which will be analyzed later). Suppose there is a symmetric bidding strategy p(q) such that s(q) = q − p(q) is strictly increasing. Note that firm i wins the auction if and only if his score offer is the highest one, i.e., qi − p(qi ) ≥ qj − p(qj ) for all j 6= i and j ∈ N . Hence, the firm seeks p(qi ) that maximizes its expected payoff p(qi ) Prob(qi − p(qi ) ≥ qj − p(qj ), ∀j 6= i . Observe that Prob qi − p(qi ) ≥ qj − p(qj ), ∀j 6= i = Prob qj ≤ s−1 (si ), ∀j 6= i = F (s−1 (si )|x)n−1 , where the first equality holds since s(·) is strictly increasing and the last equality follows from the symmetry. Since p = q − s, the firm’s problem is reduced to finding an optimal score offering strategy: q ∈ arg max q − s(˜ q ) F (˜ q |x)n−1 . q˜ 12
Since the innovation quality is unverifiable, the only credible scoring rule is s = q − p which reflects the buyer’s incentive to select the firm that brings him the highest surplus. Second-score auction, for example, in which the winner is required to match the highest rejected score, does not work because the qualities are not verifiable. See Che (1993) and Che and Gale (2003). See also Asker and Cantillon (2008) for studying score auctions when firms have multidimensional private information.
6
Maximizing the objective function yields that 1 s(q) = F (q|x)n−1
Z
q
t dF (t|x)
n−1
Z
q
=q− q(x)
q(x)
F (t|x) F (q|x)
n−1 dt.
(2)
It is evident that s(·) is strictly increasing in q, and the bidding strategy is13 Z
q
p(q) ≡ q − s(q) = q(x)
F (t|x) F (q|x)
n−1 dt.
(3)
Note that once a firm draws q, it can be interpreted as the firm’s “type” and so the equilibrium score offering in (2) is equal to the symmetric equilibrium “bid” in the standard first-price auction in independent private values setup. The equilibrium bid p(q) in (3) is given by the difference between the quality and score, and this corresponds to the amount of “bid shading” in the standard first-price auction. In such a standard auction, increasing the number of bidders is always desirable for the buyer since it intensifies competition among them. In the current model, however, it is not necessarily desirable for the buyer to invite many firms since the investment level decreases with the number of participants, as will be shown later. Hence, the optimal number of firms to be invited is unclear a priori. Next, consider the investment stage. Suppose all firms except firm i invest the same level x. Then, firm i’s expected payoff from investing xi is Z
q(xi )
πi (xi ) =
n−1
p(q)F (q|x)
Z
q(xi )
dF (q|xi ) − ψ(xi ) =
q(xi )
F (q|x)n−1 1 − F (q|xi ) dq − ψ(xi ),
q(xi )
where the last equality follows from (3) and after some rearrangement. Note that F (q|x)n−1 is the probability that other firms except firm i have qualities smaller than q, and 1 − F (q|xi ) is the probability that firm i’s quality is greater than q. Differentiating πi and evaluating it at xi = x yield that Z q(x) − F (q|x)n−1 Fx (q|x)dq − ψ 0 (x) = 0. (4) q(x)
Let x∗ be the symmetric investment strategy satisfying (4). We have so far assumed that all firms exert the same level of investment. We now show that each firm has no incentive to unilaterally deviate from x∗ so that the proposed investment level is an equilibrium indeed. To this end, suppose all firms except firm i employ the same investment strategy x∗ , and consider firm i’s payoffs when it invests xi < x∗ and 13
If the qualities are observable by the other firms, the equilibrium bidding strategy is p(qi ) = qi − qj if qi > qj = maxk∈N \{i} {qk }. See Sch¨ ottner (2008) for this treatment.
7
(a) n = 2, δ = 1
(b) n = 5, δ = 1
Figure 1: Firm’s payoff in Example 1 xi > x∗ . In the former case, firm i’s payoff is Z
q(xi )
πi (xi ) =
F (q|x∗ )n−1 (1 − F (q|xi ))dq − ψ(xi ),
q(x∗ )
since it would be defeated by other firms if qi < q(x∗ ). Consider now the case that xi > x∗ . Note that if qi > q(x∗ ), firm i’s quality is the highest one, so it would offer the score ∗
Z
q(x∗ )
s(qi ) = s(q(x )) =
q dF (q|x∗ )n−1 ,
q(x∗ )
where the last equality follows from (2).14 Using this, firm i’s payoff from the deviation is Z
q(x∗ )
Z
!
q
πi (xi ) =
∗ n−1
F (t|x ) q(xi )
Z
q(xi )
dt dF (q|xi ) +
q(x∗ )
(q − s(q(x∗ ))dF (q|xi ) − ψ(xi ).
q(x∗ )
The following proposition states that in the case of large randomness, firm i cannot benefit by decreasing or increasing its investment level from x∗ . ˆ there exists a unique symmetric Proposition 1. There exists δˆ > 0 such that for any δ ≥ δ, equilibrium in which each firm invests x∗ according to (4) and bids p(q) according to (3). Note that the threshold value δˆ is independent on n. The following example illustrates the results in Proposition 1. Example 1. Suppose ε follows the uniform distribution over [−1, 1] and ψ(x) = 21 x2 . Suppose s(qi ) > s(q(x∗ )). Then, firm i would like to reduce the score, because there is no loss of probability of winning by doing so, but it increases the price. 14
8
In Example 1, the quality is drawn uniformly over [x − 1, x + 1]. Together with the 9 and x∗ = n1 . Figure 1 depicts a firm’s quadratic cost function, one can show that δˆ = 18 payoff πi , which is concave in its investment xi , and reveals how it varies with the investment level in the symmetric equilibrium when δ = 1. In the left panel, the number of firms is 2, and in the right panel, it is 5. Observe that in each of the cases, firm i attains the highest payoff when xi = x∗ and both x∗ and πi (x∗ ) are higher at n = 2 than at n = 5. We now investigate how the investment level changes according to the number of firms. ˆ Then, x∗ is decreasing in the number of participating firms n. Lemma 1. Suppose δ ≥ δ. As suggested by Example 1, Lemma 1 states that the equilibrium investment level decreases with the number of firms. It is intuitive since a larger number of participants decreases the chance of winning for each firm, and therefore the firms become more reluctant to make a significant investment. This captures the incentive effect related to the number of participants. Next, consider the buyer’s payoff in the equilibrium. Let m(q) denote each firm’s expected surplus offer to the buyer, i.e., ∗ n−1
m(q) := Prob(win) × s(q) = F (q|x )
1 × F (q|x∗ )n−1
Z
q
t dF (t|x∗ )n−1 .
