Optimal Procurement with Quality Concerns: Lopomo, Persico & Villa (2023)

What this is. A distilled skeleton of the paper for rapid orientation. Read the original (doi:10.1257/aer.20211437) to replicate or extend the results.

TL;DR

When quality is noncontractible and low-cost suppliers tend to be low-quality (a “lemons” problem in procurement), standard first-price or second-price auctions perform poorly. Lopomo, Persico, and Villa characterize the optimal mechanism: a lowball lottery auction (LoLA) with a floor price $p_L$ and a reserve price $p_H$. Bidders with costs below $p_L$ pool at that price and one is selected randomly; bidders with costs in $[p_L, p_H]$ compete as in a standard second-price auction; bids above $p_H$ are excluded. Under a mild regularity condition, the LoLA maximizes any weighted average of buyer surplus and social surplus subject to incentive compatibility and individual rationality. The optimal floor price is independent of the number of suppliers and rises with the severity of the lemons problem. A counterfactual calibration using Italian government procurement data from Decarolis (2018) finds buyer surplus gains of up to 15 percent over first-price auctions at high levels of quality concern.

Core results

#	Result	Locator	Magnitude as reported
R1	LoLA with optimal $p_L^*$ and $p_H^*$ solves the weighted welfare maximization problem; sincere bidding is an equilibrium in weakly dominant strategies	Theorem 1, §III, p.1514	Analytical: LoLA implements the constrained-optimal mechanism for any $\beta \in [0,1]$ under Assumption 1
R2	Optimal floor and reserve prices are independent of the number of suppliers $N$; floor price is nondecreasing in the severity of the lemons problem $\xi$ for any $\beta$; when $F$ is log-concave, the social planner’s optimal floor is at least as high as the buyer’s	Proposition 1, §III, p.1515-1516	Comparative statics; independence of $N$ follows from conditions (11)-(12), while the floor-price comparison requires log-concavity
R3	Increasing $N$ raises the weighted welfare generated by the optimal LoLA (unlike standard auctions under adverse selection, where welfare can decrease in $N$)	Proposition 2, §III, p.1516-1517	Analytical; contrast: standard FPA expected surplus $E[w(c^{(1)})]$ falls as $N$ grows under adverse selection
R4	Sincere-bidding equilibrium is unique almost surely when $p_H < c_H$, there are at least three bidders, and the cost density is positive throughout its support	Proposition 3, §III, p.1517	Almost-sure uniqueness under the proposition’s positive-density condition; follows from Blume and Heidhues (2004) Vickrey-auction uniqueness result
R5	Italian calibration: buyer surplus up to 15 percent higher in buyer-optimal LoLA than in first-price auction when $\xi = 1$; gain is approximately 2.5 percent even at $\xi \approx 0.5$	Figure 7, §VD, p.1527-1528	15% buyer surplus gain at $\xi=1$; 2.5% gain at $\xi \approx 0.5$
R6	Italian calibration: social surplus improvement up to approximately 20 percent over first-price auction at $\xi = 1$; supplier profit improvement exceeds 100 percent	Figure 7, §VD, p.1527-1528	Approximately 20% social surplus gain; supplier profit gain exceeds 100% at $\xi=1$
R7	Illustrative example (§I): buyer-optimal LoLA ($p_L^* = 3/4$) achieves buyer surplus more than 10 percent above the second-price auction ($p_L = 0$) and the random assignment mechanism ($p_L = 1$)	Figure 2, §I, p.1511	$V(3/4) \approx 0.37$; $V(0) = V(1) \approx 0.33$; gain >10%
R8	Italian calibration: supplier profit under buyer-optimal LoLA exceeds the first-price benchmark, a distinct performance outcome from buyer and social surplus	Figure 7, §VD, pp.1527-1528; text p.1528	Supplier profit improvement exceeds 100% at $\xi=1$
R9	Any LoLA allocation can be implemented by an equivalent FPLoLA with the same reserve price and a suitably chosen minimum bid; the calibrated minimum bid can exceed the LoLA floor	Proposition 4, §IVD, p.1520; text p.1528	At $\xi=0$, $b_L$ is up to 24% higher than $p_L$; the thresholds converge as $\xi$ approaches 1
R10	With $a=0$ and $\xi_2 \in (2.5,4)$, the asymmetric quality-concern optimum favors supplier 2 when both costs are low and supplier 1 when both costs are high	Figure 4, Panel A, §IVE, p.1522	Qualitative allocation split: supplier 2 wins for sure at low costs; supplier 1 wins more often at high costs
R11	With $\xi_2=4$ and $a \in (0,0.5)$, shifting supplier 1’s cost distribution upward makes the asymmetric optimum favor supplier 1 and select it for sure when both costs are low	Figure 4, Panel B, §IVE, p.1522	Qualitative allocation split: supplier 1 wins for sure when both costs are relatively low
R12	In a stylized average-bid auction that discards bids in the lowest and highest quantiles, all bidders bidding the same amount is an equilibrium because any unilateral deviation is discarded; at $b=1$ the buyer surplus is below the LoLA example	Text §I, p.1512	Buyer surplus is 0.33 at $b=1$ versus about 0.37 under LoLA with $p_L^*=3/4$; equilibrium surplus can be made arbitrarily small as $b$ increases

