Operations Research

Motivation

Social media influencers earn money by endorsing products to their followers. An influencer who endorses only products that fit her organic content keeps her audience; one who accepts every sponsorship offer earns more per follower today but loses part of her audience and grows more slowly. Practitioners describe this as the choice between growing and monetizing, and the marketing literature documents both sides of it: followers unsubscribe after poorly fitting sponsored posts (Cheng and Zhang 2022), and small influencers have deeper engagement than large ones (Bentley et al., J. Global Marketing 34(3), 2021, cited in the paper).

Nistor, Selove and Villas-Boas (Management Science, Articles in Advance 2024) turn this trade-off into a deterministic continuous-time control problem. Awareness of the influencer spreads by word of mouth as in the Bass (1969) diffusion model, and the influencer's authenticity is the control. The paper's first main result, Proposition 1, characterizes the optimal policy completely: depending on two explicit conditions, the influencer is always authentic, always inauthentic, or authentic until her awareness reaches a threshold and inauthentic forever after. This mission formalizes Proposition 1 and the lemmas of §3.3 that lead to it.

Setting

Time is $t\in[0,\infty)$ and the mass of potential followers is one. The influencer's awareness $A_t\in[0,1]$ is the fraction of potential followers who know her, starting from $0<A_0<1$. At each time she is authentic or inauthentic. Following Table 1 of the paper, with parameters $0<\gamma<1$ and $\varphi>1$ (the reduced-form parameters of §3.1, which aggregate the offer arrival rate, the share of good-fit offers and the followers' tolerance of bad-fit posts):

• an authentic influencer has $F_t=A_t$ followers and instantaneous profit $\pi_t=A_t$;
• an inauthentic influencer has $F_t=\gamma A_t$ followers and instantaneous profit $\pi_t=\gamma\varphi A_t$.

Awareness grows according to the equation of motion (1),

$$\frac{dA_t}{dt}=\beta F_t(1-A_t),\qquad\beta>0,$$

so it grows at rate $\beta A_t(1-A_t)$ when she is authentic and $\gamma\beta A_t(1-A_t)$ when she is inauthentic. With discount rate $r>0$ the influencer maximizes the discounted profit (2),

$$V_t=\int_t^\infty e^{-r(u-t)}\pi_u\,du ,$$

and $V(A)$ is the value of starting at awareness $A$ and following the optimal policy.

Two closed-form objects enter the main result. The value of always being inauthentic from awareness $A$ is (Lemma 3)

$$\underline V(A)=\int_0^\infty \frac{e^{-rs}\gamma\varphi}{1+\frac{1-A}{A}e^{-\gamma\beta s}}\,ds ,$$

and its derivative is (3)

$$m(A)=\frac{d\underline V(A)}{dA}=\int_0^\infty \frac{e^{-(\gamma\beta+r)s}\gamma\varphi}{\big[A+(1-A)e^{-\gamma\beta s}\big]^2}\,ds .$$

Condition 1 (with footnote 4) holds if $\gamma\beta\ge r$ or $\big[\frac{\beta-\gamma\beta}{r-\gamma\beta}\big]\gamma\varphi>\gamma\varphi-1$.

Formalization targets

Goal: Proposition 1

For $0<\gamma<1$, $\varphi>1$, $\beta>0$, $r>0$:

1. if $\gamma\varphi\le1$, always being authentic is optimal from every $A_0\in(0,1)$;
2. if $\gamma\varphi>1$ and Condition 1 fails, always being inauthentic is optimal from every $A_0\in(0,1)$;
3. if $\gamma\varphi>1$ and Condition 1 holds, there is $A^*\in(0,1)$ with

$$m(A^*)=\frac{\gamma\varphi-1}{(1-\gamma)\beta(1-A^*)},$$

and for all sufficiently small $A_0$ the policy "authentic while $A_t<A^*$, inauthentic once $A_t\ge A^*$" is optimal.

Milestones

In the paper's order: Lemma 1 (corrected) bounds the growth of $V$ above awareness $\tfrac12$; Lemma 2 (corrected) shows that permanent inauthenticity is optimal at large awareness; Lemma 3 gives $\underline V$ in closed form; (3) identifies $m$ as its derivative; (4) computes the limit of $m$ at zero awareness,

$$\lim_{A\to0^+}m(A)=\frac{\gamma\varphi}{r-\gamma\beta}\ \ (\gamma\beta<r),\qquad m(A)\to+\infty\ \ (\gamma\beta\ge r),$$

which is where Condition 1 comes from.

Significance

Proposition 1 is the benchmark of the paper: each of the six model extensions of §4 (celebrities, follower turnover, partial authenticity, viral content, two segments, commitment) is read against it. Its third clause explains an observed pattern in influencer marketing, that small influencers post fitting content and large ones accept most sponsorships, as an optimal policy rather than a change in preferences, and Condition 1 makes explicit when growth potential outweighs current profits; the paper notes it always holds when $\beta>r$. The numerical example of §3.5 (Table 2) places the switch at about 73% awareness.

The proofs are in an online appendix and have not been machine-checked. Formalizing them requires an optimality argument against every measurable switching pattern, not only threshold policies. The mission also records two corrections: Lemma 1 and Lemma 2 are false as printed when $\gamma\varphi>1$ (explicit counterexamples are in the milestone statements), and they are stated here with the constant $\gamma\varphi/r$ that the paper's own argument gives.

Difficulty

The natural argument compares, at each instant, the extra profit from inauthenticity, $(\gamma\varphi-1)A_t$, with the value of the extra growth from authenticity, $(1-\gamma)\beta A_t(1-A_t)V'(A_t)$. This requires knowing the value function $V$, its differentiability, and the Bellman equation it satisfies, none of which is given; replacing $V$ by $\underline V$ is only justified along the part of the trajectory after the switch. The comparison must also hold against every measurable switching pattern, under which awareness solves (1) only almost everywhere. Clause 3 additionally needs the threshold equation to have a single crossing and the optimal trajectory to cross it exactly once. Optimality of a constant policy cannot be read off from instantaneous profits alone when $\gamma\varphi>1$: the instantaneously better choice (inauthenticity) can be the wrong one for a small influencer.

Formalization scope

• Policies are measurable sets $S\subseteq\mathbb R$ of times at which the influencer is inauthentic; constant policies are $\emptyset$ and $\mathbb R$.
• Admissible paths are continuous on $[0,\infty)$, start at $A_0$, take values in $[0,1]$, and solve (1) in integral (Carathéodory) form, because a measurable control makes $dA/dt$ exist only almost everywhere. The $[0,1]$ clause is implied by the equation and makes every profit stream bounded.
• Discounted profit is (2) at $t=0$, an integral over $(0,\infty)$; the paper's shift $s=u-t$ is notation. The value function $V$ of Lemma 1 is the real supremum over admissible pairs, nonempty and bounded above by $\gamma\varphi/r$ at every $A_0\in[0,1]$.
• Optimality is against every admissible pair from the same $A_0$, and every goal clause produces the optimal pair existentially. The threshold policy is the set $\{t\ge0: A_t\ge A^*\}$ of the same path it drives; a free switching time would not express a feedback rule.
• Standing assumptions. $0<\gamma<1$, $\varphi>1$, $\beta>0$, $0<A_0<1$ (§3.1–3.2); $r>0$ is the reading of "discount rate"; $\gamma\varphi>1$ is assumed in every milestone ("for the remainder of the paper", p. 5) but not in the goal, whose first clause is $\gamma\varphi\le1$. Condition 1 is the disjunction of footnote 4. Lemma 1 and Lemma 2 carry the corrected constant $\gamma\varphi/r$.

