Optimal design of scoring auctions with multidimensional quality

Abstract

This article studies the optimal design of scoring auctions used in public and private procurement. In this auction, each supplier’s offer consists of both price and quality, and a supplier whose offer achieves the highest score wins. The environment we consider has the feature that quality is multidimensional, and the cost complementarity or cost substitutability among quality attributes significantly affects the form of a scoring rule which implements the buyer’s optimal mechanism. Our results show that the optimal scoring rule can be additively separable in the quality attributes if the degree of cost substitutability between the attributes is nonpositive, and it cannot be additively separable if the degree is sufficiently high. An example shows how to compute the buyer’s loss from using an additively separable scoring rule. We also investigate how the optimal scoring rule depends on the buyer’s weight parameter on suppliers’ profits and the number of suppliers.

Appendix

Proof of Lemma 1

Fix any $\theta \in [\underline{\theta }, \bar{\theta }]$. Note that $\arg \max _{\varvec{q} \in {\mathcal {Q}}} [s(\varvec{q}) - c(\varvec{q}, \theta ) ]$ is nonempty because ${\mathcal {Q}}$ is compact and $s - c$ is upper semicontinuous in $\varvec{q}$. Take any $\varvec{q}^{*}(\theta ) \in \arg \max _{\varvec{q} \in {\mathcal {Q}}} [s(\varvec{q}) - c(\varvec{q}, \theta )]$. First, we claim that, for supplier $i$ of type $\theta$, any offer $(p^{\prime }, \varvec{q}^{\prime })$ with $\varvec{q}^{\prime } \not \in \arg \max _{\varvec{q} \in {\mathcal {Q}}} \ [s(\varvec{q}) - c(\varvec{q},\theta )]$ is weakly dominated by $(p, \varvec{q}^{*}(\theta ))$ with $s(\varvec{q}^{*}(\theta ))-p = s(\varvec{q}^{\prime }) -p^{\prime }$. The score of $(p, \varvec{q}^{*}(\theta ))$ is equal to that of $(p^{\prime }, \varvec{q}^{\prime })$, so that both offers yield the same winning probability given the other suppliers’ strategies. Supplier $i$’s expected profit from $(p^{\prime }, \varvec{q}^{\prime })$ is not higher than his expected profit from $(p, \varvec{q}^{*}(\theta ))$ because

$$\begin{aligned}&\ \left[ p^{\prime } - c\left( \varvec{q}^{\prime }, \theta \right) \right] \ \text {Prob}\left[ i \text { wins} \mid S\left( p^{\prime }, \varvec{q}^{\prime }\right) \right] \\&\quad \le \left[ p^{\prime } - c\left( \varvec{q}^{\prime }, \theta \right) + \left( s(\varvec{q}^{*}(\theta )) - c\left( \varvec{q}^{*}(\theta ), \theta \right) \right. \right. \\&\left. \left. \quad \qquad - \left( s\left( \varvec{q}^{\prime }\right) - c(\varvec{q}^{\prime },\theta )\right) \right) \right] \;\text { Prob}\left[ i \text { wins} \mid S\left( p^{\prime }, \varvec{q}^{\prime }\right) \right] \\&\quad = \left[ p - c\left( \varvec{q}^{*}(\theta ), \theta \right) \right] \;\text { Prob}\left[ i\ \text {wins} \mid S\left( p, \varvec{q}^{*}(\theta )\right) \right] , \end{aligned}$$

where the inequality follows from the hypothesis that ${\varvec{q}}^{\prime }{\not \in } \,\arg \max _{{\varvec{q}}\, \in \,{\mathcal {Q}}} [s({\varvec{q}}) - c({\varvec{q}}, \theta )] \ni {\varvec{q}}^{*}(\theta )$. The inequality is strict if $\text { Prob}[i \ \text {wins} \mid S(p^{\prime }, \varvec{q}^{\prime })] \!>\! 0$, which occurs for some strategies of the other suppliers. This proves the first claim. The claim implies that an equilibrium bidding strategy $(p(\cdot ), \varvec{q}(\cdot ))$ must satisfy $\varvec{q}(\theta ) \in \arg \max _{\varvec{q} \,\in \, {\mathcal {Q}}} [s(\varvec{q}) - c(\varvec{q}, \theta )]$ for any $\theta$ because no supplier uses weakly dominated actions.

Second, we claim that the bidding strategy $(p^{*}, \varvec{q}^{*})$ in the lemma constitutes a symmetric equilibrium. Consider the following change of variable: $k(\theta ) := s(\varvec{q}^{*}(\theta )) - c(\varvec{q}^{*}(\theta ), \theta )$. Because $c_\theta$ is continuous on the compact set ${\mathcal {Q}} \times [\underline{\theta }, \bar{\theta }]$ and thus bounded on $[\underline{\theta }, \bar{\theta }]$, it follows from the integral form envelope theorem of Milgrom and Segal (2002) [see also Theorem 3.1 of Milgrom (2004)] that $k$ is absolutely continuous, and is given by

$$\begin{aligned} k(\theta )&= k(\underline{\theta }) - \int _{\underline{\theta }}^{\theta } c_{\theta } \left( \varvec{q}^{*}(z), z\right) dz. \end{aligned}$$

The score $s(\varvec{q}^{*}(\theta )) - p^*(\theta )$ can be rewritten as

$$\begin{aligned} s\left( \varvec{q}^{*}(\theta )\right) - p^*(\theta )&= s\left( \varvec{q}^{*}(\theta )\right) - c\left( \varvec{q}^{*}(\theta ), \theta \right) - \left[ p^*(\theta ) - c\left( \varvec{q}^{*}(\theta ), \theta \right) \right] \\&= k(\theta ) - \int _{\theta }^{\bar{\theta }} c_\theta \left( \varvec{q}^{*}(z), z\right) \frac{1-F^{(N-1)}(z)}{1-F^{(N-1)}(\theta )} dz. \end{aligned}$$

Because $k$ is continuous in $\theta$, so is $s(\varvec{q}^{*}(\cdot )) - p^*(\cdot )$. Moreover, the score $s(\varvec{q}^{*}(\cdot )) - p^*(\cdot )$ is strictly decreasing in $\theta$ because for any $\theta , \theta '$ with $\theta < \theta ^{\prime }$,

$$\begin{aligned}&\left[ s\left( \varvec{q}^{*}(\theta )\right) - p^*\left( \theta \right) \right] - \left[ s\left( \varvec{q}^{*}\left( \theta ^{\prime }\right) \right) - p^*\left( \theta ^{\prime }\right) \right] \\&\quad = \left( k(\theta )-k\left( \theta ^{\prime }\right) \right) - \left[ \int _{\theta }^{\bar{\theta }} c_\theta \left( \varvec{q}^{*}(z), z\right) \frac{1-F^{(N-1)}(z)}{1-F^{(N-1)}(\theta )} dz\right. \\&\left. \quad \quad -\, \int _{\theta ^{\prime }}^{\bar{\theta }} c_\theta \left( \varvec{q}^{*}(z), z\right) \frac{1-F^{(N-1)}(z)}{1-F^{(N-1)}(\theta ^{\prime })} dz \right] \\&\quad > \left( k(\theta )-k\left( \theta ^{\prime }\right) \right) - \int _{\theta }^{\theta ^{\prime }} c_\theta \left( \varvec{q}^{*}(z), z\right) \frac{1-F^{(N-1)}(z)}{1-F^{(N-1)}(\theta )} dz\\&\quad > \left( k(\theta )-k\left( \theta ^{\prime }\right) \right) - \int _{\theta }^{\theta ^{\prime }} c_\theta \left( \varvec{q}^{*}(z), z\right) dz\\&\quad = \int _{\theta }^{\theta ^{\prime }} \left[ c_\theta \left( \varvec{q}^{*}(z),z\right) - c_\theta \left( \varvec{q}^{*}(z), z\right) \right] dz = 0. \end{aligned}$$