q(x∗ )
Then, the buyer’s expected payoff for a given n is Z
q(x∗ )
R(n) := n
∗
Z
q(x∗ )
m(q) dF (q|x ) = q(x∗ )
q dF (q|x∗ ){2:n} ,
(5)
q(x∗ )
where F (q|x∗ ){2:n} is the distribution of the second-highest-order statistic. Given this, the buyer must decide how many firms to invite. Clearly, for a fixed investment level, the buyer wishes to invite many firms to exploit the randomness of the quality realizations, which captures the sampling effect. ˆ Fix x∗ by some x¯ independent on n. Then, R(n+1; x¯) ≥ R(n; x¯). Lemma 2. Suppose δ ≥ δ. R q(¯x) Proof. Let R(n; x¯) = q(¯x) qdF (q|¯ x){2:n} be the buyer’s payoff when there are n firms and all of them invest x¯. Note that F (q|¯ x){2:n+1} first-order stochastically dominates F (q|¯ x){2:n} . Hence, it follows that R(n + 1; x¯) ≥ R(n; x¯). Lemma 1 and Lemma 2 show the two effects depending on the number of firms. As shown in Lemma 2, the buyer benefits by inviting many firms if the investment level does not depend on the number of firms, but Lemma 1 reveals that the investment level is actually decreasing with it. Thus, the buyer must consider both sampling and incentive effects, and the optimal number of firms is decided depending on which effect outweighs the other. The 9
next result shows that it is optimal for the buyer to invite as many firms as possible whenever ˆ δ ≥ δ. ˆ Then, R(n) is increasing in n, and hence the buyer finds it Proposition 2. Suppose δ ≥ δ. optimal to invite all firms in M. Proof. For the ease of proof, treat n as a real number (instead of an integer). Totally differentiate R to obtain ∂R ∂R dx∗ (n) dR(n, x∗ (n)) = + ∗ . dn ∂n ∂x dn Note that ∂R/∂n ≥ 0 by Lemma 2 and dx∗ /dn ≤ 0 by Lemma 1. Observe also that for x0 ≥ x, F (q|x0 ){2:n} first-order stochastically dominates F (q|x){2:n} for a fixed n, showing that ∂R/∂x∗ ≥ 0. Thus, the proof is done by comparing the magnitudes of the sampling effect and the incentive effect multiplied by ∂R/∂x∗ . In Appendix A.1, we show that the former is larger than the latter so that R(n) is increasing in n. The result of Proposition 2 is intuitive. Given a large amount of randomness, the sampling effect is of first order importance compared to the incentive effect. That is, the former effect dominates the latter. To see this explicitly, consider Example 1 again. Recall that there is 9 a symmetric pure-strategy equilibrium with x∗ = n1 whenever δ ≥ 18 . The buyer’s payoff in this case is 4δ R(n) = δ − + x∗ . (6) n+1 Observe that R(n + 1) − R(n) = Decompose R(n) by RS (n) = δ −
4nδ − n − 2 1 1 ≥ 0 if and only if δ ≥ + . 3 2 n + 3n + 2 4 2n 4δ n+1
and RI (n) = x∗ . Note that
∆RS (n) := RS (n + 1) − RS (n) =
4δ (n + 1)(n + 2)
is the “marginal benefit” of inviting one more firm (i.e., the sampling effect), and ∆RI (S) := |RI (n + 1) − RI (n)| =
1 n(n + 1)
9 is the “marginal cost” of it (i.e., the incentive effect). Since n ≥ 2 and δ ≥ 18 , the former exceeds the latter, and so the buyer is better off by inviting many firms even though each firm’s investment level decreases.
10
2.2
With Small Randomness
ˆ Although, in In this section, we consider the case that the randomness is smaller than δ. Appendix A.3, we present an equilibrium for each δ < δˆ when there are two firms under Example 1, it appears difficult to fully characterize equilibrium for an arbitrary number of firms and to find the optimal number of participants. (In Section 3.3, we discuss the difficulties of the analysis.) Instead, we study the case that the randomness is sufficiently small and analyze the optimal number of firms in such a case. To this end, we first characterize equilibrium at the limit as δ vanishes and then show that it is optimal for the buyer to invite only two firms for sufficiently small δ. ˆ Recall that after getting an innovation, a firm’s, say i’s, problem is to Suppose δ < δ. choose its price pi , or equivalently, its score si (qi ) = qi − pi (qi ). For a given δ > 0, let Gδi (·) denote the cumulative distribution function of firm i’s score offer. Then, the firm solves the following problem: max (qi − si ) Gδ−i (si ), si
where Gδ−i (·) = j∈N \{i} Gδj (·) is the probability of winning and N is the set of participating firms in the auction. We must also have that qi ≥ si , or else the firm would risk winning at a negative payoff. For a given δ > 0, let Siδ be the support of Gδi (·) and S δ := ∪j∈N Sjδ . Also, let Xiδ denote the support of firm i’s investment and X δ := ∪j∈N Xjδ . We begin with the following observations. Q
ˆ the following results hold. Lemma 3. In any mixed-strategy equilibrium for 0 < δ < δ, (i) Gδ−i (·) is continuous and strictly increasing in s ∈ Siδ . (ii) For any si ∈ Siδ , si (·) is strictly increasing in qi . Part (i) implies that none of distributions have mass points, and consequently, there are at least two firms offering the same score with positive probability in S δ . Part (ii) states that the higher quality a firm gets, the higher score it offers. Using those properties, we now establish our convergence result. To be precise, let Si0 be the support of score offerings of firm i at δ = 0 and S 0 := ∪j∈N S 0 . For a given s ∈ S 0 , let I(s) := i ∈ N | si = s for s ∈ S 0 be the set of firms whose scores are the same as s. ˆ Lemma 4. Fix any s ∈ S 0 . For any sequence of equilibria along with s for each δ < δ, Gδi (s) converges to some distribution G0 (s) as δ approaches to zero for all i ∈ I(s). Proof. We sketch the proof presented in Appendix A.1. The proof involves several steps. First, we show that each firm’s score is contained in an interval which is bounded at most of 11
order δ. Formally, fix a firm’s, say i’s, investment level by x, and let Siδ (x) be the support of firm i’s score for the given x. Since si is strictly increasing in qi and qi ∈ [q(x), q(x)], it holds that si ∈ [s(x), s¯(x)] for any si ∈ Siδ (x), where s(x) := s(q(x)) and s¯(x) := s(q(x)). Then, for any x ∈ Xiδ , we show that s¯(x) − s(x) ≤ δM for some finite number M . The second step shows that S δ converges to S 0 ≡ [0, x˜ − ψ(˜ x)] as δ vanishes, where 0 x˜ := arg maxx≥0 {x − ψ(x)}. Observe that for a fixed s ∈ int(S ), this implies that there is a ˜ Thus, for any selection of equilibrium along δ˜ > 0 such that s is contained in S δ for all δ < δ. s for each δ, there always exists a subsequence such that the same firm, say i, is repeated in that sequence. For a fixed s ∈ int(S 0 ), let Xδi (s) := x ∈ Xiδ | s ∈ Siδ (x) be the set of investment levels of firm i such that s ∈ Siδ (x) for given δ > 0. The third step is to show the convergence of distribution. Using the results above, we show that for any x ∈ Xδi (s), Gδ−i (·) is bounded: Gδ−i (s + δM ) ε − δM ≤ ψ(x + ε) − ψ(x)
(7)
Gδ−i (s + δM ) δM + ε ≥ ψ(x) − ψ(x − ε).
(8)
and Next, by taking limits of (7) and (8), we have that G0−i (s) = ψ 0 (x0 (s)), where x0 (s) is the limit of x. Since this holds for all i, we have this identity for any i and i0 in I(s), meaning that both Gδi (s) and Gδi0 (s) converge to the same distribution at the limit, i.e., G0 (s) = ψ 0 (x0 (s))1/(|I(s)|−1) . Observe that if the buyer has invited only two firms, then |I(s)| = 2 for any s ∈ S 0 (by Part (i) of Lemma 3), hence there is a unique symmetric equilibrium in this case.15 We now show that the buyer benefits by inviting only two firms at the limit. Proposition 3. At the limit, the buyer’s payoff from inviting only two firms is higher than that from inviting more than two firms. Proof. Observe that for any s ∈ S 0 , |I(s)|
G0 (s)|I(s)| = ψ 0 (x0 (s)) |I(s)|−1 ≥ ψ 0 (x0 (s))2 = G0 (s)2 , where the inequality holds since ψ 0 (x0 (s)) = G0 (s)|I(s)|−1 ∈ [0, 1] and |I(s)| ≥ 2. Notice that the left-hand side of the above inequality is the cumulative distribution function of the first-order statistic of payoff with arbitrary number of firms; and the right-hand is that with two firms. Since the latter first-order stochastically dominates the former, the buyer collects the highest payoff by inviting only two firms. 15
With more than two firms, there may be an asymmetric equilibrium; that is, |I(s)| can be different from |I(s0 )| for some s 6= s0 .
12
The optimality of inviting two firms is intuitive. As the randomness vanishes, the gain from inviting many firms, i.e., the sampling effect, becomes negligible and the incentive effect prevails. Hence, to promote investment, the buyer invites only two firms. Che and Gale (2003) show in more general setup that the buyer invites only two firms when the innovation technology is deterministic. Proposition 3 shows that their result still holds even when the quality realization involves a small randomness. We conclude this section by providing an example summarizing the result in Proposition 3. Example 2. Suppose ψ(x) = 12 x2 and the buyer invites n ≥ 2 firms. In the symmetric 1 equilibrium, G0 (s) = (2s) n−1 where S 0 = [0, 21 ]. For any n0 > n, G0 (s; n) stochastically dominates G0 (s; n0 ). So, it is not profitable to invite more than two firms.