A trivializing formalization is ruled out: optimality is never assumed, the competitor class contains the constant policies' logistic paths, the threshold $A^*$ must solve the equation of (3) inside $(0,1)$, and measurability of policies keeps every profit integral meaningful. The microfoundations of §3.1 ($\mu$, $\theta$, $\omega$, $\hat U$, $c$) do not enter any statement.

The development needs Carathéodory solutions of a scalar ODE with a measurable switching coefficient, comparison and uniqueness for such solutions, differentiation under an improper integral, and a verification argument for a one-dimensional deterministic control problem with bang-bang control. The last two are reusable for other continuous-time marketing and diffusion models. Proofs of any milestone are welcome, as are proofs of the logistic closed form underlying Lemma 3.

Selected references

• C. Nistor, M. Selove, J. M. Villas-Boas, Influencer Authenticity: To Grow or to Monetize, Management Science, Articles in Advance, 2024. https://doi.org/10.1287/mnsc.2023.03479
• F. M. Bass, A new product growth for model consumer durables, Management Science 15(5):215–227, 1969. https://doi.org/10.1287/mnsc.15.5.215
• M. Cheng, S. Zhang, Reputation burning: Analyzing the impact of brand sponsorship on social influencers, preprint, 2022. https://doi.org/10.2139/ssrn.4071188

Motivation

A portfolio manager who observes today's value $x_0$ of a covariate (a market signal, a macroeconomic indicator) wants to choose a portfolio that performs well conditionally on that observation: the relevant risk is $\mathbb E[\ell(Y,\alpha,\beta)\mid X=x_0]$ of the asset return $Y$, not its unconditional mean. With finitely many historical pairs $(\hat x_i,\hat y_i)$, the empirical conditional distribution at a new $x_0$ is usually empty or rests on one or two samples, so estimating it directly is unreliable. Nguyen, Zhang, Wang, Blanchet, Delage and Ye (arXiv:2103.16451v3, Operations Research 2025) hedge it with distributionally robust optimization: they take the worst case of the conditional expectation over all joint laws of $(X,Y)$ within an optimal-transport ball around the empirical measure that still put a prescribed amount of mass on the conditioning event.

The objective is a ratio of two linear functionals of the law, so the standard strong-duality theorems for Wasserstein distributionally robust optimization (Mohajerin Esfahani and Kuhn, 2018; Blanchet and Murthy, 2019; Gao and Kleywegt, 2023) do not apply directly. This mission formalizes the paper's answer in the singular case where the conditioning event is the single point $\{x_0\}$ and the event must keep mass at least $\varepsilon>0$.

Setting

Let $\mathcal X$ (covariates) and $\mathcal Y$ (returns) be spaces with ground costs $\mathbb D_{\mathcal X}$, $\mathbb D_{\mathcal Y}$, combined into the separable cost $\mathbb D((x,y),(x',y'))=\mathbb D_{\mathcal X}(x,x')+\mathbb D_{\mathcal Y}(y,y')$. The data are $N$ samples $(\hat x_i,\hat y_i)$, $i\in[N]$, with empirical measure $\hat{\mathbb P}=N^{-1}\sum_i\delta_{(\hat x_i,\hat y_i)}$.

• The optimal transport cost between laws $\mathbb Q_1,\mathbb Q_2$ is $\mathbb W(\mathbb Q_1,\mathbb Q_2)=\inf\{\mathbb E_\pi[\mathbb D(\xi_1,\xi_2)]:\pi\in\Pi(\mathbb Q_1,\mathbb Q_2)\}$, the infimum over couplings.
• The ambiguity set is $\mathbb B_\rho=\{\mathbb Q:\ \mathbb W(\mathbb Q,\hat{\mathbb P})\le\rho\}$ for a radius $\rho\ge0$.
• The fiber of radius $\gamma$ is $\mathcal N_\gamma(x_0)\times\mathcal Y$ with $\mathcal N_\gamma(x_0)=\{x:\mathbb D_{\mathcal X}(x,x_0)\le\gamma\}$; here $\gamma=0$, so the fiber is $\{x_0\}\times\mathcal Y$.
• The minimum radius $\rho_{\min}(x_0,\gamma,\varepsilon)=\inf\{\mathbb W(\mathbb Q,\hat{\mathbb P}):\mathbb Q(X\in\mathcal N_\gamma(x_0))\ge\varepsilon\}$ is the smallest radius at which some law in the ball meets the mass requirement.
• Assumption 2.4 of the paper: the costs are symmetric, non-negative and continuous, $\mathbb D$ vanishes exactly on the diagonal, each $\hat x_i$ has a unique projection onto $\mathcal N_\gamma(x_0)$ with cost $\kappa_i$, and boundary points of $\mathcal N_\gamma(x_0)$ can be left at arbitrarily small cost.

For a loss $\ell(y)=\ell(y,\alpha,\beta)$ at a fixed decision, the object of study is the worst-case conditional expected loss

$$\sup_{\mathbb Q\in\mathbb B_\rho,\ \mathbb Q(X=x_0)\ge\varepsilon}\ \mathbb E_{\mathbb Q}[\ell(Y)\mid X=x_0],\qquad \mathbb E_{\mathbb Q}[\ell(Y)\mid X=x_0]=\frac{\mathbb E_{\mathbb Q}[\ell(Y)\mathbb 1_{\{x_0\}}(X)]}{\mathbb Q(X=x_0)} .$$

Formalization targets

Goal: Theorem 3.2 (p. 15)

For $\varepsilon\in(0,1]$ and $\rho>\rho_{\min}(x_0,0,\varepsilon)$,

$$\sup_{\mathbb Q\in\mathbb B_\rho,\ \mathbb Q(X=x_0)\ge\varepsilon}\mathbb E_{\mathbb Q}[\ell(Y)\mid X=x_0] =\inf_{\lambda_1\ge0,\ \lambda_2\in\mathbb R}\Big\{\rho\lambda_1+\varepsilon\lambda_2+\frac1N\sum_{i\in[N]}\Big(\sup_{y\in\mathcal Y}\big\{\varepsilon^{-1}\ell(y)-[\mathbb D_{\mathcal X}(x_0,\hat x_i)+\mathbb D_{\mathcal Y}(y,\hat y_i)]\lambda_1-\lambda_2\big\}\Big)^+\Big\}.$$

The loss is arbitrary (measurable, real-valued), and both sides may be $+\infty$.

Milestones

1. Representation (24) (p. 47): $\mathbb Q\in\mathbb B_\rho$ iff $\mathbb Q=N^{-1}\sum_i\pi_i$ for probability measures $\pi_i$ with $N^{-1}\sum_i\mathbb E_{\pi_i}[\mathbb D(\cdot,(\hat x_i,\hat y_i))]\le\rho$.
2. Proposition 2.5 (p. 11): $\rho_{\min}$ is the value of the fractional knapsack program $\min\{N^{-1}\sum_i\kappa_i\upsilon_i:\upsilon\in[0,1]^N,\sum_i\upsilon_i\ge N\varepsilon\}$, and the problem is feasible iff $\rho\ge\rho_{\min}$.
3. Lemma A.4 (p. 36): in a linear-fractional program over $[0,1]^N$, the constraint $\sum_i\upsilon_i\ge\varepsilon N$ may be replaced by $\sum_i\upsilon_i=\varepsilon N$.
4. Proposition A.5 (p. 36): the worst case equals a supremum over weights $\upsilon$ with $\sum_i\upsilon_i=N\varepsilon$ and over laws $\mu_0^i$ on $\mathcal Y$ of the linear objective $(N\varepsilon)^{-1}\sum_i\upsilon_i\mathbb E_{\mu_0^i}[\ell]$ under one transport budget.
5. Inner duality (p. 41): for fixed strictly feasible weights, the inner supremum over $(\mu_0^i)$ equals its moment-problem dual over $\lambda_1\ge0$ and $\theta\in\mathbb R^N$.