Fix any supplier $i$ of type $\theta$. Suppose that the other suppliers follow the strategy $(p^{*}, \varvec{q}^{*})$. Take any $\hat{\theta } \in [\underline{\theta }, \bar{\theta }]$, and let $p := s(\varvec{q}^{*}(\theta )) - s(\varvec{q}^{*}(\hat{\theta })) + p^{*}(\hat{\theta })$. Supplier $i$’s expected profit from offering $(p, \varvec{q}^{*}(\theta ))$ can be written as

$$\begin{aligned}&\left[ p - c\left( \varvec{q}^{*}(\theta ), \theta \right) \right] \text {Prob}\left[ i \text { wins} \mid S\left( p, \varvec{q}^{*}(\theta )\right) \right] \\&\quad = \left[ s\left( \varvec{q}^{*}(\theta )\right) - c\left( \varvec{q}^{*}(\theta ), \theta \right) - s\left( \varvec{q}^{*}\left( \hat{\theta }\right) \right) + p^*\left( \hat{\theta }\right) \right] \\&\quad \quad \times \text { Prob}\left[ i \text { wins} \mid S\left( p^*\left( \hat{\theta }\right) , \varvec{q}^{*}\left( \hat{\theta }\right) \right) \right] \\&\quad = \left[ k(\theta ) - s\left( \varvec{q}^{*}\left( \hat{\theta }\right) \right) + p^*\left( \hat{\theta }\right) \right] \left( 1-F^{(N-1)}\left( \hat{\theta }\right) \right) \\&\quad = (k(\theta )-k(\hat{\theta })) (1-F^{(N-1)}(\hat{\theta })) + \int _{\hat{\theta }}^{\bar{\theta }} c_\theta \left( \varvec{q}^{*}(z), z\right) \left( 1-F^{(N-1)}(z)\right) dz. \end{aligned}$$

The second equality follows from the observation that the score $S(p^*(\cdot ), \varvec{q}^{*}(\cdot )) = s(\varvec{q}^{*}(\cdot )) - p^*(\cdot )$ is strictly decreasing in $\hat{\theta }$. Supplier $i$ cannot obtain a higher expected profit by deviating from $(p^*(\theta ), \varvec{q}^{*}(\theta ))$ to $(p, \varvec{q}^{*}(\theta ))$ because the difference between the expected profits is given by

$$\begin{aligned}&- \left( k(\theta )-k\left( \hat{\theta }\right) \right) \left( 1-F^{(N-1)}\left( \hat{\theta }\right) \right) + \int _{\theta }^{\hat{\theta }} c_\theta \left( \varvec{q}^{*}(z), z\right) \left( 1-F^{(N-1)}(z)\right) dz\\&\quad = \int _{\theta }^{\hat{\theta }} \left[ -c_\theta \left( \varvec{q}^{*}(z), z\right) \left( 1-F^{(N-1)}\left( \hat{\theta }\right) \right) + c_\theta \left( \varvec{q}^{*}(z), z\right) \left( 1-F^{(N-1)}(z)\right) \right] dz\\&\quad = \int _{\theta }^{\hat{\theta }} c_\theta \left( \varvec{q}^{*}(z), z\right) \left( F^{(N-1)}\left( \hat{\theta }\right) -F^{(N-1)}(z)\right) dz \ge 0, \end{aligned}$$

regardless of whether $\hat{\theta } > \theta$ or $\hat{\theta } < \theta$. It is easy to show that supplier $i$ cannot obtain a higher expected profit by deviating from $(p^{*}(\theta ), \varvec{q}^{*}(\theta ))$ to $(p^{\prime }, \varvec{q}^{*}(\theta ))$ such that $s(\varvec{q}^{*}(\theta )) - p^{\prime } \not \in [s(\varvec{q}^{*}(\bar{\theta })) - p^{*}(\bar{\theta }), s(\varvec{q}^{*}(\underline{\theta })) - p^{*}(\underline{\theta })]$. This proves the second claim. In the equilibrium, the score $s(\varvec{q}^{*}(\cdot )) - p^*(\cdot )$ is strictly decreasing in $\theta$. Thus, a supplier of type $\theta ^i$ wins only if $\theta ^i = \min \{ \theta ^1,\ldots ,\theta ^N \}$. $\square$

Proof of Lemma 2

(i) We show that a necessary and sufficient condition for the IC constraints (7) is given by a pair of conditions: envelope condition and monotonicity condition. We say that a mechanism $\rho$ satisfies the envelope condition if for any $i$ and $\theta$,

$$\begin{aligned} \Pi ^i_{\rho } (\theta ) = \Pi ^i_{\rho } \left( \bar{\theta }\right) + \int _{\theta }^{\bar{\theta }} E_{\varvec{\theta }^{-{\varvec{i}}}} \left[ X^i\left( z, \varvec{\theta }^{-{\varvec{i}}}\right) c_{\theta }\left( \varvec{Q}^{\varvec{i}}\left( z,\varvec{\theta }^{-{\varvec{i}}}\right) , z\right) \right] dz. \end{aligned}$$

Here, $\Pi ^i_{\rho } (\theta ) := \Pi ^i_{\rho } (\theta \mid \theta )$. We say that a mechanism $\rho$ satisfies the monotonicity condition if for any $i$, $\theta$ and $\hat{\theta }$ with $\hat{\theta } > \theta$,

$$\begin{aligned} \int _{\theta }^{\hat{\theta }} E_{\varvec{\theta }^{-{\varvec{i}}}} \left[ \!X^i\left( \!z, \varvec{\theta }^{-{\varvec{i}}}\right) c_{\theta }\left( \!\varvec{Q}^{\varvec{i}}\left( z,\varvec{\theta }^{-{\varvec{i}}}\right) ,z\right) - X^i\left( \hat{\theta }, \varvec{\theta }^{{-\varvec{i}}}\right) c_{\theta }\left( \!\varvec{Q}^{\varvec{i}}\left( \hat{\theta },\varvec{\theta }^{-{\varvec{i}}}\right) , z\right) \right] dz \!\ge \! 0. \end{aligned}$$

By Assumption 1, it must hold that for any $\varvec{q}, \varvec{q}^{\prime } \in {\mathcal {Q}}$ with $\varvec{q}^{\prime } \ge \varvec{q}$ and any $z,z^{\prime } \in [\underline{\theta }, \bar{\theta }]$ with $z^{\prime } \ge z$, $c(\varvec{q}^{\prime }, z^{\prime }) - c(\varvec{q}, z^{\prime }) \ge c(\varvec{q}^{\prime }, z) - c(\varvec{q}, z)$, and thus $c_{\theta }(\varvec{q}^{\prime }, z) \ge c_{\theta }(\varvec{q}, z) \ge 0$. Therefore, if $X^i(\cdot , \varvec{\theta }^{-{\varvec{i}}})$ and $Q^i_m(\cdot ,\varvec{\theta }^{-{\varvec{i}}})$ are decreasing in $\theta ^i$ for any $i,m, \varvec{\theta }^{-{\varvec{i}}}$, then the monotonicity condition is automatically satisfied.

First, we prove sufficiency. Suppose, on the contrary, that supplier $i$’s IC constraint is not satisfied. Then, there exist $\theta$ and $\hat{\theta }$ such that $\Pi ^i_{\rho }(\hat{\theta } \mid \theta ) > \Pi ^i_{\rho }(\theta )$. Hence, $E_{\varvec{\theta }^{-{\varvec{i}}}} [ X^i(\hat{\theta }, \varvec{\theta }^{-{\varvec{i}}}) c(\varvec{Q}^{\varvec{i}}(\hat{\theta },\varvec{\theta }^{-{\varvec{i}}}), \hat{\theta }) - X^i(\hat{\theta }, \varvec{\theta }^{-{\varvec{i}}}) c(\varvec{Q}^{\varvec{i}}(\hat{\theta },\varvec{\theta }^{-{\varvec{i}}}), \theta ) ] > \Pi ^i_{\rho }(\theta ) - \Pi ^i_{\rho }(\hat{\theta })$ by definition of $\Pi ^i_{\rho }$. Rewriting the left-hand side as the definite integral and applying the envelope condition to the right-hand side, we obtain

$$\begin{aligned}&\int _{\theta }^{\hat{\theta }} E_{\varvec{\theta }^{-{\varvec{i}}}} \left[ X^i\left( \hat{\theta }, \varvec{\theta }^{-{\varvec{i}}}\right) c_\theta \left( \varvec{Q}^{\varvec{i}}\left( \hat{\theta },\varvec{\theta }^{-{\varvec{i}}}\right) , z\right) \right] dz\\&\quad > \int _{\theta }^{\hat{\theta }} E_{\varvec{\theta }^{-{\varvec{i}}}} \left[ X^i\left( z, \varvec{\theta }^{-{\varvec{i}}}\right) c_\theta \left( \varvec{Q}^{\varvec{i}}\left( z,\varvec{\theta }^{-{\varvec{i}}}\right) , z\right) \right] d z. \end{aligned}$$

This contradicts the monotonicity condition.

