Optimization

Motivation

Optimal control of partial differential equations under uncertainty arises whenever a physical system is steered by a control while its sources, coefficients or loads are only known in distribution. A typical requirement is a state constraint: the state (a temperature, a concentration, a displacement) must stay below a threshold everywhere in the domain. When the source term is random, demanding this for every realization is often impossible or too conservative, and the natural relaxation is a probabilistic (chance) constraint: the state must stay below the threshold uniformly on the domain with probability at least $p$.

Probabilistic constraints are classical in operations research (Prékopa, Stochastic Programming, 1995), but in a PDE setting the constraint is an infinite family of random inequalities, one per point of the domain, and the probability function is not differentiable in general. C. Geiersbach and R. Henrion (arXiv:2306.03965v2, Math. Oper. Res. 2025) derive optimality conditions for such problems in terms of the Clarke subdifferential, building on the spherical-radial decomposition of Gaussian vectors developed by Hantoute, Henrion and Pérez-Aros (Math. Program. 2019) and van Ackooij and Henrion. This mission formalizes the PDE half of §2 of that paper: explicit formulas for the subdifferential of the probability function and the resulting optimality conditions.

Setting

Let $D\subseteq\mathbb R^d$, $d\in\{1,2,3\}$, be a nonempty, open, bounded set of class S: there are $\gamma\in(0,1)$ and $r_0>0$ with $\operatorname{meas}(B_r(x)\setminus D)\ge\gamma\operatorname{meas}(B_r(x))$ for all $x\in\partial D$ and $0<r<r_0$. For $F\in L^2(D)$ the Dirichlet problem $-\Delta y=F$ in $D$, $y=0$ on $\partial D$ has a unique weak solution in $H^1_0(D)\cap C(\bar D)$; the solution operator $A^{-1}:L^2(D)\to C(\bar D)$ is linear and continuous.

Fix $f_0,\varphi_1,\dots,\varphi_m\in L^2(D)$ and a threshold $\alpha\in\mathbb R$. The control-to-state operator is

$$S(u,z)=A^{-1}\Big(u+f_0+\sum_{i=1}^m z_i\varphi_i\Big)=P(u,z)+y_0,\qquad P(u,z)=A^{-1}u+\sum_i z_iA^{-1}\varphi_i,\ y_0=A^{-1}f_0.$$

The random vector $\xi$ on a probability space $(\Omega,\mathcal F,\mathbb P)$ is Gaussian $\mathcal N(0,\Sigma)$ with $\Sigma$ positive definite, and $y(\cdot,\omega)=S(u,\xi(\omega))$ is the random state. With $g(u,z)=\max_{x\in\bar D}[S(u,z)](x)-\alpha$ and the probability function $\varphi(u)=\mathbb P(\omega\mid g(u,\xi(\omega))\le0)$, the problem is

$$\min_{u\in L^2(D)}F(u)\quad\text{s.t.}\quad \varphi(u)=\mathbb P\big(y(x,\omega)\le\alpha\ \ \forall x\in D\big)\ge p,\tag{24}$$

with $F$ convex and Fréchet differentiable and $p\in(0,1]$.

The analysis uses the spherical-radial decomposition. Let $\mu_\zeta$ be the uniform distribution on the unit sphere $\mathbb S^{m-1}$, $\mu_\eta$ the chi distribution with $m$ degrees of freedom and density $\chi(t)=Kt^{m-1}e^{-t^2/2}$, $\Sigma^{1/2}$ the square root of $\Sigma$. The radial probability function is $e(u,v)=\mu_\eta\{r\ge0\mid g(u,r\Sigma^{1/2}v)\le0\}$ and the radius function $\rho(u,v)\in[0,\infty]$ is the supremum of that set. For $x\in\bar D$, $u_x\in L^2(D)$ is defined by $(u_x,h)_{L^2(D)}=(A^{-1}h)(x)$, and

$$M^*(\bar u,v)=\{x\in\bar D\mid[P(0,\Sigma^{1/2}v)](x)>0,\ \bar y(x)+\rho(\bar u,v)[P(0,\Sigma^{1/2}v)](x)=\alpha\},\qquad \bar y=S(\bar u,0).$$

$\partial^C$ is the Clarke subdifferential, with $L^2(D)^*$ identified with $L^2(D)$, and integrals of set-valued maps are Aumann integrals.

Formalization targets

Goal: Theorem 2.16

Assume a control $\hat u$ with $\varphi(\hat u)>p$ (41), and let $u^*$ be feasible with expected state $\tilde y=S(u^*,0)$ satisfying $\tilde y(x)<\alpha$ on $\bar D$ (42). If $u^*$ solves (24), there is $\lambda\ge0$ with

$$DF(u^*)\in-\lambda\int_{\{v\mid\rho(u^*,v)<\infty\}}\chi(\rho(u^*,v))\cdot\operatorname{clco}\Big\{\frac{u_x}{[P(0,\Sigma^{1/2}v)](x)}\ \Big|\ x\in M^*(u^*,v)\Big\}\,d\mu_\zeta(v),$$ $$\lambda\big(\mathbb P(y^*(x,\omega)\le\alpha\ \forall x\in D)-p\big)=0,$$

and these conditions are sufficient when $M^*(u^*,v)$ is a singleton for $\mu_\zeta$-a.e. $v$ with $\rho(u^*,v)<\infty$, or the set $\{z\mid g(u^*,z)\le0\}$ is bounded. The goal has no hard-coded constants; it asserts the form of the optimality system.

Milestones

In attack order: well-posedness of the state equation with the bound $\|\hat y\|_{C(\bar D)}\le C(\|z\|_2+\|u+f_0\|_{L^2(D)})$ (Lemma 1.2); measurability of the random state (Lemma 1.4); convexity and global Lipschitz continuity of $g$ (Lemma 2.9); moderate growth (Lemma 2.10); the Lipschitz and subdifferential conclusions of the general theory under (26) (Corollary 2.11); the optimality conditions (21), (22) for (24) (Corollary 2.12); $\partial^C_ue(\bar u,v)=\{0\}$ when $\rho(\bar u,v)=\infty$ (Lemma 2.13); the explicit formula $\partial^C_ue(\bar u,v)=-\chi(\rho(\bar u,v))\cdot\operatorname{clco}\{u_x/[P(0,\Sigma^{1/2}v)](x)\mid x\in M^*(\bar u,v)\}$ when $\rho(\bar u,v)<\infty$ (Lemma 2.14); and the upper estimate (40) of $\partial^C\varphi(\bar u)$ with its equality cases (Theorem 2.15).

Significance

The result turns an abstract statement — $DF(u^*)$ lies in a multiple of the Clarke subdifferential of a probability function that has no closed form — into a condition written in the problem data: Green-type functions $u_x$, the expected state, and the radius function along each direction of the sphere. This is what makes the conditions usable for checking candidate controls and for designing algorithms, and Example 2.18 of the paper computes a solution this way. The sufficiency half identifies when the nonconvex chance-constrained problem behaves like a convex one.

The paper's proofs are on paper only; to our knowledge none of these results, nor the spherical-radial decomposition, Clarke calculus in Hilbert spaces, or Aumann integrals in $L^2$, has a machine-checked proof. Formalizing Theorem 2.16 requires connecting elliptic regularity, Gaussian measure theory and nonsmooth analysis, and the milestones are reusable on their own: Lemma 1.2 is a regularity statement for the Poisson equation, and Lemmas 2.13 and 2.14 are max-function subdifferential formulas in an infinite-dimensional space.