Overall. The LoLA is a practical mechanism (a reverse second-price auction with a price floor) that is simultaneously optimal for the buyer and for the social planner, differing only in the level of the optimal floor price. The theoretical characterization generalizes both Myerson (1981) (standard auctions optimal when no lemons problem) and Manelli and Vincent (1995) (random assignment optimal under extreme lemons problem) as limiting cases.

Theory / model

The two-supplier illustration (§I, p.1509) sets costs independently and uniformly on $[0,1]$ and specifies buyer value as:

v(c_i) \equiv 4c_i - 2c_i^2 \tag{1}

For a LoLA floor $p_L$, the paper computes expected buyer surplus from the regions where a bidder wins by undercutting above the floor and the square where both costs are below the floor (equation (2), pp.1510-1511):

V(p_L) ={}& \int_{p_L}^{1}\left\{\int_{0}^{c_2}[v(c_1)-c_2] dc_1\right\} dc_2 + \int_{p_L}^{1}\left\{\int_{0}^{c_1}[v(c_2)-c_1] dc_2\right\} dc_1 \\ &+ \int_{0}^{p_L}\int_{0}^{p_L}\left[\frac{1}{2}v(c_1)+\frac{1}{2}v(c_2)-p_L\right] dc_1 dc_2 = \frac{1}{3}+\frac{1}{3}p_L^3(1-p_L) \end{aligned} \tag{2}$$ Expected social surplus in the same illustration is (equation (3), p.1511): $$\begin{aligned} S(p_L) ={}& 2\int_{p_L}^{1}\int_{0}^{c_2}[v(c_1)-c_1] dc_1 dc_2 \\ &+ \int_{0}^{p_L}\int_{0}^{p_L}\left\{\frac{1}{2}[v(c_1)-c_1]+\frac{1}{2}[v(c_2)-c_2]\right\} dc_1 dc_2 \\ ={}& \frac{2}{3}+\frac{1}{3}\left(\frac{3}{2}-p_L\right)p_L^3. \end{aligned} \tag{3}$$ The model (§II, p.1512) has one buyer with known type $$\xi$$ and $$N > 1$$ symmetric suppliers. Supplier $$i$$ has privately known cost $$c_i$$ drawn i.i.d. from density $$f$$ on $$[c_L, c_H]$$. Costs are private and quality is noncontractible. The buyer's value from procuring from a supplier with cost $$c$$ is $$v(c, \xi)$$, which is assumed to be increasing in $$c$$ (the lemons problem: higher-cost suppliers provide higher expected quality). The parameter $$\xi$$ encodes the severity of quality concerns, with $$v_{c\xi}(c, \xi) \geq 0$$. The supplier's profit when selected at payment $$m$$ is $$m - c$$; the buyer's surplus is $$v(c, \xi) - m$$. The revelation-principle direct mechanism assigns each supplier a winning probability and expected payment as functions of reported costs (equation (5), p.1513): $$q_i(c_i,c_{-i}),\quad m_i(c_i,c_{-i}) \tag{5}$$ The **virtual valuation function** (equation (4), p.1512) is: $$w(c; \xi, \beta) \equiv v(c; \xi) - c - \beta \frac{F(c)}{f(c)} \tag{4}$$ The ratio $$F(c)/f(c)$$ is the information rent earned by a type-$$c$$ supplier. The parameter $$\beta \in [0,1]$$ encodes the designer's weight on buyer surplus relative to social surplus: $$\beta = 1$$ gives buyer-surplus maximization (Myerson (1981) in reverse) and $$\beta = 0$$ gives social surplus maximization. **Assumption 1 (Regularity):** $$w(c; \xi, \beta)$$ is quasiconcave in $$c$$. This allows $$w$$ to first decrease then increase in $$c$$ (i.e., a lemons problem can be present) while remaining single-peaked. It is satisfied