All missions

Returns and the retailer's incentive for sales effort

A manufacturer that sells through a retailer can accept returns: it pays a return credit $b$ for every unit the retailer has not sold at the end of the season. Returns policies are common in publishing, software and computer hardware, and since Pasternack (Marketing Science, 1985) they have been studied as a way to make the retailer order more. Retailers also raise demand through sales effort: merchandising, shelf space, point-of-sale advertising. A recurring view in the marketing and operations literature is that returns weaken this incentive. Padmanabhan and Png (1995, p. 70) write that "by reducing the risk of losses due to excess inventory, a returns policy lessens some of the retailer's incentive to invest in such efforts", and Kandel (1996, p. 348) makes the same argument for consignment.

T. A. Taylor, Supply Chain Coordination Under Channel Rebates with Sales Effort Effects (Management Science 48(8), 2002), studies a newsvendor retailer who chooses an order quantity and a sales effort level before demand is observed. Its Proposition 4 shows that in this model the conventional view is reversed: a larger return credit makes the retailer exert strictly more effort. This mission formalizes that proposition and the solution of the retailer's problem it rests on.

Setting

Prices satisfy $0<c<w<p$ and $s<c$ (Assumption A1): $p$ is the retail price, $w$ the wholesale price, $c$ the manufacturing cost and $s$ the salvage value, which may be negative. A demand factor $\xi\ge 0$ has a density $\varphi$ with $\varphi(\xi)>0$ for every $\xi\ge0$ (Assumption A4), distribution function $\Phi(Q)=\int_0^Q\varphi$ and partial mean $\Gamma(Q)=\int_0^Q\xi\,d\Phi(\xi)$.

The retailer chooses an effort level $e\ge0$, and demand is $e\xi$. Effort costs $V(e)$, where $V$ is strictly convex and strictly increasing with $V(0)=0$ (Assumption A5; the paper declares every convexity and monotonicity statement strict). Write $\Lambda(\gamma)=\gamma V'(\gamma)-V(\gamma)$.

Under a returns-only contract $(w,b)$ with $b\in[s,w)$, the retailer orders $Q\ge0$, exerts effort $e\ge 0$, and earns the expected profit

$$\underline R_b(Q,e)=-wQ+p\,E\min(Q,e\xi)+b\,E(Q-e\xi)^+-V(e).$$

This is the integrated channel's profit $\Pi(Q,e)=-cQ+pE\min(Q,e\xi)+sE(Q-e\xi)^+-V(e)$ with $w$ in place of $c$ and $b$ in place of $s$. An optimal pair is a maximizer of $\underline R_b$ over $Q\ge0$, $e\ge0$. The paper's $\underline e$ is the effort of such a pair, and $\underline Q_0$ is the critical fractile $\Phi(\underline Q_0)=(p-w)/(p-b)$.

Formalization targets

Goal: Proposition 4 (p. 1004)

If $s\le b_2<b_1<w$, $(Q_1,e_1)$ is an optimal pair under the credit $b_1$, $(Q_2,e_2)$ is an optimal pair under $b_2$, and $e_2>0$, then

$$e_2<e_1 .$$

The paper writes this as $\partial\underline e/\partial b>0$. Its proof shows strict monotonicity, and that is the statement here.

Milestone 1: the order for a given effort (§4.2, p. 1000)

For a fixed $e>0$, the unique maximizer of $Q\mapsto\underline R_b(Q,e)$ over $Q\ge0$ is $Q_2=e\,\underline Q_0$, and

$$\max_{Q\ge0}\underline R_b(Q,e)=e\,(p-b)\,\Gamma(\underline Q_0)-V(e).$$

Milestone 2: the optimal effort (§4.2, p. 1000)

An optimal pair $(Q,\underline e)$ with $\underline e>0$ satisfies

$$V'(\underline e)=(p-b)\,\Gamma(\underline Q_0),\qquad Q=\underline e\,\underline Q_0,\qquad \underline R_b(Q,\underline e)=\Lambda(\underline e),$$

and conversely every $e>0$ solving this first-order condition gives the optimal pair $(e\underline Q_0,e)$.

Significance

The result. Proposition 4 separates two effects of a return credit. For a fixed order quantity, a larger credit lowers the marginal value of effort: it pays the retailer for unsold units, and effort only matters through sold units. This is the fixed-quantity comparison on which the conventional view rests; Cachon's survey states it for buy-backs as Eq. (19) (on Prove2Me as CachonCoord.EffortNewsvendor.sec_6_4_1_effort_coordination, clause 1). Once the order quantity is chosen together with the effort, the larger credit raises the order, and the effort follows. For the multiplicative demand model the net effect is unambiguous for every demand density and every convex effort cost. A manufacturer that wants more retailer effort can therefore use the return credit as a lever, and a model that ignores the quantity response gets the sign wrong.

Formalizing it. The result is proved in the paper, in a four-line appendix argument that relies on the §4.2 solution of the retailer's problem; neither is machine-checked anywhere. The mission produces a checked newsvendor-with-effort solution (the scaling $Q_2=e\underline Q_0$ and the profit $\Lambda(\underline e)$), which the other missions of this series and any multiplicative-effort supply-chain model can reuse, and a checked statement of the comparative statics in the return credit.

Difficulty

The obvious argument differentiates the retailer's first-order condition in $b$ with the order held fixed, and it gives the wrong sign; the order quantity must be re-optimized. The correct comparison runs through the reduced problem in the effort alone, and it needs the §4.2 solution in full: the scaling of the optimal order with the effort, which turns $E\min(Q,e\xi)$ and $E(Q-e\xi)^+$ into newsvendor expectations of $\xi$ at $Q/e$; the value of the optimal order as a function of $e$; and the first-order characterization of the optimal effort. The comparison across credits then involves two objects: the function $\Lambda$, and the retailer's reduced profits under the two credits at a common effort level. The statement compares maximizers of two different optimization problems, not roots of an equation, so a formal proof must connect optimality of the pair to the first-order condition, which uses the strict convexity of $V$ and the positivity of the density.

Formalization scope

Everything lives in the namespace ChannelRebate.ReturnsEffort. The definitions file holds:

• Demand: a measurable density $\varphi$, zero on $(-\infty,0)$, positive on $[0,\infty)$, integrating to $1$, with finite mean;
• the law of $\xi$;
• $\Phi$ and $\Gamma$ as interval integrals;
• EffortCost: A5 with strict convexity and strict monotonicity on $[0,\infty)$ and $V(0)=0$;
• $\Lambda$;
• the retailer's profit returnsProfit;
• optimal orders and optimal pairs as maximizers over $Q\ge0$ (and $e\ge0$).