Difficulty

The obvious route — differentiate $\varphi$ and use a Lagrange multiplier — fails because $\varphi$ is generally not differentiable even for smooth data (Example 2.17 of the paper), and because $g$ is only Lipschitz, being a maximum over $\bar D$. The general subdifferential estimate (Theorem 2.2, from Hantoute–Henrion–Pérez-Aros) needs a growth condition and only gives an inclusion, since the radial probability function is not Clarke regular; the radius function $\rho(\cdot,v)$ is an infimum over the open set where $P(0,\Sigma^{1/2}v)>0$, which is neither compact nor independent of $u$, so standard max-function formulas do not apply directly. Directions with $\rho=\infty$ require a separate limiting argument through the Mordukhovich subdifferential (Lemma 2.13). The regularity input, continuity of solutions on $\bar D$ for class-S domains (Kinderlehrer–Stampacchia 1980), is not in Mathlib.

Formalization scope

All declarations are in the namespace ChanceStateCtrl.PDE. $\mathbb R^d$ and $\mathbb R^m$ are EuclideanSpace ℝ (Fin d) and EuclideanSpace ℝ (Fin m), indices are Fin m, $L^2(D)$ is Lp ℝ 2 (volume.restrict D). Weak derivatives are the published HunterPDE.Shared.HasWeakDeriv; a weak solution is a function continuous on $\bar D$, zero on $\partial D$, in $L^2(D)$ with weak first derivatives in $L^2(D)$, satisfying the weak form against $C_c^\infty(D)$.

Standing hypotheses on every theorem: $D$ open, bounded and nonempty, $d\in\{1,2,3\}$, class S, and CompactSpace (closure D) (a consequence of boundedness). Nonemptiness of $D$ is the one added hypothesis: Lean's supremum over an empty domain is $0$, not $-\infty$, and Lemma 2.9's identification of events fails for $D=\emptyset$. $A^{-1}$ is a continuous linear map to $C(\bar D)$ together with the hypothesis that it returns the weak solution; it is never an arbitrary operator. $m\ge1$, as the uniform distribution on $\mathbb S^{m-1}$ requires. $\Sigma^{1/2}$ is the positive semidefinite root; $\|\Sigma^{1/2}\|$ is its operator norm. The Clarke directional derivative is $\limsup_{u'\to u,t\downarrow0}(f(u'+th)-f(u'))/t$ (the page's denominator $h$ is a misprint). $\rho$ takes values in $[0,\infty]$. clco is the norm closure of the convex hull. The integral in (40)/(43) is over the restriction of $\mu_\zeta$ to $\{\rho<\infty\}$, not a renormalization. A solution is a global minimizer over $\{u\mid\varphi(u)\ge p\}$; the page's "solution to (9)" refers to this instance of the abstract problem (9).

Trivializing formalizations are ruled out: every Clarke subdifferential is taken at a point where the function is locally Lipschitz (proved in Corollary 2.11 and Theorem 2.15), the state operator is tied to the PDE, the multiplier carries its sign and complementarity, Aumann selections must be scalarly integrable, and optimality is global.

Needed infrastructure: elliptic regularity in $C(\bar D)$ for class-S domains; the spherical-radial decomposition of $\mathcal N(0,\Sigma)$; Clarke calculus (chain rule, max functions) on Hilbert spaces; Clarke's theorem on subdifferentials of integral functionals. Contributions of any of these as stand-alone theorems are welcome.

Selected references

• C. Geiersbach, R. Henrion, Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form, arXiv:2306.03965v2, 2024; Math. Oper. Res., 2025. https://arxiv.org/abs/2306.03965
• A. Hantoute, R. Henrion, P. Pérez-Aros, Subdifferential characterization of probability functions under Gaussian distribution, Math. Program. 174 (2019), 167–194. https://doi.org/10.1007/s10107-018-1237-9
• F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, 1983; SIAM reprint 1990. https://doi.org/10.1137/1.9781611971309
• D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980; SIAM reprint 2000. https://doi.org/10.1137/1.9780898719451
• A. Prékopa, Stochastic Programming, Kluwer, 1995. https://doi.org/10.1007/978-94-017-3087-7

Motivation

The European Union's General Data Protection Regulation (GDPR, 2018) requires a firm to obtain a consumer's explicit consent before it collects personal data. The common expectation, in policy debate and in much of the empirical literature, is that consent requirements cost firms money: they shrink the pool of consumer data that can be monetized through advertising, personalization or resale. Choe, Matsushima and Shekhar (ISER Discussion Paper No. 1246, 2024; published as Management Science, 2025) build a monopoly pricing model in which consent creates a new purchase option. A consumer who would not have bought the service under compulsory data sharing can now buy it without sharing. They then ask how the firm's equilibrium price, profit and consumer surplus change.

This mission formalizes the paper's analytic profit result. If the firm's revenue is mostly usage based, meaning its revenue per unit of collected data is small, then the firm earns strictly more in equilibrium after the GDPR than before. The result is the source of the paper's title. Regulation that is usually read as a cost to firms raises the profit of one well-defined class of firms.

Setting

A monopolist sells an online service at zero marginal cost and sets a price $p\in\mathbb R$. Negative prices are allowed: they pay consumers to use the service. A consumer type is a pair $(v,c)\in[0,1]^2$, where $v$ is the valuation of the service and $c$ the privacy cost of sharing data. Types are independent and uniform on the unit square, so the mass of a set of consumers is its area. A consumer who shares data receives an opt-in benefit $x\in(0,1)$. The utilities of opt-in purchase (buy and share) and opt-out purchase (buy without sharing) are

$$U_S(v,c)=v+(x-c)-p,\qquad U_N(v,c)=v-p .$$

The firm earns data revenue $\alpha\ge0$ per unit of collected data. Its profit is $\Pi=pD+\alpha D_S$, where $D$ is total demand and $D_S$ the mass of consumers who share data.

• Before the GDPR opt-out purchase is unavailable. A consumer buys, and shares, iff $U_S\ge0$. The demand $\widehat D(p)$ is the area of $\{(v,c)\in[0,1]^2: U_S\ge 0\}$, and the profit is $\widehat\Pi(p)=(p+\alpha)\widehat D(p)$.
• After the GDPR the opt-in buyers are $\{(v,c)\in[0,1]^2 : c\le v+x-p,\ c\le x\}$ and the opt-out buyers are $\{(v,c)\in[0,1]^2: v\ge p,\ c>x\}$. Their areas are $\widetilde D_S(p)$ and $\widetilde D_N(p)$, and the profit is $\widetilde\Pi(p)=p(\widetilde D_S(p)+\widetilde D_N(p))+\alpha\widetilde D_S(p)$.

An equilibrium price is a global maximizer of the profit over $p\in\mathbb R$: the firm moves first and consumers respond. The paper's thresholds are

$$\alpha_1(x)=\max\Big\{\frac{1-2x}{2},0\Big\},\qquad \alpha_2(x)=\max\Big\{\frac{2-4x+x^2}{2x},0\Big\}.$$

"Case 1a" (before) and "case 2a" (after) are the equilibria with $x<p^\star<1+x$, in which low-valuation consumers stay out of the market.