Significance

Theorem 3.2 reduces an infinite-dimensional, fractional worst-case problem to a two-multiplier convex program whose only infinite-dimensional pieces are $N$ suprema over $\mathcal Y$. For piecewise-linear losses and norm costs the paper turns this into the conic programs of Propositions 3.3–3.4, which are what make the robust conditional portfolio model solvable in practice. It also compares the formulation with the trimming approach of Esteban-Pérez and Morales (Mathematical Programming, 2022), whose adversary may only pick laws with fiber mass exactly $\varepsilon$; Proposition A.5 shows that, at $\gamma=0$, the more powerful adversary allowed $\mathbb Q(X=x_0)\ge\varepsilon$ attains the same value.

The result is proved in the paper; nothing here is open mathematics. None of it is formalized. A complete development would add to the platform a first conditional (fractional) distributionally robust duality, a decomposition of optimal-transport balls around empirical measures (milestone 1, reusable by every finite-sample Wasserstein result), and a moment-problem duality with one budget constraint (milestone 5).

Difficulty

The ratio objective is the obstacle: neither the numerator nor the denominator alone determines the worst case, and a law that is optimal for the numerator may violate the mass constraint. The step "fix the fiber mass at $\varepsilon$" (Proposition A.5) is where the obvious argument fails: an optimal law may put more than $\varepsilon$ on $\{x_0\}$, and reducing that mass requires moving sample points off $x_0$ at vanishing transport cost, which needs points of $\mathcal X$ arbitrarily close to $x_0$. Milestone 1 needs the existence of optimal couplings, which Mathlib lacks. Milestone 5 is an infinite-dimensional Lagrangian duality whose Slater point degenerates on the boundary of the weight set, so the final interchange of supremum and infimum must avoid that boundary.

Formalization scope

$\mathcal X$ and $\mathcal Y$ are Polish spaces with their Borel σ-algebras; samples are indexed by Fin N with $N\ge1$, repeats allowed. The optimal transport cost is the published RWPI.SqrtLasso.transportCost (an infimum of lower integrals of the $[0,\infty]$-valued cost; attainment is not built in), and every expectation of the loss is the published ModelRiskOT.Duality.extIntegral, $\int\ell^+-\int\ell^-$ in $[-\infty,\infty]$. Worst-case values and duals are EReal suprema and infima. The loss is a measurable L : Y → ℝ standing for $\ell(\cdot,\alpha,\beta)$ at an arbitrary decision.

Hypotheses beyond the page, all disclosed in the statements:

• Assumption 2.4 is a hypothesis of every statement, as the paper makes it "throughout".
• NotIsolated: for every $r>0$ there is $x\ne x_0$ with $\mathbb D_{\mathcal X}(x,x_0)\le r$. This is Assumption 2.4(iii) at $\gamma=0$ read with the ambient boundary of $\{x_0\}$, as the paper's remark on p. 11 does; with the topological frontier in $\mathcal X$, (iii) is vacuous at an isolated $x_0$, and Theorem 3.2 is then false.
• Measurability of the loss, used because the dual side integrates $\ell$ against arbitrary laws on $\mathcal Y$.
• Milestone 5 requires the strict budget inequality $\sum_i\upsilon_i\mathbb D_{\mathcal X}(x_0,\hat x_i)<N\rho$; the page's Slater argument covers only that case.

Trivializing formalizations are ruled out: conditional expectations are only taken at laws with fiber mass at least $\varepsilon>0$ (never at mass $0$, where a real division would return $0$); expectations are never Bochner integrals, which would be $0$ for a non-integrable loss; the radius enters through ENNReal.ofReal ρ only where $\rho\ge0$ or $\rho>\rho_{\min}$ forces $\rho>0$; and every supremum ranges over exactly the paper's feasible set.

Needed infrastructure: existence of optimal couplings on Polish spaces (Villani, Theorem 4.1), a Lagrangian duality for moment problems, and Mathlib's Sion minimax theorem for the interchange. Contributions to any of these, and proofs of the milestones in any order, are welcome.

Selected references

• V. A. Nguyen, F. Zhang, S. Wang, J. Blanchet, E. Delage, Y. Ye, Robustifying conditional portfolio decisions via optimal transport, Operations Research, 2025; arXiv:2103.16451v3, 2024. https://arxiv.org/abs/2103.16451
• P. Mohajerin Esfahani, D. Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric, Mathematical Programming 171, 2018. https://doi.org/10.1007/s10107-017-1172-1
• J. Blanchet, K. Murthy, Quantifying distributional model risk via optimal transport, Mathematics of Operations Research 44, 2019. https://doi.org/10.1287/moor.2018.0936
• R. Gao, A. Kleywegt, Distributionally robust stochastic optimization with Wasserstein distance, Mathematics of Operations Research 48, 2023. https://doi.org/10.1287/moor.2022.1275
• A. Esteban-Pérez, J. M. Morales, Distributionally robust stochastic programs with side information based on trimmings, Mathematical Programming, 2022.
• A. Shapiro, On duality theory of conic linear problems, in Semi-Infinite Programming, Kluwer, 2001, pp. 135–165.
• M. Sion, On general minimax theorems, Pacific Journal of Mathematics 8, 1958, pp. 171–176.
• C. Villani, Optimal Transport: Old and New, Springer, 2009. https://doi.org/10.1007/978-3-540-71050-9

Motivation

An investor who does not know the expected return of a stock faces two kinds of uncertainty: risk (the randomness of returns given the model) and ambiguity (uncertainty about which model is true). The smooth ambiguity criterion of Klibanoff, Marinacci and Mukerji (Econometrica 2005) separates the two: the investor first computes the expected utility $\mathbb E[U(X_T)\mid Z=z]$ under each candidate model $z$, then averages a transformation $\phi$ of these values over her belief about $Z$. Risk aversion lives in $U$, ambiguity aversion in the curvature of $\phi$.

When the investor also learns $Z$ from prices, her belief changes over time and the nonlinear $\phi$ makes the problem time-inconsistent: a strategy that is optimal today is no longer optimal tomorrow. Following Strotz and the continuous-time game-theoretic approach of Björk, Khapko and Murgoci (Finance and Stochastics 2017), the paper of Guan, Liang and Xia (arXiv:2302.08181, Mathematics of Operations Research, 2025) looks for an equilibrium strategy: a rule that no short deviation can improve to first order. It derives a verification theorem and, for a Gaussian prior with CARA utility and power or logarithmic $\phi$, the equilibrium in closed form.

Setting

A bond pays the constant rate $r$. The stock satisfies $dS_t/S_t=Z\,dt+\sigma\,dW_t$ with volatility $\sigma>0$, a standard Brownian motion $W$, and an unknown constant drift $Z$ with prior $N(\beta_0,\sigma_0^2)$, $\sigma_0>0$. The horizon is $T>0$. Observing prices, the investor's posterior at time $t$ is $N(\beta_t,\zeta(t))$ with

$$\zeta(t)=\frac{\sigma^2\sigma_0^2}{\sigma^2+t\sigma_0^2}.$$

A strategy is a feedback function $\pi(t,x,\beta)$: the amount held in the stock given time, wealth $x$ and posterior mean $\beta$.