The expectations are the published CachonCoord.Newsvendor.expSales and expLeftover applied to the law of $e\xi$ (the image of the law of $\xi$ under $x\mapsto ex$).

The formalization makes the following commitments, each recorded on its item:

• $\Phi^{-1}$ is never an inverse function: $\underline Q_0$ is a positive solution of $\Phi(\underline Q_0)=(p-w)/(p-b)$.
• The paper writes $(\partial/\partial e)V$ without stating that $V$ is differentiable. Differentiability on $(0,\infty)$, with derivative $V'$, is a hypothesis.
• The paper restricts attention to strictly positive effort (p. 1000). The goal therefore assumes $e_2>0$; without it both optimal efforts could be $0$ when $V'(0^+)$ is large, and the strict inequality would fail.
• Existence of optimal pairs is a hypothesis, as on p. 999 ("assume the cost of effort function and demand distribution are chosen such that the existence of an optimal solution is assured").
• The credit satisfies $b<w$ strictly; at $b=w$ the critical fractile is undefined.

The goal is about optimal pairs of the joint problem. Defining $\underline e$ as the root of the first-order condition, or fixing the order quantity and varying $b$, would give a different and, in the second case, false statement; neither is the target.

Contributions are welcome:

• a proof of the scaling identity $E\min(Q,e\xi)=e\,E\min(Q/e,\xi)$;
• a proof of the newsvendor value identity $(p-b)E\min(q,\xi)-(w-b)q=(p-b)\Gamma(q)$ at the critical fractile, reusable for any newsvendor with a density;
• proofs of the two milestones;
• the goal theorem.

The two facts the paper's proof uses — that $\Lambda$ is strictly increasing on $(0,\infty)$, and that the retailer's reduced profit at a fixed effort increases with the credit — may be posted as supporting theorems.

Selected references

• T. A. Taylor, Supply Chain Coordination Under Channel Rebates with Sales Effort Effects, Management Science 48(8):992–1007, 2002. https://doi.org/10.1287/mnsc.48.8.992.168
• B. A. Pasternack, Optimal Pricing and Return Policies for Perishable Commodities, Marketing Science 4(2):166–176, 1985. https://doi.org/10.1287/mksc.4.2.166
• V. Padmanabhan, I. P. L. Png, Returns Policies: Make Money by Making Good, Sloan Management Review 37(1):65–72, 1995.
• E. Kandel, The Right to Return, Journal of Law and Economics 39(1):329–356, 1996. https://doi.org/10.1086/467352
• G. P. Cachon, Supply Chain Coordination with Contracts, in Handbooks in OR & MS 11, Elsevier, 2003, §6.4.1. https://doi.org/10.1016/S0927-0507(03)11006-7

Motivation

Manufacturers of computer hardware, software and automobiles routinely pay their retailers channel rebates: a payment per unit the retailer sells to end customers. A target rebate pays only for units sold beyond a target level. These industries also commonly offer returns: a credit for each unsold unit. Both instruments are used to change the retailer's behaviour, and the retailer controls two things that matter to the manufacturer: how much stock she orders, and how much sales effort she exerts to raise demand. Effort cannot be observed or written into a contract, but sales can.

T. A. Taylor's paper (Management Science 48(8), 2002) asks whether a contract built on sales and returns can make an independent retailer choose the effort level and the order quantity that maximize the profit of the whole supply chain, while splitting that profit in any desired proportion. Its Proposition 2 shows that returns alone, linear rebates alone, or target rebates alone cannot do this. Its Theorem 2, the goal of this mission, shows that a target rebate combined with returns can, when demand is uniform and effort cost is quadratic.

Setting

A manufacturer with unit production cost $c$ sells to a retailer at wholesale price $w$; the retailer sells at the fixed retail price $p$, and unsold units have salvage value $s$. The standing assumption is $0<c<w<p$ and $s<c$ ($s$ may be negative). Before demand is seen, the retailer chooses an order quantity $Q\ge0$ and an effort level $e\ge0$. Demand is $e\xi$, where $\xi$ is uniform on $[0,1]$, with distribution function $\Phi$ and $\Gamma(x)=\int_0^x\xi\,d\Phi(\xi)$. Effort costs $V(e)=ae^2/2$, $a>0$.

The integrated channel, which owns both firms, earns

$$\Pi(Q,e)=-cQ+pE\min(Q,e\xi)+sE(Q-e\xi)^+-V(e).$$

Its optimum uses the critical fractile $\bar Q_0$, defined by $\Phi(\bar Q_0)=(p-c)/(p-s)$, the effort $\bar e=(p-s)\Gamma(\bar Q_0)/a$ and the order $\bar Q=\bar e\bar Q_0$. Its optimal profit is $\Pi=\Lambda(\bar e)$, where $\Lambda(\gamma)=\gamma V'(\gamma)-V(\gamma)$.

A target rebate and returns contract $(w,u,b,T)$ pays the retailer $u>0$ for every unit sold beyond the target $T$, and credits her $b\in[s,w)$ for every unsold unit. Her profit is

$$R(Q,e\mid T)=-wQ+pE\min(Q,e\xi)+uE(\min(Q,e\xi)-T)^+ +bE(Q-e\xi)^+-V(e).$$

The manufacturer earns $M(Q,e\mid T)=(w-c)Q-uE(\min(Q,e\xi)-T)^+-(b-s)E(Q-e\xi)^+$. The other quantities the statements use are:

• the fractiles $\underline Q_0$ and $\underline Q_1$, with $\Phi(\underline Q_0)=(p-w)/(p-b)$ and $\Phi(\underline Q_1)=(p+u-w)/(p+u-b)$;
• the returns-only effort $\underline e=(p-b)\Gamma(\underline Q_0)/a$;
• a threshold $\tau\in[\underline Q_0,\underline Q_1]$ that separates the retailer's low and high orders;
• the retailer's best profit at effort $e$, $A(e\mid T)=\max_{Q\ge0}R(Q,e\mid T)$.

Contract terms of Theorem 2. Put $\zeta(T)=4a^2(p-s)^3T^2$ and

$$u(T)=(w-c)\frac{(p-c)^5}{(p-c)^5-\zeta(T)},\qquad b(T)=s+(w-c)\frac{(p-c)^6-(p-s)\zeta(T)}{(p-c)^6-(p-c)\zeta(T)},\qquad T_3=\frac{(p-c)^3}{2a(p-s)^2}.$$

$T_1$ and $T_2$ are the fixed points on $[0,T_3]$ of $T\mapsto\underline e\tau$ and $T\mapsto\bar e\tau$, with $\underline e$ and $\tau$ evaluated at $u(T)$, $b(T)$. $\underline L(T,w)$ and $\bar L(T)$ are the retailer's profits at effort $\underline e$ and at effort $\bar e$.

