Poisson-MNL Model Overview

• The Poisson-MNL model is a hybrid framework that integrates a Poisson arrival process with multinomial logit choice behavior for dynamic assortment and pricing.
• It decomposes revenue into components driven by decision-dependent arrivals and per-customer choices, enabling separate maximum likelihood estimation for each part.
• The model underpins UCB-based algorithms with near-optimal regret bounds and offers a versatile approach applicable to both dynamic revenue management and multinomial regression.

Searching arXiv for the specified papers and closely related work on Poisson-MNL and multinomial logit formulations. The Poisson-MNL model denotes a class of models that combine a Poisson arrival process with a multinomial logit (MNL) choice mechanism. In the formulation studied for dynamic joint assortment and pricing, the seller chooses an assortment and prices at each period, the number of arrivals in that period is Poisson with a mean that depends on those decisions, and each arriving customer then makes an MNL choice among offered products and a no-purchase option (Cai et al., 18 Feb 2026). In a distinct but historically related statistical usage, the expression also refers to the “Poisson trick,” under which an MNL likelihood can be represented through a Poisson surrogate model with observation-specific exposure terms, yielding exact fixed-effects MNL estimation for independent responses (Lee et al., 2017). These two usages are related by the interaction between Poisson and multinomial structure, but they address different problems: the former is a dynamic revenue-management and bandit model with decision-dependent arrivals, whereas the latter is a likelihood-equivalence device for fitting multinomial regression.

1. Formal definition and scope

In the dynamic assortment-pricing setting, time is indexed by periods $t=1,\dots,T$. At the start of each period, the decision-maker chooses an assortment $S_t \subseteq [N]$ with $|S_t|=K$ and prices $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$ (Cai et al., 18 Feb 2026). Product $j$ has time-varying features $z_{jt} \in \mathbb{R}^{d_z}$. Within period $t$, a random number of customers arrives, and each customer either purchases one item in the assortment or does not purchase.

The arrival process is specified as

$$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$

where $\Lambda>0$ is a known base arrival-rate constant and $x(S,p)\in\mathbb{R}^{d_x}$ is a vector of sufficient statistics describing how the offered assortment and prices affect arrivals (Cai et al., 18 Feb 2026). The unknown parameter $S_t \subseteq [N]$0 satisfies $S_t \subseteq [N]$1, and the feasible feature span has rank $S_t \subseteq [N]$2.

Conditional on arrival, customer choice follows a contextual MNL model with no-purchase option. For $S_t \subseteq [N]$3,

$S_t \subseteq [N]$4

and

$S_t \subseteq [N]$5

where $S_t \subseteq [N]$6 is unknown and satisfies $S_t \subseteq [N]$7 (Cai et al., 18 Feb 2026).

The expected revenue per arriving customer is

$S_t \subseteq [N]$8

and the expected total revenue in period $S_t \subseteq [N]$9 is

$|S_t|=K$0

This decomposition is central: the model separates the effect of decisions on how many customers arrive from their effect on what those customers buy (Cai et al., 18 Feb 2026).

In the older multinomial-regression literature, the Poisson-MNL connection takes a different form. There, for observations $|S_t|=K$1 and alternatives $|S_t|=K$2, one introduces Poisson counts

$|S_t|=K$3

with observation-specific exposure $|S_t|=K$4. Conditional on the total $|S_t|=K$5, the vector $|S_t|=K$6 is multinomial with probabilities

$|S_t|=K$7

so maximizing the profiled Poisson likelihood yields the same fixed-effects MLE and asymptotic variance as the multinomial MNL model (Lee et al., 2017). This is not a dynamic arrival model, but a reparameterization for inference.

2. Decision-dependent arrivals and model structure

The principal innovation of the dynamic Poisson-MNL formulation is the explicit dependence of the arrival rate on the seller’s decisions. Classical MNL bandit models treat arrivals as fixed and therefore optimize only the expected revenue per customer $|S_t|=K$8. By contrast, in period-based operations, total period revenue is $|S_t|=K$9, so the arrival elasticity with respect to assortment and price affects the optimal action directly (Cai et al., 18 Feb 2026).

