Revenue Management and Choice Models

Motivation

A retailer who sells several products from shared, non-replenishable inventory over a finite season must set prices without knowing how demand responds to them. Every price posted is both a sale and an experiment. This is the network revenue management problem with demand learning, and it sits between two literatures: dynamic pricing with inventory, where demand is known and the fluid linear program of Gallego and van Ryzin (1997) is the standard benchmark, and multi-armed bandits, where learning is the whole problem but there are no resource constraints.

Ferreira, Simchi-Levi and Wang (Oper. Res. 2018) combine Thompson sampling with a linear-programming step: sample a demand model from the posterior, solve the fluid LP for that model, and randomize prices according to its solution. The same paper extends the scheme to continuous price sets, contextual pricing and bandits with knapsacks.

Timeline of the relevant results:

• 1997: Gallego and van Ryzin introduce the fluid LP upper bound for network revenue management with known demand.
• 2012: Besbes and Zeevi give a non-Bayesian network pricing algorithm with worst-case regret $O(K^{5/3}T^{2/3}\sqrt{\log T})$.
• 2013: Badanidiyuru, Kleinberg and Slivkins (bandits with knapsacks) give worst-case regret $O(\sqrt{KT\log T})$.
• 2013–2014: Bubeck and Liu and Russo and Van Roy give prior-free Bayesian regret bounds for Thompson sampling in unconstrained bandits.
• 2018: Ferreira, Simchi-Levi and Wang prove the $O(\sqrt{TK\log K})$ Bayesian regret bound for TS-fixed (Theorem 1), the target of this mission.

Setting

There are $N$ products and $M$ resources. One unit of product $i$ consumes $a_{ij}\ge0$ units of resource $j$, and resource $j$ starts with inventory $I_j\ge0$ that is never replenished. The season has $T$ periods. In each period the retailer posts one of $K$ price vectors $p_k=(p_{1k},\dots,p_{Nk})$ or a shut-off price $p_\infty$ under which demand is zero.

Given the posted price $p_k$, the demand vector $D(t)\in\mathbb R^N_+$ has law $F(\cdot\,;p_k,\theta)$, where $\theta\in\Theta$ is unknown and drawn from a known, arbitrary prior $\mu_0$. Demand is independent of the past given the posted price and $\theta$, and is bounded: $D_i(t)\in[0,\bar d_i]$. Write $d_{ik}(\rho)$ for the mean demand of product $i$ under $p_k$ and parameter $\rho$, and $d=d(\theta)$.

When inventory covers all demand, all demand is sold. Otherwise the satisfied demand $\tilde D(t)$ satisfies $0\le\tilde D_i(t)\le D_i(t)$, leaves every inventory nonnegative, and leaves at least one resource at zero; no other rule is imposed. Revenue is $\mathrm{Rev}(T)=\sum_t\sum_i\tilde D_i(t)P_i(t)$.

For a mean-demand matrix $d$ and capacities $c_j=I_j/T$, the linear program $\mathrm{LP}(d)$ is

$$\max_{x\ge0}\ \sum_{k=1}^K\Bigl(\sum_{i=1}^N p_{ik}d_{ik}\Bigr)x_k\quad\text{s.t.}\quad\sum_{k=1}^K\Bigl(\sum_{i=1}^N a_{ij}d_{ik}\Bigr)x_k\le c_j\ \ \forall j,\qquad\sum_{k=1}^K x_k\le1,$$

with optimal value $\mathrm{OPT}(d)$.

TS-fixed (Algorithm 1): in each period, sample $\theta(t)$ from the posterior of $\theta$ given the history of posted prices and observed demands; let $x(t)$ be an optimal solution of $\mathrm{LP}(d(\theta(t)))$; post $p_k$ with probability $x_k(t)$ and $p_\infty$ with the remaining probability; observe demand and update the posterior.

Finally $p_{\max}=\max_k\sum_ip_{ik}\bar d_i$ and $p^j_{\max}=\max_{i:a_{ij}\neq0,\,k}p_{ik}/a_{ij}$.

Formalization targets

Goal: Theorem 1 against the LP benchmark

For $K\ge2$, $T\ge1$, every prior, every bounded demand family, every admissible fulfilment rule and every run of TS-fixed,

$$\mathbb E\bigl[\mathrm{OPT}(d)\bigr]\cdot T-\mathbb E\bigl[\mathrm{Rev}(T)\bigr]\ \le\ \Bigl(18\,p_{\max}+37\sum_{i=1}^N\sum_{j=1}^M p^j_{\max}a_{ij}\bar d_i\Bigr)\sqrt{TK\log K}.$$

The paper prints this bound for $\mathrm{BayesRegret}(T)=\mathbb E[\mathrm{Rev}^*(T)]-\mathbb E[\mathrm{Rev}(T)]$, where $\mathrm{Rev}^*$ is the revenue of the optimal policy that knows $\theta$; see Formalization scope for why the LP benchmark is stated instead.

Milestones

The article states Theorem 1 and says that its proof is in the online appendix (Supplemental Material at the DOI). The article itself contains no numbered lemma. The milestone list is therefore empty; the appendix's lemmas will be added as milestones once the appendix is held.

Significance

The bound is prior-free and has explicit constants that depend only on prices, consumption rates and demand bounds. Its dependence on $T$ matches the $\Omega(\sqrt{KT})$ lower bound for Bayesian regret in unconstrained bandits with rewards in $[0,1]$, a special case of the model with no inventory constraints (Bubeck and Cesa-Bianchi 2012, Theorem 3.5). It shows that the posterior-sampling principle survives the addition of resource constraints, lost sales and randomized LP-based pricing, and it is the template for the paper's later results (TS-update, contextual pricing, bandits with knapsacks).

The theorem is proved on paper but, as far as a platform search shows, not formalized anywhere. The platform has a formal proof of the unconstrained Bayesian Thompson sampling bound $\sqrt{kn\log k/2}$ (BanditAlgorithm.thompson_sampling_bayesian_regret, Lattimore–Szepesvári Theorem 36.5) and an open single-product deterministic upper bound in revenue management (RevenueManagement.deterministic_upper_bound). Neither has inventory, an LP subroutine, or lost sales. A formal proof here would supply the first machine-checked analysis of Thompson sampling under resource constraints and would check the paper's constants.

Difficulty

In an unconstrained bandit, Thompson sampling's regret reduces to a sum of per-period gaps between an upper confidence bound and the sampled reward, because the sampled optimal arm and the true optimal arm are identically distributed given the history. Here the action is a randomized mixture $x(t)$ from an LP, the reward is not additive in the prices chosen, and revenue is lost when inventory runs out. Two quantities must be controlled: the revenue the algorithm would collect if all demand could be served, and the revenue lost to stock-outs. The second depends on the random time at which each resource is exhausted under a pricing rule that was optimized for a sampled, not the true, demand, and on an arbitrary fulfilment rule once some resource is empty. Standard bandit arguments do not bound such lost sales, which are a nonlinear function of the whole trajectory.

Formalization scope

Lean representation. Products, resources and price vectors are indexed by Fin N, Fin M, Fin K; the posted price is an Option (Fin K) with none the shut-off price. Periods are 0-based ($t=0,\dots,T-1$ stands for the paper's $1,\dots,T$). $\Theta$ is a standard Borel space with a probability measure $\mu_0$; demand is a Markov kernel $F$ from $\Theta\times$Fin K to $\mathbb R^N$, bounded in $[0,\bar d_i]$ for every parameter. A run of TS-fixed is a family of random variables on a probability space satisfying, almost surely and via conditional expectations: $\theta\sim\mu_0$; the posterior-sampling property of $\theta(t)$ given everything before period $t$; the price draw with probabilities $x(\theta(t))$ for a measurable optimal LP selection $x$; the demand law given the past, $\theta(t)$ and the posted price; and fulfilment rules (a)/(b). The logarithm is natural. Prices, consumption and inventory are nonnegative (implicit in the paper). $\mathrm{OPT}(d)$ is a supremum over a nonempty bounded feasible set, so it has no junk value.

Corrections to the printed statement.