3 3.1
Discussions Innovation Technology
The innovation technology postulated in (1) helps us analyze the optimal number of firms. Specifically, the randomness can prevail without any investment, and so for a large randomness, the buyer invites many firms even though each firm’s investment level diminishes by doing so. For a sufficiently small randomness, however, the quality (almost entirely) depends on the investment level, making the buyer invite only two firms to promote their investments. In this section, we provide an alternative innovation technology where the randomness depends on firm’s investment level. We show that the results with large randomness still carry over—that is, our results in Section 2.1 are not restricted to the environment with the additively separable innovation technology. However, under such an alternative innovation technology, the results with small randomness in Section 2.2 do not follow. This reveals the convenience of our baseline model that disentangles the investment and the randomness. Consider the following innovation technology: given its investment x, a firm’s quality is drawn according to F (q|x) = G(q)x , where G(·) is a distribution with continuous density g(·) and support [q, q] (independent on x).16 Similar to the innovation technology (1), it also satisfies that Fx (q|x) ≤ 0 and Fxx (q|x) ≥ 0, while the randomness vanishes as the investment goes to zero.17 Applying the same analysis in Section 2.1, it is ready to see that firm i’s 16
Such an innovation technology is frequently used in the contest literature. See Fullerton and McAfee (1999) and Giebe (2014), for example. x 17 For example, if G(q) = q with support [0, 1], then E[q|x] = 1+x and V ar(q|x) = (1+x)x2 (2+x) . Hence, both the expected quality and its variance go to zero as the investment level x goes to zero.
13
payoff from investing xi when all other firms invest x is q
Z
G(q)(n−1)x (1 − G(q)xi ) dq − ψ(xi )
πi (xi ) = q
Differentiating πi and evaluating it at xi = x, we have Z
q
−
G(q)nx ln G(q)dq − ψ 0 (x) = 0.
(9)
q
Let xa be the investment level satisfying (9). Similar to (5), the buyer’s payoff is given by Z R(n) =
q a {2:n}
q dF (q|x )
Z
q
= q¯ − n
q
(n−1)xa
G(q) q
q
Z
a
G(q)nx dq.
dq + (n − 1) q
Note that for a fixed investment level, inviting more firms leads a higher buyer’s payoff by Lemma 2. The next lemma shows that R(n) is increasing in n, while xa is decreasing in n, and so the buyer benefits by inviting all firms in M. Lemma 5. There exists a symmetric equilibrium with investment level xa given by (9). In such equilibrium, xa is decreasing in n and R(n) is increasing in n.
3.2
Auction vs. Tournament
We have so far focused on the auction mechanism. However, fixed-prized tournaments, in which the best innovator receives a fixed prize, are also frequently used in reality.18 In this section, we consider a fixed-prize tournament and compare this with the first-score auction. We show that the tournament outperforms the auction when the randomness is large. Consider a fixed-prize tournament as follows. The timing is the same as before, but the price each firm can bid is fixed by P , which is determined by the buyer before firms’ making investments. Thus, a firm’s score is s = q − P , and the firm who draws the highest quality wins and is paid P . Similar as Section 2.1, consider a symmetric pure-strategy equilibrium. For fixed P and n, firm i’s payoff from investing xi when all other firms invest x is Z
q(xi )
πi (xi ) = P
F (q|x)n−1 dF (q|xi ) − ψ(xi )
q(xi )
" = P F (q(xi )|x)n−1 −
Z
#
q(xi )
F (q|x)n−2 f (q|x)F (q|xi ) dq − ψ(xi ),
q(xi ) 18
For instance, in 1820, Liverpool and Manchester Railway announced £500 for the best performing engine for the first passenger line between two British cities (Fullerton and McAfee, 1999). More recently, Netflex offers $1,000,000 for the best algorithm to predict user ratings for films (Giebe, 2014).
14
where the last equality follows from integration by parts. Differentiating πi and evaluating it at xi = x yield that Z
q(x)
(n − 1)F (q|x)n−2 f (q|x)Fx (q|x) dq − ψ 0 (x) = 0.
−P
(10)
q(x)
Denote the integration term of (10) by I. The buyer’s payoff, denoted by Rt , is given by t
Z
q(x)
q(x)
Z
n
F (q|x)n dq −
q dF (q|x) − P = q¯(x) −
R = q(x)
q(x)
ψ 0 (x) , I
where the last equality follows from integration by parts and the fact that P = ψ 0 (x)/I. Observe that any prize P induces a unique investment level x according to (10),19 and so choosing P is equivalent to choosing x for the buyer. Thus, the optimal investment level, xt , can be determined by the first- and second-order conditions of Rt with respect to x. The following proposition establishes the existence of equilibrium and characterize it ˆ It also compares the buyer’s payoff with that from the auction R(n) in (5). when δ ≥ δ. ˆ There exists a unique symmetric equilibrium with xt = 0. Proposition 4. Suppose δ ≥ δ. ˜ ˆ In such equilibrium, Rt (n) is increasing in n. Moreover, for each n there exists δ(n)(≥ δ) ˜ such that for any δ ≥ δ(n), Rt (n) > R(n). Proof. We sketch the proof presented in Appendix A.1. We show that Rt is concave in x ˆ which entail the corner solution xt = 0 and so P = 0. and ∂Rt /∂x is negative for δ ≥ δ, Thus, the buyer’s payoff in the tournament is t
Z
q(0)
q dF (q|0)n ,
R (n) = q(0)
which is increasing in n. Next, recall that the buyer’s payoff in the auction is Z
q(x∗ )
R(n) =
qdF (q|x∗ ){2:n} =
Z
q(x∗ )
q(x∗ )
qdF (q|x∗ )n −
"Z
q(x∗ )
q(x∗ )
qdF (q|x∗ )n −
q(x∗ )
Z
#
q(x∗ )
qdF (q|x∗ ){2:n}
q(x∗ )
where x∗ is given by (4) and the term in the square bracket is the information rent paid to the winner. Observe that expected quality the buyer obtains is higher in the auction than in the tournament, i.e., Z
q(x∗ ) ∗ n
Z
q(0)
q dF (q|x ) > q(x∗ ) 19
q dF (q|0)n ,
q(0)
In fact, it can be shown that the investment level x satisfying (10) is strictly increasing in P .
15
but the price paid to the winner is smaller in the tournament than in the auction, i.e., Z
q(x∗ )
Z
∗ n
q(x∗ )
q dF (q|x∗ ){2:n} > 0 = P.
q dF (q|x ) − q(x∗ )
q(x∗ )
˜ In Appendix A.1, we show that for δ ≥ δ(n), the gain from saving the information rent outweighs the loss from obtaining a smaller quality in the tournament (for a given n). In the tournament, the buyer does not elicit any effort from firms but relies entirely on the sampling effect when the randomness is large. Moreover, by using the tournament instead of the auction, the buyer saves the information rent at the cost of reduced quality. For a large randomness, the information rent is so large that the gain from the payment becomes larger than the loss from the quality. To see this concretely, consider Example 1. Recall that x∗ = 1/n. The quality difference between the auction and the tournament is Z
1 q( ) n
1 q( ) n
Z
qdF (q| n1 )n
q(0)
qdF (q|0)n =
− q(0)
1 , n
and the payment difference is Z
1 q( ) n
1 q( ) n
q
dF (q| n1 )n
Z −
1 q( ) n
1 q( ) n
q dF (q| n1 ){2:n} =
2δ . n+1
2δ ˜ Thus, Rt (n) − R(n) = − n1 + n+1 is positive if and only if δ ≥ n+1 =: δ(n). Note that since 2n t ˜ both R (n) and R(n) are increasing in n and δ(n) is decreasing in n, the buyer invites all firms in M and prefers the tournament over the auction for a large randomness. Our result is similar to Sch¨ottner (2008) who shows that the buyer may prefer the tournament over the auction when the technology involves a large amount of randomness. The main difference is that the number of firms is given exogenously in Sch¨ottner (2008), while it is determined endogenously in our model. Che and Gale (2003) show that when the innovation technology is deterministic, the first-score auction with two firms is optimal among various contest mechanisms including fixed-prize tournaments. However, Proposition 4 reveals the suboptimality of auctions for a large randomness, similar as Sch¨ottner (2008). In Appendix A.2, we analyze the tournament under the innovation technology used in Section 3.1 and show that the results still carry over.