when $$v$$ is concave and $$F/f$$ is convex (which holds for power, Pareto, and exponential distributions of costs). ## Method The **weighted welfare maximization problem** (equations (6)-(10), §III, p.1513-1514) is: $$\max_{q,m} \int_{[c_L,c_H]^N} \left\{ \sum_{i=1}^N \left[(v(c_i, \xi) - (1-\beta) \cdot c_i) \cdot q_i(c_i, c_{-i}) - \beta \cdot m_i(c_i, c_{-i})\right] \right\} \prod_{j=1}^N f(c_j)\, dc_j \tag{6}$$ subject to feasibility $$\sum_i q_i \leq 1$$, non-negativity $$q_i \geq 0$$, interim IC (equation (9)), and interim IR (equation (10)). The paper shows that this problem is solved by a **Lowball Lottery Auction (LoLA)**: a reverse second-price sealed-bid auction with floor price $$p_L$$ and reserve price $$p_H \geq p_L$$, in which bids below $$p_L$$ and above $$p_H$$ are not allowed, and ties at $$p_L$$ are broken uniformly at random (Definition, p.1514). The pointwise feasibility and nonnegative-allocation restrictions are equations (7)-(8), p.1514: $$\sum_{i=1}^{N}q_i(c_i,c_{-i}) \leq 1 \tag{7}$$ $$q_i(c_i,c_{-i}) \geq 0 \tag{8}$$ The interim incentive-compatibility constraint compares truthful reporting with any report $$c_i'$$, integrating over opponents' costs; interim individual rationality is equation (10) (equations (9)-(10), p.1514): $$\begin{aligned} &\int_{[c_L,c_H]^{N-1}}[m_i(c_i,c_{-i})-c_i q_i(c_i,c_{-i})]\prod_{j\ne i}f(c_j)dc_j \\ &\qquad\geq \int_{[c_L,c_H]^{N-1}}[m_i(c_i',c_{-i})-c_i q_i(c_i',c_{-i})]\prod_{j\ne i}f(c_j)dc_j, \quad \forall c_i,c_i',c_{-i} \end{aligned} \tag{9}$$ $$\int_{[c_L,c_H]^{N-1}}[m_i(c_i,c_{-i})-c_i q_i(c_i,c_{-i})]\prod_{j\ne i}f(c_j)dc_j \geq 0 \tag{10}$$ **Theorem 1** (Optimality of LoLA, p.1514): Under Assumption 1, the LoLA implements the solution to the optimization problem (6)-(10) when the reserve price and floor price are set to: $$p_H^* = \sup\{c \in [c_L, c_H] \text{ such that } w(c; \xi, \beta) > 0\} \tag{11}$$ $$p_L^* = \max\{p \in [c_L, c_H] \text{ such that } w(p; \xi, \beta) \geq E[w(c; \xi, \beta) \mid c \leq p]\} \tag{12}$$ The reserve price $$p_H^*$$ is the type at which the virtual valuation turns negative (identical to Myerson's reserve price). The floor price condition (12) equates the virtual valuation at $$p_L^*$$ to the average virtual valuation conditional on costs being at or below $$p_L^*$$: this reflects the optimal way to offer the same interim allocation to all types in $$[c_L, p_L^*]$$ simultaneously. For comparative statics, the authors rewrite the floor-price condition as the integral test in equation (13), p.1516: $$p_L^* = \max\left\{p\in[c_L,c_H]:\int_{c_L}^{p}w_c(c;\xi,\beta)F(c)dc\geq 0\right\} \tag{13}$$ Equilibrium bidding is sincere: suppliers with cost $$c \in [p_L^*, p_H^*]$$ bid their cost $$c$$; suppliers with $$c < p_L^*$$ bid $$p_L^*$$; suppliers with $$c > p_H^*$$ do not bid. The proof builds on `mechanism-design` duality methods and explicitly solves for the shadow prices of the monotonicity constraints (via Lemma 4 in the online appendix), because standard approaches that sidestep monotonicity