The paper gives two examples of the arrival-rate specification. One captures price sensitivity and “variety”: $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$0 A second incorporates individual and pairwise effects, including complements, substitutes, and price variation: $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$1 Both fit within the log-linear template $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$2 (Cai et al., 18 Feb 2026).

This specification is designed to avoid confounding. The Poisson component accounts for arrival intensity, while the MNL component models conditional choice among products given arrival. A plausible implication is that the model is particularly suited to settings where assortment and pricing affect both customer traffic and conversion. The paper states that ignoring decision-dependent arrival rates can drive algorithms toward high prices that reduce arrivals and therefore lower total revenue (Cai et al., 18 Feb 2026).

The model also imposes standard regularity conditions. Assortments must have cardinality exactly $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$3; prices lie in $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$4 with $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$5; the arrival features $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$6 are bounded and full-rank over the feasible set; and MNL normalization is provided by setting the no-purchase utility to $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$7 (Cai et al., 18 Feb 2026). Product features $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$8 are i.i.d. across periods with $p_t=(p_{jt})_{j=1}^N \in [p_l,p_h]^N$9.

3. Likelihood decomposition and estimation principles

A notable structural feature of the dynamic Poisson-MNL model is that the log-likelihood decomposes additively: $j$0 The Poisson term is

$j$1

and the MNL term is

$j$2

(Cai et al., 18 Feb 2026).

This separation permits independent maximum-likelihood estimation of the arrival parameter $j$3 and the choice parameter $j$4. The paper emphasizes that estimation is via MLE and that “no EM [is] needed” in this dynamic model (Cai et al., 18 Feb 2026). In the learning algorithm, an initial exploration phase produces pilot MLEs,

$j$5

followed by local MLE updates constrained to Euclidean balls around the pilots (Cai et al., 18 Feb 2026).

The “Poisson trick” literature also relies on a Poisson/MNL likelihood factorization, but in a different way. For independent multinomial data, the joint Poisson likelihood factorizes into a Poisson likelihood for totals and a multinomial conditional likelihood: $j$6 Profiling out the nuisance exposure $j$7 yields

$j$8

which is exactly the multinomial log-likelihood up to constants (Lee et al., 2017). Thus, the fixed-effects MLEs and their asymptotic variance coincide between the Poisson surrogate and the standard MNL model.

The relation between the two literatures is structural rather than substantive. Both exploit Poisson and multinomial compatibility, but the dynamic Poisson-MNL model uses Poisson structure to model decision-dependent arrivals in a sequential optimization problem, whereas the Poisson trick uses it as a computational representation for multinomial regression.

4. PMNL algorithm and sequential decision-making

The learning policy proposed for the dynamic model is PMNL, an efficient algorithm “based on the idea of upper confidence bound (UCB)” (Cai et al., 18 Feb 2026). The algorithm is divided into two stages.

In Stage 1, the policy conducts short pure exploration for

$j$9

periods, using design sequences $z_{jt} \in \mathbb{R}^{d_z}$0 satisfying the isotropy assumption. The purpose is to obtain informative pilot MLEs for both arrival and choice parameters and to guarantee bounded pilot errors with high probability (Cai et al., 18 Feb 2026). The stated pilot-error bounds $z_{jt} \in \mathbb{R}^{d_z}$1 and $z_{jt} \in \mathbb{R}^{d_z}$2 are both $z_{jt} \in \mathbb{R}^{d_z}$3 when $z_{jt} \in \mathbb{R}^{d_z}$4, with explicit formulas involving $z_{jt} \in \mathbb{R}^{d_z}$5, $z_{jt} \in \mathbb{R}^{d_z}$6, $z_{jt} \in \mathbb{R}^{d_z}$7, $z_{jt} \in \mathbb{R}^{d_z}$8, and other constants (Cai et al., 18 Feb 2026).

In Stage 2, for each period $z_{jt} \in \mathbb{R}^{d_z}$9, the policy computes local MLEs constrained around the pilots: $t$0 It then evaluates, for each candidate assortment-price pair $t$1, an optimistic revenue

$t$2

which equals the plug-in revenue under $t$3 plus two UCB bonuses: one for uncertainty in the Poisson arrival model and one for uncertainty in the MNL choice model (Cai et al., 18 Feb 2026). The policy chooses

$t$4

The confidence bonuses are built from Fisher-information matrices. For arrivals,

$t$5

For choices,

$t$6

with

$t$7

Because $t$8 depends on the unknown $t$9, the algorithm introduces data-dependent bounds $$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$0 and $$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$1 satisfying

$$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$2

which provide a “sandwich” construction for the MNL confidence term (Cai et al., 18 Feb 2026).