The inside system fixes a hypothetical drift value $z$ and starts at $(t,x,\beta)$: the posterior mean evolves by $d\beta^z_s=\frac{\zeta(s)}{\sigma^2}[(z-\beta^z_s)ds+\sigma dW_s]$ and the wealth by

$$dX_s=rX_s\,ds+\pi(s,X_s,\beta^z_s)(z-r)\,ds+\pi(s,X_s,\beta^z_s)\,\sigma\,dW_s .$$

Utility is CARA, $U(x)=-\frac1ke^{-kx}$ with $k>0$, and $\phi(u)=-\frac1\alpha(-u)^\alpha$ for $\alpha\ne0$, $\phi(u)=-\log(-u)$ for $\alpha=0$, on $u<0$; $\phi$ is concave for $\alpha>1$, linear for $\alpha=1$ and convex for $\alpha<1$. The inside expected utility is $g^{z,\pi}(t,x,\beta)=\mathbb E[U(X^{z,\pi}_T)\mid X_t=x,\beta^z_t=\beta]$, and the objective is

$$J(t,x,\beta,\pi)=\int\phi\big(g^{z,\pi}(t,x,\beta)\big)\,N(\beta,\zeta(t))(dz).$$

A strategy is admissible ($\pi\in\Pi$, Definition 5.1) if the wealth equation is well posed and certain exponential and power moments are bounded by $Ce^{\epsilon z^2}$ over short horizons. $\pi^*\in\Pi$ is an equilibrium strategy (Definition 2.1) if for every $(t,x,\beta)$ and $\pi\in\Pi$, the spike $\pi_{t,h}$ (equal to $\pi$ on $[t,t+h)$ and $\pi^*$ afterwards) satisfies

$$\limsup_{h\to0^+}\frac{J(t,x,\beta,\pi_{t,h})-J(t,x,\beta,\pi^*)}{h}\le0 .$$

Formalization targets

Goal: Theorem 5.5

Let $\alpha^*<0$ be the unique solution of $\big(1+\sigma_0^{-2}\sigma^2T^{-1}\big)\int_0^\infty e^{-x}(1+(1-\alpha^*)x)^{\alpha^*/(1-\alpha^*)}dx=1$ and assume $\alpha>\max\{\alpha^*,1-0.5\sigma_0^{-2}\sigma^2T^{-1}\}$. Let $m_1,\dots,m_6$ solve the ODE system (5.10) and put $a_1(t,m_2,m_3)=\frac{(1+\zeta m_2)(\zeta^{-1}+\alpha m_2)}{\zeta^{-1}-\alpha m_3}$. Then

$$\pi^*(t,x,\beta)=\frac{e^{-r(T-t)}}{k\sigma^2}\big\{a_1(t,m_2(t),m_3(t))+\zeta(t)m_1(t)\big\}(\beta-r)$$

is an equilibrium strategy, and its inside expected utility is

$$g^z(t,x,\beta)=-\frac1k\exp\Big\{-ke^{r(T-t)}x+\tfrac12m_1\beta^2+m_2\beta z+\tfrac12m_3z^2+m_4\beta+m_5z+m_6\Big\}.$$

Milestones

The milestones follow the paper's proof: the Gaussian moment formula (Lemma C.2) and short-horizon bounds on $\beta^z$ (Lemma C.1); admissibility under a growth bound (Proposition 5.2) and of $\pi^*$; the critical exponent, unique solvability and closed form of the reduced system (5.16) with its sign bounds (Proposition 5.4); the closed-form remaining coefficients (5.18); the extended HJB conditions (Lemma 5.3); and three verification steps of Appendix G (a martingale property, uniform integrability, and continuity (G.5)).

Significance

Theorem 5.5 gives an explicit, time-consistent investment rule under learning and smooth ambiguity. The rule is linear in the posterior mean, and its coefficient splits into the myopic Merton demand and two hedging demands, one against ambiguity in $Z$ and one against changes in $\beta$; the paper's numerical section uses this decomposition to study how ambiguity attitude changes the demand for the risky asset. At $\alpha=1$ the criterion is expected utility and $\pi^*$ is the Bayesian-learning Merton rule. The critical exponent $\alpha^*$ marks how much ambiguity seeking the closed form can tolerate.

The result is proved in the paper (Appendices C–G) but not formalized. The formalization would add to the platform its first smooth-ambiguity preference, its first equilibrium notion for a time-inconsistent continuous-time control problem with feedback strategies, and its first Bayesian drift-learning model. Lemma C.2 and the posterior-mean process are reusable beyond this paper.

Difficulty

Every quantity in the equilibrium inequality is a law-level functional of a controlled SDE: $g^{z,\pi}$ is an expectation over the wealth under the hypothetical drift, and $J$ integrates $\phi(g^{z,\pi})$ against the Gaussian posterior over $z$. The obvious route, a Taylor expansion of $J(\pi_{t,h})$ in $h$, needs to interchange the limit $h\to0^+$ with two nested expectations, one of which ranges over an unbounded drift $z$. Exponential utility makes $g^z$ grow like $e^{cz^2}$, so every estimate has to be uniform in $z$ up to a factor $e^{\epsilon z^2}$ that the Gaussian posterior can absorb, and the convex case $\alpha<1$ needs the extra lower bound on $\alpha$. On the deterministic side, the six-dimensional Riccati-type system (5.10) is nonlinear; its solvability on all of $[0,T]$ is not automatic and fails beyond the critical exponent $\alpha^*$.

Formalization scope

Time is real, with $t\in[0,T]$ explicit; Mathlib's Brownian motion IsBrownianReal is indexed by $\mathbb R_{\ge0}$ and read at s.toNNReal. Every statement holds on every probability space carrying a Brownian motion. Standing hypotheses: $\sigma,\sigma_0,T,k>0$; $Z$ ranges over $\mathbb R$.

Conventions committed to:

1. Conditioning on a state $\mathbb E[\cdot\mid X_t=x,\beta_t=\beta]$ is the inside system started at $(t,x,\beta)$ and driven by $W_s-W_t$, as the paper's appendices use it; $\mathbb E[\cdot\mid\beta_t=\beta]$ for functions of $Z$ is the posterior $N(\beta,\zeta(t))$, cited from Bismuth, Guéant and Pu (2019), not re-derived.
2. $\beta^z$ is the explicit formula (C.1). The wealth is a strong solution of (5.2) with the published Itô integral EthierKurtz.HasBrownianItoIntegral against the shifted Brownian motion.
3. Definition 5.1(i) is read on the inside system: for every $z$ and every start, the wealth equation is uniquely solvable.
4. The equilibrium predicate requires $J$ to be finite (for $\pi^*$, and for spikes with small $h$), the $\limsup$ is in $\epsilon$-form, and $t\in[0,T)$.
5. Every ODE solution keeps $\zeta^{-1}-\alpha m_3>0$ on $[0,T]$; time derivatives are taken within $[0,T]$.
6. Suprema claimed to equal $0$ are least upper bounds; $\Psi^{-1}$ is written as an equation for $\Psi$.

Each of these rules out a trivializing formalization: a Bochner expectation of a non-integrable utility is $0$ in Lean, which would make $\phi(0)$ and $J$ meaningless; a ⨆ of an unbounded set is $0$, the claimed value; an ODE "solution" through a zero of the denominator would solve a junk system; and a goal over an empty set of $\alpha^*$ or of ODE solutions would be vacuous, which Proposition 5.4 and (5.18) exclude. Corrected readings of printed slips (the conditioning $X^{\pi^*}_T=x$ in Theorems 3.5/4.5 read as $X^{\pi^*}_t=x$; Step 1 stated from every start) are disclosed in the items.