Formalization targets

Goal: Theorem 2 (p. 1002)

For every $\kappa\in(0,\Pi)$ there is $\varepsilon_0>0$ such that for every $\varepsilon\in(0,\min(\kappa,\varepsilon_0))$ the following holds. Pairs $(w^*,T^*)$ with

$$w^*\in(c,p),\qquad T^*\in(T_1,T_2),\qquad \underline L(T^*,w^*)=\kappa-\varepsilon,\qquad \bar L(T^*)=\kappa$$

exist. Every such pair gives $u^*=u(T^*)>0$, $b^*=b(T^*)\in(s,w^*)$ and $T^*>0$. The pair $(\bar Q,\bar e)$ is the retailer's unique optimum under $(w^*,u^*,b^*,T^*)$, and

$$R(\bar Q,\bar e\mid T^*)=\kappa,\qquad M(\bar Q,\bar e\mid T^*)=\Pi-\kappa.$$

Milestones

1. §4.1: $(\bar Q,\bar e)$ is the unique maximizer of $\Pi$, with value $\Lambda(\bar e)$.
2. §4.2: under returns alone, $(\underline e\,\underline Q_0,\underline e)$ is the retailer's unique optimum, with value $\Lambda(\underline e)$.
3. Lemma 2: at effort $e$ the retailer orders $e\underline Q_0$ if $e<T/\tau$, $e\underline Q_1$ if $e>T/\tau$, and either if $e=T/\tau$.
4. The two-branch formula for $A(e\mid T)$. Its derivative jumps up at $T/\tau$, so $T/\tau$ is never optimal.
5. Lemma 3: $A(\cdot\mid T)$ is concave on $[0,T/\tau)$, and either convex then concave, or concave, on $(T/\tau,\infty)$.
6. Lemma 4: a unique threshold $\Upsilon$ decides whether the optimal effort lies above $T/\tau$ (at $\hat e$) or equals $\underline e$.
7. Lemma 5: the retailer's optimal $(Q,e)$ in the three cases $T<\Upsilon$, $T>\Upsilon$, $T=\Upsilon$.
8. Lemma 6: $T_1$ and $T_2$ exist, are unique, and $0<T_1<T_2<T_3$.

Significance

The result. Theorem 2 shows that two contractible instruments can align two decisions, one of them not contractible. The rebate pushes effort and quantity up when sales are high, and the return credit raises both when demand is low. Together they reproduce the integrated channel's optimum $(\bar Q,\bar e)$, and the free parameter $\kappa$ allocates the profit $\Pi$ between the firms in any proportion. Proposition 2 of the same paper rules out each instrument alone. This makes Theorem 2 the basis for the paper's recommendation that rebates and returns be used together. The uniform instance is the only one in which the paper proves this; for normal demand it gives only numerical evidence.

Formalizing it. The theorem is proved in the paper; it has not been formalized. The printed proof treats the wholesale price as fixed while it varies: $u(T)$ and $b(T)$ contain $w$, so $T_1$, $T_2$ and $\bar L$ move with $w^*$. A machine-checked proof therefore has to repair the argument, not only transcribe it. A numerical check (p = 10, c = 4, s = 1, a = 1; for example κ = 1.2, ε = 0.012 gives w* ≈ 4.889, T* ≈ 1.095) found the statement itself consistent.

Difficulty

The retailer's problem is not concave. For fixed effort, $R(\cdot,e\mid T)$ has a kink at $Q=T$, and the optimal order jumps from $e\underline Q_0$ to $e\underline Q_1$ at $e=T/\tau$. After the order is optimized out, the profit in effort, $A(\cdot\mid T)$, has an upward kink at $T/\tau$ and can be convex and then concave beyond it. Checking the first-order condition at $\bar e$ is therefore not enough. Coordination is a statement about the global maximum of a kinked, non-concave function, and comparing the two local candidates is what the threshold $\Upsilon$ and the profits $\underline L$, $\bar L$ do. The contract is then pinned down by two equations in $(w,T)$ whose coefficients themselves depend on $w$ through $u(T)$ and $b(T)$.

Formalization scope

Everything is stated for $\xi\sim\mathrm{Uniform}(0,1)$, taken as Lebesgue measure on $[0,1]$, and for $V(e)=ae^2/2$, as in all of Lemmas 3–6 and Theorem 2. Lemma 2 and the §4.1–4.2 solutions, which the paper states for general demand, are specialised to this instance. The uniform law has bounded support, so the paper's Assumption A4 (positive density on $[0,\infty)$) is not imposed; the paper allows that relaxation on p. 995.

The expectations use the published expSales and expLeftover (Cachon's newsvendor definitions) of the image law of $\xi$ under $x\mapsto ex$. $\Phi$ and $\Gamma$ are integrals of the uniform density, not hard-coded closed forms. Every inverse ($\bar Q_0$, $\underline Q_0$, $\underline Q_1$), the threshold $\tau$, the function $j$ and its root $\Upsilon$, the fixed points $T_1$, $T_2$ and the profits $\underline L$, $\bar L$ are stated by their defining equations, never by a choice function. "Optimal" means a maximizer over $Q\ge0$, $e\ge0$. Channel coordination means that the retailer's set of maximizers is exactly $\{(\bar Q,\bar e)\}$. The manufacturer's profit $M$, which the paper does not display, is written from the contract's cash flows: units bought back at $b$ are salvaged at $s$.

A trivializing formalization is ruled out: coordination is global optimality of $(\bar Q,\bar e)$, not a first-order condition at $\bar e$. $\varepsilon_0$ may depend on $\kappa$ but not on $\varepsilon$. $T_1$ and $T_2$ are tied to their fixed-point equations and are not free variables.

A complete development needs:

• the closed forms of $\Phi$, $\Gamma$ and the three expectations under the uniform law;
• scaling identities in $e$;
• maximization of piecewise concave functions with a kink;
• one-dimensional intermediate value arguments, with the monotonicity needed to repair the proof of Theorem 2.

The uniform newsvendor identities and the kinked-maximization lemmas are reusable beyond this mission. Contributions to any milestone, and alternative proofs of the existence part of Theorem 2, are welcome.