The paper characterizes the computational problem in each period as a combinatorial-continuous optimization over $$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$3 and $$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$4. It notes that practical solvers can combine combinatorial search over assortments with concave or local search in prices when tractable, or generic solvers otherwise, while the theoretical development centers on the objective $$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$5 rather than a particular implementation (Cai et al., 18 Feb 2026).

5. Regret bounds and optimality claims

The dynamic Poisson-MNL formulation is embedded in a bandit problem whose benchmark is the period-by-period oracle maximizing true expected revenue: $$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$6 The regret of a policy $$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$7 over horizon $$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$8 is

$$n_t \sim \mathrm{Poisson}(\Lambda \lambda_t), \qquad \lambda_t := \lambda(S_t,p_t;\theta^*) = \exp\!\big(\theta^{*T}x(S_t,p_t)\big),$$9

(Cai et al., 18 Feb 2026).

Under Assumptions 1–5, the main upper bound states that there exist universal positive constants $\Lambda>0$0 such that, for all sufficiently large $\Lambda>0$1,

$\Lambda>0$2

where $\Lambda>0$3 depends only on $\Lambda>0$4; $\Lambda>0$5 only on $\Lambda>0$6; and $\Lambda>0$7 only on $\Lambda>0$8 (Cai et al., 18 Feb 2026). The paper summarizes this as a rate of

$\Lambda>0$9

which is nearly minimax optimal in $x(S,p)\in\mathbb{R}^{d_x}$0 up to a $x(S,p)\in\mathbb{R}^{d_x}$1 factor.

The role of the arrival model is evident in the dimension dependence. When $x(S,p)\in\mathbb{R}^{d_x}$2 is constant, corresponding to $x(S,p)\in\mathbb{R}^{d_x}$3, the bound reduces to $x(S,p)\in\mathbb{R}^{d_x}$4 (Cai et al., 18 Feb 2026). This shows that the cost of learning decision-dependent arrivals appears explicitly through the arrival-feature dimension $x(S,p)\in\mathbb{R}^{d_x}$5.

The lower-bound results are nontrivial and partly matching. The non-asymptotic lower bound states that under mild structural conditions, for any policy $x(S,p)\in\mathbb{R}^{d_x}$6 there exists an instance satisfying the assumptions such that

$x(S,p)\in\mathbb{R}^{d_x}$7

for a constant $x(S,p)\in\mathbb{R}^{d_x}$8 depending on $x(S,p)\in\mathbb{R}^{d_x}$9 (Cai et al., 18 Feb 2026). Additional asymptotic lower bounds exhibit explicit dimension dependence, including $S_t \subseteq [N]$00, $S_t \subseteq [N]$01, and $S_t \subseteq [N]$02 terms under different regimes (Cai et al., 18 Feb 2026). The paper concludes that PMNL is nearly minimax optimal in $S_t \subseteq [N]$03 and optimal or near-optimal in dimension dependence in common regimes.

The proof methodology reflects the hybrid structure of the model. Poisson arrivals produce sub-exponential rather than sub-Gaussian fluctuations, so the analysis uses martingale Bernstein inequalities to control the Poisson log-likelihood deviations. The MNL component is controlled via Fisher information, and regret is bounded through the cumulative UCB bonuses, which are handled by determinant or log-barrier arguments (Cai et al., 18 Feb 2026).

6. Comparison with classical MNL models and the Poisson trick

The most immediate comparison is with classical MNL bandits that assume a fixed arrival process. In those models, decisions affect only conditional choice, not traffic volume, so optimization is over per-customer revenue rather than per-period revenue. The dynamic Poisson-MNL model generalizes this by allowing the arrival rate to depend on assortment and price (Cai et al., 18 Feb 2026). The paper argues that ignoring this dependence can yield linear regret when arrivals decline with price.