1. $K\ge2$ is added. At $K=1$ the printed right-hand side is $0$, yet on a one-price instance with Bernoulli$(0.8)$ demand, $I=T/2$ and a point-mass prior, TS-fixed loses about $0.2p\sqrt T$ in expectation.
2. The LP benchmark replaces $\mathbb E[\mathrm{Rev}^*(T)]$. Section 3.1.1 bounds $\mathbb E[\mathrm{Rev}^*(T)\mid d]$ by $\mathrm{OPT}(d)\cdot T$, citing Gallego–van Ryzin. Under the paper's fulfilment rule this fails when products use disjoint resources: with two products, $I=(T,1)$, $p_1=(1,0)$, $p_2=(1/2,0)$ and deterministic demand $(1,1)$, the known-$\theta$ policy earns at least $T$ while $\mathrm{OPT}(d)\cdot T=1$. The paper states that its proof bounds the gap to "the LP benchmark defined in Section 3.1.1" (p. 1594), and the last display of Section 3.1.1 bounds $\mathrm{BayesRegret}(T)$ by exactly $\mathbb E[\mathrm{OPT}(d)]\cdot T-\mathbb E[\mathrm{Rev}(T)]$. Wherever the Gallego–van Ryzin bound holds, the corrected goal implies the printed one.

Ruled out. A bound for the "ideal" revenue $\sum_iD_i(t)P_i(t)$ instead of the satisfied revenue, or for an arbitrary policy whose prices are merely close to the LP solution, is not Theorem 1; the goal carries the full TS-fixed run and the lost-sales accounting.

Infrastructure needed. Posterior-sampling identities for general (standard Borel) priors, a Hoeffding/Azuma-type concentration for bounded demand along the price-selection process, LP sensitivity with respect to the mean-demand matrix, and a pathwise bound on lost sales under an arbitrary fulfilment rule. The LP and fluid-benchmark definitions are reusable for later missions on TS-update (Theorem 2), contextual pricing (Theorem 4) and bandits with knapsacks (Theorem 5). Contributions welcome: proofs of the goal, and formal statements of the online appendix's lemmas.

Selected references

• K. J. Ferreira, D. Simchi-Levi, H. Wang, Online Network Revenue Management Using Thompson Sampling, Operations Research 66(6):1586–1602, 2018. https://doi.org/10.1287/opre.2018.1755
• G. Gallego, G. van Ryzin, A Multiproduct Dynamic Pricing Problem and Its Applications to Network Yield Management, Operations Research 45(1):24–41, 1997. https://doi.org/10.1287/opre.45.1.24
• O. Besbes, A. Zeevi, Blind Network Revenue Management, Operations Research 60(6):1537–1550, 2012. https://doi.org/10.1287/opre.1120.1057
• A. Badanidiyuru, R. Kleinberg, A. Slivkins, Bandits with Knapsacks, FOCS 2013. https://arxiv.org/abs/1305.2545
• S. Bubeck, C.-Y. Liu, Prior-free and Prior-dependent Regret Bounds for Thompson Sampling, NeurIPS 2013. https://arxiv.org/abs/1311.0466
• D. Russo, B. Van Roy, Learning to Optimize via Posterior Sampling, Mathematics of Operations Research 39(4):1221–1243, 2014. https://doi.org/10.1287/moor.2014.0650
• S. Bubeck, N. Cesa-Bianchi, Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems, Foundations and Trends in Machine Learning 5(1), 2012. https://arxiv.org/abs/1204.5721
• T. Lattimore, C. Szepesvári, Bandit Algorithms, Cambridge University Press, 2020, Chapter 36. https://doi.org/10.1017/9781108571401

Motivation

The conditional logit model is the workhorse of discrete choice analysis in econometrics, transportation planning, marketing and revenue management. An individual facing a finite set of alternatives picks alternative $i$ with probability proportional to $e^{z_i\theta}$, where $z_i$ is a vector of observed attributes and $\theta$ an unknown parameter vector. Daniel McFadden's 1974 chapter, Conditional Logit Analysis of Qualitative Choice Behavior, derived this model from a theory of random utility maximization and set out how to estimate $\theta$ by maximum likelihood. McFadden received the 2000 Nobel Memorial Prize in Economic Sciences for his development of theory and methods for analyzing discrete choice.

Every confidence interval and hypothesis test computed from a fitted logit model rests on the large-sample theory in §III of that chapter: the maximum likelihood estimator exists with probability tending to one, converges to the true parameter, and is approximately normal with covariance given by the inverse information matrix. This mission formalizes that theory, Lemmas 5 and 6 of the paper, as proved in its Appendix.

Setting

Observations are indexed serially, $m = 0, 1, 2, \dots$, as in the paper's Appendix ("Let m be a serial index of trials and repetitions"). Observation $m$ offers $J_m \ge 1$ alternatives, and alternative $i$ carries a vector $z_{im} \in \mathbb R^K$ of independent variables. For a parameter $\theta \in \mathbb R^K$ the selection probabilities are

$$P_{im}(\theta) = \frac{e^{z_{im}\theta}}{\sum_{j=1}^{J_m} e^{z_{jm}\theta}}, \qquad \bar z_m(\theta) = \sum_{i} P_{im}(\theta)\, z_{im}.$$

The data are generated at a true parameter $\theta^0$: the chosen alternatives $Y_0, Y_1, \dots$ are independent random variables with $\Pr(Y_m = i) = P_{im}(\theta^0)$. The log-likelihood of the first $q$ observations is $L^q(\theta) = \sum_{m<q}\log P_{Y_m m}(\theta)$. The moment matrix of observation $m$ is

$$\Omega_m = \sum_{i} P_{im}(\theta^0)\,(z_{im}-\bar z_m)(z_{im}-\bar z_m)', \qquad \bar z_m = \bar z_m(\theta^0).$$

Axiom 7 asks that $J_m \le J_*$ and $|z_{im}| \le M$ uniformly, and that $\frac1q\sum_{m<q}\Omega_m$ converge to a positive definite matrix $\Omega$. Axiom 6, for a given sample, asks that no nonzero $\gamma$ satisfy $(z_{jm} - z_{Y_m m})\gamma \le 0$ for all observed $m$ and all $j$. A maximum likelihood estimator $\hat\theta^q$ is a measurable choice of a maximizer of $L^q$, wherever one exists.

Formalization targets

Goal: Lemma 6

$$\hat\theta^q \xrightarrow{\Pr} \theta^0 \quad\text{and}\quad \sqrt q\,\Omega^{1/2}(\hat\theta^q - \theta^0) \xrightarrow{d} N(0, I_K) \qquad (q \to \infty).$$

Milestones

1. Axiom 7 implies Axiom 5 (the full-rank condition) in all sufficiently large samples.
2. Equation (42): $P_{im}(\theta) \ge 1/(J_* e^{2M|\theta|})$.
3. Lemma 5: $\Pr(\text{Axiom 6 holds and } L^q \text{ attains its maximum}) \to 1$.
4. Equation (43): the first three derivatives of $\log P_{im}$ are bounded by $2M$, $4M^2$, $8M^3$.
5. Equation (46): each score $\nabla\log P_{Y_m m}(\theta^0)$ has mean zero.
6. Equation (47): each expected Hessian equals $-\Omega_m$.
7. Consistency of $\hat\theta^q$.
8. Equation (58): $q^{-1/2}\,\Omega^{-1/2}\sum_{m<q}\nabla\log P_{Y_m m}(\theta^0) \xrightarrow{d} N(0, I_K)$.

Significance

The result. Lemma 6 is what licenses reading $\hat\theta^q$ as approximately $N(\theta^0, q^{-1}\Omega^{-1})$, so that the diagonal of the inverse information matrix estimates the sampling variances and $q(\hat\theta^q-\theta^0)'\Omega(\hat\theta^q-\theta^0)$ is asymptotically $\chi^2_K$. Lemma 5 complements it: in finite samples the likelihood can fail to have a maximum (the observations are then "explained" by a direction $\gamma$ of Axiom 6), and the lemma shows this failure is asymptotically negligible. The data are not identically distributed (each observation has its own alternatives), so the result is not an instance of the textbook i.i.d. maximum likelihood theorem.

Formalizing it. The results are proved in the paper, in outline. A machine-checked version adds: a complete proof of the existence part (Lemma 5), whose published argument is a sketch by induction over an infinite index set; a precise treatment of the estimator where no maximizer exists; the correction of two misprints in the published proof (the normalization $1/q$ in (58), which must be $1/\sqrt q$, and a constant in (51)); and a multivariate Lindeberg–Feller central limit theorem for bounded, independent, non-identically distributed vectors, which the proof invokes and which is reusable well beyond this paper. No machine-checked proof of these results is known.