Selected references

• T. A. Taylor, Supply Chain Coordination Under Channel Rebates with Sales Effort Effects, Management Science 48(8):992–1007, 2002. https://doi.org/10.1287/mnsc.48.8.992.168
• G. P. Cachon, Supply Chain Coordination with Contracts, in Handbooks in Operations Research and Management Science, Vol. 11, 2003. https://doi.org/10.1016/S0927-0507(03)11006-7
• B. A. Pasternack, Optimal Pricing and Return Policies for Perishable Commodities, Marketing Science 4(2):166–176, 1985. https://doi.org/10.1287/mksc.4.2.166

Motivation

An insurance company that collects premiums and pays random claims is ruined when its surplus becomes negative. The probability of ultimate ruin is the classical solvency criterion of risk theory, going back to Lundberg and Cramér. Real insurers have two levers on this probability: they can invest part of the surplus in a risky asset, and they can cede part of every claim to a reinsurer, at a price. The question of how to use both levers dynamically, as a function of the current surplus, so as to make the probability of ruin as small as possible, is a stochastic control problem for a jump-diffusion.

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab. 12 (2002), solves this problem for the Cramér–Lundberg model with proportional reinsurance and investment in a Black–Scholes asset. The main results are a verification theorem (Theorem 1, the subject of this mission) and an existence theorem for the associated integral equation (Theorem 2, the companion mission).

Timeline. Hipp and Plum (2000) treated investment in the Cramér–Lundberg model through the Hamilton–Jacobi–Bellman equation, assuming a bounded solution. Schmidli (2001) treated optimal proportional reinsurance alone. The 2002 paper combines investment and reinsurance, and proves the verification theorem without assuming the solution to be bounded.

Setting

Claims arrive at the jumps $T_1<T_2<\dots$ of a Poisson process $N$ with rate $\lambda>0$; the claim sizes $Y_1,Y_2,\dots$ are i.i.d. with continuous distribution function $G$, $G(0)=0$, independent of $N$. The insurer receives premiums at rate $c>0$. A risky asset $Z_t=\exp\{\sigma W_t+(\mu-\frac12\sigma^2)t\}$, with $\mu,\sigma>0$ and $W$ a standard Brownian motion independent of the claims, is available.

A strategy is a pair of processes $(A_t,b_t)$, predictable for the smallest right-continuous filtration generated by the claims and $W$: $A_t\in\mathbb R$ is the amount invested in the risky asset (locally bounded), and $b_t\in[0,1]$ is the retention level, meaning the insurer pays $b_tY$ of a claim $Y$ occurring at time $t$, at a reinsurance premium rate $c(b_t)$. The surplus $X=X^{Ab}$ starting from $u\ge0$ solves

$$dX_t=\big(c-c(b_t)+\mu A_t\big)\,dt+\sigma A_t\,dW_t-b_t\,dS_t,\qquad X_0=u,$$

where $S_t=\sum_{i\le N_t}Y_i$ is the aggregate claims process. The ruin time is $\tau=\inf\{t\ge0:X_t<0\}$, the survival probability is $\delta^{Ab}(u)=\mathbb P[\tau=\infty]$, and the value function is $\delta(u)=\sup_{A,b}\delta^{Ab}(u)$.

The premium function $c(b)$ is decreasing and continuous on $[0,1]$, $c(1)=0$, $\liminf_{b\uparrow1}c(b)/(1-b)>0$, and there is $\underline b>0$ with $c(b)>c$ for $b<\underline b$ and $c(b)\le c$ for $b\ge\underline b$ (full reinsurance is too expensive).

The Hamilton–Jacobi–Bellman equation for $\delta$ is, with the convention $f(u)=0$ for $u<0$,

$$\sup_{b\in[0,1]}\sup_{A\ge0}\Big[\tfrac12\sigma^2A^2f''(u)+\big(c-c(b)+\mu A\big)f'(u)+\lambda\big(\mathbb E[f(u-bY)]-f(u)\big)\Big]=0.\tag{1}$$

For strictly concave $f$ the maximum over $A$ is attained at $A^*(u)=-\mu f'(u)/(\sigma^2f''(u))$ (equation (2)); $b^*(u)$ denotes a measurable maximiser in $b$.

Formalization targets

Goal: Theorem 1 (p. 896)

Let $f$ be strictly increasing and nonnegative on $[0,\infty)$, continuous there, twice continuously differentiable on $(0,\infty)$, zero on $(-\infty,0)$, and a solution of (1) at every $u>0$. Then $f$ is bounded, $f(\infty)=\lim_{x\to\infty}f(x)\in(0,\infty)$,

$$\delta(u)=\frac{f(u)}{f(\infty)}\qquad(u\ge0),$$

and every surplus process from $u>0$ following $A_t=A^*(X_{t-})$, $b_t=b^*(X_{t-})$ has survival probability $\delta(u)$.

Milestones

The proof of Theorem 1 runs through: Lemma 3 (for small surplus the optimal retention is $b^*=1$); the identity $\mathbb E[f(X^*_{\tau^*\wedge t})]=f(u)$ along the feedback strategy; the inequality $\mathbb E[f(X_{t\wedge\tau})]\le f(u)$ for every strategy; Lemma 1 (almost surely, either ruin occurs or $X_t\to\infty$); the existence of a strategy with $\mathbb P[\tau=\infty]>0$; the comparison $\delta^{Ab}(u)\le f(u)/f(\infty)$ with equality for the feedback strategy; and the absence of ruin by creeping, $\mathbb P[\tau^*<\infty,X^*_{\tau^*}=0]=0$.

Companion statements

The second sentence of Theorem 1 (that $f(x)=1+\int_0^xg(z)\,dz$ satisfies the hypotheses whenever $g$ is a decreasing solution of the integral equation (5)) and its last sentence (at most one such solution of (1) with $f(0)=1$) are separate items.

Significance

Theorem 1 reduces a control problem over all predictable strategies to an analytic question: find one increasing smooth solution of (1). Combined with Theorem 2 (existence of a solution of (5)), it shows that the optimal survival probability is $f(u)/f(\infty)$ and that the optimal strategy is a Markov feedback rule in the current surplus. It also yields that this solution is bounded, which Hipp and Plum (2000) had to assume, and closes the gap they left open on ruin by creeping (Remark (iii), p. 897). Numerical procedures for the optimal strategy (§5 of the paper) rest on this identification.

The theorem is proved in the paper; nothing in it is formalized. The mission produces a machine-checked verification theorem for a controlled jump-diffusion with an Itô integral against Brownian motion, a Poisson claim stream and a predictable control, together with a formal model of the Cramér–Lundberg process with investment and reinsurance that later results in risk theory can reuse.