The proofs use, beyond Mathlib: Itô's formula, the Burkholder–Davis–Gundy inequality, Novikov's criterion, the generalized Gronwall inequality (Willett–Wong 1965, Theorem 2), and the Gaussian filter. Contributions of these tools as separate results are welcome. Not included: Lemmas C.3 and D.1, Steps 2, 4 and 5 of Appendix G, and the verification Theorems 3.5 and 4.5.

Selected references

• G. Guan, Z. Liang, J. Xia, Equilibrium Portfolio Selection for Smooth Ambiguity Preferences, Mathematics of Operations Research, 2025; preprint arXiv:2302.08181v1, 2023. https://arxiv.org/abs/2302.08181
• P. Klibanoff, M. Marinacci, S. Mukerji, A Smooth Model of Decision Making under Ambiguity, Econometrica 73(6), 2005. https://doi.org/10.1111/j.1468-0262.2005.00640.x
• T. Björk, M. Khapko, A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance and Stochastics 21(2), 331–360, 2017. https://arxiv.org/abs/1612.03650
• A. Bismuth, O. Guéant, J. Pu, Portfolio choice, portfolio liquidation, and portfolio transition under drift uncertainty, Mathematics and Financial Economics 13(4), 661–719, 2019. https://arxiv.org/abs/1611.07843
• D. Willett, J. S. W. Wong, On the discrete analogues of some generalizations of Gronwall's inequality, Monatshefte für Mathematik 69(4), 362–367, 1965. https://doi.org/10.1007/BF01297622

Motivation

Ranking and selection (R&S) is the problem of identifying, by simulation, the alternative with the largest mean performance among $k$ candidates, when each alternative can only be observed with noise. It is a basic tool of simulation optimization, with roots in Bechhofer's indifference-zone formulation (Bechhofer 1954). Cloud and parallel computing have made problems with thousands to millions of alternatives routine, and for such large-scale problems the relevant question is how the total sampling budget must grow with $k$ to keep the probability of correct selection (PCS) from collapsing.

Li, Fan and Hong (arXiv:2303.02951, Management Science 2025) call a procedure sample optimal if a budget linear in $k$, $B=ck$, keeps $\liminf_{k\to\infty}\mathrm{PCS}>0$. Order $k$ is the least any procedure can use, since every alternative needs at least one observation. As the paper recalls, Zhong & Hong (2022) and Hong et al. (2022) show that well-known procedures such as KN (Kim & Nelson 2001) and OCBA (Chen et al. 2000) need a budget of order $k\log k$; the procedures previously proved sample optimal, median elimination (Even-Dar et al. 2006) and the knockout-tournament procedures, rely on a rigid halving structure with carefully planned allocations in each round. The paper's surprising observation is that the simplest procedure of all, the greedy procedure, which always samples the alternative that currently looks best, is already sample optimal.

Setting

There are $k$ alternatives. Alternative $i$ yields independent observations $X_{i1},X_{i2},\dots\sim N(\mu_i,\sigma_i^2)$, independent across alternatives, and alternative $1$ is the unique best: $\mu_1>\mu_i$ for $i\ge2$. Write $\bar X_i(n)=\frac1n\sum_{j=1}^nX_{ij}$ for the running average of alternative $i$.

The greedy procedure (Procedure 1) with total budget $B$:

1. takes one observation from each alternative;
2. while the total number of observations is below $B$, takes one more observation from the alternative with the largest current running average $\bar X_i(n_i)$;
3. selects the alternative with the largest final running average.

The PCS is the probability that alternative $1$ is selected. The asymptotic regime is fixed by Assumption 1: there are constants $\gamma>0$ and $\bar\sigma^2>0$ with $\mu_1-\max_{i\ge2}\mu_i\ge\gamma$ and $\max_i\sigma_i^2\le\bar\sigma^2$, however large $k$ is.

For an i.i.d. $N(0,1)$ sequence with running average $\bar Z(n)$, the crossing time of a level $x$ is $\tilde N(x)=\inf\{n\ge1:\bar Z(n)<x\}$. With $\Phi$ the standard normal distribution function, the paper's key function is

$$C(x)=\exp\Bigl(\sum_{n=1}^\infty\frac1n\,\Phi(-\sqrt n\,x)\Bigr),\qquad x>0,$$

which equals $\mathrm E[\tilde N(x)]$ (Lemma 2). Finally, $\bar X_1^*=\min_{n\ge1}\bar X_1(n)$ is the boundary, $N_1^*$ the first time it is attained, and $N_i(b)=\inf\{n\ge1:\bar X_i(n)<b\}$ the boundary-crossing time of alternative $i$.

Formalization targets

Goal: Theorem 1 (p. 15)

Under Assumption 1, if $B=ck$ with $c>C(\gamma/\bar\sigma)$, then there is $\gamma_0\in(0,\gamma)$ with $C\bigl((\gamma-\gamma_0)/\bar\sigma\bigr)=c$, and

$$\liminf_{k\to\infty}\mathrm{PCS}\ \ge\ \Pr\Bigl\{\min_{n\ge1}\bar X_1(n)>\mu_1-\gamma_0\Bigr\}=\Bigl[C\Bigl(\frac{\gamma_0}{\sigma_1}\Bigr)\Bigr]^{-1}>0.$$

The constant $c$ and the gap $\gamma$ are free; the bound depends only on the shape of $C$.

Milestones

1. Lemma 1 (p. 13): $\Pr\{\arg\min_{n\ge1}\bar Z(n)<\infty\}=1$.
2. Lemma 2 (pp. 13–14): for $x>0$, $\Pr\{\tilde N(x)<\infty\}=1$ and $\mathrm E[\tilde N(x)]=C(x)$, continuous and strictly decreasing; for $x<0$, $\mathrm E[\tilde N(x)]=\infty$ and $\Pr\{\tilde N(x)<\infty\}=1-\exp(-\sum_n\frac1n\Phi(\sqrt n\,x))$; for $x=0$, probability $1$ and infinite mean.
3. (4.3) (p. 14): for $x<0$, $\Pr\{\min_{n\ge1}\bar Z(n)>x\}=[C(-x)]^{-1}$.
4. (3.2) (p. 12): $\mathrm{PCS}\ge\Pr\{\sum_{i=2}^kN_i(\bar X_1^*)+N_1^*\le B\}$.
5. (4.4) (p. 15): on $\{\bar X_1^*>\mu_1-\gamma_0\}$, $N_i(\bar X_1^*)\le\tilde N_i((\gamma-\gamma_0)/\bar\sigma)$, pathwise.
6. (EC.1.5) (p. ec3): $\liminf_k\mathrm{PCS}\ge\Pr\{\min_n\bar X_1(n)>\mu_1-\gamma_\epsilon^-\}-\Pr\{\arg\min_n\bar X_1(n)=\infty\}$, where $C((\gamma-\gamma_\epsilon^-)/\bar\sigma)=c-2\epsilon$.

Significance

The result. Theorem 1 shows that a procedure with no tuning parameters, no elimination and no knowledge of $\gamma$ or $\bar\sigma$ achieves the optimal order of budget growth for large-scale R&S. It gives an explicit, computable lower bound on the limiting PCS. The companion Theorem 2 of the paper shows that this bound is the exact limit under the slippage configuration with a common variance, so the constant cannot be improved in general. The same boundary-crossing analysis underlies the paper's explore-first variant, which turns the positive limit into an arbitrarily high one.

Formalizing it. The result is proved in the paper; nothing here is claimed open. As far as is known, no part of it has a machine-checked proof. A formalization adds:

• a precise, total specification of the greedy procedure as a function of an observation array, including tie-breaking and budget rounding;
• the pathwise inclusion behind (3.2), which the paper argues in prose, and whose naive reading is false before $N_1^*$;
• Spitzer-type identities for the Gaussian random walk (Lemma 2), which Mathlib does not have and which are reusable well beyond this paper.