Next, we prove necessity. Using the integral form envelope theorem of Milgrom and Segal (2002), the IC constraints (7) imply that for any $i$ and $\theta$, the interim payoff $\Pi ^i_{\rho }(\theta ) = \max _{\hat{\theta }} E_{\varvec{\theta }^{-{\varvec{i}}}} [P^i(\hat{\theta }, \varvec{\theta }^{-{\varvec{i}}}) - X^i(\hat{\theta }, \varvec{\theta }^{-{\varvec{i}}}) c (\varvec{Q}^{\varvec{i}}(\hat{\theta },\varvec{\theta }^{-{\varvec{i}}}), \theta )]$ is given by

$$\begin{aligned} \Pi ^i_{\rho }(\theta )&= \Pi ^i_{\rho }\left( \bar{\theta }\right) - \int _{\theta }^{\bar{\theta }} \frac{\partial \Pi ^i_{\rho }}{\partial \theta }(z \mid z) dz \\&= \Pi ^i_{\rho }\left( \bar{\theta }\right) + \int _{\theta }^{\bar{\theta }} E_{\varvec{\theta }^{-{\varvec{i}}}} \left[ X^i\left( z, \varvec{\theta }^{-{\varvec{i}}}\right) c_\theta \left( \varvec{Q}^{\varvec{i}}\left( z,\varvec{\theta }^{-{\varvec{i}}}\right) ,z\right) \right] dz. \end{aligned}$$

We thus obtain the envelope condition. Also, the IC constraints (7) imply that for any $i$, $\theta$ and $\hat{\theta }$, $E_{\varvec{\theta }^{-{\varvec{i}}}} [X^i(\hat{\theta }, \varvec{\theta }^{-{\varvec{i}}}) c(\varvec{Q}^{\varvec{i}}(\hat{\theta },\varvec{\theta }^{-{\varvec{i}}}), \hat{\theta }) - X^i(\hat{\theta }, \varvec{\theta }^{-{\varvec{i}}}) c(\varvec{Q}^{\varvec{i}}(\hat{\theta },\varvec{\theta }^{-{\varvec{i}}}), \theta )] \le \Pi ^i_{\rho }(\theta ) - \Pi ^i_{\rho }(\hat{\theta })$. Rewriting the left-hand side as the definite integral and applying the envelope condition to the right-hand side, we obtain the monotonicity condition.

(ii) We solve the optimization problem. The IC constraints (7) imply that for each $i$, $\Pi ^i_{\rho } (\cdot )$ is decreasing in $\theta$ because $\Pi ^i_{\rho } (\theta ) \ge \Pi ^i_{\rho } (\theta ^{\prime } \mid \theta ) \ge \Pi ^i_{\rho } (\theta ^{\prime })$ for any $\theta < \theta ^{\prime }$. The second inequality follows from the assumption that $c$ is strictly increasing in $\theta$. Hence, the IR constraints (8) are replaced by $\Pi ^i_{\rho } (\bar{\theta }) \ge 0$ for each $i$. Using the result (i), the IC constraints (7) are replaced by the envelope and monotonicity conditions. By the envelope condition and the interchange of the order of integration, $E[\Pi ^i_{\rho } (\theta ^i)]$ is given by

$$\begin{aligned}&\int _{\underline{\theta }}^{\bar{\theta }} \Pi ^i_{\rho } \left( \theta ^i\right) f\left( \theta ^i\right) d \theta ^i \\&\quad = \Pi ^i_{\rho }\left( \bar{\theta }\right) + \int _{\underline{\theta }}^{\bar{\theta }} \int _{\theta ^i}^{\bar{\theta }} E_{\varvec{\theta }^{-{\varvec{i}}}} \left[ X^i\left( z, \varvec{\theta }^{-{\varvec{i}}}\right) c_\theta \left( \varvec{Q}^{\varvec{i}}\left( z,\varvec{\theta }^{-{\varvec{i}}}\right) , z\right) \right] dz f\left( \theta ^i\right) d \theta ^i\\&\quad = \Pi ^i_{\rho }\left( \bar{\theta }\right) + \int _{\underline{\theta }}^{\bar{\theta }} E_{\varvec{\theta }^{-{\varvec{i}}}} \left[ X^i\left( z, \varvec{\theta }^{-{\varvec{i}}}\right) c_\theta \left( \varvec{Q}^{\varvec{i}}\left( z,\varvec{\theta }^{-{\varvec{i}}}\right) , z\right) \right] \frac{F(z)}{f(z)} f(z) d z. \end{aligned}$$

Hence, the buyer’s objective function is rewritten as

$$\begin{aligned}&\sum _{i=1}^N E \left[ X^i(\varvec{\theta }) v\left( \varvec{Q}^{\varvec{i}}(\varvec{\theta })\right) - P^i(\varvec{\theta }) + \alpha \Pi ^i_{\rho } \left( \theta ^i\right) \right] \\&\quad = \sum _{i=1}^N E \left[ X^i(\varvec{\theta })\left[ v\left( \varvec{Q}^{\varvec{i}}(\varvec{\theta })\right) - c\left( \varvec{Q}^{\varvec{i}}(\varvec{\theta }), \theta ^i\right) \right] - (1-\alpha ) \Pi ^i_{\rho } \left( \theta ^i\right) \right] \\&\quad = \sum _{i=1}^N E \left[ X^i(\varvec{\theta }) \left[ v\left( \varvec{Q}^{\varvec{i}}(\varvec{\theta })\right) - c\left( \varvec{Q}^{\varvec{i}}(\varvec{\theta }), \theta ^i\right) - (1-\alpha ) c_{\theta }\left( \varvec{Q}^{\varvec{i}}(\varvec{\theta }), \theta ^i\right) \frac{F\left( \theta ^i\right) }{f\left( \theta ^i\right) } \right] \right. \\&\quad \left. \quad - (1-\alpha ) \Pi ^i_{\rho } \left( \bar{\theta }\right) \right] \\&\quad = \sum _{i=1}^N E \left[ X^i(\varvec{\theta }) \Phi \left( \varvec{Q}^{\varvec{i}}(\varvec{\theta }), \theta ^i\right) - (1-\alpha ) \Pi ^i_{\rho } \left( \bar{\theta }\right) \right] . \end{aligned}$$

Note that $\arg \max _{\varvec{q} \in {\mathcal {Q}}} \Phi (\varvec{q}, \theta ^i)$ is nonempty for any $\theta ^i$ because ${\mathcal {Q}}$ is compact and $\Phi$ is continuous in $\varvec{q}$. The above objective function is maximized when $\Pi ^i_{\rho } (\bar{\theta }) = 0$, and $\varvec{Q}^{\varvec{i}}(\varvec{\theta })$ and $X^i(\varvec{\theta })$ are respectively given by $\varvec{Q}^{*}(\theta ^i)$ and $X^{i*}(\varvec{\theta })$ in the lemma. This is because $\varvec{Q}^{*}(\theta ^i)$ maximizes $\Phi (\varvec{q}, \theta ^i)$ and the maximized value $\Phi (\varvec{Q}^{*}(\theta ^i), \theta ^i)$ is decreasing in $\theta ^i$. The latter fact follows from $\Phi (\varvec{Q}^{*}(\theta ), \theta ) \ge \Phi (\varvec{Q}^{*}(\theta ^{\prime }), \theta ) > \Phi (\varvec{Q}^{*}(\theta ^{\prime }), \theta ^{\prime })$ for any $\theta < \theta ^{\prime }$, where the second inequality follows from the assumptions that $c$ is strictly increasing in $\theta$ (Assumption 1), and both $c_\theta$ and $F/f$ are increasing in $\theta$ (Assumptions 2, 3).

Finally, we show that the direct mechanism $\rho ^* = (P^{i*}, \varvec{Q}^{*}, X^{i*})_{i = 1,\ldots ,N }$ satisfies the ignored monotonicity condition. Now, $\Phi$ is quasisupermodular in $\varvec{q}$ by Assumption 4, and has strictly increasing differences in $(\varvec{q}, -\theta ^i)$ from Assumptions 1–3. It then follows from Theorem 4$^{\prime }$ of Milgrom and Shannon (1994) that $\varvec{Q}^{*}(\theta ) \ge \varvec{Q}^{*}(\theta ^{\prime })$ for any $\theta < \theta ^{\prime }$. Also, $X^{i*}$ is decreasing in $\theta ^i$. These facts imply that $\rho ^*$ satisfies the monotonicity condition. $\square$

Proof of Lemma 3

(i) This was shown in the last paragraph of the proof of Lemma 2.