Difficulty

The standard verification argument applies Itô's formula to $f(X_t)$ and uses (1) to show that $f(X_{t\wedge\tau})$ is a supermartingale, and a martingale under the feedback strategy. Here $f$ is not $C^2$ at $0$ ($f''(0+)=-\infty$), is discontinuous at $0$ after the extension by $0$, and is not assumed bounded, so the stochastic integral is only a local martingale and the passage $t\to\infty$ needs both Lemma 1 and a separate argument that the feedback process does not creep through $0$. Lemma 1 itself is only sketched in the paper ("We will just describe the argument"). The optimality part needs a surplus process following the feedback rule; its existence is not proved in the paper.

Formalization scope

The model is formalized on a measurable space with a probability measure: i.i.d. exponential interarrival times (realising the Poisson process), i.i.d. claim sizes with law $\nu$, and a real Brownian motion (IsBrownianReal), the three families mutually independent. Independence of $W$ from the claims is not printed in the paper but is used by (1) and its proof; it is a disclosed standing assumption. The filtration is $\mathcal F_t=\bigcap_{s>t}\sigma(S_r,W_r:r\le s)$, not completed. The Itô integral is the published relation EthierKurtz.HasBrownianItoIntegral, and its paths are required to be continuous, so that the pathwise ruin event is well defined. The surplus equation holds for every $t$ and every sample point.

Conventions: the premium $c(b)$ is real valued (so $c(0)<\infty$, which the paper allows to fail when $\mathbb E[Y]<\infty$); the supremum in (1) is stated as a least upper bound equal to $0$, so an unbounded family never counts as a solution; $f$ is $C^2$ on $(0,\infty)$ and (1) is required for $u>0$; $\delta$ is a supremum over all admissible triples (strategy and surplus process); the controls of the feedback rule at non-positive pre-jump surplus are left free, and optimality is stated for $u>0$. Expectations of $f(X_{t\wedge\tau})$ are lower Lebesgue integrals of nonnegative functions, so no default value of a non-integrable expectation satisfies an inequality. The optimality clause is stated for every process following the feedback rule; it would be vacuous if no such process existed, and its existence is not part of the target.

Infrastructure needed: Itô's formula for a semimartingale with Brownian and compound-Poisson parts, the compensator of the compound Poisson random measure, localisation, and optional stopping. The model definitions, the HJB predicate and the comparison lemmas are reusable for other ruin-minimisation and dividend problems. Contributions to any milestone, and to general stochastic-calculus lemmas they need, are welcome.

Selected references

• H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab. 12(3) (2002), 890–907. https://doi.org/10.1214/aoap/1031863173
• C. Hipp, M. Plum, Optimal investment for insurers, Insurance Math. Econom. 27 (2000), 215–228. https://doi.org/10.1016/S0167-6687(00)00049-4
• H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J. (2001), 55–68. https://doi.org/10.1080/034612301750077338
• P. Brémaud, Point Processes and Queues: Martingale Dynamics, Springer (1981). https://doi.org/10.1007/978-1-4684-9477-8

Motivation

Manufacturers in computer hardware, software and automobiles routinely pay channel rebates to their retailers: a payment per unit the retailer sells to end consumers, as opposed to per unit the retailer buys. A target rebate pays only for units sold beyond a target level, a linear rebate pays for every unit sold. T. A. Taylor, Supply Chain Coordination Under Channel Rebates with Sales Effort Effects, Management Science 48(8) (2002), doi:10.1287/mnsc.48.8.992.168, asks whether such payments can make an independent retailer act in the interest of the whole supply chain.

The question belongs to the supply-chain contracting literature built on the newsvendor model. A wholesale price above cost makes the retailer under-order relative to the integrated channel (double marginalization, Spengler 1950). Returns contracts (Pasternack 1985) and revenue sharing (Pasternack 1999; Cachon and Lariviere 2000, as cited by Taylor) are known remedies. Cachon's survey, Supply Chain Coordination with Contracts (Handbooks in OR & MS 11, 2003), discusses sales-rebate contracts through a first-order condition. Taylor's Theorem 1 is the global statement for the target rebate, with an explicit contract and an arbitrary profit split, and without any lump-sum side payment.

This mission covers §3 of the paper, the model without sales effort. Missions II–IV of the series cover §4, where the retailer's effort also affects demand.

Setting

A manufacturer sells to a retailer who places one order of size $Q \ge 0$ before observing demand $\xi$. Demand has a density $\varphi$ with $\varphi(x) = 0$ for $x < 0$ and $\varphi(x) > 0$ for every $x \ge 0$ (Assumption A4), and a finite mean. Write

$$\Phi(Q) = \int_0^Q \varphi(x)\,dx, \qquad \Gamma(Q) = \int_0^Q x\,\varphi(x)\,dx .$$

The retail price $p$, the production cost $c$ and the salvage value $s$ are fixed with $0 < c < p$ and $s < c$; $s$ may be negative. A contract specifies a wholesale price $w$, a rebate $u > 0$ and a target $T \ge 0$ (Assumption A1: $0 < c < w < p$, $s < c$, $u > 0$, $T \ge 0$). No other transfer is allowed (Assumption A3).

• The integrated channel earns $\pi(Q) = -cQ + pE\min(Q,\xi) + sE(Q-\xi)^+$. Its optimal order $\bar Q_0$ solves $\Phi(\bar Q_0) = (p-c)/(p-s)$, and its optimal profit is $\pi = (p-s)\Gamma(\bar Q_0)$.
• Under the target rebate $(w,u,T)$ the retailer earns

$$r(Q\mid T) = -wQ + pE\min(Q,\xi) + sE(Q-\xi)^+ + uE(\min(Q,\xi)-T)^+,$$

and the manufacturer earns $m(Q\mid T) = (w-c)Q - uE(\min(Q,\xi)-T)^+$.

• Under the wholesale price-only contract ($u = 0$) the retailer orders $Q_0$ with $\Phi(Q_0) = (p-w)/(p-s)$ and earns $\underline r = (p-s)\Gamma(Q_0)$.

The contract coordinates the channel when $\bar Q_0$ is the retailer's unique optimal order. Finally $\hat u(w) = (w-c)(p-s)/(c-s)$.

Formalization targets

Goal: Theorem 1 (p. 996)

Fix $\kappa \in (0,\pi)$. For every sufficiently small $\varepsilon \in (0,\kappa)$ there is exactly one triple $(w^*,u^*,T^*)$ with $\underline r(w^*) = \kappa-\varepsilon$, $u^* = \hat u(w^*)$, $T^* \ge 0$ and

$$(p+u^*-s)\Gamma(\bar Q_0) - u^*\big(\Gamma(T^*) + T^*[1-\Phi(T^*)]\big) = \kappa, \qquad (1)$$

and this triple has $w^* \in (c,p)$, $u^*>0$, $T^*>0$, coordinates the channel, and gives the retailer $r^* = \kappa$ and the manufacturer $m^* = \pi-\kappa$.