Formalization targets

Goal: Proposition A3, first bullet (p. 34)

If $\alpha<\alpha_1(x)$, both profit functions attain their maxima over $\mathbb R$, and

$$\max_{p\in\mathbb R}\widetilde\Pi(p)\;>\;\max_{p\in\mathbb R}\widehat\Pi(p),$$

that is, $\widetilde\Pi^\star_{2a}>\widehat\Pi^\star_{1a}$. The hypothesis forces $x<1/2$, because $\alpha\ge0$.

Milestones

1. Lemma A3, first bullet (p. 33): for $x\le 1/2$ and $\alpha<\alpha_1(x)$, every equilibrium price before the GDPR lies in $(x,1+x)$ (case 1a), and so does every equilibrium price after it (case 2a).
2. Equilibrium profit before the GDPR (p. 31): for $\alpha<\alpha_1(x)$, $\widehat\Pi^\star=2(1+x+\alpha)^3/27$.
3. Equilibrium profit after the GDPR (p. 32): for $\alpha<\alpha_2(x)$, $\widetilde\Pi^\star=\big(4+4x(x+2\alpha+2\alpha x)+x^2(x-2\alpha)^2\big)/16$.
4. Online Appendix §F, first display (OA p. 9): for $\alpha<\alpha_1(x)$, $\widetilde\Pi^\star-\widehat\Pi^\star=J(x,\alpha)/432$, with $J$ the explicit quartic polynomial printed there.
5. Online Appendix §F, conclusion (OA p. 9): $J(x,\alpha)>0$ whenever $0<x<1/2$ and $0\le\alpha<(1-2x)/2$.

Significance

The result. Proposition A3 separates firms by revenue model. A firm whose revenue is mostly usage based gains from the GDPR. The opt-out option brings in consumers with high privacy costs who did not buy before, and this extra demand lets the firm both raise its price and sell to more consumers. Together these outweigh the lost data revenue. The other two bullets of Proposition A3 treat firms with larger $\alpha$, for which the comparison can go either way. Their thresholds $\alpha_5(x)$ and $\alpha_6(x)$ are defined only as "the value of $\alpha$ that solves" an equation, with existence and uniqueness shown in plots, so they are not posed here. The paper's Propositions 1–4 are its main-text results at $x=1/4$. The price comparison (Proposition 1) is the $x=1/4$ instance of the general Proposition A1, the subject of the sibling mission in this series. Propositions 2–4, A2, A3 (bullets 2–3) and A4 rest on numerically located thresholds and are not posed.

Formalizing it. The paper's proof is in print, but no part of it has been machine-checked. The model is fully explicit (a uniform distribution, two areas, two profit functions on $\mathbb R$), so a complete formal proof would certify every link from the primitives to the inequality. That covers the computation of the polygon areas, the global maximization of a piecewise polynomial over all real prices, and the final polynomial inequality. The printed proof of the last step contains a slip: the value it gives for "$J(x,\alpha_1(x))$" is the value of $J/432$. A formalization fixes such details for good.

Difficulty

The closed forms are easy to verify once the equilibrium is known to be interior to case 1a and case 2a. The work lies elsewhere. Computing the demands requires the Lebesgue area of a family of polygons whose shape changes with $p$. Over all of $\mathbb R$ each demand is piecewise polynomial with several pieces, and the profits are piecewise cubic. Identifying the global maximizer means checking every piece, including negative prices before the GDPR and the corner pieces where demand saturates at $0$ or $1$. A first-order condition on one piece gives only a candidate. The comparison of the two maximal values then reduces to a two-variable quartic inequality on a triangle, which holds strictly on the open region but becomes tight at its corner $x\to0$, $\alpha\to1/2$, where $J(0,1/2)=0$.

Formalization scope

• Model. One definitions file, GDPRPrivacy.PriceRise.Model, shared with the sibling mission on the equilibrium price. A type is q : ℝ × ℝ with q.1 = v and q.2 = c. Demands are (volume S).toReal for the explicit sets $S\subseteq[0,1]^2$ above, with the opt-in and opt-out sets of p. 13 copied literally. Ties are assigned to opt-in, which changes only null sets. The file also defines $\alpha_1,\alpha_2,\beta,\Gamma$.
• Standing assumptions. Every theorem assumes $0<x<1$ and $\alpha\ge0$ (§3, pp. 7–8). No other hypothesis is added. Milestone 1 keeps the paper's $x\le 1/2$, and milestone 5 keeps its printed region $\alpha<(1-2x)/2$, $x<1/2$.
• Equilibria. An equilibrium price is any p with IsMaxOn (profitBefore x α) Set.univ p (respectively profitAfter). Every statement asserts that a maximizer exists and then quantifies over all maximizers.
• Ruled-out trivializations. The goal compares the maximal values of the two model profit functions. It does not compare the closed forms $2(1+x+\alpha)^3/27$ and $\widetilde\Pi^\star_{2a}$ (that is milestones 4–5 alone), it does not compare the profits at a common price, and it does not restrict the price range. The existence conjuncts prevent a vacuous "for every maximizer".
• Needed infrastructure. Areas of polygons in $\mathbb R^2$ via MeasureTheory.volume (Fubini or regionBetween), maximization of piecewise polynomials on $\mathbb R$, and polynomial inequalities. Lemmas computing the area of $\{(v,c)\in[0,1]^2: c\le v+t\}$ as a function of $t$ are reusable for other uniform-type screening and pricing models. Contributions of such lemmas, and of the explicit demand formulas (4)–(6) of the paper, are welcome.

Selected references

• C. Choe, N. Matsushima and S. Shekhar, The Bright Side of the GDPR: Welfare-Improving Privacy Management, ISER Discussion Paper No. 1246, Osaka University, June 2024. The artifact formalized here; theorem and page numbers refer to it.
• C. Choe, N. Matsushima and S. Shekhar, The Bright Side of the GDPR: Welfare-Improving Privacy Management, Management Science, 2025. https://doi.org/10.1287/mnsc.2024.06653
• S. P. Anderson and J. S. Gans, Platform Siphoning: Ad-Avoidance and Media Content, American Economic Journal: Microeconomics 3(4), 2011. https://doi.org/10.1257/mic.3.4.1
• Regulation (EU) 2016/679 (General Data Protection Regulation), Official Journal of the European Union L 119, 2016. https://eur-lex.europa.eu/eli/reg/2016/679/oj

Motivation

A retailer often knows more about her own demand than the supplier who must produce for her. In a decentralized supply chain the supplier sets a production quantity before demand is realized, and an under-informed supplier produces too little or too much. The retailer can shape what the supplier believes by committing in advance to a disclosure policy, a rule that maps what she knows to messages she sends. This is the Bayesian persuasion or information design framework of Kamenica and Gentzkow (2011), applied by Candogan and Gurkan (2024) to a newsvendor supplier.

The natural expectation is that the retailer either tells the supplier everything or nothing. The paper shows the answer depends on the supplier's critical ratio, and that it can be non-monotone. For nested demand supports (the setting of the first missions in this series) partial disclosure appears for low critical ratios. This mission formalizes the paper's §4 extension to supports that overlap without being nested. There, under an explicit condition on the supports, the optimal policy moves from partial disclosure to no disclosure to full disclosure as the critical ratio rises.

Setting

The state is $\omega\in\{0,1\}$ with prior $p_0=\mathbb P(\omega=0)\in[0,1]$. In state $\omega$ the retailer's demand $D_\omega$ is uniform on $[a_\omega,b_\omega]$. The supports intersect:

$$b_0>b_1>a_0>a_1\ge 0 .$$

Write $\delta_u=b_0-b_1$, $\delta_d=a_0-a_1$, $\delta_{\mathsf{IS}}=b_1-a_0$; all three are positive.

