Pair-Demand Valuations in Markets

• Pair-demand valuations are models where the value of any bundle is determined by its best pair, generalizing unit-demand to capture complementarities.
• They include variants such as k-demand (k=2), bi-demand, sharp multi-unit demand, and two-level private demand, each with unique rules and economic implications.
• These models influence welfare maximization and pricing strategies by enabling polynomial-time solutions in some cases while introducing NP-hard challenges in equilibrium computation.

Pair-demand valuations are valuation models in which bundles of cardinality two are the relevant objects of choice. In combinatorial markets, they arise as the $k=2$ instance of $k$-demand valuations, where a bundle’s value is determined by its best 2-subset; in bi-demand markets, each buyer may take up to two items and values a bundle as the sum of the two highest single-item values in it; in sharp multi-unit demand models, each buyer wants exactly two items or none; and in Bayesian multi-unit pricing with private demands, the analogous “pair-demand” case is the first non-trivial setting with two possible demand levels (Deligkas et al., 2020, Bérczi et al., 2021, Chen et al., 2012, Devanur et al., 2017). Across these formulations, pair-demand valuations form a natural extension of unit-demand preferences, but they also mark a sharp change in equilibrium existence, algorithmic tractability, and pricing structure.

1. Formal models of pair-demand

The term “pair-demand” does not denote a single canonical valuation class. Instead, it refers to several closely related models in which demand is bounded by, concentrated on, or parameterized by the number two.

Setting	Formal rule	Interpretation
$k$-demand, $k=2$	$v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$, or equivalently $\max\{\,v(T):T\subseteq S,\ |T|=2\}$ with the convention that if $|S|<2$ then $v_2(S)=v(S)$	The value of a bundle is the value of its best contained pair
Bi-demand valuations	$$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$	The value is the sum of the two highest single-item values in $X$
Sharp multi-unit demand with $k$0	Each buyer receives either exactly two items or none	A bundle of size less than $k$1 has no value
Two-demand private-demand model	Type $k$2 with $k$3 and value for $k$4 units equal to $k$5	Demand is private and can take two possible levels

For $k$6-demand valuations, Deligkas, Melissourgos, and Spirakis define the extension on a finite set of indivisible items $k$7 by

$k$8

and for $k$9 this specializes to the pair-demand rule

$k$0

They illustrate the definition with pair-only value tables, including cases in which several pairs are simultaneously “demand sets” (Deligkas et al., 2020).

The bi-demand model in dynamic pricing is more structured. There is a ground set $k$1 of items and a set $k$2 of buyers. Each buyer $k$3 has a value $k$4 for each item $k$5, a demand bound $k$6, and bundle value

$k$7

which is explicitly described as the sum of the two highest single-item values in $k$8 (Bérczi et al., 2021).

The sharp-demand model of Briest et al. is different in both objective and semantics. Items are $k$9, each with quality $k=2$0. Buyers are $k=2$1, each with sharp demand $k=2$2 and unit-quality value $k=2$3, so item values take the correlated form $k=2$4. Allocations are disjoint, with each $k=2$5 either empty or of size $k=2$6, and utility is

$k=2$7

or $k=2$8 if $k=2$9 (Chen et al., 2012).

In the unlimited-supply Bayesian pricing model of Devanur, Haghpanah, and Psomas, the buyer’s type is $v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$0, where $v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$1 is the private per-unit value and $v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$2 is the private demand. If allocated $v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$3 units at price $v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$4, quasi-linear utility is

$v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$5

Here the “pair-demand” perspective refers to the simplest non-trivial case with exactly two possible demands, rather than to a combinatorial pair of distinct items (Devanur et al., 2017).

2. Boundary between unit-demand and complementarity

Pair-demand valuations generalize unit-demand, but they do not preserve the same structural guarantees. In the terminology of Deligkas–Melissourgos–Spirakis, 1-demand = unit-demand valuations form a strict subclass of 2-demand: any 1-demand function is 2-demand but not vice versa. Unit-demand valuations satisfy Gross Substitutes (GS) and always admit a Walrasian equilibrium. General 2-demand valuations, by contrast, can exhibit complementarities and need not be GS (Deligkas et al., 2020).