Difficulty

The obvious route, "the log-likelihood is concave, so its maximizer converges", needs a maximizer to exist, and in a finite sample it may not; the estimator is defined only on an event whose probability must first be shown to tend to one. Consistency then needs a uniform law of large numbers for the gradient on a sphere around $\theta^0$, controlled by the third-derivative bound (43). Asymptotic normality needs a central limit theorem for independent but not identically distributed score vectors, with covariances $\Omega_m$ that converge only on average; the i.i.d. central limit theorem does not apply. Finally the random Hessian at an intermediate point must be shown to converge in probability, which ties the consistency result into the normality argument.

Formalization scope

• Vectors live in EuclideanSpace ℝ (Fin K); $z\theta$ is the inner product, and all norms are Euclidean (footnote 11's sum-of-absolute-values norm is equivalent and gives the same qualitative axiom); derivative bounds use operator norms.
• The paper's $N$ trials with $R_n$ repetitions are the special case of the serial indexing in which consecutive observations repeat their data; the sample size $\sum_n R_n$ is $q$.
• Axiom 7's limit (27) is taken in its serial form (48), with $P$ evaluated at $\theta^0$.
• The estimator is any measurable selection that maximizes $L^q$ whenever $L^q$ has a maximum, and is unconstrained otherwise. Requiring a maximizer for every sample would be unsatisfiable, since Axiom 6 fails with positive probability, and would make the goal vacuous; this convention rules that out.
• Consistency is TendstoInMeasure. Asymptotic normality is TendstoInDistribution to a random vector whose law is stdGaussian. $\Omega^{1/2}$ is the positive semidefinite square root CFC.sqrt.
• Needed infrastructure: derivatives of log-sum-exp, a law of large numbers for bounded independent vectors, and a multivariate Lindeberg–Feller theorem. Mathlib provides the one-dimensional i.i.d. central limit theorem only. Contributions of these general results as separate theorems are welcome.

Selected references

• D. McFadden, Conditional logit analysis of qualitative choice behavior, in P. Zarembka (ed.), Frontiers in Econometrics, Academic Press, New York, 1974, pp. 105–142.
• W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, 1966 (Lindeberg–Feller theorem, pp. 256–258).
• C. R. Rao, Linear Statistical Inference and Its Applications, Wiley (cited by McFadden as Rao (1968), pp. 347–351, for the asymptotic $\chi^2$ test).

Motivation

The conditional logit model assigns to an alternative $i$ in a finite choice set the probability $e^{V_i}/\sum_j e^{V_j}$, where $V_j$ is a "representative utility" built from observed attributes of the alternative and the decision maker. It is the workhorse of discrete choice econometrics, transportation demand forecasting, marketing and revenue management, where it underlies multinomial logit assortment and pricing models. Its appeal for applied work is computational; its appeal for economics is that it can be read as the aggregate behaviour of a population of utility maximizers. This mission formalizes the result that makes that reading exact: Lemmas 1 and 2 of D. McFadden, Conditional logit analysis of qualitative choice behavior (1974), which show that, under a mild regularity condition, logit choice probabilities arise from random utility maximization exactly when the idiosyncratic taste shocks follow the extreme value (Gumbel) distribution.

Timeline:

• 1959. J. Marschak gives a nonconstructive proof that i.i.d. extreme value shocks yield logit probabilities; R. D. Luce's choice axiom appears the same year.
• 1965. Luce and Suppes publish the constructive argument, attributed to E. Holman and A. Marley, that is reproduced as the proof of Lemma 1.
• 1974. McFadden proves the converse (Lemma 2): if i.i.d. shocks with a translation complete distribution produce logit probabilities, the distribution is extreme value.
• Later. The random utility characterization was extended to correlated shocks (generalized extreme value models, McFadden 1978), which are not part of this mission.

Setting

An individual faces $J \ge 1$ alternatives with representative utilities $V_1, \dots, V_J \in \mathbb{R}$. The utility of alternative $j$ is $U_j = V_j + \varepsilon_j$, where the taste shocks $\varepsilon_1, \dots, \varepsilon_J$ are independent and identically distributed with a common law $\mu$ on $\mathbb{R}$ and distribution function $G(t) = \mu((-\infty, t])$. The individual chooses the alternative of highest utility, so the selection probability of $i$ is (Equation (2) of the paper)

$$P_i(V) = \Pr\big[\varepsilon_j - \varepsilon_i < V_i - V_j \ \text{ for all } j \ne i\big],$$

computed under the product law of the shocks. The logit formula (Equation (12)) is $L_i(V) = e^{V_i}/\sum_{j=1}^J e^{V_j}$. The extreme value law (Equation (13)) is $G(\varepsilon) = e^{-e^{-\varepsilon}}$.

A law $\mu$ is translation complete if for every function $h$ of bounded total variation on $\mathbb{R}$ with $h(\pm\infty) = 0$, the condition $\int h(e + a)\, d\mu(e) = 0$ for every real $a$ forces $h = 0$ outside a Lebesgue-null set. Laws whose characteristic function never vanishes, the extreme value law among them, are translation complete (footnote 5 of the paper).

In Lean the law is μ : Measure ℝ with [IsProbabilityMeasure μ], $G$ is ProbabilityTheory.cdf μ, the selection probability is selProb μ V i for V : Fin J → ℝ, and the logit formula is logitProb V i.

Formalization targets

Goal: the characterization

Fix a universe $X$ of alternatives with a representative utility map $u:X\to\mathbb R$ onto the real line. For a translation complete law $\mu$ normalized by $G(0) = e^{-1}$,

$$\Big(\text{for every finite }B\subseteq X,\ i\in B:\ P_i(B) = \frac{e^{u(i)}}{\sum_{j\in B} e^{u(j)}}\Big) \iff \Big(\forall \varepsilon \in \mathbb{R}:\ G(\varepsilon) = e^{-e^{-\varepsilon}}\Big).$$

The normalization only fixes the location of the shocks: without it the conclusion is the one-parameter family of Lemma 2 below.

Milestones

1. Equation (3) for i.i.d. shocks without atoms: $P_i(V) = \int \prod_{j \ne i} G(\varepsilon + V_i - V_j)\, dG(\varepsilon)$.
2. The integrand of Lemma 1's proof: under (13), the density times the other distribution functions equals $e^{-\varepsilon} \exp\big(-e^{-\varepsilon} \sum_j e^{V_j - V_i}\big)$.
3. Lemma 1: extreme value shocks give $P_i(V) = L_i(V)$ for every $J$ and $V$.
4. The functional equation of Lemma 2's proof: $G(v - \log K) = G(v)^K$ for every positive integer $K$ and real $v$.
5. Values at logarithms of rationals: with $\alpha = -\log G(0)$, $\alpha > 0$ and $G(\log(K/L)) = e^{-\alpha L / K}$ for positive integers $K, L$.
6. Lemma 2: $G(\varepsilon) = e^{-\alpha e^{-\varepsilon}}$ for some $\alpha > 0$, and $G(0) = e^{-1}$ gives (13).

Significance

The result separates two readings of the logit formula. Lemma 1 shows that it is consistent with utility maximization; Lemma 2 shows that, within the class of i.i.d. additive random utility models with translation complete shocks, the extreme value law is the only one consistent with it. Consequences drawn from the random utility reading, such as the log-sum formula for expected maximum utility used in welfare analysis, therefore apply to logit models without further distributional assumptions inside that class. The same reading supports the interpretation of multinomial logit demand in assortment optimization and revenue management.

Both lemmas are proved in the paper and in later textbooks; neither is open. As far as a search of the Prove2Me catalog shows, neither has a machine-checked proof. The mission provides Lean statements of the random utility model with i.i.d. shocks, of translation completeness and of the extreme value law that later discrete choice formalizations can reuse.