Difficulty

The obvious argument fails at its first step. The greedy procedure's sample sizes depend on all past observations of all alternatives, so the PCS has no tractable expression, and the classical route through pairwise comparisons at fixed sample sizes does not apply. The boundary-crossing perspective sidesteps this, at three costs. First, (3.2) requires a careful pathwise argument: before the best alternative's running average reaches its minimum, it can lead, be sampled, and lose the lead, so "once best, always best" is false until $N_1^*$. Second, Lemma 2 is a fluctuation-theory identity (Spitzer's formula for the mean ladder epoch of a Gaussian walk), not an elementary computation. Third, the limit $k\to\infty$ combines a strong law over i.i.d. crossing times with a continuity argument in $\gamma_0$, which needs the continuity and strict monotonicity of $C$.

Formalization scope

• Model. One observation array $X_{ij}$ indexed by $i,j\in\mathbb N$, measurable, with $X_{ij}\sim N(\mu_i,v_i)$, and mutually independent as a whole family. Problem $k$ uses alternatives $0,\dots,k-1$; Lean's alternative $0$ is the paper's alternative $1$. One sequence $(\mu_i,v_i)$ serves every $k$, not a triangular array. Variances are positive ($v_i>0$), as the paper's normalizations by $\sigma_i$ require.
• Procedure. A deterministic map from the array to the selected index. Ties in every arg max go to the smallest index (a null event under normality). The budget $B=ck$ is $\lfloor ck\rfloor$; the number of greedy steps uses truncated subtraction, so the procedure is total.
• Quantities. Crossing times and argmins take values in $\mathbb N\cup\{\infty\}$; $\mathrm E[\tilde N(x)]$ is a lower Lebesgue integral in $[0,\infty]$; probabilities and the liminf are in $[0,\infty]$. "$\min_n y(n)>a$" means bounded below with infimum $>a$. $C$ is applied only at positive arguments.
• Ruled out. Crossing times typed in $\mathbb N$ (where an empty infimum is $0$), the junk value of a divergent tsum ($C(x)=1$ for $x\le0$), an unbounded infimum read as $0$, and a "for every $\gamma_0$ with $C(\cdot)=c$" statement that holds vacuously: the goal asserts the existence of $\gamma_0$.
• Infrastructure. Running averages and crossing times of random walks, the Gaussian Spitzer identity, the strong law for i.i.d. $\mathbb N\cup\{\infty\}$-valued variables, and a pathwise analysis of the greedy procedure. The random-walk results are reusable for any boundary-crossing or sequential-testing development. Contributions on any milestone are welcome, and proofs of Lemma 2 by a route other than Siegmund's are equally useful.
• Source. The PDF is the arXiv v3 preprint (11 Dec 2023); every page and equation number refers to it.

Selected references

• Z. Li, W. Fan, L. J. Hong, The (Surprising) Sample Optimality of Greedy Procedures for Large-Scale Ranking and Selection, arXiv:2303.02951v3, 2023; Management Science, 2025. https://arxiv.org/abs/2303.02951
• R. E. Bechhofer, A Single-Sample Multiple Decision Procedure for Ranking Means of Normal Populations with known Variances, Annals of Mathematical Statistics 25(1), 1954. https://doi.org/10.1214/aoms/1177728845
• D. Siegmund, Sequential Analysis: Tests and Confidence Intervals, Springer, 1985. https://doi.org/10.1007/978-1-4757-1862-1
• Y. Zhong, L. J. Hong, Knockout-Tournament Procedures for Large-Scale Ranking and Selection in Parallel Computing Environments, Operations Research 70(1), 2022. https://doi.org/10.1287/opre.2020.2065

Motivation

Reward-based crowdfunding and venture capital are two of the main sources of early-stage finance, and the empirical record on how they interact points both ways: some studies find that more crowdfunding activity is followed by more VC investment, others find the opposite. Hu, Jin and Keppo (SSRN 3512638; Management Science, 2024, doi:10.1287/mnsc.2022.00994) explain both signs with one firm-level model. A crowdfunding campaign has two roles: it raises a modest amount of money, and its outcome is a public signal about whether the product will succeed. Two VC firms with different priors compete for the deal, and the more optimistic one may prefer to wait for the campaign's outcome rather than invest at once.

The paper builds on earlier work on the informational value of crowdfunding for professional investors (Roma, Gal-Or and Chen, 2018; Babich, Marinesi and Tsoukalas, 2021) and on crowdfunding as a mechanism (Strausz, 2017). Its main result, Proposition 6, says exactly when the mere option of crowdfunding lowers the expected VC investment and when it raises it.

Setting

A startup needs at least $R_0$ to launch a project whose revenue is $R_s$ if it succeeds and $R_f$ if it fails, with $R_f < R_0 < R_s$; each state has probability $1/2$. It offers an investing VC firm an equity share $\alpha \in [0,1]$. VC$_1$ and VC$_2$ believe the startup succeeds with probabilities $p_1 \ge p_2$, the startup itself with probability $p_a$; under a belief $p$ the expected revenue is $pR_s + (1-p)R_f$, giving $e_1, e_2, e_a$. A VC's valuation of the share is $v_i = \alpha e_i$.

The investment level at which VC$_1$ acquires the share is

$$\kappa(v_1, v_2, R) = \begin{cases} 0 & v_1 < R,\\ \beta v_1 + (1-\beta)\max\{v_2, R\} & v_1 \ge R,\end{cases}$$

where $\beta \in [0,1]$ is the startup's negotiation power. VC$_2$ never invests; it matters only as the startup's outside option.

Timeline. At $t_0$ VC$_1$ may invest. If it does not, the startup runs a crowdfunding campaign with target $K \in (0, R_0)$, which produces a signal $S$ (success, $K$ collected) or $F$ (failure) with $P(S\mid s) = P(F\mid f) = \tfrac12 + \tfrac\rho2$, where $\rho \in [0,1]$ is the signal's accuracy. All parties update by Bayes' rule, with posteriors $\tilde p|S$, $\tilde p|F$ and expected revenues $\tilde e_i|S$, $\tilde e_i|F$. At $t_1$ VC$_1$ reconsiders: after $S$ the requirement is $R_0 - K$ and the share is scaled to $\alpha' = (R_0 - K)\alpha/R_0$; after $F$ both stay as they were. Assumption (I) requires $\tilde e_1|S \ge R_0$.

VC$_1$ compares its surplus from investing now, $\pi_1 = v_1 - \kappa(v_1, v_2, R_0)$, with its expected surplus from waiting, $\tilde\pi_1$; the value of waiting is $\Delta_1 = \tilde\pi_1 - \pi_1$, and VC$_1$ invests at $t_0$ when $v_1 \ge R_0$ and $\Delta_1 \le 0$. The paper shows that this happens exactly when $\alpha \ge \alpha_0$, for an explicit threshold $\alpha_0$ (Proposition 2). The startup chooses $\alpha$ to maximize its expected payoff. Without crowdfunding it offers $\alpha^o = R_0/e_1$. With crowdfunding it offers $\alpha^c = \alpha_0$ or $\alpha^c = R_0/\tilde e_1|S$, according to the sign of its own value of waiting $\Delta_a$ (Proposition 4). The expected VC investment level $I^o$ or $I^c$ is the level along VC$_1$'s decisions, averaged over the signal under the unconditional belief $P(\theta = s) = 1/2$.