(ii) Fix any $\theta \in (\underline{\theta }, \bar{\theta }]$ and any $\alpha , \alpha ^{\prime }$ with $\alpha ^{\prime } > \alpha$. From Theorem 4$^{\prime }$ of Milgrom and Shannon (1994) together with Assumption 4, it is enough to show that the virtual surplus $\Phi$ has strictly increasing differences in $(\varvec{q}, \alpha )$. To show this, take any $\varvec{q}, \varvec{q}^{\prime }$ with $\varvec{q}^{\prime } > \varvec{q}$. Since the first two terms in $\Phi$ does not depend on $\alpha$ and $\theta > \underline{\theta }$ implies $F(\theta )/f(\theta ) > 0$, we will complete the proof by showing that $c_\theta (\varvec{q}^{\prime }, \theta ) > c_\theta (\varvec{q}, \theta )$. Since $c_\theta$ has increasing differences in $(\varvec{q}, \theta )$, $c_\theta (\varvec{q}^{\prime }, \theta ) - c_\theta (\varvec{q}, \theta ) \ge c_\theta (\varvec{q}^{\prime }, z) - c_\theta (\varvec{q}, z)$ for any $z \in [\underline{\theta }, \theta ]$. This implies

$$\begin{aligned} \left[ c_\theta \left( \varvec{q}^{\prime }, \theta \right) - c_\theta (\varvec{q}, \theta )\right] \left( \theta - \underline{\theta }\right)&= \int _{\underline{\theta }}^{\theta } \left[ c_\theta \left( \varvec{q}^{\prime }, \theta \right) - c_\theta (\varvec{q}, \theta ) \right] dz\\&\ge \int _{\underline{\theta }}^{\theta } \left[ c_\theta \left( \varvec{q}^{\prime }, z\right) - c_\theta (\varvec{q}, z) \right] d z > 0, \end{aligned}$$

where the last inequality follows from $c(\varvec{q}^{\prime }, \theta ) - c(\varvec{q}^{\prime }, \underline{\theta }) > c(\varvec{q}, \theta ) - c(\varvec{q}, \underline{\theta })$. Thus, $c_\theta (\varvec{q}^{\prime }, \theta ) > c_\theta (\varvec{q}, \theta )$, and Theorem 4$^{\prime }$ of Milgrom and Shannon (1994) implies $\varvec{Q}^{*}(\theta ; \alpha ^{\prime }) \ge \varvec{Q}^{*}(\theta ; \alpha )$.

(iii) Fix any $\theta \in (\underline{\theta }, \bar{\theta }]$ and any $\alpha , \alpha ^{\prime }$ with $\alpha ^{\prime } > \alpha$. Under the assumptions in part (iii), we only show that $Q_m^*(\theta ; \alpha ^{\prime }) > Q_m^*(\theta ; \alpha )$. The proof of the first claim is analogous. Let $\varvec{q}^{\prime } := \varvec{Q}^{*}(\theta ; \alpha ^{\prime })$ and $\varvec{q} := \varvec{Q}^{*}(\theta ; \alpha )$, so that $\varvec{q}^{\prime } \ge \varvec{q}$ from part (ii). The assumptions in part (iii) avoid a corner solution so that $q_m \in (0, \bar{q}_m)$. Let $\Phi (\cdot ; \alpha ^{\prime })$ and $\Phi (\cdot ; \alpha )$ be the virtual surplus for $\alpha ^{\prime }$ and $\alpha$. Since $\varvec{q}$ maximizes the virtual surplus $\Phi (\cdot , \theta ; \alpha )$, $\Phi (\varvec{q}, \theta ; \alpha ) \ge \Phi (\hat{q}_m, (q_l)_{l \not = m}, \theta ; \alpha )$ for any $\hat{q}_m \in [0, q_m)$. This, together with Assumption 4, implies that $\Phi (q_m, (q_l^{\prime })_{l \not = m}, \theta ; \alpha ) \ge \Phi (\hat{q}_m, (q_l^{\prime })_{l \not = m}, \theta ; \alpha )$ for any $\hat{q}_m \in [0, q_m)$. Hence, by the differentiability of $\Phi$ in $q_m$, $\frac{\partial \Phi }{\partial q_m}(\hat{\varvec{q}}, \theta ; \alpha ) \ge 0$, where $\hat{\varvec{q}} := (q_m, (q_l^{\prime })_{l \not = m})$. Since $q_m < \bar{q}_m$ and $F(\theta )/f(\theta ) > 0$, we will complete the proof by showing that $\frac{\partial c_\theta }{\partial q_m}(\hat{\varvec{q}}, \theta ) > 0$ and thus $\frac{\partial \Phi }{\partial q_m}(\hat{\varvec{q}}, \theta ; \alpha ^{\prime }) > \frac{\partial \Phi }{\partial q_m}(\hat{\varvec{q}}, \theta ; \alpha ) \ge 0$. Since $c_\theta$ has increasing differences in $(\varvec{q}, \theta )$, $\frac{\partial c_\theta }{\partial q_m}(\hat{\varvec{q}}, \theta ) \ge \frac{\partial c_\theta }{\partial q_m}(\hat{\varvec{q}}, z)$ for any $z \in [\underline{\theta }, \theta ]$. This, together with Assumption 1, implies that

$$\begin{aligned} \frac{\partial c_\theta }{\partial q_m}\left( \hat{\varvec{q}}, \theta \right) (\theta - \underline{\theta })&= \int _{\underline{\theta }}^{\theta } \frac{\partial c_\theta }{\partial q_m}\left( \hat{\varvec{q}}, \theta \right) dz \ge \int _{\underline{\theta }}^{\theta } \frac{\partial c_\theta }{\partial q_m}\left( \hat{\varvec{q}}, z\right) dz\\&= \frac{\partial c}{\partial q_m}\left( \hat{\varvec{q}}, \theta \right) - \frac{\partial c}{\partial q_m}\left( \hat{\varvec{q}}, \underline{\theta }\right) > 0. \end{aligned}$$

(iv) This follows from the fact that the virtual surplus $\Phi$ does not depend on the number $N$ of suppliers, as shown in (9). $\square$

Proof of Proposition 1

To construct an optimal scoring rule, we begin with some preliminary definitions. Let $\mathcal {D} \subset [\underline{\theta }, \bar{\theta }]$ be the set of (countably many) discontinuous points of the optimal quality schedule $\varvec{Q}^{*} : [\underline{\theta }, \bar{\theta }] \rightarrow {\mathbb {R}}_+^M$. We define the following set in ${\mathbb {R}}_+^M$:

$$\begin{aligned} {\mathcal {Q}}^* := \left\{ \varvec{Q}^{*}(\theta ) \mid \theta \in \left[ \underline{\theta }, \bar{\theta }\right] {\setminus } \mathcal {D}\right\}&\cup \left\{ (1-\lambda ) \varvec{Q}^{*}(\theta +) + \lambda \varvec{Q}^{*}(\theta ) \mid \theta \in \mathcal {D}, \lambda \in [0,1]\right\} \nonumber \\&\cup \{ (1-\lambda ) \varvec{Q}^{*}(\theta ) + \lambda \varvec{Q}^{*}(\theta -) \mid \theta \in \mathcal {D}, \lambda \in [0,1] \} \end{aligned}$$

(18)

Here, $\varvec{Q}^{*}(\theta +)\,\, {:=}\,\, (\lim _{\theta ^{\prime } \searrow \theta } Q_m^*(\theta ^{\prime }))_{m=1,\ldots ,M}$ and $\varvec{Q}^{*}(\theta -) \!\!:=\!\! (\lim _{\theta ^{\prime } \nearrow \theta } Q_m^*(\theta ^{\prime }))_{m=1,\ldots ,M}$ denote the vectors of the right limit and the left limit of $Q_m^*$ at $\theta$, respectively; the one-sided limits exist because $Q_m^*$ is increasing by Lemma 3 (i). Note that ${\mathcal {Q}}^*$ may not be a subset of ${\mathcal {Q}}$. It is easy to verify that ${\mathcal {Q}}^*$ is a compact connected chain in ${\mathbb {R}}_+^M$. For each $m$ and $q \in [Q_m^*(\bar{\theta }), Q_m^*(\underline{\theta })]$, the set $\{ (q_1,\ldots ,q_M) \in {\mathcal {Q}}^* \mid q_m = q \}$ is a nonempty compact chain, and thus it has a greatest element. We denote the greatest element by $\varvec{q}^{\varvec{m}}(q) = (q_1^m(q),\ldots ,q_M^m(q))$. Let $\varvec{q}^{\varvec{m}}(q) := \varvec{Q}^{*}(\underline{\theta })$ for $q \ge Q_m^*(\underline{\theta })$, and $\varvec{q}^{\varvec{m}}(q) := \varvec{Q}^{*}(\bar{\theta })$ for $q \le Q_m^*(\bar{\theta })$. For each $m$, let $\theta ^m(q) := \inf \{ \theta \in [\underline{\theta }, \bar{\theta }] \mid q \in [Q_m^*(\theta +), Q_m^{*}(\theta -)] \}$ for $q \in [Q_m^*(\bar{\theta }), Q_m^*(\underline{\theta })]$ and let $\theta ^m(q) := \bar{\theta }$ for $q \in [0, Q_m^*(\bar{\theta }))$. The functions $\varvec{q}^{\varvec{m}} : {\mathbb {R}}_+ \rightarrow {\mathcal {Q}}^{*}$ and $\theta ^m : [0,Q_m^*(\underline{\theta })] \rightarrow [\underline{\theta }, \bar{\theta }]$ possess the following properties: $q_l^m(\cdot )$ is increasing in $q$ for each $l$, and $\theta ^m(\cdot )$ is decreasing and continuous in $q$. We will often invoke these properties.