The supplier holds a belief $p=\mathbb P(\omega=0)$ and chooses a quantity $Q\ge 0$ that minimizes the newsvendor cost (2),

$$p\,\mathbb E\big[\xi(D_0-Q)^+ + (1-\xi)(Q-D_0)^+\big]+(1-p)\,\mathbb E\big[\xi(D_1-Q)^+ + (1-\xi)(Q-D_1)^+\big],$$

where $\xi=C_s/(C_s+C_e)\in(0,1)$ is the critical ratio of the shortage cost $C_s$ and the excess cost $C_e$. A best-response profile $Q^*$ assigns an optimal quantity $Q^*_p$ to every belief $p\in[0,1]$.

A policy $\Gamma=(\mathcal M,\pi)$ consists of a finite message space $\mathcal M$ and probabilities $\pi(m\mid\omega)$. After message $m$ the supplier's belief is the Bayes posterior (1),

$$p_m=\frac{\pi(m\mid 0)\,p_0}{\pi(m\mid 0)\,p_0+\pi(m\mid 1)(1-p_0)} .$$

The retailer orders the realized demand and bears only the shortage. Her design problem (3) minimizes, over policies $\Gamma$ and best-response profiles $Q^*$, the expected shortage

$$\mathbb E\Big[p(D_0-Q^*_p)^+ + (1-p)(D_1-Q^*_p)^+\ \Big|\ \Gamma\Big],$$

where $p$ is the induced posterior. Full disclosure reveals $\omega$. No disclosure sends one uninformative message. The function

$$K_{\mathsf{IS}}(p)=p\,\mathbb E[(D_0-Q^*_p)^+]+(1-p)\,\mathbb E[(D_1-Q^*_p)^+]$$

of (B.1) is the retailer's cost when the posterior is $p$, and $\kappa_{\mathsf{IS}}$ denotes its lower convex envelope on $[0,1]$.

Formalization targets

Goal: Proposition 3 (p. 15)

Assume $\delta_d>2\delta_u$ and $\delta_{\mathsf{IS}}>\max\big(\delta_d-2\delta_u,\ \delta_d\delta_u/(\delta_d-2\delta_u)\big)$. Let $\breve\xi=(\delta_d-2\delta_u)/(\delta_d-\delta_u)$ and $\underline\tau_{\mathsf{IS}}(p)=(1-p)(a_0-a_1)/(b_1-a_1)$. Then:

$$\begin{aligned} \xi\in[\breve\xi,1) &\implies \text{full disclosure is optimal},\\ \xi\in[\underline\tau_{\mathsf{IS}}(p_0),\breve\xi) &\implies \text{no disclosure is optimal},\\ \xi\in(0,\underline\tau_{\mathsf{IS}}(p_0)) &\implies \Gamma^*_{\mathsf{IS}} \text{ of (7) is optimal}, \end{aligned}$$

where $\Gamma^*_{\mathsf{IS}}$ has two messages, sends $m_0$ in state $0$ with probability 1, and sends $m_0$ in state $1$ with probability $\frac{\xi(b_1-a_1)}{(a_0-a_1)-\xi(b_1-a_1)}\cdot\frac{p_0}{1-p_0}$.

Milestones (the paper's proof, App. C.1)

1. Proposition 2 (p. 15): the supplier's optimal quantity $Q^*_p$ is unique and given by the three-branch formula (6), with thresholds $\underline\tau_{\mathsf{IS}}(p)\le\bar\tau_{\mathsf{IS}}(p)=1-p(b_0-b_1)/(b_0-a_0)$.
2. (B.2) (p. 25): the optimal value of (3) is $\kappa_{\mathsf{IS}}(p_0)$, the Kamenica–Gentzkow reduction.
3. Equivalence of hypotheses (p. 32): the hypotheses of Proposition 3 hold if and only if $\bar\tau_{\mathsf{IS}}(1)>\breve\xi>\underline\tau_{\mathsf{IS}}(0)$.
4. $K_{\mathsf{IS}}$ in closed form (pp. 32–33), in the three regimes of (6).
5. Lemma 2 (p. 31): $\kappa_{\mathsf{IS}}$ is the chord $(1-p)K_{\mathsf{IS}}(0)+pK_{\mathsf{IS}}(1)$ for $\xi\ge\breve\xi$; it is $K_{\mathsf{IS}}$ itself for $\xi\in[\underline\tau_{\mathsf{IS}}(0),\breve\xi)$; and for smaller $\xi$ it is $K_{\mathsf{IS}}$ above $\breve p=1-\xi(b_1-a_1)/(a_0-a_1)$ and the chord from $0$ to $\breve p$ below.
6. The posteriors of $\Gamma^*_{\mathsf{IS}}$ (p. 32): $m_0$ has probability $p_0/\breve p$ and induces posterior $\breve p$, while $m_1$ induces posterior $0$.

Significance

The result. With nested supports, more information can only hurt the retailer at low critical ratios. With overlapping supports, Proposition 3 shows the regimes of full, no and partial disclosure can occur in a different order. It also gives a sufficient condition on the shape of the supports for that pattern. Together with the paper's Theorem 1 (nested) and Theorem 3 (disjoint), it gives the optimal policy for the main configurations of two uniform supports. The explicit thresholds tell a practitioner which disclosure regime applies from four interval endpoints and a cost ratio.

Formalizing it. The paper's argument is in its Online Appendix (C.1, pp. 30–35). It cites Kamenica–Gentzkow for the value of the design problem and proves the envelope lemma through second-derivative sign computations. None of this is machine-checked. A complete development yields a formal concavification theorem for a finite-state persuasion problem over all finite message spaces, which is reusable well beyond this paper. It also yields a checked closed form for the newsvendor quantile under a two-uniform mixture. The goal and every milestone are open on the platform; only the published convex-envelope definition is reused.

Difficulty

The obvious argument compares full disclosure, no disclosure and one candidate partial policy, and picks the best of the three. That proves nothing, because optimality is claimed against every policy on every finite message space. The reduction to the convex envelope (milestone 2) is what makes the comparison finite, and it is itself a theorem about splitting a prior into posteriors.

The envelope of $K_{\mathsf{IS}}$ is the second obstacle. $K_{\mathsf{IS}}$ is piecewise rational in $p$, with two regime boundaries that move with $p$. Its curvature changes sign across the boundaries, and the sign of its second derivative depends on $\xi$ relative to $\breve\xi$. Verifying that the claimed $\kappa_{\mathsf{IS}}$ is convex at the kink $\breve p$ requires a one-sided slope inequality whose sign rests on the hypotheses on $\delta_d,\delta_u,\delta_{\mathsf{IS}}$. The equivalence of milestone 3 is where those hypotheses enter.

Formalization scope

All objects live in the namespace SCInfoDesign.Intersecting. States are Fin 2; $\omega=0$ is the state with demand on $[a_0,b_0]$, and beliefs are always $\mathbb P(\omega=0)$. The uniform expectation is $\frac{1}{b-a}\int_a^b$. Policies are stochastic maps from Fin 2 to any finite type; the paper does not bound the message space, and finite spaces are the standard Kamenica–Gentzkow setting. The supplier's cost is normalized by $C_s+C_e$, which leaves its minimizers unchanged, so $\xi\in(0,1)$ is the only cost parameter. A message of probability zero carries the default posterior $0$ and weight $0$. No disclosure is the one-message policy. $K_{\mathsf{IS}}$ uses the largest optimal quantity, which is the unique one here. The envelope is the published BiconvexProg.BranchBound.convexEnvelope on $[0,1]$.

