Dynamic Pricing under Nested Logit Demand

• The paper introduces a dynamic pricing framework that uses a nested logit model to capture substitution effects and computes market-clearing prices via convex optimization.
• The paper employs gradient-based methods, including Nesterov acceleration, to achieve convergence rates of O(1/t) and O(1/t²) for efficient price updates.
• The paper demonstrates that incorporating consumer information frictions and quadratic supplier adjustment costs enhances price stability and algorithmic performance.

Dynamic pricing under nested logit demand addresses the allocation of prices in online marketplaces where buyer preferences are structured by correlated random utility models and suppliers incur adjustment costs. The framework models product demand using a nested logit structure to capture substitution patterns between products within and across groups (“nests”), incorporates information-processing frictions among consumers, and introduces quadratic adjustment costs on the supply side. Market-clearing prices are computed by convex optimization of a total expected revenue function, leveraging both primal and dual formulations. Rigorous gradient-based algorithms yield convergence guarantees for price updates, with implications for digital market operators seeking algorithmic equilibrium between supply and demand (Müller et al., 2021).

1. Model Specification

The marketplace consists of $n$ products, partitioned into $L$ disjoint nests $N_1, \ldots, N_L$. Consumer heterogeneity is explicitly modeled: each type $j$ has deterministic intrinsic values $a_{j,i}$ for each product and faces prices $p=(p(1),\ldots,p(n))^\top \in \mathbb{R}^n_{\ge 0}$. The random utility for type $j$ individual choosing product $i$ is given by

$U_{j,i}(p)=a_{j,i}-p(i)+\varepsilon_{j,i},$

with $$\varepsilon_j=(\varepsilon_{j,1},\ldots,\varepsilon_{j,n})$$ following the nested-logit distribution:

$$f_{\varepsilon_j}(z) = \prod_{\ell=1}^L \exp\Big\{-z_{i}/u_{j,\ell}\Big\} \exp\Big\{-M_{j,\ell}^{-1}\sum_{k\in N_\ell}e^{-z_k/u_{j,\ell}}\Big\},$$

where $u_{j,\ell}\in (0,1]$ control nest-specific choice correlation (within-nest correlation $1-u_{j,\ell}^2$, no correlation across nests).

The expected surplus for a type $j$ consumer is

$$E_j(p) = \mathbb{E}[\max_{1\leq i \leq n} \{ a_{j,i} - p(i) + \varepsilon_{j,i} \}].$$

This surplus function $E_j$ is convex and differentiable; its negative partial derivatives correspond to choice probabilities:

$\frac{\partial E_j}{\partial p(i)} = -x_{j,i}(p),$

where $x_{j,i}(p)$ is the probability a consumer of type $j$ chooses product $i$. The closed-form for $x_{j,i}(p)$ under nested logit (for $i\in N_\ell$) is:

$$x_{j,i}(p) = \frac{e^{(a_{j,i}-p(i))/u_{j,\ell}}}{\sum_{k\in N_\ell}e^{(a_{j,k}-p(k))/u_{j,\ell}}} \, \cdot\, \frac{ (\sum_{k\in N_\ell}e^{(a_{j,k}-p(k))/u_{j,\ell}})^{u_{j,\ell}}} {\sum_{m=1}^L (\sum_{k\in N_m} e^{(a_{j,k}-p(k))/u_{j,m}})^{u_{j,m}} }$$

For $N_j$ consumers of type $j$, aggregate demand is $D(p) = \sum_{j=1}^J N_j\; x_j(p)$.

2. Total Expected Revenue and Market Clearing

Suppliers ($K$ in total) optimize their profit given prices. Each supplier $k$ solves

$$\pi_k(p) = \max_{y_k\in Y_k}\{ p^\top y_k - C_k(y_k) \},$$

with $$C_k(y_k) = C_k^0(y_k) + \frac{1}{2}I_k \|y_k - y_k^0\|_2^2$$ expressing convex cost and an adjustment penalty (strong convexity $I_k>0$); $\nabla \pi_k(p) = y_k(p)$ by the envelope theorem.

The total expected revenue (TER) function, central to pricing, is

$$\mathrm{TER}(p) = \sum_{k=1}^K \pi_k(p) + \sum_{j=1}^J N_j E_j(p),$$

which is convex and $C^1$ in $p$. Its gradient,

$$\nabla \mathrm{TER}(p) = \sum_{k=1}^K y_k(p) - \sum_{j=1}^J N_j x_j(p),$$

balances aggregate supply and demand.

The market-clearing price vector is then given by

$p^* = \arg\min_{p \geq 0} \mathrm{TER}(p),$

with equilibrium characterized by coordinatewise balance of supply and demand:

$$p^* \geq 0, \quad \nabla \mathrm{TER}(p^*) \geq 0, \quad {p^*}^\top \nabla \mathrm{TER}(p^*) = 0.$$

Convexity of $\mathrm{TER}$ is guaranteed since $\pi_k(p)$ (supremum of affine maps) and $E_j(p)$ (logit surplus) are convex.