Simulation results in the paper support this distinction. In dynamic pricing with varying assortments, PMNL shows sublinear regret, whereas a baseline policy FM23, which assumes a fixed arrival rate per period and uses a learn-then-earn scheme, exhibits linear regret when $S_t \subseteq [N]$04 depends on price or assortment. When arrivals are truly constant, $S_t \subseteq [N]$05, PMNL matches FM23 performance. In a joint assortment-pricing experiment with $S_t \subseteq [N]$06 and $S_t \subseteq [N]$07, PMNL again shows sublinear regret, while a naive UCB that ignores $S_t \subseteq [N]$08-dependence shows linear regret (Cai et al., 18 Feb 2026). These are simulation findings rather than universal guarantees, but they illustrate the operational significance of decision-dependent arrivals.

A different comparison concerns the statistical Poisson trick. That method establishes an exact equivalence between fixed-effects multinomial logit estimation and a Poisson GLM with nuisance exposures. Specifically, with

$S_t \subseteq [N]$09

conditioning on the total count $S_t \subseteq [N]$10 yields the multinomial likelihood, and profiling out $S_t \subseteq [N]$11 recovers the MNL objective exactly (Lee et al., 2017). This is a fitting device for multinomial regression, not a model of arrivals and sequential revenue optimization.

The distinction is important because the phrase “Poisson-MNL model” may be interpreted differently across literatures. In revenue management and online learning, it refers to a two-stage generative mechanism: Poisson arrivals followed by MNL choices. In generalized linear modeling, it may refer to the Poisson surrogate representation of MNL or to Gamma-Poisson extensions for correlated multinomial responses (Lee et al., 2017). The shared terminology reflects mathematical kinship, but the inferential goals and substantive interpretations differ.

7. Extensions, limitations, and related formulations

The dynamic Poisson-MNL framework is explicitly parametric. Its theoretical analysis assumes a correctly specified Poisson log-linear arrival model and MNL choice probabilities, bounded features and parameters, and information growth conditions $S_t \subseteq [N]$12 (Cai et al., 18 Feb 2026). The paper identifies several extensions: non-Poisson arrival models with Bernstein-type tails, alternative choice models such as nested logit, inventory constraints, fully continuous pricing with specialized solvers, exogenous time effects in $S_t \subseteq [N]$13, and richer contexts (Cai et al., 18 Feb 2026). This suggests a broader research agenda in which the Poisson-MNL construction serves as a baseline hybrid demand model.

The practical limitations are also stated clearly. Joint optimization over combinatorial assortments and continuous prices can be computationally heavy, and the paper leaves practical solver design application-dependent (Cai et al., 18 Feb 2026). The model’s robustness under misspecification is discussed cautiously: misspecification of $S_t \subseteq [N]$14 or $S_t \subseteq [N]$15 affects performance, though the UCB mechanism remains sensible, and finer price grids may improve search resolution at computational cost (Cai et al., 18 Feb 2026).

In the multinomial-regression literature, an important extension is the Gamma-Poisson model for correlated responses. There, group-by-alternative random effects $S_t \subseteq [N]$16 are given independent gamma distributions, yielding a closed-form marginal likelihood for the Poisson surrogate and allowing exact inference relative to that approximating model via an ECM algorithm (Lee et al., 2017). This extension captures alternative-specific random intercept heterogeneity across clusters but does not provide random slopes unless further augmented. It differs from standard mixed logit, which introduces random tastes through random coefficients in the utility index and usually requires approximation of intractable integrals (Lee et al., 2017).

A common misconception is that all “Poisson-MNL” formulations are interchangeable. The evidence from these two papers indicates otherwise. One formulation is a sequential decision model with decision-dependent customer arrivals, regret analysis, and UCB learning (Cai et al., 18 Feb 2026). The other is a likelihood-equivalent representation of multinomial regression and its Gamma-Poisson extension for correlated data (Lee et al., 2017). Their common ground lies in Poisson–multinomial factorization, but their objects of inference, assumptions, and applications are materially different.

Taken together, these formulations show that the Poisson-MNL interface is a versatile modeling principle rather than a single canonical model. In one direction, it supports nearly optimal online learning for dynamic assortment and pricing with endogenous traffic (Cai et al., 18 Feb 2026). In another, it provides an exact and computationally convenient route to fixed-effects MNL estimation, along with tractable Gamma-Poisson generalizations for clustered multinomial data (Lee et al., 2017).