16
3.3
With Intermediate Randomness
In our main analysis, we have considered two polar cases—when the randomness is large and when it is sufficiently small—and show that those two cases have opposite implications on the number of firms the buyer invites. A natural question is what is the optimal number of ˆ However, it appears difficult to answer firms for intermediate levels of randomness, δ < δ. this question, and we want to briefly discuss the complications. Note that for an intermediate level of randomness, an equilibrium might have mixedstrategies in the investment stage or the bidding stage or both. First, consider the bidding stage. Recall that in Section 2.2, the distribution of score offering is obtained from (7) and (8), the upper and lower bounds of the distribution, but those bounds are depending on δ. Hence, the exact score distribution is obtained only for sufficiently small δ but not for an intermediate level of δ. Next, consider the investment stage. Allowing for mixed-strategies in investments, let Ki (x) denote the distribution of firm i’s investment with support Xi and Hi (q) be the induced quality distribution.20 From the firm i’s payoff when it invests xi ∈ Xi , we have the following first-order condition Z
q(xi )
− q(xi )
Y
Hj (q)Fx (q|xi )dq − ψ 0 (xi ) = 0,
j6=i
and the corresponding second-order condition together with indifference conditions, i.e., πi (xi ) = πi (x0i ) for all xi , x0i ∈ Xi . Using those conditions, however, we are not able to derive the equilibrium investment strategy in a closed form. The situation is not as simple even Q under the restriction of symmetric equilibrium. In this case, j6=i Hj (q) = H(q)n−1 and so the first-order condition becomes an (n − 1)th order equation, which does not have a closed form solution again. By fixing n = 2 and further assuming a specific distribution of F , we can obtain a closed form solution and characterize an equilibrium as shown in Appendix A.3.
4
Conclusion
This paper studies a procurement problem in which firms undertake investments and the resulting qualities of the innovative activities are ex ante uncertain. In such an environment, the buyer’s payoff depends on the number of firms as well as the degree of randomness in the quality of innovation. We show that incentive and sampling effects affect the buyer’s payoff in opposite ways, and the optimal number of firms is determined endogenously: for a large randomness, the buyer does not limit the number of firms; while for a sufficiently small 20
R
F (q|x)dKi (x); and if Ki is a discrete Pk Pk distribution with mass points xt , t = 1, . . . , k for some k, then Hi (q) = t=1 mt F (q|xt ), where t=1 mt = 1. For instance, if Ki is a continuous distribution, then Hi (q) =
17
Xi
randomness, the buyer invites only two firms. Although we have focused on the first-score auction, we also show that when the randomness is large, the buyer prefers the fixed-prize tournament to the auction and invites many firms in the tournament. This shows that the importance of the randomness in innovation technologies and the associated sampling effect are not restricted in the first-price auction. In fact, the suboptimality of the auction for a large randomness opens a natural question on an optimal procurement mechanism when the randomness plays a role and the buyer decides the number of solicitations. We leave this question for future research.
A A.1
Appendix Omitted Proofs
Proof of Proposition 1. We show that πi is strictly concave in xi so that x∗ is the unique equilibrium. To this end, consider the case xi < x∗ at first. Firm i’s payoff from such a deviation is Z q(xi ) F (q|x∗ )n−1 1 − F (q|xi ) dq − ψ(xi ). πi (xi ) = q(x∗ )
Note that if xi ≤ x∗ − δ∆ε so that q(xi ) ≤ q(x∗ ), then πi (xi ) = −ψ(xi ), which is strictly concave. Suppose now xi ∈ (x∗ − δ∆ε, x∗ ) so that q(x∗ ) < q(xi ). Then, dπi (xi ) =− dxi
Z
q(xi )
F (q|x∗ )n−1 Fx (q|xi )dq − ψ 0 (xi ),
q(xa )
which becomes zero at xi = x∗ by (4). Observe that Z q(xi ) d2 πi (xi ) ∗ n−1 = −F (q(xi )|x ) Fx (q(xi )|xi ) − F (q|x∗ )n−1 Fxx (q|xi )dq − ψ 00 (xi ) dx2i ∗ q(x ) Z q(xi ) h(ε) − F (q|x∗ )n−1 Fxx (q|xi )dq − ψ 00 (xi ) = F (q(xi )|x∗ )n−1 δ q(x∗ ) ≤
h(ε) − ψ 00 (xi ), δ
where the second equality holds since Fx (q|x) = − 1δ h q−x and the last inequality follows δ ∗ ∗ ∗ from the facts that F (q(xi )|x ) < F (q(x )|x ) = 1 and Fxx (q|xi ) ≥ 0. Now, let δˆ := 89 ψh(ε) 00 (0) . ˆ Then, for any δ ≥ δ, h(ε) h(ε) h(ε) − ψ 00 (xi ) ≤ − ψ 00 (0) ≤ − ψ 00 (0) < 0, ˆ δ δ δ 18
(A.1)
where the first inequality holds since ψ 000 (·) ≥ 0 and xi ≥ 0. Next, consider the case xi > x∗ . Suppose xi ∈ (x∗ , x∗ + δ∆ε) so that q(xi ) < q(x∗ ). Firm i’s payoff from such a deviation is Z
q(x∗ )
Z
!
q ∗ n−1
πi (xi ) =
F (t|x ) ∗ n−1
F (q|x )
Z
∗
q(xi )
q − q(x∗ ) +
q(x∗ ) ∗ n−1
F (q|x )
F (q|x∗ )n−1 dq dF (q|xi ) − ψ(xi ),
Z
q(x∗ )
F (q|x∗ )n−1 F (q|xi )dq + q(xi ) − q(x∗ )
dq −
q(x∗ )
Z
!
q(x∗ )
Z
q(x∗ )
q(x∗ )
=
F (q|x∗ )n−1 F (q|xi )dq
q(xi )
+ Z
q(x∗ )
dq F (q(x )|xi ) −
q(x∗ )
Z
(q − s(q(x∗ ))dF (q|xi ) − ψ(xi )
q(x∗ )
q(x∗ )
=
q(xi )
dt dF (q|xi ) +
q(x∗ )
q(xi )
Z
Z
q(xi )
q(xi )
−
F (q|xi )dq − ψ(xi ), q(x∗ )
where the second equality holds since s(q(x∗ )) = q(x∗ ) − dπi (xi ) =− dxi
Z
q(x∗ ) ∗ n−1
F (q|x )
Z
R q(x∗ ) q(x∗ )
q(xi )
Fx (q|xi )dq −
F (q|x∗ )n−1 dq. Therefore,
Fx (q|xi )dq − ψ 0 (xi ),
q(x∗ )
q(xi )
which is zero at xi = x∗ by (4). Note also that Z q(x∗ ) d2 πi (xi ) ∗ n−1 F (q|x∗ )n−1 Fxx (q|xi )dq = F (q(xi )|x ) Fx (q(xi )|xi ) − 2 dxi q(xi ) Z q(xi ) − Fx (q(xi )|xi ) − Fxx (q|xi )dq − ψ 00 (xi ) q(x∗ )
h(ε) h(ε) F (q(xi )|x∗ )n−1 + − ψ 00 (xi ) δ δ h(ε) ≤ − ψ 00 (xi ) < 0, δ ≤−
where the first inequality holds since Fxx (q|x) ≥ 0 and Fx (q|x) = − 1δ h q−x , the second δ ∗ ∗ ∗ inequality holds since F (q(xi )|x ) > F (q(x )|x ) = 0, and the last inequality follows from ˆ Lastly, note that for xi ≥ x∗ + δ∆ε (so that q(xi ) ≥ q(x∗ )), firm i’s payoff is (A.1) for δ ≥ δ. Z
q(xi )
πi (xi ) =
(q − s(q(x∗ )))dF (q|xi ) − ψ(xi ),
q(xi )
which is concave in xi for δ ≥ δˆ similar as before. 19
Proof of Lemma 1. Note that since Fx (q|x) = −f (q|x), the integration term of (4) is Z
q(x∗ ) ∗ n−1
F (q|x )
Z
∗
q(x∗ )
(n − 1)F (q|x∗ )n−1 f (q|x∗ )dq,
f (q|x )dq = 1 −
q(x∗ )
q(x∗ )
where the equality is obtained by integration by parts. This implies that q(x∗ )
Z
F (q|x∗ )n−1 f (q|x∗ )dq =
q(x∗ )
and hence ψ 0 (x∗ (n)) = n1 > i.e., x∗ is decreasing in n.