Standing assumptions, added and disclosed because the paper uses them implicitly: $0<\xi<1$; $a_1\ge 0$ (nonnegative demand, so the constraint $Q\ge 0$ never binds); $p_0\in[0,1]$. Nondegeneracy $a_\omega<b_\omega$ follows from the support ordering. Two printed slips are corrected in the statements and quoted as printed in the milestones: $b_h$ for $b_0$ in the $K_{\mathsf{IS}}$ display, and the label $m_1$ for $m_0$ on p. 32. The overlapping indicators of (C.2) at $p=\breve p$ are read piecewise.

Trivializing encodings are excluded. Optimality is over all finite policies and all best-response profiles, not over the three named policies. The supplier's response is an argmin of (2), never the closed form (6), which is a milestone. Each named policy is also asserted to be a policy.

Contributions are welcome on any milestone. The concavification theorem (milestone 2) and the uniform-mixture newsvendor quantile (milestone 1) are the reusable pieces. Proposition 5 (shift case) and the second-derivative display (C.3) are not posed here.

Selected references

• O. Candogan, H. Gurkan, The Value of Information Design in Supply Chain Management, Management Science, 2024. https://doi.org/10.1287/mnsc.2023.03004 (statements cited from the SSRN preprint, abstract 4378997, https://ssrn.com/abstract=4378997)
• E. Kamenica, M. Gentzkow, Bayesian Persuasion, American Economic Review 101(6), 2590–2615, 2011. https://doi.org/10.1257/aer.101.6.2590
• F. A. Al-Khayyal, J. E. Falk, Jointly Constrained Biconvex Programming, Mathematics of Operations Research 8(2), 273–286, 1983 (source of the published convex-envelope definition). https://doi.org/10.1287/moor.8.2.273

Motivation

A retailer usually knows more about her future demand than the supplier who has to produce for her. If the supplier must commit to a production quantity before the order arrives, the retailer bears the cost of every unit he fails to produce. She cannot pay him or contract on quantities, but she can decide how much of her demand information to share, and she can commit to that disclosure rule in advance. Which information should she share?

Candogan and Gurkan (Management Science, 2024; preprint SSRN 4378997) answer this question in a two-state model with uniform demand, using the Bayesian persuasion framework of Kamenica and Gentzkow (2011). Their main text treats nested demand supports. This mission formalizes their extension to disjoint supports, where the high-demand state's support lies entirely above the low-demand state's (Theorem 3). It is the fourth of four missions on the paper; the others cover nested supports (Theorem 1), the value of information (Theorem 2) and intersecting supports (Proposition 3).

Setting

There are two states $\omega\in\{0,1\}$ with prior $p_0=\mathbb P(\omega=0)\in[0,1]$. In state $i$ the retailer's demand is $D_i\sim U[a_i,b_i]$ with $a_i<b_i$ and $a_i\ge0$. Here the supports are disjoint, $b_1<a_0$, so state $0$ is the high-demand state.

A disclosure policy $\Gamma=(\pi,\mathcal M)$ is a finite message space $\mathcal M$ and, for each state $\omega$, a probability distribution $\pi(\cdot\mid\omega)$ on $\mathcal M$. The retailer commits to $\Gamma$, observes $\omega$, and sends a message $m\sim\pi(\cdot\mid\omega)$. Message $m$ has probability $\mathbb P(m)=p_0\pi(m\mid0)+(1-p_0)\pi(m\mid1)$, and the supplier's posterior is given by Bayes' rule (1):

$$p_m=\frac{p_0\,\pi(m\mid0)}{\mathbb P(m)}.$$

The supplier has unit shortage cost $C_s>0$ and unit excess cost $C_e>0$, with critical ratio $\xi=C_s/(C_s+C_e)\in(0,1)$. At belief $p$ he solves the newsvendor problem (2), here normalized by $C_s+C_e$:

$$\min_{Q\ge0}\ p\,\mathbb E\big[\xi(D_0-Q)^++(1-\xi)(Q-D_0)^+\big]+(1-p)\,\mathbb E\big[\xi(D_1-Q)^++(1-\xi)(Q-D_1)^+\big].$$

A best-response profile $Q^*$ assigns to every belief $p\in[0,1]$ an optimal quantity $Q^*_p$; the set of them is $\mathcal B(\Gamma)$.

The retailer's cost at belief $p$ and quantity $Q$ is her expected shortage $R(p,Q)=p\,\mathbb E[(D_0-Q)^+]+(1-p)\,\mathbb E[(D_1-Q)^+]$. Her design problem (3) is

$$\min_{\Gamma,\ Q^*\in\mathcal B(\Gamma)}\ \sum_{m\in\mathcal M}\mathbb P(m)\,R\big(p_m,Q^*_{p_m}\big).$$

A policy is optimal if, paired with some best-response profile, it attains this minimum against every finite-message policy and every best-response profile. Full disclosure sends $m_i$ exactly in state $i$; no disclosure sends one message. The retailer's cost function (B.1) is $K_{DS}(p)=R(p,\bar Q_p)$, where $\bar Q_p$ is the largest optimal quantity at $p$, and $\kappa_{DS}$ is its lower convex envelope on $[0,1]$, the greatest convex function on $[0,1]$ below $K_{DS}$.

Formalization targets

Goal: Theorem 3 (p. 16)

Let $\tilde p=\max\big(1-\xi,\ 2(b_0-a_0)/(b_1-a_1)\big)$. For every $p_0\in[0,1]$:

$$2(b_0-a_0)\ge b_1-a_1\ \Longrightarrow\ \text{full disclosure is optimal;}$$ $$b_1-a_1>2(b_0-a_0)\ \Longrightarrow\ \begin{cases}\text{no disclosure is optimal}, & p_0\ge\tilde p,\\ \Gamma^*_{DS}\text{ is optimal}, & p_0<\tilde p,\end{cases}$$

where $\Gamma^*_{DS}$ has messages $m_0,m_1$ with $\pi^*_{DS}(m_0\mid\omega=0)=1$ and $\pi^*_{DS}(m_0\mid\omega=1)=\frac{1-\tilde p}{\tilde p}\frac{p_0}{1-p_0}$ (8). The paper writes "there exists $\tilde p$" and then gives its value; the goal states the value.

Milestones (App. C.2 and App. B)

1. Proposition 6 (p. 37): the supplier's optimal quantity, $Q^*_p=\xi(b_1-a_1)/(1-p)+a_1$ for $p<1-\xi$ and $(\xi-1+p)(b_0-a_0)/p+a_0$ for $p>1-\xi$; at $p=1-\xi$ every $Q\in[b_1,a_0]$ is optimal.
2. The value claim after (B.2) (p. 25): the optimal value of (3) is $\kappa_{DS}(p_0)$.
3. (C.9) (p. 38): $K_{DS}$ in closed form.
4. (C.13) (p. 39): $\lim_{x\to(1-\xi)^-}K_{DS}(x)-K_{DS}(1-\xi)=(1-\xi)(a_0-b_1)>0$.
5. Lemma 3 (p. 38): $\kappa_{DS}$ is the chord $pK_{DS}(1)+(1-p)K_{DS}(0)$ when $2(b_0-a_0)/(b_1-a_1)\ge1$, and otherwise the chord from $(0,K_{DS}(0))$ to $(\tilde p,K_{DS}(\tilde p))$ followed by $K_{DS}$ itself on $[\tilde p,1]$.