Difficulty

Lemma 1 is a computation with the extreme value density; its formal cost lies in passing from the product-measure probability (2) to the iterated integral (3) and evaluating an improper integral. Lemma 2 is harder. The natural first idea is to differentiate the logit identity in the utilities and solve a differential equation for $G$; this requires a density, which Lemma 2 does not assume. Without a density, the only handle on $G$ is the logit identity itself, an equality of integrals against $dG$ that holds for every utility vector; turning such integral identities into pointwise information about $G$ is where the hypothesis of translation completeness enters, and it yields statements only outside a Lebesgue-null set, so one-sided continuity of distribution functions is needed to recover identities at every point. A second subtlety is that the paper's (14) is written with $G$ while the event (2) is strict, so with a general law the integrals involve left limits of $G$.

Formalization scope

Alternatives are indexed by Fin J; the model is indexed by the utility vector, so the individual attributes $s$ and alternative attributes $x_j$ enter only through $V$. The shocks have joint law Measure.pi (fun _ => μ), which is what "independently identically distributed" means; a general joint law is not allowed. The event in (2) uses strict inequalities, and the selection probability is defined for every law, with no density. Translation completeness quantifies over BoundedVariationOn h Set.univ with limits $0$ at both ends; such $h$ are bounded and measurable, so the integrals are genuine, and "measure zero" is Lebesgue measure.

Lemma 2's hypothesis is stated on every finite subset of the paper's alternative universe, with a surjective utility map. Distinct alternatives may have the same utility. The printed proof uses $K$ equal-utility alternatives, which surjectivity alone need not supply; proving the stated theorem requires an additional continuity argument. A trivializing formalization is ruled out: the selection probability is a genuine product-measure probability, and the hypotheses of the goal are met by the extreme value law, which is translation complete with $G(0) = e^{-1}$.

A complete development needs: Fubini for Measure.pi over Fin J split at one coordinate; the Gumbel density and the improper integral $\int e^{-\varepsilon} e^{-c e^{-\varepsilon}} d\varepsilon = 1/c$; the facts that bounded-variation functions are bounded and measurable and that distribution functions are right-continuous with left limits. The integral representation (milestone 1) and the Gumbel computations are reusable in any random utility formalization. Proofs of the milestones, of the footnote-5 fact that the extreme value law is translation complete, and alternative proofs of Lemma 2 are welcome.

Selected references

• D. McFadden, Conditional logit analysis of qualitative choice behavior, in P. Zarembka (ed.), Frontiers in Econometrics, Academic Press, New York, 1974, pp. 105–142.
• J. Marschak, Binary choice constraints and random utility indicators, in K. Arrow, S. Karlin, P. Suppes (eds.), Mathematical Methods in the Social Sciences, Stanford University Press, 1960 (Stanford Symposium, 1959).
• R. D. Luce and P. Suppes, Preference, utility, and subjective probability, in R. D. Luce, R. Bush, E. Galanter (eds.), Handbook of Mathematical Psychology, Vol. III, Wiley, 1965.
• R. D. Luce, Individual Choice Behavior: A Theoretical Analysis, Wiley, 1959.
• W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, 1966, p. 479.
• D. McFadden, Modelling the choice of residential location, in A. Karlqvist et al. (eds.), Spatial Interaction Theory and Planning Models, North-Holland, 1978, pp. 75–96.

Motivation

A retailer choosing which products to display, or an airline choosing which fare classes to open, solves an assortment optimization problem: pick the set of offered products that maximizes expected revenue when customers choose among what is offered according to a discrete choice model. Under the multinomial logit model the answer has a simple form: an optimal assortment consists of the few highest-revenue products (Talluri and van Ryzin, 2004). The multinomial logit model, however, forces every pair of products to compete in the same way. The nested logit model relaxes this by grouping products into nests (brands, store sections, departure times) and letting a customer first choose a nest and then a product inside it.

Davis, Gallego and Topaloglu (Operations Research, 2014; DOI 10.1287/opre.2014.1256) study how much of the multinomial logit structure survives under the nested logit model. Their first answer is the theorem this mission targets: when the nest dissimilarity parameters are at most one and no customer who chose a nest leaves it without buying, offering the top products of each nest is still optimal. The other missions of this series treat the cases where this fails: dissimilarity parameters above one (the problem becomes NP-hard) and nests with their own no-purchase option.

Setting

There are $m$ nests $M = \{1, \dots, m\}$ and, in each nest, $n$ products $N = \{1, \dots, n\}$. Product $j$ of nest $i$ has a revenue $r_{ij} \ge 0$ and a preference weight $v_{ij} > 0$; products are ordered so that $r_{i1} \ge r_{i2} \ge \dots \ge r_{in}$. Nest $i$ carries a dissimilarity parameter $\gamma_i > 0$ and a within-nest no-purchase weight $v_{i0} \ge 0$, and $v_0 \ge 0$ is the weight of choosing no nest at all.

An assortment is a tuple $(S_1, \dots, S_m)$ of subsets $S_i \subseteq N$. Write

$$V_i(S_i) = v_{i0} + \sum_{j \in S_i} v_{ij}, \qquad R_i(S_i) = \frac{\sum_{j \in S_i} r_{ij} v_{ij}}{V_i(S_i)}, \quad R_i(\emptyset) = 0.$$

A customer picks nest $i$ with probability $Q_i = V_i(S_i)^{\gamma_i} / (v_0 + \sum_{l \in M} V_l(S_l)^{\gamma_l})$ and then, inside the nest, product $j$ with probability $v_{ij}/V_i(S_i)$. The expected revenue is

$$\Pi(S_1, \dots, S_m) = \sum_{i \in M} Q_i\, R_i(S_i) = \frac{\sum_{i \in M} V_i(S_i)^{\gamma_i} R_i(S_i)}{v_0 + \sum_{i \in M} V_i(S_i)^{\gamma_i}},$$

and problem (2) asks for $Z^* = \max \Pi(S_1, \dots, S_m)$ over all assortments. The nested-by-revenue assortment $N_{ij} = \{1, \dots, j\}$ collects the $j$ highest-revenue products of nest $i$, with $N_{i0} = \emptyset$ and $N_+ = \{0, 1, \dots, n\}$.

This mission works under the standing assumptions of §3 of the paper: competitive products, $\gamma_i \le 1$, and fully-captured nests, $v_{i0} = 0$, for every nest $i$.

Formalization targets

Goal: Theorem 4 (p. 15)

If $\gamma_i \le 1$ and $v_{i0} = 0$ for all $i \in M$, there exists an optimal solution $(S^*_1, \dots, S^*_m)$ of problem (2) such that

$$S^*_i = N_{ij} \ \text{ for some } j \in N_+, \qquad \text{for all } i \in M.$$

Milestones

1. The case $v_0 = 0$ (p. 14). Offering only the single product with the largest revenue $\max_{i} r_{i1}$ is optimal.
2. Proposition 2 (p. 14). If $S^*$ is optimal and $S^*_i \ne \emptyset$, then $R_i(S^*_i) \ge Z^*$.
3. Lemma 3 (p. 14). If $Z = \Pi(S)$, $R_i(S_i) \ge Z$ and some $j \in S_i$ has $r_{ij} < \gamma_i Z + (1-\gamma_i) R_i(S_i)$, removing $j$ strictly increases the expected revenue.
4. $g(\alpha) \le \gamma$ (p. 15). For $0 < \gamma \le 1$ and $0 < \alpha < 1$: $(1 - \alpha^{\gamma})/(\alpha^{\gamma-1} - \alpha^{\gamma}) \le \gamma$.
5. Revenue threshold (p. 15). Every $j \in S^*_i$ of an optimal $S^*$ has $r_{ij} \ge \gamma_i Z^* + (1-\gamma_i) R_i(S^*_i)$.
6. $h(\alpha) \ge \gamma$ (p. 16). For $0 < \gamma \le 1$ and $0 < \alpha < 1$: $(1 - \alpha^{\gamma})/(1 - \alpha) \ge \gamma$.
7. Exchange step (p. 15). If $S^*$ is optimal, $j \in S^*_i$, $k \notin S^*_i$ and $k < j$, then adding $k$ to $S^*_i$ keeps the assortment optimal.

A companion item (not a milestone) states the algorithmic consequence at the end of §3: solving the linear program (4) over the candidates $\{N_{ij} : j \in N_+\}$ and choosing in each nest a maximizer of problem (5) gives an optimal solution of (2).