1 , n
(A.2)
= ψ 0 (x∗ (n + 1)) . Since ψ 00 (·) > 0, we have x∗ (n) > x∗ (n+1),
1 n+1
Proof of Proposition 2. Write R(n) from (5), q(x∗ )
Z
qn(n − 1)(1 − F (q|x∗ ))F (q|x∗ )n−2 f (q|x∗ )dq
R(n) = q(x∗ )
Z
∗
q(x∗ )
= q(x ) − n
∗ n−1
F (q|x )
Z
q(x∗ )
dq + (n − 1)
q(x∗ )
F (q|x∗ )n dq,
q(x∗ )
where the last equality follows from integration by parts. Regarding n as a real number, ! Z q(x∗ ) Z q(x∗ ) ∗ dR(n) dq(x dq(x∗ ) ) d F (q|x∗ )n−1 dq − n = − + F (q|x∗ )n−1 dq dn dn dn dn ∗ ∗ q(x ) q(x ) ! Z Z q(x∗ ) ∗ q(x ) ∗ ) dq(x d + F (q|x∗ )n dq + F (q|x∗ )n dq + (n − 1) dn dn ∗ ∗ q(x ) q(x ) Z q(x∗ ) h i dx∗ =− F (q|x∗ )n−1 1 − F (q|x∗ ) + n − (n − 1)F (q|x∗ ) ln F (q|x∗ ) dq + dn q(x∗ ) Z q(x∗ ) Z q(x∗ ) Z q(x∗ ) ∗ n−1 ∗ n =− F (q|x ) dq + F (q|x ) dq − n F (q|x∗ )n−1 ln F (q|x∗ )dq q(x∗ )
q(x∗ )
Z
q(x∗ )
q(x∗ )
+ (n − 1)
F (q|x∗ )n ln F (q|x∗ )dq −
q(x∗ )
1 n2 ψ 00 (x∗ )
,
(A.3)
where the last equality holds since dx∗ /dn = −1/(n2 ψ 00 (x∗ )), which is obtained by applying the implicit function theorem to ψ 0 (x∗ ) − n1 ≡ 0. Now, observe that Z
q(x∗ ) ∗ n−1
F (q|x ) q(x∗ )
∗
Z
q(x∗ )
ln F (q|x )dq =
F (q|x∗ )n−1
q(x∗ )
20
ln F (q|x∗ ) f (q|x∗ )dq f (q|x∗ )
∗ ∗ q(x ) ∗ n ln F (q|x ) F (q|x ) f (q|x∗ ) q(x∗ )
=
q(x∗ )
Z = −
q(x∗ )
Z
q(x∗ )
− q(x∗ )
∗ ∗ n−1 ln F (q|x ) F (q|x ) f (q|x∗ )
0
F (q|x∗ )dq
0 ∗ ∗ n−1 ∗ ∗ n−1 ∗ n ∗ f (q|x ) (n − 1)F (q|x ) ln F (q|x ) + F (q|x ) − F (q|x ) ln F (q|x ) dq, f (q|x∗ )2
implying that Z
q(x∗ ) ∗ n−1
F (q|x )
n
Z
∗
q(x∗ )
ln F (q|x )dq = −
∗ n−1
F (q|x )
q(x∗ )
q(x∗ )
f 0 (q|x∗ ) + F (q|x ) ln F (q|x ) dq f (q|x∗ )2 ∗ n
∗
Similarly, we also have that Z
q(x∗ ) ∗ n
Z
∗
q(x∗ )
F (q|x ) ln F (q|x )dq = −
(n+1) q(x∗ )
∗ n
∗ n+1
F (q|x ) + F (q|x ) q(x∗ )
f 0 (q|x∗ ) ln F (q|x ) dq f (q|x∗ )2 ∗
Substituting these into (A.3), dR(n) 2 = dn n+1
Z
q(x∗ ) ∗ n
Z
q(x∗ )
F (q|x∗ )n ln F (q|x∗ )
F (q|x ) dq − q(x∗ )
q(x∗ )
f 0 (q|x∗ ) dq f (q|x∗ )2
Z ∗ 1 f 0 (q|x∗ ) n − 1 q(x ) dq − 2 00 ∗ F (q|x∗ )n+1 ln F (q|x∗ ) + ∗ 2 n + 1 q(x∗ ) f (q|x ) n ψ (x ) Z q(x∗ ) Z q(x∗ ) f 0 (q|x∗ ) 2 2 1 ≥ F (q|x∗ )n dq − F (q|x∗ )n ln F (q|x∗ ) dq − 2 00 ∗ ∗ 2 n + 1 q(x∗ ) n + 1 q(x∗ ) f (q|x ) n ψ (x ) Z q(x∗ ) 1 2 F (q|x∗ )n dq − 2 00 ∗ ≥ n + 1 q(x∗ ) n ψ (x ) ≥
2δ 1 − 2 00 ∗ , 2 (n + 1) h(¯ ε) n ψ (x )
where the first inequality holds since f 0 (q|x∗ ) ≥ 0 and the last inequality holds since 1 = n+1
Z
q(x∗ ) ∗ n
∗
∗
∗
Z
q(x∗ )
F (q|x ) f (q|x )dq ≤ f (q(x )|x ) q(x∗ )
q(x∗ )
h(¯ ε) F (q|x ) dq = δ ∗ n
Z
q(x∗ )
F (q|x∗ )n dq.