Formalization targets

Goal: Proposition 6

Under the standing assumptions,

$$I^c < I^o \quad\text{if } e_1 \ge R_0 \text{ and } \Delta_a > 0, \qquad I^o \le I^c \quad\text{if } e_1 \ge R_0 \text{ and } \Delta_a \le 0,$$

and if $e_1 < R_0$ no share in $[0,1]$ draws an investment without crowdfunding, while $I^c > 0$.

Milestones

1. Proposition 1: the no-crowdfunding benchmark $\alpha^o = R_0/e_1$, $I^o \in \{0, R_0\}$.
2. Proposition 2(i): VC$_1$ invests at $t_0$ iff $\alpha \ge \alpha_0$, with $\alpha_0$ given by the printed closed forms $T_1$, $T_2$.
3. Proposition 2(ii): if VC$_1$ waits, it invests only after $S$, and then iff $\alpha \ge R_0/\tilde e_1|S$.
4. Corollary 1 and Proposition 3(i): $R_0/\tilde e_1|S \le \alpha_0$ and $R_0/e_1 \le \alpha_0 \le R_0/e_2$.
5. The startup's objective with crowdfunding (§5.2) equals its anticipated payoff.
6. Proposition 4: $\alpha^c$ is optimal.
7. The closed form of $I^c$ (proof of Proposition 6).
8. Companions: Corollary 2 (the effect on the equity share), Proposition 3(ii) and Proposition 5 ($\alpha_0$ and $\Delta_a$ decrease in $p_2$).

Significance

Proposition 6 separates two populations of startups. For a startup that VC$_1$ would fund anyway ($e_1 \ge R_0$) and that prefers to induce a delay ($\Delta_a > 0$), crowdfunding lowers expected investment: after success the requirement drops to $R_0 - K$, and after failure nobody invests. For a startup that VC$_1$ would not fund on its prior ($e_1 < R_0$), crowdfunding creates investment that would otherwise not exist. If the startup prefers to keep the investment before the campaign ($\Delta_a \le 0$), the mere option of crowdfunding raises the level even though no campaign is run. Together these account for both signs of the empirical crowdfunding–VC relationship.

The paper's proofs are case analyses on closed forms. No part of this model has been machine-checked, and nothing on the platform models crowdfunding, VC competition or this signal structure. A formalization checks the seven-subcase comparison behind Proposition 2. It also pins down hypotheses the paper uses without stating them: nonnegative revenue, $\beta < 1$, $e_a > 0$, $\rho > 0$ and $p_1 < 1$. Each is needed, and each has an explicit counterexample without it.

Difficulty

The hard step is Proposition 2(i). Both $\pi_1$ and $\tilde\pi_1$ are piecewise linear in $\alpha$, with breakpoints where $v_2$, $\tilde v_2|S$, $\tilde v_1|S$ and $\tilde v_1|F$ cross their requirements. The claim that $\{\alpha : \Delta_1 \le 0\} \cap \{v_1 \ge R_0\}$ is a single up-ray $[\alpha_0, 1]$ needs the relative order of these breakpoints. Which closed form ($T_1$ or $T_2$) is the threshold depends on where VC$_2$'s post-signal valuation sits. The threshold is therefore not read off a single formula: each ordering of the breakpoints is a separate case, and the paper's proof runs through seven of them. Everything downstream, including the startup's optimum and the comparison of investment levels, needs the threshold to be exact.

Formalization scope

Everything is elementary real arithmetic, with no measure theory. The parameters are a record of nine reals. "Expected" means the two-term average over $S$ and $F$, with $P(S\mid p) = p(\tfrac12 + \tfrac\rho2) + (1-p)(\tfrac12 - \tfrac\rho2)$, the mixture implied by the signal accuracies (the paper does not print it). The posteriors, $T_1$, $T_2$, $\alpha_0$, $\pi_a$, $\tilde\pi_a$, $\Delta_a$ and $\alpha^c$ are the printed closed forms. VC$_1$'s decisions, the startup's payoffs and the investment levels are defined from the surpluses and from $\kappa$. A VC that cannot meet the requirement after a signal gets surplus $0$, as in the paper's proof (the printed $\max\{\cdot, 0\}$ would give it its valuation). VC$_1$'s ties go to investing now and the startup's ties to $\alpha_0$. "Optimal" means a global maximizer over $[0,1]$. "Raises" is weak where equality can occur ($\beta = 0$, $p_2 = p_1$), and comparative statics in $p_2$ are weak.

Standing assumptions on every statement: $R_f < R_0 < R_s$, $0 < K < R_0$, $\rho \le 1$, $0 \le \beta$, $0 \le p_2 \le p_1 \le 1$, $0 \le p_a \le 1$, $\tilde e_1|S \ge R_0$. Added hypotheses:

• $0 \le R_f$ on every statement;
• $\beta < 1$ on Propositions 2 and 4, the startup's objective, the $I^c$ formula and the goal;
• $e_a > 0$ on Proposition 4, the $I^c$ formula and the goal;
• $\rho > 0$ and $p_1 < 1$ only on the branch $\Delta_a > 0$: the $I^c$ formula's second case, Corollary 2's "reduces" and the goal's "reduces" case.

Proposition 3(i)'s bound $\alpha_0 \le R_0/e_2$ is stated as $\alpha_0 e_2 \le R_0$, because $e_2 = 0$ is allowed.

The goal compares investment levels computed from VC$_1$'s decisions, not the paper's closed forms $I^o$, $I^c$, whose comparison is three lines of algebra. The threshold $\alpha_0$ is not defined as the infimum of the shares at which VC$_1$ invests, which would make Proposition 2(i) a tautology. The goal is stated at the paper's selected share $\alpha^c$, not at every maximizer.

Not formalized: Online Appendix A (the startup re-optimizes $\alpha'$ after the campaign), Appendix B (profitability for the startup and VC$_1$), and Appendix C (comparative statics in $K$ and $\rho$). Proofs of any milestone are welcome. The case analysis of Proposition 2(i), stated as lemmas about the piecewise-linear surpluses, would be reusable by every downstream item.

Selected references

• M. Hu, Y. Jin, J. Keppo, The Crowdfunding Effects on Venture Capital Investment, Management Science, 2024. SSRN author manuscript: https://ssrn.com/abstract=3512638; journal: https://doi.org/10.1287/mnsc.2022.00994
• V. Babich, S. Marinesi, G. Tsoukalas, Does crowdfunding benefit entrepreneurs and venture capital investors?, Manufacturing & Service Operations Management 23(2), 508–524, 2021.
• P. Roma, E. Gal-Or, R. R. Chen, Reward-based crowdfunding campaigns: Informational value and access to venture capital, Information Systems Research 29(3), 679–697, 2018.
• R. Strausz, A theory of crowdfunding: A mechanism design approach with demand uncertainty and moral hazard, American Economic Review 107(6), 1430–1476, 2017. https://doi.org/10.1257/aer.20151700

Motivation

Large online retailers such as Amazon run two businesses on one storefront. They sell products themselves, and they host a marketplace on which third-party sellers sell, paying the platform a percentage of their revenue. A retailer that also runs the marketplace can decide which products to sell itself, and it controls which offer most consumers see. Competition authorities have asked whether this dual role lets the platform appropriate third-party sellers' business, and empirical work (Zhu and Liu, 2018) finds that Amazon enters only a small, high-performing subset of marketplace products.

Hervas-Drane and Shelegia (Retailer-Led Marketplaces, Management Science, 2025) give a model in which these choices can be computed. Its main result, Proposition 2, says which products the platform sells itself when its assortment is limited. This mission formalizes that proposition together with the pricing and entry results it rests on.