3. Dual Formulation and Strong Convexity

Primal-dual analysis leverages convex conjugates:

$$E_j^*(q) = \sup_{p \in \mathbb{R}^n} \{ p^\top q - E_j(p) \}, \qquad C_k^*(\lambda) = \sup_{y_k} \{ \lambda^\top y_k - C_k(y_k) \}$$

The nested-logit entropy formula shows $E_j^*$ is $\beta_j$-strongly convex with respect to $\ell_1$, where $\beta_j = \min_{\ell} u_{j,\ell}$. $C_k$ is $I_k$-strongly convex in $\ell_2$, so $C_k^*$ is $1/I_k$-smooth.

The dual convex program (by strong duality and linearity of the constraints) is:

$$\text{Minimize} \quad \sum_{k=1}^K C_k(y_k) + \sum_{j=1}^J N_j E_j^*(q_j)$$

subject to

$$\sum_{k=1}^K y_k = \sum_{j=1}^J N_j q_j, \quad y_k \in Y_k, \quad q_j \in \Delta$$

The dual objective is $\mu$-strongly convex in the block norm,

$$\| (y,q) \|^2 = \sum_{k=1}^K \|y_k\|_2^2 + \sum_{j=1}^J \|q_j\|_2^2$$

where

$$\mu = \min\left\{ \min_k I_k,\; \min_j N_j \beta_j \right\}$$

4. Gradient-Based Dynamic Pricing Algorithms

Projected gradient and accelerated gradient schemes enable practical computation of dynamic prices.

Basic projected gradient (rate $O(1/t)$):

For stepsize $h \leq 1/L_{\mathrm{TER}}$ with

$$L_{\mathrm{TER}} = \sum_{j=1}^J N_j \beta_j + \sum_{k=1}^K I_k,$$

iterate:

$$p_{t+1} = \max \{ 0,\; p_t - h \sum_k y_k(p_t) + h \sum_j N_j x_j(p_t) \}$$

Convergence rate:

$$\mathrm{TER}(p_t) - \mathrm{TER}(p^*) \leq \frac{ \|p_0 - p^* \|_2^2 }{2 h t } = O(1/t)$$

Nesterov-accelerated method (rate $O(1/t^2)$):

With $q_0 = p_0$, $\theta_0 = 1$, iterate:

• $$p_{t+1} = \max\{0, q_t - h \nabla \mathrm{TER}(q_t)\}$$
• $\theta_{t+1} = (1 + \sqrt{1 + 4\theta_t^2})/2$
• $$q_{t+1} = p_{t+1} + ((\theta_t - 1)/\theta_{t+1})(p_{t+1} - p_t)$$

Convergence result:

$$\mathrm{TER}(p_t) - \mathrm{TER}(p^*) \leq \frac{2\|p_0 - p^*\|_2^2}{ h (t+1)^2 } = O(1/t^2)$$

Table: Algorithmic Convergence in Dynamic Pricing

Algorithm	Iteration Update	Rate
Gradient-projection	$$p_{t+1} = \max\{0, p_t - h \nabla \mathrm{TER}(p_t)\}$$	$O(1/t)$
Accelerated (Nesterov)	See above (three-step update)	$O(1/t^2)$

5. Interpretation, Implications, and Limitations

• Information-processing frictions: The nested logit structure ($u_{j,\ell}<1$) captures limited consumer rationality, making the surplus $E_j$ less steep and allowing larger stepsizes in gradient updates. This “smoothing” effect dampens demand swings and enhances numerical stability.
• Supplier adjustment costs: Quadratic penalties ($I_k\|y_k-y_k^0\|_2^2$) on suppliers’ deviations from reference outputs ensure that supply reacts smoothly to price shifts, further contributing to increased algorithmic stability.
• Rate improvements: Both consumer frictions and supply-side penalties increase the smoothness constant $L_{\mathrm{TER}}$, yet convex-analytic optimization achieves optimal $O(1/t^2)$ convergence—improving over the $O(1/\sqrt{t})$ typical in fully rational, frictionless systems.
• Implementation implications: Online market operators can apply these updates by measuring realized or expected demands and supplies and adjusting prices by the prescribed algorithm. Convergence to equilibrium is guaranteed under mild assumptions about $u_{j,\ell}$ and $I_k$.
• Model limitations: The mechanism requires known stepsize estimates ($\beta_j$, $I_k$); assumes price-taking behavior, exogenous and fixed nests, and deterministic consumer values $a_{j,i}$; population sizes $N_j$ are fixed; only nonnegative nest correlations are permitted. Richer substitution structures require more general models (e.g., generalized nested logit).

6. Context within Market Design and Discrete Choice

The dynamic pricing scheme extends classical market-clearing optimization by incorporating behavioral and operational frictions. The nested logit model advances over basic multinomial logit by allowing positive within-nest correlation, modeling realistic consumer substitution effects. The dual formulation links supply, demand, and price evolution to convex program structure, enabling rigorous rate guarantees for large-scale computational applications in electronic platforms and online marketplaces (Müller et al., 2021).

A plausible implication is that online-market intermediaries, through systematic application of the studied schemes, can achieve efficient and stable price-discovery even in settings characterized by consumer indecision and supplier inertia, provided the necessary convex-analytic prerequisites are met.