We are now ready to construct an optimal scoring rule. For each $m$, we define a function $\sigma _m : {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+$ as $\sigma _m(q_m) := - c(\varvec{Q}^{*}(\bar{\theta }), \bar{\theta })$ for $q_m < Q_m^*(\bar{\theta })$, and

$$\begin{aligned} \sigma _m(q_m) := \sum _{l = 1}^M \int _{0}^{q_l^m(q_m)} \frac{\partial c}{\partial q_l} \left( \varvec{q}^{\varvec{l}}(q), \theta ^l(q)\right) dq \end{aligned}$$

(19)

for $q_m \ge Q_m^*(\bar{\theta })$; the integral is well-defined and finite because $q^l_{l^{\prime }}(q)$ and $\theta ^l(q)$ are monotonic and thus Lebesgue-measurable, and $\partial c/\partial q_l$ is continuous. Since $q_l^m(\cdot )$ is increasing in $q_m$ for any $l$, so is the function $\sigma _m(\cdot )$. We define a function $s : {\mathbb {R}}_+^M \rightarrow {\mathbb {R}}$ as

$$\begin{aligned} s(q_1,\ldots ,q_M) := \min \left\{ \sigma _1(q_1),\ldots ,\sigma _M(q_M)\right\} . \end{aligned}$$

By construction, $\sigma _m$ is upper semicontinuous on ${\mathbb {R}}_+$, and thus $s$ is upper semicontinuous on ${\mathbb {R}}_+^M$. We define a scoring rule $s^* : {\mathcal {Q}} \rightarrow {\mathbb {R}}$ as the restriction of $s$ to ${\mathcal {Q}}$, that is, $s^*(\varvec{q}) := s(\varvec{q})$ for $\varvec{q} \in {\mathcal {Q}}$. By construction, the rule $s^*$ has the property in the proposition.

To prove that the scoring rule $s^*$ implements the optimal mechanism, it is enough to show that, for any $\theta$, $\varvec{Q}^{*}(\theta ) \in \arg \max \{ s(\varvec{q}) - c(\varvec{q}, \theta ) \mid \varvec{Q}^{*}(\bar{\theta }) \le \varvec{q} \le \varvec{Q}^{*}(\underline{\theta }) \}$. We now show that, for any $\varvec{q}^{\prime } \not \in {\mathcal {Q}}^*$ with $\varvec{q}^{\prime } > \varvec{Q}^{*}(\bar{\theta })$, there exists $\varvec{q} \in {\mathcal {Q}}^*$ such that $\varvec{q} < \varvec{q}^{\prime }$ and $s(\varvec{q}) = s(\varvec{q}^{\prime })$. Fix any such $\varvec{q}^{\prime }$. Observe that $\{ \varvec{q}^{1}(q_1^{\prime }),\ldots ,\varvec{q}^{\varvec{M}}(q_M^{\prime }) \}$ are totally ordered, and thus it has a least element. Let $\varvec{q}$ be the least element. Then, $\varvec{q} \le (q_1^1(q_1^{\prime }),\ldots ,q_M^M(q_M^{\prime })) = (q_1^{\prime },\ldots ,q_M^{\prime }) = \varvec{q}^{\prime }$. Since $\varvec{q} \in {\mathcal {Q}}^*$ and $\varvec{q}^{\prime } \not \in {\mathcal {Q}}^*$, $\varvec{q} < \varvec{q}^{\prime }$. By construction of $s$, $s(\varvec{q}^{\prime }) = \min _m \{ \sigma _m(q_m^{\prime }) \} = s(\varvec{q})$.

To avoid some trivial cases, we assume that $Q_m^*(\underline{\theta }) > Q_m^*(\bar{\theta })$ for each $m$. We now define a function $z : {\mathcal {Q}}^* \rightarrow [0,1]$ as

$$\begin{aligned} z(q_1,\ldots ,q_M) := \frac{1}{M} \sum _{m =1}^M \frac{ q_m - Q_m^*\left( \bar{\theta }\right) }{Q_m^*\left( \underline{\theta }\right) - Q_m^*\left( \bar{\theta }\right) }. \end{aligned}$$

Since $z$ is continuous and the domain ${\mathcal {Q}}^*$ is connected, the range $\{z(\varvec{q}) \in [0,1] \mid \varvec{q} \in {\mathcal {Q}}^*\}$ is equal to $[0,1]$ by the Intermediate Value Theorem (see Ok 2007). The function $z(\cdot )$ is bijective and thus invertible. We denote the inverse of $z(\cdot )$ by $\varvec{Q} : [0,1] \rightarrow {\mathcal {Q}}^*$, that is, $\varvec{Q}(z) := \varvec{q}$ with $z(\varvec{q}) = z$. The function $\varvec{Q} = (Q_1,\ldots ,Q_M)$ is continuous by the Homeomorphism Theorem (see Ok 2007). By the previous argument with the assumption that $c$ is increasing in $\varvec{q}$, we will complete the proof by showing that, for any $\theta$, $\varvec{Q}^{*}(\theta ) \in \arg \max \{ s(\varvec{Q}(z)) - c(\varvec{Q}(z), \theta ) \mid z \in [0,1] \}$.

Fix any $\theta$. We claim that $s(\varvec{Q}(z)) - c(\varvec{Q}(z), \theta )$ is increasing in $z$ on $\{ z \in [0,1] \mid \varvec{Q}(z) \le \varvec{Q}^{*}(\theta ) \}$, and decreasing in $z$ on $\{ z \in [0,1] \mid \varvec{Q}(z) \ge \varvec{Q}^{*}(\theta ) \}$. We will only prove the first claim; the proof of the second claim is symmetric. We first show that, for any $\varvec{q}^{\prime } \in {\mathcal {Q}}^*$, $s(\varvec{q}^{\prime })$ is given by

$$\begin{aligned} s(\varvec{q}^{\prime }) = \sum _{m = 1}^M \int _{0}^{q_m^{\prime }} \frac{\partial c}{\partial q_m} \left( \varvec{q}^{\varvec{m}}(q), \theta ^m(q)\right) dq. \end{aligned}$$

(20)

Fix any $\varvec{q}^{\prime } = (q_1^{\prime },\ldots ,q_M^{\prime }) \in {\mathcal {Q}}^*$, and let $\varvec{q}^{\varvec{m}}(q_m^{\prime })$ be the least element of $\{ \varvec{q}^\mathbf{1 }(q_1^{\prime }),\ldots ,\varvec{q}^{\varvec{M}}(q_M^{\prime })\}$. By construction of $\sigma _m(\cdot )$, $s(\varvec{q}^{\prime }) = \sigma _m(q_m^{\prime })$. Now, $\varvec{q}^{\prime } \le \varvec{q}^{\varvec{m}}(q_m^{\prime }) \le (q_1^1(q_1^{\prime }),\ldots ,q_M^M(q_M^{\prime })) = \varvec{q}^{\prime }$ because $\varvec{q}^{\prime } \in {\mathcal {Q}}^*$. Thus, $\varvec{q}^{\varvec{m}}(q_m^{\prime }) = \varvec{q}^{\prime }$, which implies that the score $s(\varvec{q}^{\prime })$ is given by (20).