constraints do not apply under the lemons problem. **Proposition 4** (FPLoLA equivalence, §IVD, p.1520): The sincere equilibrium of any LoLA can also be implemented by a first-price LoLA (FPLoLA) with the same reserve price and a suitably chosen minimum bid $$b_L \geq p_L$$, where $$b_L$$ is up to 24 percent higher than $$p_L$$ in the Italian calibration. Two extensions clarify the interpretation of the baseline model. When quality $$x$$ is stochastically related to cost, the one-dimensional value function is expected quality conditional on cost (equation (14), §IVA, p.1518): $$v(c,\xi)=\int x\,d\Psi(x\mid c,\xi) \tag{14}$$ In the post-auction performance example, a type-$$t$$ supplier chooses effort to minimize effort cost plus expected fine; the resulting effort, total cost, and fine are equations (15)-(17), §IVB, pp.1518-1519: $$e^*(t)=\arg\min_{e}\{\gamma(e,t)+\phi(e)\}=1-\frac{t}{2} \tag{15}$$ $$\gamma(e^*(t),t)+\phi(e^*(t))=1-\frac{t^2}{4} \tag{16}$$ $$\phi(e^*(t))=\frac{t^2}{4} \tag{17}$$ The asymmetric-bidder extension (§IVE, pp.1521-1522) has no closed-form optimal mechanism: with $$a=0$$ and $$\xi_2\in(2.5,4)$$, the paper's numerical allocation favors supplier 2 when both costs are low and supplier 1 when both are high. With $$\xi_2=4$$ and $$a\in(0,0.5)$$, a higher cost distribution for supplier 1 shifts the allocation toward supplier 1 (Figure 4). ## Empirical specifications The Italian calibration (§V, pp.1523-1528) illustrates the gains from using the LoLA relative to the first-price auction (the format Italian government procurement actually uses). The authors do not estimate new regression specifications: they use the structural cost-density estimate and supplied outcome distributions in Figure 5, then solve the mechanism counterfactual. In Decarolis's structural cost model, auction-specific common cost $$y$$ and privately known idiosyncratic cost $$z_i$$ combine as follows (footnote 22, p.1523): $$c_i=y+z_i$$ The calibration normalizes $$y=0$$ and uses the estimated density $$\hat{f}$$ of $$z_i$$, together with empirical marginal distributions for delays and overruns. Consequently, fixed effects and standard errors do not apply to these counterfactual computations; the reported performance comparisons are Figure 7, not regression coefficients. **Buyer payoff function (equation (18), p.1523):** $$v(c, \xi) = \text{const} - K E[D(c, \xi) + O(c, \xi)] \tag{18}$$ where $$D(c, \xi)$$ is the delivery delay ratio, $$O(c, \xi)$$ is the cost overrun ratio, and both are unobserved random variables that depend on the winning supplier's cost $$c$$ and the quality-concern parameter $$\xi$$. After calibration using the empirical marginal distributions $$g_D$$ and $$g_O$$ (Figure 5, p.1524), the calibrated payoff simplifies to: $$\hat{v}(c, \xi) = \text{const}(\xi) - \xi K[\delta(c) + \omega(c)] \tag{19}$$ where $$\delta(c)=G_D^{-1}([1-\hat{F}(c)]^N)$$ and $$\omega(c)=G_O^{-1}([1-\hat{F}(c)]^N)$$ (equation (19), p.1525). Here $$G_D$$ and $$G_O$$ are the CDFs of the delay and overrun marginals