Significance

The result shows that the shape of the optimal disclosure policy is decided by the relative width of the two supports and, below a threshold, by the prior alone: when the high-demand support is at least half as wide as the low-demand one, the retailer reveals everything, whatever the supplier's cost structure. Otherwise the critical ratio enters only through $\tilde p$, and the optimal partial policy pools the high state with part of the low state so that the supplier's posterior lands exactly at $\tilde p$. Together with the nested and intersecting cases, it shows that the non-monotone dependence of informativeness on $\xi$ found under nested supports is not universal.

The paper proves the result in its Online Appendix (App. C.2, pp. 37–40), by citation of Kamenica–Gentzkow for the reduction to a convex envelope and by a case analysis for the envelope. To our knowledge none of it is machine-checked. This mission produces a formal model of persuasion with a newsvendor receiver, a formal proof of the concavification step for finite message spaces, and the envelope computation. Formalization also pins down a boundary case the printed proof leaves implicit (see Formalization scope).

Difficulty

The reduction to the envelope is cited from Kamenica and Gentzkow in the paper, so it has to be established from scratch for this model, with its joint choice of policy and best response and its finite message spaces. The envelope itself requires handling a function that is concave on $[0,1-\xi)$, convex on $(1-\xi,1]$ and discontinuous at $1-\xi$, where it jumps down. The obvious argument — treat $K_{DS}$ as continuous and compare one-sided derivatives — fails at that point, and the jump is what makes the chord to $\tilde p=1-\xi$ admissible.

The newsvendor step is not just the critical-fractile formula: at $p=1-\xi$ the mixture c.d.f. is flat at level $\xi$ on $[b_1,a_0]$, so the supplier's best response is not unique there.

Formalization scope

The Lean development lives in the namespace SCInfoDesign.Disjoint. Conventions:

• States are Fin 2, beliefs are $\mathbb P(\omega=0)$, and uniform expectations are interval integrals divided by the interval length.
• The supplier's cost (2) is divided by $C_s+C_e$, so $\xi\in(0,1)$ is the parameter; this does not change his optimal quantities.
• Message spaces are finite types; optimality is over all finite policies and all best-response profiles. The paper does not bound $\mathcal M$; binary messages already attain the optimum.
• No disclosure is the one-message policy; every uninformative policy has the same cost.
• A message of probability $0$ gets posterior $0$ and weight $0$ in (3).
• Standing hypotheses, implicit in the paper and stated in every item: $a_0<b_0$, $a_1<b_1$, $0\le a_0$, $0\le a_1$, $0<\xi<1$, $b_1<a_0$, $p_0\in[0,1]$.
• The boundary $p=1-\xi$. Proposition 6 and (C.9) print the quantity $b_1$ there. The paper's own design problem (3) and its footnote 9 let the retailer select the supplier's best response, and she prefers the largest one, $a_0$. $K_{DS}$ is defined with the largest optimal quantity; with $b_1$, (C.13) and Lemma 3 are false whenever $\tilde p=1-\xi$. Proposition 6 is therefore stated with the optimal set $[b_1,a_0]$ at the boundary.

The formalization is not trivial by construction: the supplier's response is an argmin of (2), not the closed form of Proposition 6, and optimality quantifies over every finite policy with every best response, not over the three named policies.

Infrastructure needed: interval integrals of piecewise-linear functions, the convex envelope (the published BiconvexProg.BranchBound.convexEnvelope), and a Jensen/splitting argument for Bayes-plausible posteriors. The concavification step and the newsvendor argmin for uniform mixtures are reusable beyond this mission, by the three sibling missions in particular. Proofs of any milestone are welcome.

Selected references

• O. Candogan, H. Gurkan, The Value of Information Design in Supply Chain Management, Management Science, 2024. https://doi.org/10.1287/mnsc.2023.03004 (preprint: https://ssrn.com/abstract=4378997)
• E. Kamenica, M. Gentzkow, Bayesian Persuasion, American Economic Review 101(6), 2011. https://doi.org/10.1257/aer.101.6.2590
• R. Levi, G. Perakis, J. Uichanco, The data-driven newsvendor problem: new bounds and insights, Operations Research 63(6), 2015. https://doi.org/10.1287/opre.2015.1422
• S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511804441

Motivation

A retailer often knows more about her own demand than the supplier who produces for her. When the supplier must commit to a production quantity before demand is realized, what the retailer tells him shapes how much he produces, and therefore how much demand she can serve. Information design (Bayesian persuasion, Kamenica and Gentzkow, 2011) asks how a sender should commit in advance to a randomized disclosure rule to steer a receiver's action.

O. Candogan and H. Gurkan, The Value of Information Design in Supply Chain Management (SSRN 4378997; Management Science, 2024, DOI 10.1287/mnsc.2023.03004) solve this problem for a supplier facing a newsvendor problem with uniform demand. Their Theorem 1 identifies the retailer's optimal policy: full disclosure, no disclosure, or an explicit two-message partial disclosure, according to thresholds on the critical ratio. Their Theorem 2, the goal of this mission, quantifies how much the optimal policy is worth compared with the two naive policies a practitioner would use by default, and how this gain moves with the retailer's prior and the supplier's cost structure.

Setting

The state is $\omega\in\{0,1\}$ with prior $p_0=\mathbb P(\omega=0)$. Demand is $D_0\sim U[a_0,b_0]$ in state $0$ and $D_1\sim U[a_1,b_1]$ in state $1$, with nested supports $a_0\le a_1<b_1\le b_0$.

Supplier. At belief $p=\mathbb P(\omega=0)$ the supplier chooses $Q\ge0$ to minimize the expected newsvendor cost

$$S_\xi(p,Q)=p\,\mathbb E\big[\xi(D_0-Q)^++(1-\xi)(Q-D_0)^+\big]+(1-p)\,\mathbb E\big[\xi(D_1-Q)^++(1-\xi)(Q-D_1)^+\big],$$

where $\xi=C_s/(C_s+C_e)\in(0,1)$ is the critical ratio of the unit shortage cost $C_s$ to the total unit cost $C_s+C_e$.

Retailer. The retailer learns $\omega$ and sends a message $m$ from a finite set $\mathcal M$ according to a committed policy $\pi(m\mid\omega)$. Message $m$ has probability $\mathbb P(m)=p_0\pi(m\mid0)+(1-p_0)\pi(m\mid1)$ and induces the posterior $p_m=p_0\pi(m\mid0)/\mathbb P(m)$. The supplier best-responds with an optimal quantity $Q^*(p_m)$, and the retailer pays her expected shortage

$$R(p,Q)=p\,\mathbb E[(D_0-Q)^+]+(1-p)\,\mathbb E[(D_1-Q)^+].$$

Her optimal expected cost $V_\xi(p_0)$ is the infimum of $\sum_m\mathbb P(m)\,R(p_m,Q^*(p_m))$ over all finite message sets, all policies and all best responses of the supplier.