Since $\varvec{Q}(z) = (Q_1(z),\ldots ,Q_M(z)) \in {\mathcal {Q}}^*$ for any $z$, the objective function is given by

$$\begin{aligned} s(\varvec{Q}(z)) - c(\varvec{Q}(z), \theta ) = \sum _{m = 1}^M \int _{0}^{Q_m(z)} \frac{\partial c}{\partial q_m} \left( \varvec{q}^{\varvec{m}}(q), \theta ^m(q)\right) dq - c\left( \varvec{Q}(z), \theta \right) . \end{aligned}$$

It is continuous in $z \in [0,1]$ because $\varvec{Q}(\cdot )$ is continuous. Fix any $z \in (0,1)$ with $\varvec{Q}(z) < \varvec{Q}^{*}(\theta )$. Then, it is enough to show that the following lower right derivative of the objective function at $z$, which may be infinity, is nonnegative (see, e.g., Royden and Fitzpatrick 2010):

$$\begin{aligned} \lim _{\delta \rightarrow 0} \inf _{0 < h < \delta } \left[ \frac{s\left( \varvec{Q}(z+h)\right) - s\left( \varvec{Q}(z)\right) }{h} - \frac{c\left( \varvec{Q}(z+h), \theta \right) - c\left( \varvec{Q}(z),\theta \right) }{h} \right] \end{aligned}$$

(21)

The first term in the bracket in (21) is given by

$$\begin{aligned} \sum _{m =1}^M \left[ \frac{\int _{Q_{m}(z)}^{Q_{m}(z+h)} \frac{\partial c}{\partial q_m} \left( \varvec{q}^{\varvec{m}}(q), \theta ^m(q)\right) dq}{Q_m(z+h) - Q_m(z)} \right] \frac{Q_m(z+h) - Q_m(z)}{h}. \end{aligned}$$

(22)

The second term in the bracket in (21) can be rewritten as

$$\begin{aligned}&\sum _{m = 1}^M \left[ \frac{c\left( (Q_l(z+h))_{l \ge m},(Q_l(z))_{l < m}, \theta \right) - c\left( (Q_l(z+h))_{l > m},(Q_l(z))_{l \le m}, \theta \right) }{Q_m(z+h) - Q_m(z)} \right] \nonumber \\&\quad \qquad \times \frac{Q_m(z+h) - Q_m(z)}{h}. \end{aligned}$$

(23)

Fix any $m$. Assume without loss of generality that $Q_m(z+h) > Q_m(z)$ for any $h > 0$; if not, then the terms in the summation in (22) and (23) would become zero for small enough $h > 0$. By the assumption, $\varvec{q}^{\varvec{m}}(q)$ is continuous from the right at $q = Q_m(z)$, and $\lim _{h \searrow 0} \varvec{q}^{\varvec{m}}(Q_m(z+h)) = \varvec{q}^{\varvec{m}}(Q_m(z)) = \varvec{Q}(z)$. Also, $\theta ^m(\cdot )$ is always continuous. Hence, the bracketed term in (22) converges to $\frac{\partial c}{\partial q_m} (\varvec{Q}(z), \theta ^m(Q_m(z)))$ as $h \searrow 0$. Since $c(\cdot )$ is continuously differentiable and $Q_l(\cdot )$ is continuous for each $l$, the bracketed term in (23) converges to $\frac{\partial c}{\partial q_m} (\varvec{Q}(z), \theta )$ as $h \searrow 0$.

We now consider two cases. First, consider the case where $Q_m(z) < Q_m^*(\theta +)$. Then, $\theta ^m(Q_m(z)) > \theta$ and thus

$$\begin{aligned} \frac{\partial c}{\partial q_m} \left( \varvec{Q}(z), \theta ^m(Q_m(z))\right) > \frac{\partial c}{\partial q_m} (\varvec{Q}(z), \theta ) \end{aligned}$$

(24)

by Assumption 1. Second, consider the case where $Q_m^{*}(\theta +) \le Q_m(z) < Q_m^{*}(\theta )$, which occurs only if $\theta \in \mathcal {D}$. Then, $\theta ^m(Q_m(z)) = \theta$. A simple computation also shows that

$$\begin{aligned} \frac{Q_{m}(z+h) - Q_{m}(z)}{h} = \frac{M}{\sum _{l =1}^M \frac{ Q_l^*(\theta -) - Q_l^*(\theta +)}{Q_l^*(\underline{\theta }) - Q_l^*(\bar{\theta })}} \in (0, \infty ) \end{aligned}$$

(25)

for any $h>0$ with $Q_m(z+h) \le Q_m^*(\theta )$. Therefore, applying these facts to each $m$, we can find a small enough $\delta > 0$ such that the bracketed term in (21) is nonnegative for any $h$ with $h \in (0, \delta )$. $\square$

Proof of Theorem 1

We will construct an optimal scoring rule which is additively separable, in a similar manner to Proposition 1. Let $\mathcal {D} \subset [\underline{\theta }, \bar{\theta }]$ and ${\mathcal {Q}}^* \subset {\mathbb {R}}_+^M$ be the sets defined in the proof of Proposition 1. For each $m$, let $\varvec{q}^{\varvec{m}} : {\mathbb {R}}_+ \rightarrow {\mathcal {Q}}^*$ and $\theta ^m : [0,Q_m^*(\underline{\theta })] \rightarrow [\underline{\theta }, \bar{\theta }]$ be the functions defined in the proof of Proposition 1. For each $j=1,2$ and each $m \in {\mathcal {M}}_j$, we define a function $\sigma _m^j : {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+$ as $\sigma _m^j(q_m) := - c(\varvec{Q}^{*}(\bar{\theta }), \bar{\theta })$ for $q_m < Q_m^*(\bar{\theta })$, and

$$\begin{aligned} \sigma _m^j(q_m) := \sum _{l \in {\mathcal {M}}_j} \int _{0}^{q_l^m(q_m)} \frac{\partial c}{\partial q_l} \left( \varvec{q}^{\varvec{l}}(q), \theta ^l(q)\right) dq \end{aligned}$$

for $q_m \ge Q_m^*(\bar{\theta })$. For each $j=1,2$, we define a function $s_j : {\mathbb {R}}_+^{|{\mathcal {M}}_j|} \rightarrow {\mathbb {R}}$ as

$$\begin{aligned} s_j\left( (q_m)_{m \in {\mathcal {M}}_j}\right) := \min \left\{ \sigma _m^j(q_m) \mid m \in {\mathcal {M}}_j\right\} . \end{aligned}$$

(26)

We define a scoring rule $s^* : {\mathcal {Q}} \rightarrow {\mathbb {R}}$ as the restriction of $s_1+s_2$ to ${\mathcal {Q}}$, that is, $s^*(\varvec{q}_\mathbf{1 }, \varvec{q}_\mathbf{2 }) := s_1(\varvec{q}_\mathbf{1 }) + s_2(\varvec{q}_\mathbf{2 })$ for $(\varvec{q}_\mathbf{1 }, \varvec{q}_\mathbf{2 }) \in {\mathcal {Q}}$.

Let $\varvec{Q} = (Q_1,\ldots ,Q_M) : [0,1] \rightarrow {\mathcal {Q}}^*$ be the function defined in the proof of Proposition 1, and let $\varvec{Q}_\mathbf{1 } := (Q_m)_{m \in {\mathcal {M}}_1}$ and $\varvec{Q}_\mathbf{2 } := (Q_m)_{m \in {\mathcal {M}}_2}$. As in the proof of Proposition 1, we will complete the proof by showing that, for any $\theta$, $\varvec{Q}^{*}(\theta ) \in \arg \max \{ s_1(\varvec{Q}_\mathbf{1 }(z)) + s_2(\varvec{Q}_\mathbf{2 }(z^{\prime }) - c(\varvec{Q}_\mathbf{1 }(z), \varvec{Q}_\mathbf{2 }(z^{\prime }), \theta ) \mid z, z^{\prime } \in [0,1] \}$.