This distinction is especially important because many tractability and existence results in market design hinge on GS. The data explicitly place 2-demand valuations inside the usual hierarchy

$v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$6

and note that intersections with the 2-demand restriction produce subclasses such as “2-demand XOS” and “2-demand budget-additive.” Even within these restricted subclasses, one loses GS and equilibrium existence becomes hard to decide (Deligkas et al., 2020).

At the same time, pair-demand does not automatically imply arbitrary complementarity. If the pair-values $v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$7 arise from a collection of additive menus,

$v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$8

with each $v_2(S)=\max\{\,v(T):T\subseteq S,\ |T|\le 2\}$9 supported on at most two items, then $\max\{\,v(T):T\subseteq S,\ |T|=2\}$0 is fractionally subadditive (XOS), hence subadditive. However, not all 2-demand valuations are subadditive unless additional structure is imposed (Deligkas et al., 2020).

A useful conceptual consequence is that pair-demand valuations are best viewed as a threshold class. They remain close enough to unit-demand for matching-based and LP-based techniques to remain relevant in several special cases, yet they are already rich enough to produce complementarities, equilibrium non-existence, and NP-hardness. This suggests that “allowing only pairs” is not a minor perturbation of the unit-demand model, but a structurally significant change.

3. Welfare maximization and dynamic pricing in bi-demand markets

For bi-demand valuations, maximum social welfare admits a clean LP formulation as a maximum-weight $\max\{\,v(T):T\subseteq S,\ |T|=2\}$1-matching problem. The market is viewed as a complete bipartite graph $\max\{\,v(T):T\subseteq S,\ |T|=2\}$2 with edge weights $\max\{\,v(T):T\subseteq S,\ |T|=2\}$3. An integral optimum corresponds to a maximum-weight $\max\{\,v(T):T\subseteq S,\ |T|=2\}$4-matching with $\max\{\,v(T):T\subseteq S,\ |T|=2\}$5 for all items and $\max\{\,v(T):T\subseteq S,\ |T|=2\}$6 for all buyers. Relaxing integrality yields the primal LP

$\max\{\,v(T):T\subseteq S,\ |T|=2\}$7

Its dual introduces $\max\{\,v(T):T\subseteq S,\ |T|=2\}$8 for item-capacity constraints and $\max\{\,v(T):T\subseteq S,\ |T|=2\}$9 for buyer-capacity constraints: $|S|<2$0 By complementary slackness, edges with equality $|S|<2$1 are exactly those used in some optimum $|S|<2$2-matching; these edges are called “tight” or “legal” (Bérczi et al., 2021).

A key contribution is the selection of an optimal dual solution with distinguished structural properties. By a perturbation argument, one can choose an optimum dual $|S|<2$3 such that: (a) an edge $|S|<2$4 is tight if and only if it appears in at least one maximum-weight $|S|<2$5-matching; (b) $|S|<2$6 if and only if there is a maximum-weight $|S|<2$7-matching that leaves $|S|<2$8 strictly under-saturated, and analogously for $|S|<2$9; and (c) this solution can be found in polynomial time by alternating modifications of the weights and re-computing a minimum cover. The tight-edge subgraph $v_2(S)=v(S)$0 therefore captures exactly the feasible allocations at each residual stage (Bérczi et al., 2021).

The dynamic-pricing scheme maintains the residual instance $v_2(S)=v(S)$1, an optimal dual cover $v_2(S)=v(S)$2 for that instance, the tight-edge graph $v_2(S)=v(S)$3, and an “adequate ordering” $v_2(S)=v(S)$4 of $v_2(S)=v(S)$5 with respect to $v_2(S)=v(S)$6. Adequacy means that whenever a buyer $v_2(S)=v(S)$7 arrives, her top-2 neighbors in $v_2(S)=v(S)$8 induce a feasible set of two items for her. The prices are then set by

$v_2(S)=v(S)$9

where

$$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$0

The next buyer picks her best bundle of size $$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$1 according to these prices, with ties broken arbitrarily; by construction she chooses exactly two legal items and thus a feasible set. The process then removes that buyer and her chosen items and repeats (Bérczi et al., 2021).

The proof sketch rests on two ideas. Complementary slackness ensures that legal edges support all maximum-weight matchings, while the tiny lexicographic perturbation via $$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$2 forces each buyer to pick exactly the two smallest-indexed legal neighbors. Adequacy of $$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$3 prevents infeasible pair selections that would destroy optimal completability. Since every buyer picks a feasible pair, after $$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$4 steps all items are allocated and the resulting allocation is a global optimum of the primal LP (Bérczi et al., 2021).