Significance

Theorem 4 reduces problem (2), a search over $2^{mn}$ assortments, to $(n+1)^m$ nested-by-revenue combinations, and the paper then finds the best one with a linear program with $1 + m$ variables and $1 + m(1+n)$ constraints. It marks the exact boundary of the classical multinomial logit structure inside the nested logit model: the paper's §4 shows that a single nest with $\gamma_i > 1$ already breaks it, and that the general problem is NP-hard. The structural statement is also the base case for the approximation guarantees of §§5–6, which compare against nested-by-revenue assortments.

The theorem is proved in the paper; to our knowledge it has no machine-checked proof. A formal proof would supply a verified reduction from a combinatorial revenue maximization over the nested logit model to a polynomial-size search, with every boundary case (empty nests, $v_0 = 0$, ties in revenues) handled explicitly.

Difficulty

The obvious argument copies the multinomial logit proof: take an optimal assortment and swap a low-revenue product for a missing higher-revenue one. Under the nested logit model this exchange changes the nest's attraction $V_i(S_i)^{\gamma_i}$ non-linearly, so the revenue of the modified assortment is not an affine function of the change, and a simple swap can lower the expected revenue. The argument must instead control how adding or removing one product moves the nest weight relative to the nest revenue, and this is exactly where $\gamma_i \le 1$ enters, through two scalar inequalities in the ratio $\alpha$ of nest weights. With $\gamma_i > 1$ these inequalities fail and so does the theorem.

A second subtlety is ties: several optimal assortments may exist, and only some of them are nested by revenue. The statement asserts existence, not that every optimum has this form.

Formalization scope

All statements live in the namespace NestedLogitVariants.Competitive and share one definition file. Nests form a finite type ι with decidable equality; products are Fin n, indexed $0, \dots, n-1$, so $N_{ij}$ is nbr n j $= \{k : k < j\}$ with $j \le n$, and $j = 0$ gives $\emptyset$. Powers are Real.rpow, and $x / 0 = 0$, which gives $R_i(\emptyset) = 0$. Optimality of an assortment means its revenue is at least that of every assortment.

Standing assumptions carried as hypotheses: $v_0 \ge 0$, $v_{i0} \ge 0$, revenues ordered within each nest, and §3's $\gamma_i \le 1$ and $v_{i0} = 0$. Three pins are disclosed: $v_{ij} > 0$ (the paper allows zero-weight padding products, under which Proposition 2 fails), $r_{ij} \ge 0$, and $\gamma_i > 0$ (the paper's $\gamma_i \ge 0$; its convention $V_i(\emptyset)^{\gamma_i} = 0$ fails at $\gamma_i = 0$). The section's "without loss of generality $v_0 > 0$" is a hypothesis of Proposition 2, Lemma 3, the threshold, the exchange step and the LP item; the goal itself only assumes $v_0 \ge 0$, and the case $v_0 = 0$ is milestone 1. The two scalar inequalities are stated as inequalities, not as monotonicity claims.

The goal is not trivialized by any hypothesis: it assumes none of the milestones, and stating "some nested-by-revenue assortment exists" (always true) or "every optimal assortment is nested by revenue" (false under ties) would be a different theorem.

Needed infrastructure is light: finite sums, real powers, and concavity of $x \mapsto x^{\gamma}$ for $\gamma \le 1$. The scalar lemmas are reusable for other nested logit results. Proofs of any milestone are welcome independently.

Selected references

• J. M. Davis, G. Gallego, H. Topaloglu, Assortment optimization under variants of the nested logit model, Operations Research 62(2), 250–273, 2014. https://doi.org/10.1287/opre.2014.1256 (cited from the authors' revised manuscript of June 18, 2013)
• K. Talluri, G. van Ryzin, Revenue management under a general discrete choice model of consumer behavior, Management Science 50(1), 15–33, 2004. https://doi.org/10.1287/mnsc.1030.0147
• D. McFadden, Modelling the choice of residential location, in A. Karlqvist et al. (eds.), Spatial Interaction Theory and Planning Models, North-Holland, 75–96, 1978.

Motivation

Assortment optimization asks which set of products a firm should offer when customers choose among the offered products according to a discrete choice model; it underlies shelf-space planning in retail and fare-class control in airline revenue management (Talluri and van Ryzin, 2004). Under the nested logit model products are grouped into nests, and a customer first picks a nest and then a product inside it. Davis, Gallego and Topaloglu (Operations Research, 2014; DOI 10.1287/opre.2014.1256) map out how hard this problem is across variants of the model.

When every nest dissimilarity parameter is at most one and a customer who chose a nest always buys there, offering the top-revenue products of each nest is optimal (Theorem 4 of the paper, the subject of an earlier mission of this series). Once a customer may leave a nest without buying — a partially-captured nest — that structure breaks and the problem becomes NP-hard (Theorem 8). This mission targets the paper's response: a small, explicitly constructed family of candidate assortments per nest from which a linear program recovers a solution within a factor of two of optimal.

Setting

There are $m$ nests $M$ and, in each nest, $n$ products $N = \{1, \dots, n\}$. Product $j$ of nest $i$ has a revenue $r_{ij} \ge 0$ and a preference weight $v_{ij} > 0$, with $r_{i1} \ge \dots \ge r_{in}$. Nest $i$ has a dissimilarity parameter $\gamma_i > 0$ and a within-nest no-purchase weight $v_{i0} \ge 0$; $v_0 \ge 0$ is the weight of choosing no nest. For an assortment $S_i \subseteq N$,

$$V_i(S_i) = v_{i0} + \sum_{j \in S_i} v_{ij}, \qquad R_i(S_i) = \frac{\sum_{j \in S_i} r_{ij} v_{ij}}{V_i(S_i)},\quad R_i(\emptyset)=0,$$

and the expected revenue of $(S_1, \dots, S_m)$ is $\Pi = \sum_i V_i(S_i)^{\gamma_i} R_i(S_i) / (v_0 + \sum_i V_i(S_i)^{\gamma_i})$. The optimal expected revenue $Z^*$ is the optimal value of the linear program

$$\text{(3)}\quad \min\ x \quad\text{s.t.}\quad v_0 x \ge \sum_{i \in M} y_i,\qquad y_i \ge V_i(S_i)^{\gamma_i}\big(R_i(S_i) - x\big)\ \ \forall S_i \subseteq N,\ i \in M,$$

and problem (4) is the same program with the second family of constraints imposed only for a chosen collection of candidate assortments in each nest.

Throughout, $\gamma_i \le 1$ for every nest and the $v_{i0}$ are arbitrary. For a capacity $\epsilon_i \ge 0$, the knapsack value $K_i(\epsilon_i)$ is the largest $\sum_{j \in S} r_{ij} v_{ij}$ over assortments $S$ with $\sum_{j \in S} v_{ij} \le \epsilon_i$ (display (9)). Its continuous relaxation (11) allows fractional $z_{ij} \in [0, \mathbf 1(v_{ij} \le \epsilon_i)]$ under the same capacity. The greedy solution $\hat z_i(\epsilon_i)$ of (11) fills the capacity with the products of weight at most $\epsilon_i$ in revenue order, each fully while it fits and the next one fractionally, and

$$\hat S_i(\epsilon_i) = \{ j \in N : \hat z_{ij}(\epsilon_i) = 1 \}.$$

Problem (10) replaces the per-assortment constraints of (3) by $y_i \ge \max_{\epsilon_i \ge 0} (v_{i0}+\epsilon_i)^{\gamma_i}[K_i(\epsilon_i)/(v_{i0}+\epsilon_i) - x]$.

Formalization targets

Goal: Theorem 10 (p. 24)

Let $(\hat x, \hat y)$ be an optimal solution of (4) with candidate collections $\{\hat S_i(\epsilon_i) : \epsilon_i \in [0,\infty]\} \cup \{\{j\} : j \in N\}$. Then

$$(2\hat x,\ 2\hat y) \ \text{ is feasible for (3).}$$

Milestones

1. Per-nest identity (proof of Lemma 9, p. 23). For $x \ge 0$, $\max_{S_i} V_i(S_i)^{\gamma_i}(R_i(S_i) - x) = \max_{\epsilon_i \ge 0}(v_{i0}+\epsilon_i)^{\gamma_i}[K_i(\epsilon_i)/(v_{i0}+\epsilon_i) - x]$.
2. Lemma 9 (p. 23). Problems (3) and (10) have the same optimal solutions.
3. Relaxation (p. 23). Every feasible point of (9) is feasible for (11), so $\hat K_i(\epsilon_i) \ge K_i(\epsilon_i)$.
4. Greedy solution (pp. 23–24). $\hat z_i(\epsilon_i)$ is optimal for (11) and has at most one fractional component.
5. Sign (A.3, p. 45). $\hat x \ge 0$.
6. Inequalities (28) and (29) (A.3, pp. 45–46). In both cases — $\hat z_i(\epsilon)$ with and without a fractional component — $2\hat y_i \ge (v_{i0}+\epsilon)^{\gamma_i}[K_i(\epsilon)/(v_{i0}+\epsilon) - 2\hat x]$.