Milestones

1. §3.1, p. 995. The newsvendor solutions: $\bar Q_0$ and $Q_0$ are the unique optimal orders, with values $(p-s)\Gamma(\bar Q_0)$ and $(p-s)\Gamma(Q_0)$, and $Q_0 < \bar Q_0$.
2. §3.2, p. 995. The piecewise form of $r(\cdot\mid T)$, its derivative $p-w-(p-s)\Phi(Q)$ below $T$ and $p+u-w-(p+u-s)\Phi(Q)$ above, strict concavity on $[0,T)$ and $(T,\infty)$, and an upward kink of the derivative at $T$.
3. Lemma 1, p. 995. With $\Phi(Q_1) = (p+u-w)/(p+u-s)$, the equation $r(Q_0\mid\tau_0) = r(Q_1\mid\tau_0)$ has exactly one root $\tau_0 \in [Q_0,Q_1]$, lying in $(Q_0,Q_1)$; the retailer's optimal order set is $\{Q_1\}$ for $T<\tau_0$, $\{Q_0\}$ for $T>\tau_0$, and $\{Q_0,Q_1\}$ for $T=\tau_0$.
4. §3.2, p. 996. The retailer's optimal profit is $(p+u-s)\Gamma(Q_1) - u(\Gamma(T)+T[1-\Phi(T)])$ if $T<\tau_0$ and $(p-s)\Gamma(Q_0)$ if $T \ge \tau_0$.
5. Proposition 1, p. 996. If $T = 0$, channel coordination requires $m^* < 0$.

Significance

Theorem 1 shows that a single sales-based instrument, a target rebate, achieves what a wholesale price alone cannot: the retailer orders the channel-optimal quantity, and the channel profit $\pi$ is split in any proportion $\kappa : \pi-\kappa$. The parameter $\kappa$ can be set to the retailer's opportunity cost, so a manufacturer with all the bargaining power extracts the rest of the channel profit without a side payment. Proposition 1 explains why the target is needed: a linear rebate that coordinates the channel leaves the manufacturer with a loss. The quantity-only analysis is also the base case of §4, where the paper shows that a target rebate combined with returns coordinates effort and quantity as well.

The results are proved in the paper (Appendix, p. 1005). No machine-checked proof of any of them is known to exist. The mission produces a Lean statement of the model and of each result, and invites formal proofs of the newsvendor solution, the shape of the kinked profit, the threshold lemma and the coordination theorem.

Difficulty

The retailer's profit $r(\cdot\mid T)$ is not concave: its derivative jumps up at the target. The usual newsvendor argument, a stationary point of a concave function is a global maximum, therefore fails. A first-order condition at $\bar Q_0$ shows only that $\bar Q_0$ is a local candidate on $(T,\infty)$. The retailer may still prefer the smaller candidate $Q_0$ on $[0,T]$, and coordination fails exactly when she does. Theorem 1(b) requires comparing the two branches globally, through the threshold $\tau_0$ of Lemma 1, and then showing that the contract's target lies strictly below $\tau_0$.

The contract itself is defined implicitly. $w^*$ solves an equation involving the critical fractile $Q_0(w)$, and $T^*$ solves (1). Existence and uniqueness of both, and the placement of $T^*$ relative to $\tau_0$, are part of the claim.

Formalization scope

All definitions are in ChannelRebate.Quantity.Setting. Demand is a structure Demand carrying the density $\varphi$ with the properties listed under Setting; its law is Lebesgue measure with density $\varphi$. $E\min(Q,\xi)$ and $E(Q-\xi)^+$ are expSales and expLeftover of the published definition CachonCoord_Newsvendor_Contracts, applied to this law. $E(\min(Q,\xi)-T)^+$ is the local rebateUnits. $\Phi$ and $\Gamma$ are interval integrals from $0$.

Conventions:

• $\Phi^{-1}$ is never a function. Each critical fractile ($\bar Q_0$, $Q_0$, $Q_1$) is a hypothesis giving its defining equation, and $\tau_0$ is any solution of $f_0(\tau_0) = 0$ in $[Q_0,Q_1]$.
• An optimal order maximizes over $Q \ge 0$. "Coordination" and the order sets of Lemma 1 are equalities of the set of maximizers, so a tie with $Q_0$ does not count as coordination.
• "For $\varepsilon$ sufficiently small" is "there exists $\varepsilon_0 > 0$ such that for all $\varepsilon \in (0,\min(\varepsilon_0,\kappa))$".
• In Theorem 1 the uniqueness ranges over all real triples satisfying the specification, with $w$ unrestricted; $w^* \in (c,p)$ is a conclusion.
• The manufacturer's profit is its own formula, never channel profit minus retailer profit; $m^* = \pi-\kappa$ is a conclusion.
• The standing assumptions A1, A3 and A4 appear as hypotheses or in Demand; no continuity of $\varphi$ is assumed.

A formalization that reads coordination as the first-order condition at $\bar Q_0$, or that defines $T^*$ or $w^*$ by a choice function, would make Theorem 1 vacuous or weaker, and is ruled out by the statements above.

A complete development needs the newsvendor calculus for a density (derivatives of $E\min(Q,\xi)$ and $E(Q-\xi)^+$ in $Q$, strict monotonicity of $\Phi$ on $[0,\infty)$), the intermediate value theorem for the two implicit equations, and the global comparison of the two concave branches. The newsvendor calculus is reusable for missions II–IV of this series and for other newsvendor contracts. Proofs of the milestones in any order are welcome.

Selected references

• T. A. Taylor, Supply Chain Coordination Under Channel Rebates with Sales Effort Effects, Management Science 48(8):992–1007, 2002. doi:10.1287/mnsc.48.8.992.168
• G. P. Cachon, Supply Chain Coordination with Contracts, in Handbooks in Operations Research and Management Science 11, 2003. doi:10.1016/S0927-0507(03)11006-7
• B. A. Pasternack, Optimal Pricing and Return Policies for Perishable Commodities, Marketing Science 4(2):166–176, 1985. doi:10.1287/mksc.4.2.166
• J. J. Spengler, Vertical Integration and Antitrust Policy, Journal of Political Economy 58(4):347–352, 1950. doi:10.1086/256964

Motivation

Two-stage stochastic programs are solved with an approximation of the true distribution of the random data: an empirical measure, a scenario tree, or a discretization. Whether the computed decisions mean anything depends on how the optimal value and the solution set react when the underlying probability measure is replaced by a nearby one, and on which distance between measures "nearby" refers to. Rachev and Römisch (preprint, edoc.hu-berlin.de; published in Math. Oper. Res. 27 (2002) 792–818, doi:10.1287/moor.27.4.792.304) organize this question in two steps. First, a minimal information distance $d_{\mathcal F_{\mathcal U}}$, built from the integrands of the problem itself, controls optimal values and solution sets (their Theorems 2.2 and 2.3). Second, for a given class of models, $d_{\mathcal F_{\mathcal U}}$ is bounded by a canonical probability metric that depends only on the analytical properties of the integrands.