q(x∗ )
Note that the first equality of the above equation follows by the same reason as (A.2). Note also that ε) 9 h(¯ ε) 9 h(¯ ε) (n + 1)2 h(¯ δˆ = ≥ ≥ , 00 00 ∗ 2 00 8 ψ (0) 8 ψ (x ) 2n ψ (x∗ ) where the penultimate inequality holds since ψ 000 (·) ≥ 0 and the last inequality becomes ˆ we have that dR(n)/dn ≥ 0. equality when n = 2. Since δ ≥ δ, Proof of Lemma 3. Proof of (i). We first show that Gδi does not have a mass point for all 21
i and S δ is an interval. Suppose to the contrary that Gδi jumps at some s ∈ Siδ . Note that no other firm offers score equal to or slightly lower than s, since this would be dominated by offering slightly more than s because the latter strategy would give a discrete jump in the probability of winning with a slight decrease in the payoff conditional on winning. But then, firm i would be strictly better off by slightly lowering its score. Hence, we reach a contradiction. Next, to see that S δ is an interval, suppose there is a gap in S δ . Then, there are s, s0 ∈ S δ , where s < s0 , and i such that s0 ∈ Siδ , Gδ−i (s) = Gδ−i (s0 ) since there are no point masses. But then, firm i would be strictly better off by offering s instead of s0 , a contradiction. Lastly, note that since none of Gδi ’s have mass points and S δ is an interval, it is immediate that Gδ−i is continuous for all i. Moreover, since there is no positive measure of interval in which Gδ−i is a constant, it is strictly increasing in s ∈ Siδ . Proof of (ii). Let τi (s, q) := (q − s)Gδ−i (s). We first show that si ∈ arg maxs∈Siδ τi (s, qi ) is nondecreasing in qi . To do this, let s0 > s and q 0 > q. Suppose τi (s0 , q) − τi (s; q) ≥ 0, or equivalently, q[Gδ−i (s0 ) − Gδ−i (s)] ≥ s0 Gδ−i (s0 ) − sGδ−i (s). Note that since q 0 > q and Gδ−i (s0 ) − Gδ−i (s) > 0 (because Gδ−i (·) is strictly increasing in s), we have q 0 [Gδ−i (s0 ) − Gδ−i (s)] > q[Gδ−i (s0 ) − Gδ−i (s)] ≥ s0 Gδ−i (s0 ) − sGδ−i (s), implying that τi (ˆ s; qˆ) − τi (s; qˆ) > 0. By the Monotone Selection Theorem (Migrom and Shannon, 1994), we have that si is nondecreasing in qi . Next, note that Gδ−i (·) is differentiable almost everywhere and that ∂τ (s, q) = −Gδ−i (s) + (q − s)(Gδ−i (s))0 ∂s is strictly increasing in q since (Gδ−i (s))0 > 0 (because Gδ−i is strictly increasing in s). Thus, si is strictly increasing in qi by Edlin and Shannon (1998, Theorem 1), Proof of Lemma 4. Before proceeding, we first establish a useful lemma. Lemma A.1. If L : Rn → Rm is a linear map (i.e., additive and homogenous of degree 1), then there exists a constant M0 > 0 such that kL(x)k ≤ M0 kxk for all x ∈ Rn . Proof. Let M = sup {kL(e1 )k , · · · , kL(en )k}, where e1 , . . . , en is the standard basis for Rn . Letting x = (x1 , . . . , xn ), kL(x)k = kx1 L(e1 ) + · · · + xn L(en )k ≤ |x1 | kL(e1 )k + · · · + |xn | kL(en )k ≤ M (|x1 | + · · · + |xn |) ≤ M n kxk
22
Taking M0 = nM , we get the result. We now prove Lemma 4. Let s(x) := s(q(x)) and s(x) := s(q(x)) for x ∈ X δ . Step 1. For any x ∈ Xiδ , s(x) − s(x) ≤ δM for some constant M > 0. Proof. Fix any x ∈ Xiδ . It is clear that for any q, q˜ ∈ [q(x), q(x)], |q − q˜| ≤ δ∆ε. Note that by the definition of derivative, we also have |s(q) − s(˜ q ) − s0 (˜ q )(q − q˜)| ≤ |q − q˜| Since |s(q) − s(˜ q )| − |s0 (˜ q )(q − q˜)| ≤ |s(q) − s(˜ q ) − s0 (˜ q )(q − q˜)|, we get |s(q) − s(˜ q )| ≤ |s0 (˜ q )(q − q˜)| + |q − q˜| Now set n = m = 1 in Lemma A.1 and take L = s0 (˜ q ). Then, there is M0 such that 0 |s (˜ q )(q − q˜)| ≤ M0 |q − q˜|. Therefore, |s(q) − s(˜ q )| ≤ (M0 + 1) |q − q˜| Define M1 = M0 + 1. Then |s(q) − s(˜ q )| ≤ M1 δ∆ε for any q, q˜ ∈ [q(x), q(x)]. By letting M = M1 ∆ε, we have the desired result. ˜ S δ converges to S 0 ≡ [0, x˜ − ψ(˜ Step 2. There exists δ˜ > 0 such that for all δ < δ, x)], where x˜ = arg maxx≥0 {x − ψ(x)}. Proof. Let x := inf(X δ ) and x := sup(X δ ). And let s := s(x), s := s(x). We first show that q(0) ≤ s ≤ q(0). Suppose s < q(0). If firm i invests xi = 0 and sets si (0) = q(0), then for any si ∈ (s, q(0)], its payoff is (qi − si )Gδ−i (si ) > 0, which is a contradiction. Suppose now that s > q(0). This implies that x > 0, since otherwise s(0) = q − p = δε − p ≤ δε − p < s − p ≤ s. A firm’s, say i’s, payoff from investing x is Z
q(x)
(q − s(q))Gδ−i (s(q)) dF (q|x) − ψ(x).
q(x)
Note that the first part of the above equation shrinks to zero as δ goes to zero (since the integrand is bounded), while the second part remains positive. Therefore, for sufficiently small δ, x should be close to zero. Since the highest quality at x = 0 is q(0) and q ≥ s(q), it follows that s ≤ q(0). Next, we show that q(˜ x) − ψ(˜ x) ≤ s ≤ q(˜ x) − ψ(˜ x), where x˜ := arg maxx≥0 {x − ψ(x)}. Note that q(˜ x) − ψ(˜ x) is the maximum value that s = q − p can take given that the firm will
23
be paid its cost, since s = q − p ≤ q − ψ(x) ≤ x + δε − ψ(x) ≤ x˜ + δε − ψ(˜ x), where the first inequality holds since p ≥ ψ(x) in equilibrium, and the last inequality follows x) − ψ(˜ x). Suppose now s < q(˜ x) − ψ(˜ x) on the from the optimality of x˜. Hence, s ≤ q(˜ ¯ x) − ψ(˜ x)), contrary. Then, there exists a profitable (˜ x, si ) for some i such that si ∈ (s, q(˜ since q(˜ x) − si ≥ q(˜ x) − si > q(˜ x) − (q(˜ x) − ψ(˜ x)) = ψ(˜ x), and so the firm gets a positive expected payoff, whereas its payoff from offering ¯s is zero. Finally, observe that as δ vanishes, s and ¯s converge to 0 and x˜ − ψ(˜ x), respectively. Since S δ is an interval, the result follows. Step 3. For any s ∈ int(S 0 ), let x ∈ Xδi (s). For any i ∈ N , Gδi (s) converges to G0 (s) = 1 ψ 0 (x0 ) |I(s)|−1 , where x0 is the limit of x. Proof. Fix s ∈ int(S 0 ), and for any x ∈ Xδi (s) = x ∈ Xiδ | s ∈ Siδ (x) , let s(q) be the optimally chosen score for the given δ > 0. Since s, s(q) ∈ Siδ and s(x) − s(x) ≤ δM , it follows that for all q ∈ [q(x), q(x)], (q − s(q))Gδ−i (s(q)) ≤ (q − (s − δM ))Gδ−i (s + δM ), implying that Z
q(x)
(q − s(q))Gδ−i (s(q)) dF (q|x) − ψ(x)
q(x)
Z
(q − (s − δM ))Gδ−i (s + δM ) dF (q|x) − ψ(x).
≤
q(x)
q(x)
(A.4) Suppose now the firm increases its investment level by η > 0. We then have Z
q(x)
(q−s(q))Gδ−i (s(q)) dF (q|x)−ψ(x)
Z
q(x)+η
(q−s)Gδ−i (s) dF (q|x+η)−ψ(x+η), (A.5)
≥
q(x)
q(x)+η
where the inequality follows from the optimality of s(q). Combining (A.4) and (A.5), we have Z
q(x)+η
q(x)+η
(q−s)Gδ−i (s+δM )dF (q|x+η)−
Z
q(x)
(q−(s−δM ))Gδ−i (s+δM )dF (q|x) ≤ ψ(x+η)−ψ(x).
q(x)
24
Note that the LHS of the above inequality is the same as Gδ−i (s + δM )
Z
q(x)+η
!
q(x)
Z qdF (q|x + η) −
qdF (q|x) − δM
q(x)+η
= Gδ−i (s + δM ) (η − δM ) .
q(x)
That is, we have Gδ−i (s + δM ) (η − δM ) ≤ ψ(x + η) − ψ(x).
(A.6)
Next, applying the optimality of s(q) again, we have Z
q(x)
(q−s(q))Gδ−i dF (q|x)−ψ(x)
Z
q(x)−η
(q−s)Gδ−i (s+δM )dF (q|x−η)−ψ(x−η). (A.7)
≥
q(x)
q(x)−η
Combining (A.4) and (A.7), we have Z
q(x)
Z (q−(s−δM ))G−i (s+δM )dF (q|x)−
q(x)
q(x)−η
(q−s)Gδ−i (s+δM )dF (q|x−η) ≥ ψ(x)−ψ(x−η),
q(x)−η
equivalently, Gδ−i (s + δM ) (η + δM ) ≥ ψ(x) − ψ(x − η).