Two further statements accompany the goal: the factor-two revenue guarantee obtained from Theorem 10 and Theorem 1 of the paper, and the fact that every $\hat S_i(\epsilon_i)$ is one of the at most $1 + n^2$ assortments $N^k_{ij}$, the first $j$ products by revenue among the $k$ lightest.

Significance

Theorem 10 turns an NP-hard assortment problem into a linear program with $1 + m$ variables and $1 + m(1 + n + n^2)$ constraints whose solution is within a factor of two of optimal. The construction is explicit: the candidates are defined by a greedy rule, not by an optimization oracle. The same template, a restricted linear program whose doubled optimum is feasible for the full one, is reused in §6 of the paper for the most general instances, and Lemma 9's knapsack reformulation is the link to the classical approximation theory of knapsack problems (Williamson and Shmoys, 2011).

The theorem is proved in the paper. No machine-checked proof of it, of Lemma 9, or of greedy optimality for the continuous knapsack with an eligibility bound exists on the platform. Formalizing it yields a checked factor-two guarantee and a reusable fractional-knapsack development.

Difficulty

The obvious argument would compare the restricted program (4) with (3) constraint by constraint. That fails: (3) has one constraint per subset of products, and most subsets are not candidates. The comparison has to pass through the knapsack reformulation (10), which requires showing that a maximum over all subsets equals a maximum over a one-dimensional capacity parameter, using $\gamma_i \le 1$ and $x \ge 0$ in an essential way. The second obstacle is that the greedy assortment $\hat S_i(\epsilon_i)$ keeps only the fully taken products, so its value can fall short of the continuous knapsack value, and no single candidate assortment need attain the knapsack bound. With dissimilarity parameters above one the monotonicity behind the reformulation is lost, and §6 of the paper needs a different factor.

Formalization scope

Products are Fin n (indices $0, \dots, n-1$), nests a finite type, and every quantity is real. Powers are Real.rpow; $x/0 = 0$, which gives $R_i(\emptyset) = 0$. An optimal solution of a linear program is a feasible pair whose $x$ is minimal among feasible pairs. The constraint "$y_i \ge \max_{\epsilon_i \ge 0}(\dots)$" of (10) is stated in constraint form, for every $\epsilon_i \ge 0$, so no real supremum is taken. $K_i(\epsilon)$ is defined for $\epsilon \ge 0$ only; its placeholder value for $\epsilon < 0$ is never used. Ties in revenue (and, for $N^k_{ij}$, in weight) are broken by index. The candidate collection is taken over real $\epsilon_i \ge 0$; $\epsilon_i = \infty$ adds nothing, since every capacity of at least $\sum_j v_{ij}$ already gives $\hat S_i = N$.

Standing assumptions and added hypotheses: $\gamma_i \le 1$ for every nest (the section's assumption) on the goal and on every model milestone; the pins $v_{ij} > 0$, $r_{ij} \ge 0$, $\gamma_i > 0$ and the revenue ordering, shared by the series; $n \ge 1$ on Lemma 9, on $\hat x \ge 0$ and on (28)/(29), the paper's nonempty $N$; and $v_0 > 0$ on the factor-two revenue guarantee, where Theorem 1 of the paper fails without it.

The greedy assortments $\hat S_i(\epsilon_i)$ are defined explicitly. Quantifying over arbitrary optimal solutions of (11) instead would change the candidate collection and is not the paper's theorem. The goal states feasibility for the full program (3) and does not mention knapsack values, the greedy solution or the case split. A formalization that weakens the conclusion to feasibility for (10), or that drops the singletons from the candidate collection, is not a solution.

Needed infrastructure: fractional knapsack optimality of the greedy rule with an eligibility bound, monotonicity of $t \mapsto t^{\gamma}$ and $t \mapsto t^{\gamma - 1}$ for $\gamma \le 1$, and finite maximization over subsets. The fractional-knapsack lemmas are reusable beyond this mission. Proofs of any milestone, and alternative decompositions of the goal, are welcome.

Selected references

• J. M. Davis, G. Gallego, H. Topaloglu, Assortment Optimization under Variants of the Nested Logit Model, Operations Research 62(2), 2014 (revised manuscript of June 18, 2013, cited here). DOI 10.1287/opre.2014.1256
• K. T. Talluri, G. J. van Ryzin, Revenue Management Under a General Discrete Choice Model of Consumer Behavior, Management Science 50(1), 15–33, 2004. DOI 10.1287/mnsc.1030.0147
• D. P. Williamson, D. B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011. DOI 10.1017/CBO9780511921735

Why myopic pricing is a problem

A seller who does not know how demand responds to price has to learn the demand curve from its own sales while it is selling. The most natural policy is certainty equivalent pricing (also called myopic pricing or passive learning): after every period, estimate the unknown demand parameters from all data collected so far, and charge the price that would be optimal if the estimates were the truth. It is simple, uses all data, and is what a price manager would do without further thought.

den Boer and Zwart (Management Science 60(3):770–783, 2014) show that this policy can fail. In the linear-demand, Gaussian-noise model, the prices it produces fail to converge to the optimal price with positive probability: the policy is not strongly consistent. The result motivates the paper's main contribution, controlled variance pricing, which adds just enough price dispersion to keep learning (treated in the companion mission of this series).

The phenomenon has a history in adaptive control:

• 1976. Anderson and Taylor study the linear system $y_t = a_0 + a_1x_t + \epsilon_t$ controlled by a certainty equivalent rule that steers $y_t$ to a target, and examine by simulation the statistical properties of the least squares estimates it produces (Econometrica 44(6), 1976).
• 1982. Lai and Robbins (Adv. Appl. Math. 3(1), 1982) prove that there are parameter values for which the certainty equivalent controls converge with positive probability to a value different from the optimal control.
• 2014. den Boer and Zwart adapt the argument to revenue maximization with linear demand, without the conditions Lai and Robbins place on the initial inputs and the input bounds: any two different initial prices in $[p_l, p_h]$ give the failure with positive probability.

Setting

A monopolist sells one product in periods $t = 1, 2, \dots$ at prices $p_t$ from an interval $[p_l, p_h]$ with $0 < p_l < p_h$. The demand in period $t$ is

$$d_t = a_0^{(0)} + a_1^{(0)} p_t + e_t ,$$

where $e_1, e_2, \dots$ are independent $N(0, \sigma^2)$ random variables. The parameters are unknown to the seller and satisfy $\sigma > 0$, $a_0^{(0)} > 0$, $a_1^{(0)} < 0$, $a_0^{(0)} + a_1^{(0)}p_h \ge 0$. The expected revenue at price $p$ is $r(p, a_0, a_1) = p(a_0 + a_1p)$, maximized at the optimal price

$$p_{\mathrm{opt}} = -\frac{a_0^{(0)}}{2a_1^{(0)}}, \qquad p_l < p_{\mathrm{opt}} < p_h .$$

Certainty equivalent pricing charges two different initial prices $p_1 \ne p_2$ in $[p_l, p_h]$. After $t \ge 2$ periods it computes the least squares estimates $\hat a_t = (\hat a_{0t}, \hat a_{1t})$, the solution of the normal equations $\sum_{i \le t}(1, p_i)^{\mathsf T}(d_i - \hat a_{0t} - \hat a_{1t}p_i) = 0$, and charges

$$p_{t+1} = \arg\max_{p \in [p_l, p_h]} p\,(\hat a_{0t} + \hat a_{1t}p),$$

with $p_{t+1} = p_h$ when the estimated slope $\hat a_{1t}$ is nonnegative.