Fix any $\theta$. For each $j = 1,2$, define ${\mathcal {Q}}_j^{*} := \{(q_m)_{m \in {\mathcal {M}}_j} \in {\mathbb {R}}_+^{|{\mathcal {M}}_j|} \mid (q_1,\ldots ,q_M) \in {\mathcal {Q}}^* \text { for some } (q_m)_{m \in \{1,\ldots ,M \} \setminus {\mathcal {M}}_j} \}$. The product ${\mathcal {Q}}_1^* \times {\mathcal {Q}}_2^*$ can be divided into three regions, as in Fig. 1 for the case $M=2$. We now claim that:

• (i) For each $i = 1,2$ and $j \not = i$, for any $\varvec{q}_{\varvec{j}} \in {\mathcal {Q}}_j^*$ with $\varvec{q}_{\varvec{j}} \le (Q_m^*(\theta ))_{m \in {\mathcal {M}}_j}$, $s_i(\varvec{Q}_{\varvec{i}}(z)) + s_j(\varvec{q}_{\varvec{j}}) - c(\varvec{Q}_{\varvec{i}}(z), \varvec{q}_{\varvec{j}}, \theta )$ is increasing in $z$ on $\{ z \in [0,1] \mid \varvec{Q}_{\varvec{i}}(z) \le \varvec{Q}_{\varvec{i}}(z^{\prime }) \text { where } z^{\prime } := \min \{ z^{\prime \prime } \mid \varvec{Q}_{\varvec{j}}(z^{\prime \prime }) = \varvec{q}_{\varvec{j}} \} \}$, and, for any $\varvec{q}_{\varvec{j}} \in {\mathcal {Q}}_j^*$ with $\varvec{q}_{\varvec{j}} \ge (Q_m^*(\theta ))_{m \in {\mathcal {M}}_j}$, $s_i(\varvec{Q}_{\varvec{i}}(z)) + s_j(\varvec{q}_{\varvec{j}}) - c(\varvec{Q}_{\varvec{i}}(z), \varvec{q}_{\varvec{j}}, \theta )$ is decreasing in $z$ on $\{ z \in [0,1] \mid \varvec{Q}_{\varvec{i}}(z) \ge \varvec{Q}_{\varvec{i}}(z^{\prime }) \text { where } z^{\prime } := \max \{ z^{\prime \prime } \mid \varvec{Q}_{\varvec{j}}(z^{\prime \prime }) = \varvec{q}_{\varvec{j}} \} \}$.

• (ii) For any $\varvec{q}_\mathbf{2 } \in {\mathcal {Q}}_2^*$ with $\varvec{q}_\mathbf{2 } \ge (Q_m^*(\theta ))_{m \in {\mathcal {M}}_2}$, $s_1(\varvec{Q}_\mathbf{1 }(z)) + s_2(\varvec{q}_\mathbf{2 }) - c(\varvec{Q}_\mathbf{1 }(z), \varvec{q}_\mathbf{2 }, \theta )$ is increasing in $z$ on $\{ z \in [0,1] \mid \varvec{Q}_\mathbf{1 }(z) \le (Q_m^*(\theta ))_{m \in {\mathcal {M}}_1} \}$. For any $\varvec{q}_\mathbf{2 } \in {\mathcal {Q}}_2^*$ with $\varvec{q}_\mathbf{2 } \le (Q_m^*(\theta ))_{m \in {\mathcal {M}}_2}$, $s_1(\varvec{Q}_\mathbf{1 }(z)) + s_2(\varvec{q}_\mathbf{2 }) - c(\varvec{Q}_\mathbf{1 }(z), \varvec{q}_\mathbf{2 }, \theta )$ is decreasing in $z$ on $\{ z \in [0,1] \mid \varvec{Q}_\mathbf{1 }(z) \ge (Q_m^*(\theta ))_{m \in {\mathcal {M}}_1} \}$.

In Fig. 1, the arrows represent the direction to which $s_1(\varvec{q}_\mathbf{1 }) + s_2(\varvec{q}_\mathbf{2 }) - c(\varvec{q}_\mathbf{1 },\varvec{q}_\mathbf{2 }, \theta )$ goes up. Region (a) corresponds to claim (i) for $i = 1$ and $j=2$, region (b) corresponds to claim (i) for $i = 2$ and $j=1$, and region (c) corresponds to claim (ii), respectively.

We will only prove the first part of claim (i); the proof of the remaining parts is analogous. Without loss of generality, we consider the case where $i = 1$ and $j=2$. Fix any $\varvec{q}_\mathbf{2 } \in {\mathcal {Q}}_2^*$ with $\varvec{q}_\mathbf{2 } \le (Q_m^*(\theta ))_{m \in {\mathcal {M}}_2}$. The same argument as in the proof of Proposition 1 implies that, for any $z$, the objective function [net of $s_2(\varvec{q}_\mathbf{2 })$] is given by

$$\begin{aligned}&s_1(\varvec{Q}_\mathbf{1 }(z)) - c(\varvec{Q}_\mathbf{1 }(z), \varvec{q}_\mathbf{2 }, \theta )\\&\quad = \sum _{m \in {\mathcal {M}}_1} \int _{0}^{Q_m(z)} \frac{\partial c}{\partial q_m} \left( \varvec{q}^{\varvec{m}}(q), \theta ^m(q)\right) dq - c(\varvec{Q}_\mathbf{1 }(z), \varvec{q}_\mathbf{2 }, \theta ). \end{aligned}$$

Fig. 1

Illustration of Theorem 1

It is continuous in $z \in [0,1]$ because $\varvec{Q}_\mathbf{1 }(\cdot ) = (Q_m(\cdot ))_{m \in {\mathcal {M}}_1}$ is continuous in $z$. Fix any $z \in (0,1)$ with $\varvec{Q}_\mathbf{1 }(z) < \varvec{Q}_\mathbf{1 }(z^{\prime })$ where $z^{\prime } := \min \{ z^{\prime \prime } \mid \varvec{Q}_\mathbf{2 }(z^{\prime \prime }) = \varvec{q}_\mathbf{2 } \}$. Note that $\varvec{Q}_\mathbf{1 }(z^{\prime }) \le (Q_m^*(\theta ))_{m \in {\mathcal {M}}_1}$ because $\varvec{Q}_\mathbf{2 }(z^{\prime }) = \varvec{q}_\mathbf{2 } \le (Q_m^*(\theta ))_{m \in {\mathcal {M}}_2}$. Then, it is enough to show that the following lower right derivative of the objective function at $z$, which may be infinity, is nonnegative (see, e.g., Royden and Fitzpatrick 2010):

$$\begin{aligned} \lim _{\delta \rightarrow 0} \inf _{0 < h < \delta } \left[ \frac{s_1\left( \varvec{Q}_\mathbf{1 }(z+h)\right) - s_1\left( \varvec{Q}_\mathbf{1 }(z)\right) }{h} - \frac{c\left( \varvec{Q}_\mathbf{1 }(z+h), \varvec{q}_\mathbf{2 }, \theta \right) - c\left( \varvec{Q}_\mathbf{1 }(z), \varvec{q}_\mathbf{2 }, \theta \right) }{h} \right] . \end{aligned}$$

(27)

The first term in the bracket in (27) is given by

$$\begin{aligned} \sum _{m \in {\mathcal {M}}_1} \left[ \frac{\int _{Q_{m}(z)}^{Q_{m}(z+h)} \frac{\partial c}{\partial q_m} \left( \varvec{q}^{\varvec{m}}(q), \theta ^m(q)\right) dq}{Q_m(z+h) - Q_m(z)} \right] \frac{Q_m(z+h) - Q_m(z)}{h}. \end{aligned}$$

(28)

Suppose without loss of generality that ${\mathcal {M}}_1 = \{ 1,\ldots , \bar{m} \}$ for some $\bar{m}$ with $1 \le \bar{m} < M$ . The second term in the bracket in (27) can be rewritten as

$$\begin{aligned}&\sum _{m \in {\mathcal {M}}_1} \left[ \frac{c\left( (Q_l(z+h))_{m \le l \le \bar{m}},(Q_l(z))_{l < m}, \varvec{q}_\mathbf{2 }, \theta \right) - c\left( (Q_l(z+h))_{m < l \le \bar{m}},(Q_l(z))_{l \le m}, \varvec{q}_\mathbf{2 }, \theta \right) }{Q_m(z+h) - Q_m(z)} \right] \nonumber \\&\quad \qquad \times \frac{Q_m(z+h) - Q_m(z)}{h}. \end{aligned}$$

(29)

Fix any $m \in {\mathcal {M}}_1$. Assume without loss of generality that $Q_m(z+h) > Q_m(z)$ for any $h > 0$; if not, then the terms in the summation in (28) and (29) would become zero for small enough $h > 0$. By the assumption, $\varvec{q}^{\varvec{m}}(q)$ is continuous from the right at $q = Q_m(z)$, and $\lim _{h \searrow 0} \varvec{q}^{\varvec{m}}(Q_m(z+h)) = \varvec{q}^{\varvec{m}}(Q_m(z)) = \varvec{Q}(z) = (\varvec{Q}_\mathbf{1 }(z), \varvec{Q}_\mathbf{2 }(z))$. Also, $\theta ^m(\cdot )$ is always continuous. Hence, the bracketed term in (28) converges to $\frac{\partial c}{\partial q_m} (\varvec{Q}_\mathbf{1 }(z), \varvec{Q}_\mathbf{2 }(z), \theta ^m(Q_m(z)))$ as $h \searrow 0$. Since $c(\cdot )$ is continuously differentiable and $Q_l(\cdot )$ is continuous for each $l$, the bracketed term in (29) converges to $\frac{\partial c}{\partial q_m} (\varvec{Q}_\mathbf{1 }(z), \varvec{q}_\mathbf{2 }, \theta )$ as $h \searrow 0$.

