Pricing Similar Products Together—and Bounding How Much Better Prices Could Be

Why products cannot be priced in isolation

A seller’s best price for one product depends on the alternatives customers can choose. Nested logit models this by placing closer substitutes into groups called “nests,” such as similar room classes or related fares. Customers can also choose an outside option: buying nothing.

The preprint studies one seller, a fixed assortment and a single selling period. Costs and demand parameters are assumed known. The objective is expected profit per potential customer, not revenue alone. The model does not learn customer preferences or track inventory over time.

Operational restrictions complicate the calculation. A ceiling may bind for one product but not its neighbors; a weighted price cap can connect several products. These constraints can invalidate the markup formulas that simplify unconstrained pricing. Finding attractive prices first and adjusting them afterward need not preserve profit.

Moving the hardest calculations to the group level

The authors’ central contribution is an exact reformulation: a different mathematical description with the same optimal prices. It neither rounds prices onto a grid nor approximates the specified demand function.

First, they subtract profit from a sufficiently large constant. Minimizing this positive “shifted loss” is equivalent to maximizing profit; the term does not mean an observed business loss. They then transform prices into exponential variables and aggregate product contributions within each nest.

For the base problem, the remaining nonconvexity is isolated in at most N + 1 bilinear links, where N is the number of nests. A bilinear link multiplies two decision variables. One handles the objective ratio; the others handle nest contributions. The remaining nonlinear constraints have exact convex representations using exponential cones, a standard mathematical structure supported by suitable solvers.

This distinction matters computationally. Adding products enlarges the convex subproblems, but does not increase the base number of difficult links. Adding nests does.

A price recommendation with an error bound

The algorithm uses spatial branch-and-bound. It solves relaxed versions of the problem to bound the best achievable objective, checks candidate prices against the original constraints, and divides promising search regions into smaller pieces.

A feasible price vector establishes an achievable profit. The relaxation supplies an upper bound on optimal profit. Their difference bounds how much additional profit could remain, even when computation stops early.

The reported percentage gap requires care: it is normalized using shifted loss, not profit. A 0.01% shifted-loss gap does not mean that profit is within 0.01% of its optimum. The absolute gap, however, directly bounds remaining profit improvement per customer.

These are numerical certificates subject to solver tolerances. They certify optimization within the assumed demand model, not whether that model accurately describes customers. The reported audit checked feasibility, objectives and gaps, but did not independently verify the solver’s underlying bounds.

What the synthetic experiments found

All experiments used generated instances, not observed sales or field trials. All 160 runs in the matched nested-logit and simpler multinomial-logit campaign met the certification test. In a separate 96-run campaign with 50–150 products, 69 certified; the others stopped at an open-node limit, with shifted-loss gaps no larger than 0.282%.

The product-count experiment certified all 28 runs, covering up to 500 products with two or four nests. At 500 products, median runtimes were about 196 seconds for two nests and 411 seconds for four. Increasing nest count produced much sharper search-tree growth.

The economic comparisons also distinguish two shortcuts. Across 80 matched pairs, prices optimized under a calibrated multinomial-logit approximation lost an average of 6.14% of profit, and up to 34.42%, when evaluated under the same nested-logit demand as the benchmark prices.

Separately, projecting unconstrained common-markup prices into the feasible region lost a median 5.56%, and up to 32.60%, across 16 cases. Further local optimization nearly matched branch-and-bound in that comparison. Multistart local optimization also matched its incumbents throughout the matched campaign. The global method’s additional benefit was the bound, not consistently higher achieved profit.

Where the guarantees stop

The base formulation covers finite price intervals and linear price caps with nonnegative coefficients—not arbitrary signed price relationships. Extensions have different costs: a minimum total purchase probability adds no bilinear links, while an upper limit adds nest-level links. Price-order rules are inexpensive when the products have equal price sensitivities; detailed individual-product share constraints can make the difficult search dimension grow with product count.

The theoretical efficiency result is conditional. For fixed nest count, polynomial arithmetic-work bounds require controlled numerical conditioning, controlled instance constants and a uniform ability to recover sufficiently good feasible prices from tight relaxations. That recovery property is assumed, not proved for every instance.

Practical limits are substantial too. Four deliberately selected, wide-range challenging instances all failed to certify, ending with shifted-loss gaps of roughly 25%–63%. The evidence therefore supports a structured optimization method that works well on the tested small-nest instances—not a universal fast pricing solver or demonstrated real-world profit improvement.