Formalization targets

Goal: Proposition 1

$$P\big(p_t \not\to p_{\mathrm{opt}}\big) > 0 .$$

The goal states only the failure of convergence, for every admissible parameter and every pair of different initial prices; it does not fix where the prices go.

Stronger: the prices stick at the boundary

$$P\big(p_t = p_h \ \text{for all } t \ge 3\big) > 0 .$$

This is what the paper's argument establishes; since $p_{\mathrm{opt}} < p_h$ it implies the goal.

Milestones

The milestones are the displayed steps of the appendix proof, in attack order: the determinant of the coefficient matrix of the linear system (12); the bound $P(\sup_{t \ge 3}|(t-2)^{-1}\sum_{i=3}^t e_i| > \epsilon) \le 8\sigma^2\epsilon^{-2} < 1$ for $\epsilon > \sqrt 8\,\sigma$; positivity of the probability of an explicit event $A_\delta$ on the noise for large $\delta$; the case $t = 2$ (the first fitted line pushes $p_3$ to $p_h$); the representation $\hat a_t - a^{(0)} = (\bar e_t - \bar p_tC_t/V_t,\ C_t/V_t)$ of the least squares error; recursive and closed forms of $V_t$ and $C_t$; and the deterministic induction that every noise path in $A_\delta$ keeps the price at $p_h$ forever.

Significance

The result is the standard counterexample to certainty equivalence in dynamic pricing. It shows that estimation and optimization cannot be separated naively: a policy that always exploits its current estimate can lock itself into a price at which the data no longer move the estimate enough to correct it. Every later policy in this literature that forces exploration (controlled variance pricing, semi-myopic policies, constrained iterated least squares) is designed against this failure, and its necessity is argued by pointing to results of this kind.

The result is proved in the paper; nothing here is open. To our knowledge it has no machine-checked proof. Formalizing it adds:

• a verified pathwise analysis of the least squares recursion along a price path, reusable for other proofs about adaptive estimation with two parameters;
• a verified maximal bound for running means of i.i.d. Gaussian noise, of the kind used in many consistency proofs;
• a clean probabilistic statement of the failure, against which consistency results for exploration policies can later be contrasted.

Difficulty

The obvious heuristic, "with positive probability the first two observations are so noisy that the fitted slope is wrong", is not enough: one bad estimate is corrected by later data unless the policy stops generating informative data. The proof has to control the whole infinite future. It does so by showing that on a single event, defined through the first two noise values and a uniform bound on all later running means, the price stays at $p_h$ forever, which requires the closed form of the least squares estimate along a price path that is constant from period 3 on. That event involves infinitely many noise variables, so its probability is positive only through a maximal inequality, and independence between $(e_1, e_2)$ and the later noise. A second subtlety is the choice of constants: the size of the band $\delta$ enters the conditions on $(e_1, e_2)$, so the order in which $\delta$ and the set of admissible $(e_1, e_2)$ are chosen matters (the printed proof picks them in a circular order; a non-circular choice exists).

Formalization scope

• Model. CVPricing.CertEquiv.Model bundles $p_l, p_h, a_0^{(0)}, a_1^{(0)}, \sigma$ with the standing assumptions of §2 as fields, including $p_l < p_{\mathrm{opt}} < p_h$ (the paper's neighbourhood assumption specialized to linear demand). The noise is the referenced published definition RobustBooking.Shared.GaussianNoise (measurable, mutually independent, each $N(0, \sigma^2)$); its Lean index $k$ is period $k+1$, so the paper's $e_i$ is ε (i - 1).
• Policy. cePrice is a deterministic recursion on a noise path, so the random price process is obtained by evaluating it at $\omega$. Periods are 1-based. The least squares estimate is the referenced KeskinZeevi.SufficientConditions.lsEstimateOf, the solution of the normal equations (4), unique whenever $p_1 \ne p_2$. The certainty equivalent rule is the projection of $-\hat a_{0t}/(2\hat a_{1t})$ onto $[p_l, p_h]$ when $\hat a_{1t} < 0$, and $p_h$ when $\hat a_{1t} \ge 0$; the latter is the convention the paper's proof adopts for wrong-signed estimates.
• Corrected slips. The definition of the event $A$ is printed with "$\delta|\bar e_t| \le \delta$" (read $|\bar e_t| \le \delta$) and with its second line missing a factor $\delta$ on the term $(2p_h - p_1 - p_2)$; both are restored as in (12) and the last display of the proof. The intercept of the first fitted line is printed without $a_0^{(0)}$; the correct intercept is stated, and the printed condition remains sufficient for $p_3 = p_h$.
• WLOG. The steps of the proof assume $p_1 < p_2$ and are stated under that ordering; the goal and the stronger statement cover $p_1 \ne p_2$.
• No trivialization. The goal is a statement about the Gaussian law of the noise: a theorem that exhibits one bad noise path, or that assumes $P(A) > 0$, does not prove it. The event in the goal is a set of outcomes whose measurability is not asserted.
• Welcome contributions. Kolmogorov's maximal inequality for sums of independent square-integrable variables; least squares identities for two-parameter regression; the independence argument separating $(e_1, e_2)$ from the later noise.

Selected references

• A. V. den Boer, B. Zwart, Simultaneously Learning and Optimizing Using Controlled Variance Pricing, Management Science 60(3):770–783, 2014. https://doi.org/10.1287/mnsc.2013.1788
• T. L. Lai, H. Robbins, Iterated least squares in multiperiod control, Advances in Applied Mathematics 3(1):50–73, 1982. https://doi.org/10.1016/S0196-8858(82)80005-5
• T. W. Anderson, J. B. Taylor, Some experimental results on the statistical properties of least squares estimates in control problems, Econometrica 44(6):1289–1302, 1976. https://doi.org/10.2307/1914261
• Y. S. Chow, H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, 3rd ed., Springer, 2003. https://doi.org/10.1007/978-1-4612-1950-7

Motivation

A firm that sets prices without knowing how demand responds to them has to learn the demand curve from its own sales. Each price it charges is both a revenue decision and an experiment. The natural policy, certainty equivalent pricing, re-estimates the demand parameters after every period and charges the price that would be optimal if the estimates were exact. den Boer and Zwart show that this policy can fail: with positive probability its prices settle at a suboptimal value, because they converge too fast for the estimates to keep improving (den Boer–Zwart 2014, Proposition 1, the subject of the companion mission). The same phenomenon was found by Lai and Robbins (1982) for a linear control problem.

Their remedy, Controlled Variance Pricing (CVP), keeps certainty equivalent pricing but forces the sample variance of the chosen prices to decay no faster than $t^{\alpha-1}$. The main result is that this small amount of enforced exploration gives regret $O(T^\alpha + T^{1-\alpha}\log T)$, hence $O(T^{1/2+\delta})$ for every $\delta > 0$, for a broad class of demand models that are specified only through their first two moments. Keskin and Zeevi (2014) later placed CVP in a larger family of semi-myopic policies with $\sqrt T\log T$ regret for linear demand.

Setting

A seller chooses in each period $t = 1, 2, \dots$ a price $p_t \in [p_l, p_h]$, with $0 < p_l < p_h$, and then observes demand $d_t$. Demand at price $p$ has mean $h(a_0^{(0)} + a_1^{(0)}p)$ and variance $\sigma^2 v(h(a_0^{(0)} + a_1^{(0)}p))$, where the link $h$ and variance function $v$ are known and $C^2$ on $[0,\infty)$, $\dot h > 0$, and the parameter $a^{(0)} = (a_0^{(0)}, a_1^{(0)})$ with $a_0^{(0)} > 0 > a_1^{(0)}$ is unknown. The noise $e_t = d_t - h(a_0^{(0)} + a_1^{(0)}p_t)$ is a martingale difference with conditional variance $\sigma^2 v(\cdot)$ and a uniformly bounded conditional moment of some order $r > 3$.