For linear two-stage programs the integrands are locally Lipschitz in the random parameter and the canonical metric is a Fortet–Mourier metric. When the second stage has integer variables, the recourse function is discontinuous, and metrics of Fortet–Mourier type no longer control the problem. This mission formalizes the paper's answer for that case (§3.2): mixed-integer two-stage programs are Hölder stable with respect to a polyhedral discrepancy $d_{1,phk}$ built from Lipschitz functions restricted to polyhedra. The structure of mixed-integer value functions that makes this possible goes back to Blair and Jeroslow (1977), Bank et al. (1982) and Schultz (1996), all cited in the paper.

Setting

Let $c\in\mathbb R^m$, let $X\subseteq\mathbb R^m$ be closed and $\Xi\subseteq\mathbb R^s$ a polyhedron, i.e. an intersection of finitely many closed half-spaces. The second stage has costs $q\in\mathbb R^{\hat m}$, $\bar q\in\mathbb R^{\bar m}$ and $(r,\hat m)$- and $(r,\bar m)$-matrices $W$, $\bar W$. Its value function is

$$\Phi(t)=\min\{qy+\bar q\bar y:\ Wy+\bar W\bar y=t,\ y\in\mathbb Z^{\hat m}_+,\ \bar y\in\mathbb R^{\bar m}_+\}\qquad(t\in\mathbb R^r),$$

and $\mathcal T$ is the set of right-hand sides $t$ for which the constraint set is nonempty. The right-hand side $h(\xi)\in\mathbb R^r$ and the technology matrix $T(\xi)$ depend affinely on $\xi\in\mathbb R^s$. For a Borel probability measure $\mu$ on $\Xi$ the program is

$$\min\Big\{\int_\Xi f_0(\xi,x)\,\mu(d\xi):\ x\in X\Big\},\qquad f_0(\xi,x)=cx+\Phi(h(\xi)-T(\xi)x). \tag{9}$$

Three conditions make (9) well defined: (B1) $W$ and $\bar W$ have rational entries; (B2) $h(\xi)-T(\xi)x\in\mathcal T$ for all $(\xi,x)\in\Xi\times X$ (relatively complete recourse); (B3) there is $u$ with $W'u\le q$ and $\bar W'u\le\bar q$ (dual feasibility).

$v(\nu)$ and $S(\nu)$ denote the optimal value and solution set of (9) under $\nu$; for an open bounded $\mathcal U\subseteq\mathbb R^m$, $v_{\mathcal U}(\nu)$ and $S_{\mathcal U}(\nu)$ are the same objects with $X$ replaced by $X\cap\operatorname{cl}\mathcal U$. $S_{\mathcal U}(\nu)$ is a complete local minimizing (CLM) set if it is nonempty and contained in $\mathcal U$. The minimal information distance is $d_{\mathcal F_{\mathcal U}}(\mu,\nu)=\sup_{x\in X\cap\operatorname{cl}\mathcal U}|\int_\Xi f_0(\xi,x)(\mu-\nu)(d\xi)|$. The moment classes are $\mathcal P_{p,K}(\Xi)=\{\nu:\int_\Xi\|\xi\|^p\nu(d\xi)\le K\}$, and the polyhedral metric is

$$d_{1,phk}(\mu,\nu)=\sup\Big\{\Big|\int_P f(\xi)(\mu-\nu)(d\xi)\Big|:\ P \text{ a polyhedron with at most } k \text{ faces},\ f \text{ 1-Lipschitz on } P,\ |f(\xi)|\le\max\{1,\|\xi\|\}\Big\}.$$

Finally, $\psi(\tau)=\inf\{\int_\Xi f_0(\xi,x)\mu(d\xi)-v(\mu): d(x,S(\mu))\ge\tau,\ x\in X\cap\operatorname{cl}\mathcal U\}$ is the growth function at $\mu$, $\psi^{-1}(t)=\sup\{\tau\ge0:\psi(\tau)\le t\}$, and $\Psi(\eta)=\eta+\psi^{-1}(2\eta)$.

Formalization targets

Goal: Theorem 3.6

Under (B1)–(B3), with $\mu\in\mathcal P_{p,K}(\Xi)$ for some $p>1$, $K>0$, $S(\mu)\ne\emptyset$ and $\mathcal U$ an open bounded neighbourhood of $S(\mu)$, there are $L>0$, $\delta>0$ and $k\in\mathbb N$ such that for all $\nu\in\mathcal P_{p,K}(\Xi)$ with $d_{1,phk}(\mu,\nu)<\delta$

$$|v(\mu)-v_{\mathcal U}(\nu)|\le L\,d_{1,phk}(\mu,\nu)^{\frac{1}{1+\frac{r}{p-1}}},\qquad \emptyset\ne S_{\mathcal U}(\nu)\subseteq S(\mu)+\Psi\Big(L\,d_{1,phk}(\mu,\nu)^{\frac{1}{1+\frac{r}{p-1}}}\Big)\mathbb B,$$

and $S_{\mathcal U}(\nu)$ is a CLM set. The constants and the number of faces are existential; only the exponent is fixed.

Milestones

1. Lemma 3.5 (i)–(ii): a countable, locally finite Borel partition of $\mathcal T$ into translates of $\operatorname{pos}\bar W$ minus finitely many translates, on each piece of which $\Phi$ is Lipschitz with a common constant.
2. Lemma 3.5, last assertion: $\Phi$ is finite and lower semicontinuous on $\mathcal T$, and $|\Phi(t)-\Phi(\tilde t)|\le\alpha\|t-\tilde t\|+\beta$.
3. Growth bound (proof of Theorem 3.6): $|f_0(\xi,x)|\le\|c\|\|x\|+\alpha(\|h(\xi)\|+\|T(\xi)\|\|x\|)+\beta$, hence every measure with finite first moment lies in the domain of $d_{\mathcal F_{\mathcal U}}$.
4. Theorem 2.2 for $d=0$: $|v(\mu)-v_{\mathcal U}(\nu)|\le d_{\mathcal F_{\mathcal U}}(\mu,\nu)$ for all admissible $\nu$, Berge upper semicontinuity of $S_{\mathcal U}$, CLM sets near $\mu$.
5. Theorem 2.3 for $d=0$: $\emptyset\ne S_{\mathcal U}(\nu)\subseteq S(\mu)+\Psi(\hat L d_{\mathcal F_{\mathcal U}}(\mu,\nu))\mathbb B$ with $\Psi(\eta)=\eta+\psi^{-1}(\eta)$.
6. Estimate (12): $d_{\mathcal F_{\mathcal U}}(\mu,\nu)\le C\,d_{1,phk}(\mu,\nu)^{1/(1+r/(p-1))}$ for small $d_{1,phk}(\mu,\nu)$.