displayed in Figure 5, p.1524. The calibrated virtual valuation used for the counterfactuals (equation (20), p.1525) is: $$\hat{w}(c;\xi,\beta)\equiv\hat{v}(c;\xi)-c-\beta\frac{\hat{F}(c)}{\hat{f}(c)} \tag{20}$$ It is positive for all $$c$$ and $$\beta$$ at the estimated parameters, implying it is optimal to set no reserve price in the LoLA. Optimal floor prices $$p_L^*$$ are then computed from condition (12) for each $$(\xi, \beta)$$ pair (Figure 6, p.1526). For each value of $$\xi \in [0,1]$$, the paper computes expected buyer surplus, supplier profit, and social surplus under the buyer-optimal LoLA and under the first-price auction (Figure 7, p.1527). The virtual valuation satisfies Assumption 1 (quasiconcavity) for all four displayed values of $$\xi \in \{0, 0.33, 0.67, 1\}$$, confirming that LoLA is optimal in the calibrated setting (Figure 6, p.1526). ## Datasets used | Dataset | Role in paper | Wiki page | |---|---|---| | Italian government procurement auctions (Decarolis 2019, "Dati Aste") | Estimated cost density $$\hat{f}$$, delay distribution $$g_D$$, and overrun distribution $$g_O$$ used to calibrate the buyer payoff function | no page yet | Sample: Italian public procurement auctions. Cost units are $$10^5$$ euros. Distributions estimated structurally by Decarolis (2018) and provided to the authors by email (May 10, 2019). Replication data including calibrated distributions are publicly available at [ICPSR E182801V1](https://doi.org/10.3886/E182801V1). ## When to read the full paper Read Lopomo, Persico, and Villa (2023) to: - **Understand the full proof** of Theorem 1, particularly the dual solution approach and the shadow prices of monotonicity constraints (online Appendix A). - **Extend the mechanism** to asymmetric bidders, descending-clock formats, or first-price implementations (Section IV, pp.1517-1522). - **Use the software** applications (available on GitHub, footnote 7, p.1507) that compute buyer-optimal procurement mechanisms given any cost distribution and value function $$v(c, \xi)$$, including non-LoLA cases. - **Calibrate the framework** to other procurement settings using the semiparametric calibration method of Section VB, which constructs $$v(c, \xi)$$ from empirical quality distributions conditional on cost. The calibration results (Figure 7, p.1527) are the entry point for policy analysis; the asymmetric-bidder numerical results (Section IVE, p.1521-1522) are the entry point for applied mechanism designers. ## Attribution and rights This page is an LLM-distilled extract; it is not human-verified and the results have not been reproduced. Cite the original: > Lopomo, Giuseppe, Nicola Persico, and Alessandro T. Villa. 2023. "Optimal Procurement with Quality Concerns." *American Economic Review* 113(6): 1505–1529. https://doi.org/10.1257/aer.20211437 Replication data: Lopomo, Persico, and Villa (2023), [ICPSR E182801V1](https://doi.org/10.3886/E182801V1), American Economic Association / ICPSR. Access: paywalled (AEA subscription). Extract-only; no PDF hosted here.