The expected revenue is $r(p, a) = p\,h(a_0 + a_1p)$. Near $a^{(0)}$ it has a unique maximizer $p(a)$ in the open interval $(p_l, p_h)$ with $\partial_p^2 r < 0$ there, and $p_{\mathrm{opt}} = p(a^{(0)})$. The regret of a policy is

$$\operatorname{Regret}(T) = \mathbb E\Big[\sum_{t=1}^T r(p_{\mathrm{opt}}, a^{(0)}) - r(p_t, a^{(0)})\Big].$$

The estimate $\hat a_t$ is the maximum quasi-likelihood estimate (MQLE), the root of the quasi-score equation (3), $\sum_{i\le t} \frac{\dot h}{\sigma^2 v(h)}(1, p_i)^\top(d_i - h(\hat a_0 + \hat a_1 p_i)) = 0$. With $\bar p_t$ and $\operatorname{Var}(p)_t$ the sample mean and variance of $p_1,\dots,p_t$, the taboo interval is $\mathrm{TI}(t) = (\bar p_t - w_t, \bar p_t + w_t)$ with $w_t = \sqrt{c[(t+1)^\alpha - t^\alpha](t+1)/t}$. CVP starts from two distinct prices $p_1, p_2$, fixes $\alpha \in (0,1)$ and $0 < c < 2^{-\alpha}(p_1-p_2)^2\min\{1,(3\alpha)^{-1}\}$, and for $t \ge 2$: if $\hat a_t$ does not exist or has the wrong signs, it charges whichever of $p_1, p_2$ is farther from $\bar p_t$; otherwise it charges $p(\hat a_t)$ if that keeps $\operatorname{Var}(p)_{t+1} \ge c(t+1)^{\alpha-1}$, and the best price outside $\mathrm{TI}(t)$ if not.

Formalization targets

Goal: Theorem 1

$$\operatorname{Regret}(T, \mathrm{CVP}) = O\big(T^\alpha + T^{1-\alpha}\log T\big) \qquad (1/2 < \alpha < 1),$$

stated as: there is $K > 0$, depending on the model, $\alpha$, $c$ and the initial prices but not on $T$, with $\operatorname{Regret}(T) \le K(T^\alpha + T^{1-\alpha}\log T)$ for all $T \ge 1$. The constant is left free, so the statement survives any sharpening of the constants.

Milestones

1. Proposition 2: $\operatorname{Var}(p)_t \ge c\,t^{\alpha-1}$ for all $t \ge 2$ along every CVP path.
2. Lemma 1: $\lambda_{\max}(P_t) \le (1+p_h^2)t$ and $t\operatorname{Var}(p)_t \le (1+p_h^2)\lambda_{\min}(P_t)$ for the design matrix $P_t = \sum_{i\le t}(1,p_i)^\top(1,p_i)$.
3. Proposition 3: $\hat a_t$ eventually exists, $\hat a_t \to a^{(0)}$ a.s., and for some $\rho_0$, $\mathbb E[T_{\rho_0}^{1/2}] < \infty$ and $\mathbb E[\|\hat a_t - a^{(0)}\|^2\mathbf 1_{t > T_{\rho_0}}] = O(\log t/t^\alpha)$.
4. Eq. (11): in the normal–linear case, $\mathbb E\|\hat a_t - a^{(0)}\|^2 = O(\log t / t^\alpha)$.
5. Eqs. (17), (18), (20): the quadratic revenue gap, the local Lipschitz bound on $p(a)$, and $|p_{t+1} - p(\hat a_t)| \le |\mathrm{TI}(t)|$ for large $t$.
6. The closing bound $\mathbb E[(p_t - p_{\mathrm{opt}})^2] = O(t^{\alpha-1} + \log t/t^\alpha)$.

Significance

The theorem shows that a policy that is certainty equivalent almost all of the time, with a single interpretable tuning parameter $\alpha$, attains regret $O(T^{1/2+\delta})$ in generalized linear demand models, without distributional assumptions beyond two moments. It explains the role of $\alpha$ precisely: $T^\alpha$ is the cost of exploration and $T^{1-\alpha}\log T$ the cost of estimation error. Proposition 2 and Lemma 1 are reusable for any policy that enforces a variance floor on its actions, and (11) is a self-contained rate for least squares under adaptively chosen designs.

The result is proved in the paper, with Proposition 3 delegated to den Boer and Zwart (2012) for general links. To our knowledge none of it has been machine-checked. A formalization would verify the delegated consistency argument, fix the conditions under which it applies (see Formalization scope), and provide a Lean development of adaptive least squares and quasi-likelihood rates that the related Keskin–Zeevi missions also need.

Difficulty

The deterministic parts are short. The difficulty is Proposition 3. The prices are chosen adaptively from past data, so the regressors are not independent of the noise, and standard rates for (quasi-)likelihood estimates do not apply. The natural argument, bounding $\|\hat a_t - a^{(0)}\|^2$ by $Q_t/\lambda_{\min}(P_t)$ with $Q_t$ a self-normalized martingale quadratic form, needs a bound $\mathbb E[Q_t] = O(\log t)$ that holds in expectation and not only almost surely, as in Lai and Wei (1982). For a non-linear link the MQLE is defined only implicitly, and its existence near $a^{(0)}$ has to be shown first, with a moment bound on the last time it fails. That is the random time $T_\rho$. Turning almost-sure consistency into a rate in expectation is where most of the work lies.

Formalization scope

All declarations live in the namespace CVPricing.Regret. Periods are 1-based. Prices, demands and parameters are real; $a = (a_0, a_1) \in \mathbb R \times \mathbb R$ with the Euclidean norm (euclidNorm), not Mathlib's sup norm. The design matrix, sample mean and $t\operatorname{Var}(p)_t$ are the published Keskin–Zeevi definitions fisherOf, avgPriceOf, infoMetricOf. Every $O(\cdot)$ is "there is $K > 0$ such that for all $t$", with $K$ quantified after the model data. Rates are stated for $t \ge 2$ and the regret for $T \ge 1$. $h$ and $v$ are total functions constrained on $[0,\infty)$. A root of (3) counts only where $\hat a_0 + \hat a_1 p_i \ge 0$ for every observed $p_i$. CVP is a predicate on a realized path that allows every maximizer in (7) and (8).

Disclosed deviations from the page:

• the model requires $a_0^{(0)} + a_1^{(0)}p_h > 0$ (printed: $\ge 0$), because in the boundary case the policy's case (c) fires infinitely often and the proof of Theorem 1 does not cover it;
• (3) is assumed to have at most one root (the page notes roots need not be unique, and the policy cannot select the root nearest $a^{(0)}$);
• the neighbourhood assumption is read as a unique maximizer over $[p_l, p_h]$ lying in $(p_l, p_h)$;
• the demand process is given by its conditional mean, its conditional variance and (2), with integrable noise moments, not by a fixed law $D(p)$;
• the initial prices are deterministic.

Corrected slips: Proposition 2 is stated for $c \le 2^{-\alpha}(p_1-p_2)^2\min\{1/2,(3\alpha)^{-1}\}$, because the printed range fails at $t=2$ ($\operatorname{Var}(p)_2 = (p_1-p_2)^2/4$, not $/2$). Theorem 1 keeps the printed range. Eq. (20) is stated for $p_{t+1}$ and for $t$ beyond an explicit threshold.

The goal does not assume the variance bound, consistency or (20). The policy contains the variance check and the taboo interval, and the regret is the expectation over the actual price process. A statement that assumed any of these, or that dropped the taboo step, would be trivial or false. Contributions are welcome on adaptive least squares (Sherman–Morrison and determinant-ratio bounds), martingale last-time moment bounds, and the implicit-function step (18).

Selected references

• A. V. den Boer, B. Zwart, Simultaneously Learning and Optimizing Using Controlled Variance Pricing, Management Science 60(3):770–783, 2014. https://doi.org/10.1287/mnsc.2013.1788
• A. V. den Boer, B. Zwart, Mean square convergence rates for maximum quasi-likelihood estimators, Stochastic Systems 4(2):375–403, 2014 (cited as 2012 working paper). https://doi.org/10.1214/12-SSY086
• T. L. Lai, C. Z. Wei, Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems, Annals of Statistics 10(1):154–166, 1982. https://doi.org/10.1214/aos/1176345697
• T. L. Lai, H. Robbins, Iterated least squares in multiperiod control, Advances in Applied Mathematics 3(1):50–73, 1982. https://doi.org/10.1016/S0196-8858(82)80005-5
• N. B. Keskin, A. Zeevi, Dynamic Pricing with an Unknown Demand Model: Asymptotically Optimal Semi-Myopic Policies, Operations Research 62(5):1142–1167, 2014. https://doi.org/10.1287/opre.2014.1294