(A.8)
Now, observe that from (A.6), we have limGδ−i (s + δM )(η − δM ) = limGδ−i (s + δM )η ≤ lim [ψ(x + η) − ψ(x)] = ψ(x0 + η) − ψ(x0 ),
δ→0
δ→0
δ→0
where x0 is the limit of x. Similarly, from (A.8), lim Gδ−i (s + δM )(η + δM ) = lim Gδ−i (s + δM )η ≥ lim [ψ(x) − ψ(x − η)] = ψ(x0 ) − ψ(x0 − η).
δ→0
δ→0
δ→0
We thus have ψ(x0 (s)) − ψ(x0 (s) − η) ≤ lim Gδ−i (s + δM ) ≤ limGδ−i (s + δM ) η→0 δ→0 η δ→0 0 0 ψ(x (s) + η) − ψ(x (s)) ≤ lim = ψ 0 (x0 (s)). η→0 η
ψ 0 (x0 (s)) = lim
This implies that Gδ−i (s) converges to G0−i (s) ≡ ψ 0 (x0 (s)) as δ approaches to 0. Finally, for the convergence of Gδi (s), note that Part (i) of Lemma 3 implies that there are at least two distinct firms in I(s). Hence, for any i, i0 ∈ I(s), we have Q 0 G0−i (s) G0i0 (s) j∈I(s),j6=i Gj (s) Q = = . 0 G0−i0 (s) G0i (s) j∈I(s),j6=i0 Gj (s) 25
Since G0−i (s) = G0−i0 (s) = ψ 0 (x0 (s)), we have that Gδi (s) and Gδi0 (s) converges to the same distribution, denoted by G0 (s). Therefore, Y
ψ 0 (x0 (s)) = G0−i (s) =
Gj (s) = G0 (s)|I(s)|−1 .
j∈I(s),j6=i 1
That is, G0 (s) = ψ 0 (x0 (s)) |I(s)|−1 .
Proof of Lemma 5. Note first that Z q d2 πi (xi ) =− G(q)(n−1)x G(q)xi (ln G(q))2 dq − ψ 00 (xi ) < 0, dx2i q showing that xa is an equilibrium. Next, to see how xa and R(n) change according to n, treat n as a real number. Let Z q a a G(q)nx ln G(q)dq + ψ 0 (xa ) = 0 F(x (n), n) ≡ q
from (9). Applying the implicit function theorem, Rq
xa G(q)nx (ln G(q))2 dq ∂F/∂n dx q =− = −R q < 0, dn ∂F/∂xa nG(q)nxa (ln G(q))2 dq + ψ 00 (xa ) q a
a
(A.9)
which proves that xa is decreasing in n. Lastly, observe that dR(n) =− dn
q¯
Z
G(q)(n−1)x
a
a
a
a
a
dq
a
a
dq
1 − G(q)x + n ln G(q)x − (n − 1)G(q)x ln G(q)x
q
Z
q
a
n(n − 1)G(q)(n−1)x 1 − G(q)x
−
a
ln G(q)
q q¯
Z
G(q)(n−1)x
>−
a
a
dxa dq dn
a
1 − G(q)x + n ln G(q)x − (n − 1)G(q)x ln G(q)x
q
Z
q
+
a a (n − 1)G(q)(n−1)x 1 − G(q)x xa ln G(q)dq
q
Z =−
q
G(q)(n−1)x
a
a
1 − G(q)x + ln G(q)x
q
26
a
dq.
(A.10)
where the inequality follows from the facts that ln G(q) ≤ 0 and Rq a Rq a a a x G(q)nx (ln G(q))2 dq x G(q)nx (ln G(q))2 dq dxa xa q q = −R q > − = − R q dn n nG(q)nxa (ln G(q))2 dq + ψ 00 (xa ) nG(q)nxa (ln G(q))2 dq q
q
from (A.9), since ψ 00 (xa ) > 0. Now, let F(q|x) ≡ 1 − G(q)x + ln G(q)x . It is easy to see that F(q|x) = 0 and dF/dq = x(1 − G(q)x )g(q)/G(q) ≥ 0, showing that the integrand in (A.10) is negative and so dR(n)/dn ≥ 0. Proof of Proposition 4. Note first that Z
q(x)
(n − 1)F (q|x)
I(x) = −
n−2
q(x) q−x δ
since Fx (q|x) = − 1δ h Z
(n − 1)F (q|x)n−2 f (q|x)2 dq > 0,
f (q|x)Fx (q|x)dq =
q(x)
d dx
q(x)
Z
= −f (q|x). We show I 0 (x) = 0. Observe that
q(x) n−2
F (q|x)
2
q(x)
Z
2
d F (q|x)n−2 f (q|x)2 dq dx
f (q|x) dq = f (q(x)|x) +
q(x)
q(x)
= f (¯ q (x)|x)2 +
Z
q(x)
h i (n − 2)F (q|x)n−3 Fx (q|x)f (q|x)2 + 2F (q|x)n−2 f (q|x)fx (q|x) dq
q(x) q(x)
Z
2
= f (¯ q (x)|x) − (n − 2)
F (q|x)
n−3
q(x)
Z
3
2F (q|x)n−2 f (q|x)fx (q|x)dq,
f (q|x) dq +
q(x)
q(x)
where the last equality holds since Fx (q|x) = −f (q|x). Note also that Z
q(x) n−3
F (q|x)
3
n−3
f (q|x) dq = F (q|x)
q(x) Z f (q|x) F (q|x) − q(x)
q(x)
= f (¯ q (x)|x)2 − (n − 3)
Z
q(x)
F (q|x)n−3 f (q|x)3 dq +
Z
q(x)
F (q|x)
q(x)
q(x)
1 0 h δ2
q−x δ
q(x)
(n − 2)
0 F (q|x)n−3 f (q|x)2 F (q|x)dq
2F (q|x)n−2 f (q|x)fx (q|x)dq,
q(x)
where the last equality holds since f 0 (q|x) = Z
q(x)
2
n−3
3
2
= −fx (q|x). We thus have
Z
q(x)
2F (q|x)n−2 f (q|x)fx (q|x)dq
f (q|x) dq = f (q(x)|x) +
q(x)
q(x)
and so I 0 (x) = 0, that is, I(x) is a constant independent on x. Hence, we denote I(x) by I. Next, observe that ∂Rt =− ∂x
Z
q(x)
nF (q|x)n−1 Fx (q|x)dq −
q(x)
27
ψ 00 (x) ψ 00 (x) =1− , I I
where the last equality follows from (A.2), and that ∂ 2 Rt /∂x2 = −ψ 000 (x)/I ≤ 0, since ψ 000 (·) ≥ 0, showing that Rt is concave in x. ˆ To this end, note that We now show that the optimal investment xt is zero for δ ≥ δ. Z
q(x)
(n − 1)F (q|x)
I=
n−2
Z
2
q(x)
f (q|x) dq < f (q(x)|x)
q(x)
(n − 1)F (q|x)n−2 f (q|x)dq =
q(x) h(¯ ε) . δ
since the integral term in the RHS of the inequality is 1 and f (q(x)|x) =
h(¯ ε) , δ
Therefore,
∂Rt ψ 00 (x) ψ 00 (0)δ =1− <1− ≤ 0, ∂x I h(¯ ε) where the first inequality holds since I < h(¯ ε)/δ and ψ 00 (x) ≥ ψ 00 (0) and the last inequality ε) t holds since δ ≥ δˆ = 89 ψh(¯ 00 (0) . Together with the concavity of R , this entails the corner solution xt = 0 and consequently P = 0. We thus have Z
t
q(0)
F (q|0)n dq,
R (n) = q(0) − q(0)
which is increasing in n, clearly. Finally, for a fixed n, comparing the buyer’s payoffs, Rt (n) − R(n) " Z
#
q(0)
= q(0) −
"
F (q|0)n dq − q¯(x∗ ) − n
= − x∗ + n
q(x∗ )
F (q|x∗ )n−1 dq + (n − 1)
Z
q(x∗ )
q(0)
Z
Z
#
q(x∗ )
F (q|x∗ )n dq
q(x∗ )
q(x∗ )
F (q|x∗ )n−1 1 − F (q|x∗ ) dq
q(x∗ )
Z q(x∗ ) n > −x + F (q|x∗ )n−1 1 − F (q|x∗ ) f (q|x∗ )dq ∗ ∗ f (q(x )|x ) q(x∗ ) nδ 1 1 δ ∗ = −x + − = −x∗ + h(¯ ε) n n + 1 (n + 1)h(¯ ε) ∗
where the second equality holds since ∂ ∂x
Z
q(x)
Z
n
q(x)
F (q|x) dq = 1 + q(x)
implying that
n−1
nF (q|x)
Z
q(x)
R q(x∗ ) q(x∗ )
F (q|x∗ )n dq =
q(x)
Fx (q|x)dq = 1 −
nF (q|x)n−1 f (q|x)dq = 0
q(x)
R q(0) q(0)
F (q|0)n dq. Now, observe that
δ > x∗ ⇔ ψ 0 (n + 1)h(¯ ε)
δ (n + 1)h(¯ ε) 28
> ψ 0 (x∗ ) =
1 , n
where the equivalence holds since ψ 00 (·) > 0 and the last equality follows from (A.2). Define ˇ δ(n) be such that δˇ 1 0 ψ = . (n + 1)h(¯ ε) n ˜ ˆ δ(n)}. ˇ ˜ and δ(n) := max{δ, Since ψ 00 (·) > 0, for any δ ≥ δ(n), Rt (n) > R(n).