Naive policies. Under no disclosure the supplier keeps the prior, and the retailer pays $K(p_0)=R(p_0,Q^*_{p_0})$. Under full disclosure he learns $\omega$, and she pays $K_F(p_0)=(1-p_0)K(0)+p_0K(1)$. The values of information are

$$VI_{ND}(\xi,p_0)=K(p_0)-V_\xi(p_0),\qquad VI_{FD}(\xi,p_0)=K_F(p_0)-V_\xi(p_0).$$

Thresholds. $\underline\tau(p)=p(a_1-a_0)/(b_0-a_0)$, $\hat p=\xi(b_0-a_0)/(a_1-a_0)$, $p_0^*=\sqrt{\hat p}$ and $\xi^*=1-\tfrac12(a_1-a_0)/(b_0-b_1+a_1-a_0)$.

Formalization targets

Goal: Theorem 2 (p. 13)

In the partial-disclosure regime $\xi\in(0,\underline\tau(p_0)]$, with $a_0<a_1$:

$$VI_{ND}\ \text{is unimodal in } p_0 \text{ with peak } p_0^*=\sqrt{\xi(b_0-a_0)/(a_1-a_0)}\ \text{and decreasing in } \xi;$$ $$VI_{FD}\ \text{is decreasing in } p_0 \text{ and unimodal in } \xi \text{ with peak } \xi^*.$$

For fixed $\xi$ the regime is $p_0\in[\hat p,1]$; for fixed $p_0$ it is $\xi\in(0,\underline\tau(p_0)]$.

Milestones

1. Proposition 1 (p. 8): the supplier's optimal quantity is unique and given in closed form.
2. (B.2) (p. 25): $V_\xi(p_0)=\kappa(p_0)$, the lower convex envelope of $K$ on $[0,1]$.
3. (B.5) (p. 28): $K$ in closed form for $\xi\le(a_1-a_0)/(b_0-a_0)$.
4. Lemma 1 (p. 25): $\kappa$ in closed form in the three regimes of $\xi$.
5. The four derivative displays (p. 30): $\partial VI_{ND}/\partial\xi$, $\partial VI_{ND}/\partial p_0$, $\partial VI_{FD}/\partial\xi$, $\partial VI_{FD}/\partial p_0$ in the regime.

Significance

The result. Theorem 2 tells a retailer when designing information is worth the effort. Against no disclosure, the gain is largest at the intermediate prior $p_0^*$ and shrinks as shortages become relatively more expensive for the supplier. Against full disclosure, the gain shrinks as the prior probability of the wide-support state $\omega=0$ grows and peaks at the intermediate critical ratio $\xi^*$. These comparative statics are what the paper's managerial discussion (pp. 13–14, Figures 4–5) rests on.

Formalizing it. The result is proved in the paper (Online Appendix B, pp. 24–30), by citing Kamenica–Gentzkow for the reduction to the convex envelope and by explicit calculus. No part of it is machine-checked. A complete formalization would contain a self-contained proof of the concavification reduction for a two-state sender–receiver game with finite message sets, the newsvendor quantile of a uniform mixture, and an exact convex-envelope computation, each reusable beyond this paper.

Difficulty

The derivative formulas are routine once $K$ and $\kappa$ are known in closed form. The work lies upstream. The optimal cost is an infimum over all finite policies and all best responses, and turning it into $\kappa(p_0)$ requires the Kamenica–Gentzkow argument: Jensen's inequality over Bayes-plausible posterior distributions for the lower bound, and a two-point splitting for attainment. $\kappa$ itself is a supremum over all convex minorants, and identifying it requires showing that $K$ is convex below $\hat p$ and concave above, and that the chord from $(\hat p,K(\hat p))$ to $(1,K(1))$ is supporting. Finally, $K$ is defined through an argmin, so its closed form needs the uniqueness of the newsvendor quantile. A solver who differentiates the closed forms of p. 28 directly proves only the last step.

Formalization scope

All objects live in the namespace SCInfoDesign.ValueInfo. The supports are bundled in a structure Params. The critical ratio $\xi$ is an explicit argument of every object, because Theorem 2 varies it.

Conventions.

• Uniform expectations are interval integrals divided by the interval length.
• The supplier's cost is the paper's cost (2) divided by $C_s+C_e$, which has the same minimizers.
• Message sets are finite (Fin n).
• A zero-probability message gets the junk posterior $0$ and weight $0$.
• $K$ uses the largest optimal quantity, which is the unique one here.
• $\kappa$ is the published BiconvexProg.BranchBound.convexEnvelope on $[0,1]$.
• The value of information is the difference, not its absolute value; the difference is nonnegative.
• "Unimodal with peak $x^*$" means strictly increasing up to $x^*$ and strictly decreasing from $x^*$, on the regime.
• The $\xi$-claims are stated for $p_0<1$: at $p_0=1$ both values are identically $0$.

Added hypotheses. These are the paper's implicit standing assumptions: $a_0<b_0$ and $a_1<b_1$, nonnegative demand ($0\le a_0$, $0\le a_1$), and $0<\xi<1$. The goal and the derivative displays also assume $a_0<a_1$: otherwise the regime is empty and $\hat p$ is $0/0$.

Ruled out. The optimal cost is not the cost of the paper's policy $\Gamma^*$, and the supplier's response is an argmin of (2), not the closed form of Proposition 1. Theorem 2 is therefore a statement about the game, not about a formula.

Infrastructure. A complete development needs the concavification theorem for finite policies (milestone 2, reusable for every two-state persuasion model), the uniform-mixture newsvendor quantile (milestone 1), the exact computation of a convex envelope (milestone 4) and one-variable calculus. Contributions of any milestone, of general lemmas on convex envelopes on an interval, or of the Kamenica–Gentzkow reduction in general form are welcome. The proofs of the paper are in its Online Appendix.

Selected references

• O. Candogan, H. Gurkan, The Value of Information Design in Supply Chain Management, Management Science, 2024 (read as SSRN 4378997). https://ssrn.com/abstract=4378997 , https://doi.org/10.1287/mnsc.2023.03004
• E. Kamenica, M. Gentzkow, Bayesian Persuasion, American Economic Review 101(6), 2011. https://doi.org/10.1257/aer.101.6.2590
• R. Levi, G. Perakis, J. Uichanco, The Data-Driven Newsvendor Problem: New Bounds and Insights, Operations Research 63(6), 2015. https://doi.org/10.1287/opre.2015.1422

Motivation

A retailer who sees her own demand before her supplier does can decide how much of that knowledge to pass upstream. Sharing everything is not obviously best: the supplier uses the information to set his production quantity for his own benefit, and the retailer bears a cost when he under-produces. Candogan and Gurkan (SSRN 4378997; published in Management Science, 2024, doi:10.1287/mnsc.2023.03004) treat this as a Bayesian persuasion (information design) problem in the sense of Kamenica and Gentzkow (AER 2011): the retailer commits in advance to a signalling policy, and the supplier best-responds to the posterior belief that each signal induces. The paper characterizes the retailer's optimal policy in closed form for uniform demand. This mission covers its first main result, the case where one demand support is nested inside the other.

The question links the supply-chain literature on information sharing between firms, where sharing is usually all-or-nothing, with the persuasion literature, where optimal signals are typically partial.

Setting

The state is $\omega\in\{0,1\}$, with prior $p_0=\mathbb P(\omega=0)\in[0,1]$. Demand is $D_\omega$, where $D_0$ is uniform on $[a_0,b_0]$ and $D_1$ is uniform on $[a_1,b_1]$. The supports are nested: $0\le a_0\le a_1<b_1\le b_0$ and $a_0<b_0$.