Companion: Corollary 3.7

For bounded $\Xi$ and every $\mu\in\mathcal P(\Xi)$ the same conclusions hold with the Lipschitz rate $L\,\alpha_{phk}(\mu,\nu)$, where $\alpha_{phk}(\mu,\nu)=\sup\{|\mu(P)-\nu(P)|\}$ over polyhedra with at most $k$ faces.

Significance

Theorem 3.6 says that, for mixed-integer recourse, closeness of the distributions in $d_{1,phk}$ is enough for closeness of optimal values and of localized solution sets, with an explicit Hölder exponent $(p-1)/(p-1+r)$ that degrades with the second-stage dimension $r$ and improves with the number of moments. Two consequences follow. Scenario reduction and discretization methods for mixed-integer models can be judged by a polyhedral discrepancy; and, through the entropy estimates of §4 of the paper, empirical approximations converge at rates governed by uniform laws for polyhedra. Corollary 3.7 shows that for bounded supports the rate becomes Lipschitz in the polyhedral discrepancy.

The results are proved in the paper, partly by reference: Lemma 3.5 is quoted from Bank et al., Schultz, and Blair–Jeroslow, and the partition argument behind (12) refers to Schultz (1996). None of it has a machine-checked proof. Formalizing the mission produces the first formal treatment of mixed-integer value functions with rational data, a formal version of the minimal-information-distance stability theorems in the objective-only case, and the polyhedral metric itself.

Difficulty

The obvious route to a stability estimate, a Lipschitz bound on $\xi\mapsto f_0(\xi,x)$ followed by a Fortet–Mourier or Wasserstein bound, fails at the first step: $\Phi$ jumps across the boundaries of the pieces $\mathcal B_i$, so $f_0(\cdot,x)$ is only piecewise Lipschitz, with pieces that move with $x$. Any metric that controls $d_{\mathcal F_{\mathcal U}}$ must therefore see the indicator functions of these pieces, which is why polyhedra enter. The second obstacle is that there are countably many unbounded pieces: a single polyhedral bound covers only a bounded region of $\mathcal T$, and the tail outside it must be paid for with moments, which is the origin of the exponent. Lemma 3.5 itself, in particular the uniform Lipschitz constant and the local finiteness of the partition, rests on rationality of $W$ and $\bar W$; without (B1) the value function need not be lower semicontinuous.

Formalization scope

$\mathbb R^m$, $\mathbb R^s$, $\mathbb R^r$ are Euclidean spaces (the paper never fixes its norm, and every constant in the statements is existential). Measures are Borel probability measures on $\mathbb R^s$ with $\nu(\mathbb R^s\setminus\Xi)=0$. The integral of the extended-real integrand $f_0$ is the published DupacovaWets.Consistency.expect ($+\infty$ when $\int f_0^+=\infty$). $\Phi$, optimal values and $\psi$ are extended-real infima, $+\infty$ on empty sets; the paper's "min" is read as an infimum. Distances and $\Psi$ take values in $[0,\infty]$, and every estimate between optimal values states explicitly that both are finite, so that no convention for $\infty-\infty$ can make a bound vacuous. $T(\xi)$ is a linear map $\mathbb R^m\to\mathbb R^r$ and $\|T(\xi)\|$ its operator norm. Integer vectors are tuples of natural numbers; $W$, $\bar W$ are real matrices with rational entries.

Disclosed reading: a "polyhedron with at most $k$ faces" is an intersection of $k$ closed half-spaces. Because $k$ is existential in Theorem 3.6 and Corollary 3.7, the theorems are equivalent under this reading and under a facet count. The clause "piecewise polyhedral" of Lemma 3.5 is not defined in the paper and is not formalized. The page states "$\mathcal P_{\mathcal F_{\mathcal U}}(\Xi)\subseteq\mathcal P_1(\Xi)$"; the argument proves and uses $\mathcal P_1(\Xi)\subseteq\mathcal P_{\mathcal F_{\mathcal U}}(\Xi)$, and that direction is the one formalized. Theorems 2.2 and 2.3 are stated in their objective-only ($d=0$) form for a general normal integrand.

The integrals in $d_{1,phk}$ are Bochner integrals over $P$, which are meaningful because measures in $\mathcal P_{p,K}(\Xi)$ with $p>1$ integrate $\max\{1,\|\xi\|\}$. A formalization that took the supremum over all of $\mathbb R^s$ instead of $P$, quantified $\nu$ over all of $\mathcal P(\Xi)$, or let a non-integrable function contribute $0$ would not state the paper's theorem, and these readings are excluded.

Needed infrastructure: the structure theory of mixed-integer value functions with rational data (the partition of Lemma 3.5 and the attainment of the minimum in (10)), lower semicontinuity of integral functionals of normal integrands via Fatou's lemma, Berge's upper semicontinuity of parametric argmin maps, and approximation of polyhedral pieces by polyhedra. The value-function results and the general $d=0$ stability theorems are reusable beyond this mission. Proofs of any milestone, and alternative arguments for Lemma 3.5, are welcome.

Selected references

• S. T. Rachev, W. Römisch, Quantitative stability in stochastic programming: The method of probability metrics, Math. Oper. Res. 27(4) (2002) 792–818. doi:10.1287/moor.27.4.792.304. Preprint used here: edoc.hu-berlin.de.
• B. Bank, J. Guddat, D. Klatte, B. Kummer, K. Tammer, Non-Linear Parametric Optimization, Akademie-Verlag, Berlin, 1982. doi:10.1007/978-3-0348-6328-5
• R. Schultz, Rates of convergence in stochastic programs with complete integer recourse, SIAM J. Optim. 6 (1996) 1138–1152. doi:10.1137/S1052623494271655
• C. E. Blair, R. G. Jeroslow, The value function of a mixed integer program: I, Discrete Math. 19 (1977) 121–138. doi:10.1016/0012-365X(77)90028-0
• R. T. Rockafellar, R. J.-B. Wets, Variational Analysis, Springer, 1998. doi:10.1007/978-3-642-02431-3