A.2
Fixed-prize tournament
Consider the fixed-prize tournament and assume the innovation technology in Section 3.1. For given P and n, firm i’s payoff from investing xi when all other firms invest x is q
Z
G(q)(n−1)x dG(q)xi − ψ(xi ) =
πi (xi ) = P q
P xi − ψ(xi ). (n − 1)x + xi
From the first-order condition at xi = x, we have P = n2 xψ 0 (x)/(n − 1). Thus, the buyer’s payoff is Z q Z q n2 xψ 0 (x) nx t q dG(q) − P = q − G(q)nx dq − . R = n−1 q q From the first-order condition of Rt with respect x, we have q
Z
nG(q)nx ln G(q)dq −
− q
n2 (ψ 0 (x) + xψ 00 (x)) = 0. n−1
(A.11)
Let xt be the investment level satisfying (A.11). The next lemma compares xt with xa , the investment under the auction given by (9), and also shows that Rt is increasing in n. Lemma 6. There exists a symmetric equilibrium with investment level xt given by (A.11). In such equilibrium, xt < xa for a fixed n and Rt (n) is increasing in n. Proof. The second order condition of Rt with respect to x is ∂ 2 Rt =− ∂x2
Z q
q
n2 2ψ 00 (x) + xψ 000 (x) n G(q) (ln G(q)) dq − < 0, n−1 2
nx
2
where the inequality holds since ψ 00 (·) > 0 and ψ 000 (·) ≥ 0, showing that xt is an equilibrium. Next, we show that xt < xa for any given n. Suppose to the contrary that xt ≥ xa so a t that ψ 0 (xt ) ≥ ψ 0 (xa ) (since ψ 00 (·) > 0). Note that we also have G(q)nx ≥ G(q)nx and so ψ 0 (xa ) = −
Z
q
nxa
G(q) q
Z ln G(q)dq ≥ −
q
nxt
G(q) q
29
n ψ 0 (xt ) + xˆt ψ 00 (xt ) ln G(q)dq = > ψ 0 (xt ), n−1
where the first equality follows from (9) and the second equality follows from (A.11), and n > 1 and ψ 00 (·) > 0. We thus have a contradiction. the last inequality holds since n−1 Lastly, to determine the optimal number of firms, note that dRt =− dn
Z
q
t
xt G(q)nx ln G(q)dq −
q
n(n − 2)xt ψ 0 (xt ) (n − 1)2
# n2 ψ 0 (xt ) + xˆt ψ 00 (xt ) dxt nG(q) ln G(q)dq + − n−1 dn q n xt ψ 0 (xt ) + (xt )2 ψ 00 (xt ) n(n − 2)xt ψ 0 (xt ) − = n−1 (n − 1)2 t 0 t t 2 00 t nx ψ (x ) n(x ) ψ (x ) = > 0, + (n − 1)2 n−1 "Z
q
nxt
where the second equality follows from the facts that the term in the square bracket in the Rq RHS of the first equality is zero and q xG(q)nx ln G(q)dq = −n xψ 0 (x) + x2 ψ 00 (x) /(n − 1) by (A.11). To compare the buyer’s payoffs under the auction and the tournament, note that the quality difference between those two is Z
q
q dG(q)
nxa
Z −
q
q
t
q dG(q)nx > 0
q
since xa > xt , and the price difference is Z
q
q dG(q) q
nxa
Z −
!
q
xa
q d G(q) q
{2:n}
−
nxt ψ 0 (xt ) . n−1
As opposed to Section 3.2, however, the sign of the price difference depends on the cost function as well as the distribution of the quality. In fact, if the sign of (A.2) is negative, then the auction outperforms the tournament. Finding an exact condition under which one mechanism outperforms the other is beyond the scope of this paper, but we provide an example showing that the auction can be suboptimal as in our baseline model. Example 3. Suppose G(q) = q, [q, q] = [0, 1] and ψ(x) = 12 x2 . Then, Rt (n) > R(n) for all n.
A.3
Equilibrium with two firms
In this section, we consider Example 1 and provide an equilibrium for any δ ≤ 12 when there are two firms. Consider the following symmetric investment strategy: both firms put masses m1 , . . . , mk on x1 , . . . , xk , where 30
(i) For r ∈ N, k = 2r for any
1 2(r+1)
≤δ<
1 . 2r
(ii) x1 = 12 (1 − 2δk + δ 2 k(k + 2), x2 = 21 (1 + 4δ − δ 2 k(k + 2)) and x + (t − 1)δ 1 xt = x2 + (t − 2)δ (iii) m1 = mk =
1 s+1
if t ≥ 3 is an odd number, if t ≥ 3 is an even number.
and for an even number t ∈ [2, k − 1], mt =
t/2 r(r + 1)
and mt+1 =
r − t/2 . r(r + 1)
Observe that xt ’s are symmetric around 1/2 and satisfy that xt ≤ xt+1 and x1 + xk = 1, x2 + xk−1 = 1, . . . , xr−1 + xr = 1. Similarly, mt ’s satisfy that m1 = mk , m2 = mk−1 , . . . , mr = mr+1 . For instance, when δ ∈ [ 41 , 12 ), k = 2 and for t = 1, 2, (mt ) = 12 , 12 and (xt ) = 12 − 2δ + 4δ 2 , 21 + 2δ − 4δ 2 , (A.12) and when δ ∈ [ 16 , 14 ), k = 4 and for t = 1, 2, 3, 4, (mt ) = 13 , 16 , 61 , 13 and (xt ) =
1 2
− 4δ + 12δ 2 , 12 + 2δ − 12δ 2 , 1 − x2 , 1 − x1
(A.13)
P and so on. Let H(q) be the induced quality distribution, H(q) := kt=1 mt F (q|xt ), where F (·|xt ) is U [xt − δ, xt + δ], and h(·) be the associated probability density function. In Figure A.1, we depict h(·), H(·) (the first and the second columns, respectively) and the firms’ payoff (the last column) for δ = 31 , 14 , 51 , 16 . In the first two cases, both firms put equal mass according to (A.12), and in the last two cases, they puts masses according to (A.13). Observe that quality distribution converges to the uniform distribution over [0, 1] as δ vanishes, as suggested by Lemma 4 and Example 2.
31
• δ=
1 3
• δ=
1 4
• δ=
1 5
• δ=
1 6
Figure A.1: h(q), H(q) and Firm payoff
32
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