Bayesian Costs Complicate Price Discrimination but Keep It Computable

Third-degree price discrimination is usually studied with the seller’s marginal cost treated as fixed. Bergemann, Brooks, and Morris gave a clean characterization of the seller-surplus and buyer-surplus outcomes attainable by segmenting an aggregate market under that assumption. This paper asks what remains true when the cost is itself private information of the seller. The answer is mixed: the achievable welfare region can lose the simple shape of the fixed-cost model, but it is still algorithmically tractable through a finite linear representation.

Core Contribution

The authors’ main claim is a characterization result rather than an empirical finding. For a finite value set $\mathcal{V}$ of size $n$ and a finite cost set $\mathcal{C}$ of size $m$, they show that the achievable welfare outcomes under Bayesian seller cost coincide with a proper projection of a polytope described by polynomially many linear constraints. That matters because the naive object is much larger: segmentations can be represented through extremal markets, but there are exponentially many such markets.

The technical bridge is a decomposition theorem. Any market can be written as a convex combination of $O(n)$ extremal markets while preserving seller surplus and bounding buyer surplus in the right direction. This extends the equal-surplus extremal-market idea from the fixed-cost setting to a Bayesian-cost setting where different costs induce different relevant price intervals.

Technical Approach

The paper first defines extremal markets $F_{q,S}$. The construction fixes a vector of minimum optimal prices $q$ and a support set $S$, then builds a market in which all supported prices between consecutive entries of $q$ give equal seller surplus for the corresponding cost. Lemma 5 identifies the optimal price set in such markets: for each cost $c_j$, the minimum optimal price is $q_j$, and the optimal prices lie in $[q_j,q_{j+1}] \cap S$ when the market has positive surplus.

Algorithm 1, DecomposeIntoExtremalMarkets(F), repeatedly subtracts the largest feasible multiple of an extremal market from the residual market. The stopping analysis is concrete: Lemma 7 states that the loop needs at most $2n$ iterations. This gives the structural theorem behind the paper’s reduction: general segmentations can be replaced by segmentations into extremal markets without changing the seller surplus and while preserving the buyer-surplus interval relevant for tie-breaking.

The second reduction is from extremal markets to fragments. Section 4 defines indifference fragments by restricting an extremal market to the interval $[q_j,q_{j+1})$ and renormalizing by its incoming tail mass. Because extremal-market decompositions are still exponential, the authors introduce undominated fragments and reduced fragment decompositions. The reduced representation has size $O(mn^2)$, with welfare quantities remaining linear in the reduced variables.

Results and Analysis

There are no simulations or datasets here; the result is a proof-backed algorithmic characterization. The paper’s evidence is the chain of reductions: arbitrary market segmentations reduce to $O(n)$ extremal markets; extremal-market decompositions reduce to $O(mn^2)$ fragment variables; and the feasible reduced fragment decompositions satisfy exact-composition and flow-conservation constraints. The “discounted” part appears in the flow constraints: transitions between fragment states scale mass by ratios determined by equal-surplus conditions across prices and costs.

The payoff is that optimizing any linear combination of seller surplus and buyer surplus becomes a polynomial-time linear-programming problem. This is stronger than merely proving existence of good segmentations: it gives a computational route to recover optimal segmentations for objectives such as maximizing seller surplus subject to buyer-surplus constraints, maximizing buyer surplus under seller-surplus targets, or optimizing a weighted welfare objective.

The appendices support the main proof obligations rather than add experiments. The visible proof pages show the preservation of seller surplus, minimum buyer surplus, and maximum buyer surplus through reduced fragment decompositions. In Proposition 1, seller surplus is written as a linear function of reduced variables $x$, while maximum and minimum social welfare are written as linear functions of $(x,z)$; buyer surplus follows by subtracting seller surplus from social welfare. The final proof of Theorem 4 closes the loop: if a segmentation exists with seller surplus $SS$ and buyer surplus $BS$, then a reduced fragment decomposition exists with the same seller surplus and with $BS$ lying between its minimum and maximum buyer-surplus bounds, and conversely Algorithm 2 reconstructs a segmentation whose welfare quantities match the reduced representation.

Limitations

The result is exact but model-specific. It assumes finite buyer values and finite seller costs, and the tractability statement is polynomial in those discrete supports. The paper does not claim a closed-form geometric description comparable to the fixed-cost case, and the constructed linear programs may still be large for fine discretizations. Its contribution is therefore best read as a computational characterization of the Bayesian-cost model, not as a simple economic formula for the welfare frontier.