Setting

There is a unit mass of products. Each product is an independent market indexed by its consumers' valuation $v$. Valuations are distributed according to a probability measure $\mu$ on $[c,\infty)$ with CDF $G(v)=\mu((-\infty,v])$. Every unit costs $c>0$ to supply. Each market has a unit mass of unit-demand consumers, who buy at a price $p$ only if $p\le v$.

The monopolist retailer $M$ sets an ad valorem fee $f\in[0,1]$ on the third-party seller's revenue. He can steer a share $\lambda$ of consumers (the inattentive ones) to the seller of his choice; the remaining $1-\lambda$ buy from the cheapest offer, with ties going to $M$. He has capacity $k\in[0,1]$: he can supply at most a fraction $k$ of all products.

The game has three stages. In Stage 1 $M$ sets $f$. In Stage 2 each product's third-party seller $T$ decides whether to enter, and then $M$ chooses which products to enter. In Stage 3, in each market both present firms price sequentially: $T$ quotes first, then $M$ quotes and steers. The solution concept is subgame perfect equilibrium.

Proposition 1 solves Stage 3. A product supplied by $M$ alone (SBM) earns him $\pi_M^{SBM}=v-c$. A product supplied by $T$ alone (SBT) earns her $(1-f)v-c$ and earns him the fee $\pi_M^{SBT}=fv$. When both supply it (SBMT), $T$ charges

$$\bar p_T=\frac{\lambda v+c(1-\lambda)}{1-f(1-\lambda)},$$

and $M$ earns $\pi_M^{SBMT}=\lambda(v-c)+(1-\lambda)f\bar p_T$. Lemma 1 says $T$ enters exactly the products $v\ge v_T=c/(1-f)$, the marketplace products. An allocation of $M$ is any measurable set $A$ of products with $\mu(A)\le k$. His total profit is the SBM profit on $A$ below $v_T$, plus the SBMT profit on $A$ above $v_T$, plus the fee on marketplace products outside $A$.

Formalization targets

Goal: Proposition 2

Assume $\lambda\ge\tfrac12$ (Assumption 1) and $f<1$. Let $\Delta\pi(v)=\pi_M^{SBMT}(v)-\pi_M^{SBT}(v)$ and

$$v^*=\frac{c\lambda}{(1-f)^2(\lambda-f(1-\lambda))},\qquad \underline k=1-G(v^*).$$

Then:

• Mode I. If $k\le\underline k$, the optimal allocation is unique up to $\mu$-null sets and equals $[v_K,\infty)$ with $1-G(v_K)=k$ (empty if $k=0$).
• Mode II. If $k>\underline k$, it is $[v_L,v_T)\cup[v_H,\infty)$, where

$$\pi_M^{SBM}(v_L)=\Delta\pi(v_H),\qquad (1-G(v_H))+(G(v_T)-G(v_L))=k.$$

Milestones

1. Proposition 1, its three supply configurations: the SBM monopoly price, the SBT equilibrium, and the SBMT subgame perfect equilibrium, which is unique for $0<\lambda<1$ and $v(1-f)>c$.
2. Lemma 1 with (1) and Property 1: the entry threshold $v_T$.
3. The closed form of $\Delta\pi$, its positivity above $v_T$, and its monotonicity in $v$ and in $f$.
4. (A.2): $\Delta\pi(v^*)=\pi_M^{SBM}(v_T)$, and $v^*\ge v_T$.
5. The $(v_L,v_H)$ system has a unique solution when $k>\underline k$.

Significance

Proposition 2 is the step that the rest of the paper builds on. Display (4) writes the monopolist's profit as a function of the fee by plugging in the thresholds $v_L$, $v_T$, $v_H$, $v_K$ of Proposition 2. The optimal fee (Lemma 2, Proposition 3) and the policy comparisons of §6 (a pure-retailer response, fee regulation, steering bans) are all statements about that function. The two modes are also the paper's empirical prediction. A platform with little capacity, effective steering and a high fee competes only in the top marketplace products. With more capacity it also takes products no third party sells.

The paper proves Proposition 2 in Appendix A. The proof is an informal capacity-allocation argument. No machine-checked version exists. A formal proof has to turn the argument about entering the products of highest marginal profitability first into a statement about all measurable allocations of mass at most $k$. It must also prove uniqueness up to null sets, and handle the corners $k=0$, $k=1$ and $f=0$ that the prose passes over.

Difficulty

The obvious argument ranks products by the marginal profitability of entry and fills capacity from the top. Two facts make this harder than it looks. First, the ranking is not monotone in $v$. Below $v_T$ the gain is $v-c$; above $v_T$ it is $\Delta\pi(v)$, which starts again from $0$ at $v_T$. So the optimal set is an upper level set of a piecewise function, and in Mode II it has two components. Second, the claim is about every measurable set of mass at most $k$, not about thresholds. Optimality needs an exchange argument against arbitrary sets, and uniqueness needs the level sets of the gain to be $\mu$-null. Locating the thresholds also needs continuity and strict monotonicity of $G$, which come from the density and the full-support hypothesis.

Proposition 1 has its own subtlety. At $p_T=\bar p_T$ the monopolist is indifferent between matching $T$'s price and charging $v$. $T$'s best price exists only if $M$'s strategy breaks this tie in $T$'s favour, so the equilibrium has to be stated for strategies defined at every price of $T$.

Formalization scope

Products form a probability measure μ : Measure ℝ. The standing hypotheses are:

• $\mu((-\infty,c))=0$, $\mu\ll$ Lebesgue (the density $g$), and Integrable (fun v => v) μ, which the paper never states but which is needed for profits to be finite;
• $\mu$ charges every nonempty open interval of $[c,\infty)$, the reading of $G^{-1}$ in (3);
• $c>0$, $f\in[0,1)$, $\lambda\in[\tfrac12,1]$, $k\in[0,1]$.

The foreclosed corner $f=1$ ($v_T=\infty$) is not covered. Marketplace products are encoded as $\{v: c\le v(1-f)\}$. Optimal allocations quantify over all measurable sets of mass at most $k$, and uniqueness is up to $\mu$-null sets. Restricting allocations to intervals or to unions of two intervals would assume the conclusion, and the formalization does not do it.

In Proposition 1, $M$'s strategy is a function of $T$'s price that is a best response at every price, and $T$'s price is optimal against it. Uniqueness of the SBMT outcome is stated only for $0<\lambda<1$ and $v(1-f)>c$, where it holds. Lemma 1's duopoly clause likewise needs $0<\lambda<1$.

Two definition files carry the model:

• Model: $G$, the marketplace set, the Proposition 1 payoffs, $\Delta\pi$, $v^*$, $\underline k$, allocations and profit;
• PricingGame: the three pricing games and their equilibria.

The bathtub-type lemma behind Proposition 2 is reusable beyond this mission: among measurable sets of mass at most $k$, an upper level set of a weight maximizes its integral. Contributions of that lemma, of the milestones, and of alternative proofs are welcome.

Selected references

• A. Hervas-Drane, S. Shelegia, Retailer-Led Marketplaces, Management Science, Articles in Advance, 2025. https://doi.org/10.1287/mnsc.2023.00315
• F. Zhu, Q. Liu, Competing with Complementors: An Empirical Look at Amazon.com, Strategic Management Journal 39:2618–2642, 2018. https://doi.org/10.1002/smj.2932
• A. Hervas-Drane, S. Shelegia, Price Competition with a Stake in Your Rival, International Journal of Industrial Organization 84:102862, 2022 (the simultaneous-pricing variant of the pricing game). https://doi.org/10.1016/j.ijindorg.2022.102862