A disclosure policy $\Gamma=(\pi,\mathcal M)$ is a finite message space $\mathcal M$ and conditional probabilities $\pi(m\mid\omega)$. After message $m$ the supplier holds the posterior

$$p=\frac{\pi(m\mid\omega=0)\,p_0}{\pi(m\mid\omega=0)\,p_0+\pi(m\mid\omega=1)(1-p_0)}.$$

Full disclosure sends message $m_i$ exactly in state $i$; no disclosure sends the same message in both states.

At belief $p$ the supplier solves the newsvendor problem

$$\min_{Q\ge 0}\; p\,\mathbb E\big[\xi(D_0-Q)^+ + (1-\xi)(Q-D_0)^+\big] + (1-p)\,\mathbb E\big[\xi(D_1-Q)^+ + (1-\xi)(Q-D_1)^+\big],$$

where $\xi=C_s/(C_s+C_e)\in(0,1)$ is his critical ratio (shortage cost $C_s$, excess cost $C_e$). A best-response profile $Q^*$ picks an optimal quantity $Q^*_p$ for every belief $p\in[0,1]$. The retailer pays the expected shortage $p\,\mathbb E(D_0-Q^*_p)^+ + (1-p)\,\mathbb E(D_1-Q^*_p)^+$ at the induced posterior. Her design problem (3) minimizes the expectation of this cost over the random posterior, jointly over policies $\Gamma$ and best-response profiles $Q^*$. The function $K(p)$ of (B.1) is her cost when both parties hold belief $p$.

The thresholds are

$$\xi_h=\frac{a_1-a_0}{b_0-a_0-b_1+a_1},\qquad \underline\tau(p)=\frac{p(a_1-a_0)}{b_0-a_0},\qquad \bar\tau(p)=1-\frac{p(b_0-b_1)}{b_0-a_0}.$$

Formalization targets

Goal: Theorem 1 (p. 9)

For every prior $p_0\in[0,1]$:

$$\xi\in[\xi_h,1)\Rightarrow\text{full disclosure is optimal};\qquad \xi\in[\underline\tau(p_0),\xi_h]\Rightarrow\text{no disclosure is optimal};$$ $$\xi\in(0,\underline\tau(p_0)]\Rightarrow\Gamma^*\text{ is optimal},\quad \pi^*(m_1\mid\omega=0)=\frac{1-p_0}{p_0}\,\frac{\xi(b_0-a_0)}{(a_1-a_0)-\xi(b_0-a_0)},\ \ \pi^*(m_1\mid\omega=1)=1.$$

Milestones

1. Proposition 1 (p. 8): the supplier's optimal quantity $Q^*_p$ is unique and has an explicit three-piece form in $(p,\xi)$.
2. (B.2) (p. 25): the optimal value of (3) is $\kappa(p_0)$, where $\kappa$ is the lower convex envelope of $K$ on $[0,1]$.
3. (B.5) (p. 28): $K$ in closed form when $\xi\le(a_1-a_0)/(b_0-a_0)$.
4. Lemma 1 (p. 25): $\kappa$ in the three regimes of $\xi$.
5. The $\Gamma^*$ display (p. 26): under $\Gamma^*$, message $m_0$ has probability $(p_0-\hat p)/(1-\hat p)$ and induces posterior $1$, and, when $p_0<1$ (so that $m_1$ has positive probability), $m_1$ induces $\hat p=\xi(b_0-a_0)/(a_1-a_0)$.

Significance

The result shows that the optimal amount of disclosure depends on the critical ratio. When $\xi$ is high the retailer reveals everything; in an intermediate band she reveals nothing; when $\xi$ is low she uses a two-message policy that always reveals the narrow-support state ($\omega=1$ always sends $m_1$) but pools part of the wide-support state with it. Corollaries 1–4 of the paper read the thresholds off this theorem (independence of the prior for large $\xi$, full disclosure when $a_0=a_1$, never full disclosure when $b_0=b_1$), and Theorem 2 of the paper, on the value of information design relative to full and no disclosure, is built on Lemma 1 and (B.5).

The paper proves the result in its Online Appendix, citing Kamenica and Gentzkow for the reduction to a convex envelope. None of it has been machine-checked. A formalization would make the persuasion reduction explicit for finite message spaces and best-response profiles, and would check the case analysis of the envelope, where the paper prints one overlapping-indicator slip (Lemma 1, case 3).

Difficulty

The obvious argument is to compare the three named policies and pick the cheapest. That is not a proof: optimality is over every policy on every finite message space, and it is only the convex-envelope reduction that cuts the comparison down to splittings of the prior into posteriors. The reduction itself requires showing that the attainable expected costs are exactly the convex combinations of values of $K$ at Bayes-plausible posteriors. The envelope computation then needs the shape of $K$: the supplier's quantity is piecewise defined, so $K$ is a piecewise rational function of $p$ whose convexity changes across the boundaries $\underline\tau(p)=\xi$ and $\bar\tau(p)=\xi$. In the low-$\xi$ regime $K$ is neither convex nor concave on $[0,1]$, so its greatest convex minorant is a genuine mixture of $K$ itself and a line segment, whose junction point $\hat p$ has to be located exactly.

Formalization scope

States are Fin 2; a policy on a finite type $M$ is a map $\pi : \mathrm{Fin}\,2\to M\to\mathbb R$ with nonnegative entries and rows summing to $1$. The paper does not bound $\mathcal M$; finite message spaces are a convention, and binary spaces already attain the optimum. Uniform expectations are interval integrals divided by the length. The supplier's cost is normalized by $C_s+C_e$ and parameterized by $\xi$, which leaves its minimizers unchanged. A message of probability zero is given posterior $0$ and weight $0$.

The supplier's response is defined as an argmin of (2), never by the closed form of Proposition 1, and optimality quantifies over all finite policies and all best-response profiles, never over the three named policies; a formalization that hard-codes either would make the goal a statement about a formula. $K$ is evaluated at the largest optimal quantity, which is the unique one under the mission's hypotheses. The envelope $\kappa$ is the published definition BiconvexProg.BranchBound.convexEnvelope (greatest convex minorant) on $[0,1]$.

Standing hypotheses, stated in every theorem: $a_0<b_0$, $a_1<b_1$, $0<\xi<1$, $p_0\in[0,1]$, and nesting $a_0\le a_1$, $b_1\le b_0$. The hypotheses $0\le a_0$, $0\le a_1$ are added: demand is nonnegative, and without them the constraint $Q\ge0$ can bind. Equal supports are allowed; the threshold $\xi_h$ is then $0/0=0$ in Lean and the goal remains true. Lemma 1's third case is formalized as the piecewise function its two indicators intend.

A complete development needs: the convex-envelope (concavification) theorem for finite-support distributions on $[0,1]$, the newsvendor quantile characterization for a mixture of uniforms, and calculus for piecewise rational functions. The first two are reusable beyond this mission. Contributions to any milestone are welcome.

Selected references

• O. Candogan, H. Gurkan, The Value of Information Design in Supply Chain Management, Management Science, 2024. SSRN preprint: https://ssrn.com/abstract=4378997; DOI: https://doi.org/10.1287/mnsc.2023.03004
• E. Kamenica, M. Gentzkow, Bayesian Persuasion, American Economic Review 101(6), 2011. https://doi.org/10.1257/aer.101.6.2590
• R. Levi, G. Perakis, J. Uichanco, The Data-Driven Newsvendor Problem: New Bounds and Insights, Operations Research 63(6), 2015. https://doi.org/10.1287/opre.2015.1422