We now consider two cases. First, consider the case where $Q_m(z) < Q_m^*(\theta +)$. Then, $\theta ^m(Q_m(z)) > \theta$ and thus

$$\begin{aligned} \frac{\partial c}{\partial q_m} \left( \varvec{Q}_\mathbf{1 }(z), \varvec{Q}_\mathbf{2 }(z), \theta ^m(Q_m(z))\right) \!>\! \frac{\partial c}{\partial q_m} \left( \varvec{Q}_\mathbf{1 }(z), \varvec{Q}_\mathbf{2 }(z), \theta \right) \ge \frac{\partial c}{\partial q_m} \left( \varvec{Q}_\mathbf{1 }(z), \varvec{q}_\mathbf{2 }, \theta \right) . \end{aligned}$$

(30)

The first inequality follows from Assumption 1, and the second inequality follows from the hypothesis that, for any $l \in {\mathcal {M}}_2$, the two attributes $m \in {\mathcal {M}}_1$ and $l \in {\mathcal {M}}_2$ are cost complements, and $\varvec{Q}_\mathbf{2 }(z) \le \varvec{q}_\mathbf{2 }$. Second, consider the case where $Q_m^*(\theta +) \le Q_m(z) < Q_m^*(\theta )$, which occurs only if $\theta \in \mathcal {D}$. Then, $\theta ^m(Q_m(z)) = \theta$, and the same equality as (25) in Proposition 1 holds. Therefore, applying these facts to each $m$, we can find a small enough $\delta > 0$ such that the bracketed term in (27) is nonnegative for any $h$ with $h \in (0, \delta )$. $\square$

Proof of Theorem 2

Let $\varvec{q}^{\prime }_\mathbf{1 } := (Q_m^*(\underline{\theta }))_{m \in {\mathcal {M}}_1}$, $\varvec{q}_\mathbf{1 } := (Q_m^*(\bar{\theta }))_{m \in {\mathcal {M}}_1}$, $\varvec{q}^{\prime }_\mathbf{2 } := (Q_m^*(\underline{\theta }))_{m \in {\mathcal {M}}_2}$, and $\varvec{q}_\mathbf{2 } := (Q_m^*(\bar{\theta }))_{m \in {\mathcal {M}}_2}$. From Lemma 3 (i), $\varvec{q}^{\prime }_\mathbf{1 } \ge \varvec{q}_\mathbf{1 }$ and $\varvec{q}^{\prime }_\mathbf{2 } \ge \varvec{q}_\mathbf{2 }$. By the assumption in the paragraph preceding Theorem 2, $Q_{m}^*(\underline{\theta }) > Q_{m}^*(\bar{\theta })$ for some $m \in {\mathcal {M}}_1$ and $Q_{l}^*(\underline{\theta }) > Q_{l}^*(\bar{\theta })$ for some $l \in {\mathcal {M}}_2$, so that $\varvec{q}^{\prime }_\mathbf{1 } > \varvec{q}_\mathbf{1 }$ and $\varvec{q}^{\prime }_\mathbf{2 } > \varvec{q}_\mathbf{2 }$. Suppose that a scoring rule $s$ implements the optimal mechanism. Then, it follows from Lemma 1 that the inequality $s(\varvec{q}^{\prime }_\mathbf{1 }, \varvec{q}_\mathbf{2 }) - c(\varvec{q}^{\prime }_\mathbf{1 }, \varvec{q}_\mathbf{2 }, \bar{\theta }) \le s(\varvec{q}_\mathbf{1 }, \varvec{q}_\mathbf{2 }) - c(\varvec{q}_\mathbf{1 }, \varvec{q}_\mathbf{2 }, \bar{\theta })$ must hold because $(\varvec{q}_\mathbf{1 }, \varvec{q}_\mathbf{2 }) = \varvec{Q}^{*}(\bar{\theta })$. This, together with the hypothesis in the theorem, implies that

$$\begin{aligned} s\left( \varvec{q}^{\prime }_\mathbf{1 }, \varvec{q}_\mathbf{2 }\right) \!-\! s(\varvec{q}_\mathbf{1 }, \varvec{q}_\mathbf{2 }) \le c\left( \varvec{q}^{\prime }_\mathbf{1 }, \varvec{q}_\mathbf{2 }, \bar{\theta }\right) - c\left( \varvec{q}_\mathbf{1 }, \varvec{q}_\mathbf{2 }, \bar{\theta }\right) < c\left( \varvec{q}^{\prime }_\mathbf{1 }, \varvec{q}^{\prime }_\mathbf{2 }, \underline{\theta }\right) - c\left( \varvec{q}_\mathbf{1 }, \varvec{q}^{\prime }_\mathbf{2 }, \underline{\theta }\right) . \end{aligned}$$

Now, if the scoring rule $s$ is additively separable, then $s(\varvec{q}^{\prime }_\mathbf{1 }, \varvec{q}_\mathbf{2 }) - s(\varvec{q}_\mathbf{1 }, \varvec{q}_\mathbf{2 }) = s_1(\varvec{q}^{\prime }_\mathbf{1 }) - s_1(\varvec{q}_\mathbf{1 }) = s(\varvec{q}^{\prime }_\mathbf{1 }, \varvec{q}^{\prime }_\mathbf{2 }) - s(\varvec{q}_\mathbf{1 }, \varvec{q}^{\prime }_\mathbf{2 })$. Then, $s(\varvec{q}^{\prime }_\mathbf{1 }, \varvec{q}^{\prime }_\mathbf{2 }) - s(\varvec{q}_\mathbf{1 }, \varvec{q}^{\prime }_\mathbf{2 }) < c(\varvec{q}^{\prime }_\mathbf{1 }, \varvec{q}^{\prime }_\mathbf{2 }, \underline{\theta }) - c(\varvec{q}_\mathbf{1 }, \varvec{q}^{\prime }_\mathbf{2 }, \underline{\theta })$, and thus a supplier of type $\underline{\theta }$ never chooses $(\varvec{q}^{\prime }_\mathbf{1 }, \varvec{q}^{\prime }_\mathbf{2 }) = \varvec{Q}^{*}(\underline{\theta })$ from Lemma 1. Therefore, there is no additively separable scoring rule which implements the optimal mechanism. $\square$

Proof of Proposition 2

Fix any $\theta \in (\underline{\theta }, \bar{\theta }]$ and any $\alpha , \alpha ^{\prime }$ with $\alpha ^{\prime } > \alpha$. With attribute $m$ which satisfies the conditions in Lemma 3 (iii), $\varvec{Q}^{*}(\theta ; \alpha ^{\prime }) > \varvec{Q}^{*}(\theta ; \alpha )$. By hypothesis, $s^{\alpha ^{\prime }}$ and $s^{\alpha }$ implement the optimal mechanism given $\alpha ^{\prime }$ and $\alpha$ respectively. Then, Lemma 1 implies that $s^{\alpha ^{\prime }}(\varvec{Q}^{*}(\theta ; \alpha ^{\prime })) - s^{\alpha ^{\prime }}(\varvec{Q}^{*}(\theta ; \alpha )) \ge c(\varvec{Q}^{*}(\theta ; \alpha ^{\prime }), \theta ) - c(\varvec{Q}^{*}(\theta ; \alpha ), \theta )$ and $s^{\alpha }(\varvec{Q}^{*}(\theta ; \alpha )) - s^{\alpha }(\varvec{Q}^{*}(\theta ; \alpha ^{\prime })) \!\ge \! c(\varvec{Q}^{*}(\theta ; \alpha ), \theta ) - c(\varvec{Q}^{*}(\theta ; \alpha ^{\prime }), \theta )$. It follows from these two inequalities that the last inequality in the proposition holds. $\square$

Proof of Proposition 3

This fact is immediate from Lemma 1 and Lemma 3 (iv). $\square$