Computationally, each phase requires solving one maximum-weight $$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$5-matching, or equivalently the primal or dual LP, in a bipartite graph of size $$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$6; this can be done in time $$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$7 or better. Computing the special dual cover requires $$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$8 perturbed reweightings plus matching-cover computations, and determining an adequate ordering uses standard matching-connectivity tests and recursion on strictly smaller instances. The resulting dynamic scheme is polynomial-time and, in the bi-demand case, yields a purely posted-price, tie-break-free mechanism that always achieves maximum social welfare (Bérczi et al., 2021).

4. Walrasian equilibrium: existence, hardness, and special cases

For pair-demand valuations, the Walrasian picture changes sharply relative to unit-demand. The decision problem “WalrasianExists” asks whether there is an allocation $$v_t(X)=\max\{\,v_t(X')\mid X'\subseteq X,\ |X'|\le 2\}$$9 and prices $X$0 such that $X$1 is a Walrasian equilibrium. Deligkas–Melissourgos–Spirakis show that WalrasianExists is strongly NP-hard even when every agent’s valuation is 2-demand XOS and each agent has positive value on at most six items (Deligkas et al., 2020).

The hardness proof proceeds from 3-bounded 3-dimensional matching (3DM). For each $X$2 one creates an agent $X$3, and for each $X$4 an item $X$5. If $X$6 belongs to triplets $X$7, the corresponding agent defines additive menus $X$8 supported on the two items $X$9, each with unit weight, and valuation

$k$00

An allocation of welfare $k$01 exists if and only if one can choose, for each $k$02, a disjoint pair corresponding to one of its triplets, that is, a 3D matching of size $k$03. Hence deciding Walrasian equilibrium existence is equivalent to solving the 3DM instance under the stated bounded-occurrence conditions (Deligkas et al., 2020).

The paper also isolates tractable special cases. For pure unit-demand markets, winner determination reduces to maximum-weight perfect matching on a bipartite graph, and prices can be recovered by solving a linear program with constraints

$k$04

together with nonnegativity on allocated items and zero prices on unallocated ones. Using the quasi-NC matching algorithm of Fenner–Gurjar–Thierauf and Lüker’s quasi-NC algorithm for 2-variable LP, the full Walrasian problem is in quasi-NC for unit-demand valuations (Deligkas et al., 2020).

For 2-demand single-minded valuations, each agent wants exactly one specific pair $k$05 for value $k$06 and has zero value for any other bundle. Winner determination then reduces to maximum-weight matching on the general graph whose vertices are items and whose edges $k$07 carry weights $k$08. Given the matching, one again recovers equilibrium prices by the 2-variable LP trick. The running time is $k$09 or better using Edmonds’ blossom algorithm plus polynomial-time LP (Deligkas et al., 2020).

Economically, the distinction is explicit: unit-demand agents enjoy GS, and prices can be adjusted without creating demand cycles; once agents can value pairs, complementarities may arise, GS fails, and equilibria may not exist. The special cases show that matching-based reductions still survive when the pair structure is sufficiently sparse or rigid, but the general 2-demand XOS setting already lies in strongly NP-hard territory (Deligkas et al., 2020).

5. Revenue maximization with sharp pair demand

In the sharp multi-unit demand model with $k$10 and correlated values $k$11, the central objects are envy-free pricing and competitive equilibrium. An allocation $k$12 consists of disjoint sets, with each $k$13 either empty or of size $k$14, and prices satisfy $k$15. The pair $k$16 is envy-free if every winning buyer prefers her assigned pair to every other 2-set and has nonnegative utility, while every losing buyer derives nonpositive utility from every 2-set. It is a competitive equilibrium if it is envy-free and every unsold item has price zero. Revenue is $k$17 (Chen et al., 2012).

For this pair-demand case, the main complexity statements are explicit. Under $k$18 and sharp $k$19 demands, a competitive equilibrium may or may not exist, but there is a polynomial-time algorithm in $k$20 that tests existence and, if one exists, returns a revenue-maximizing competitive equilibrium. For envy-free pricing, because $k$21 is constant, the general constant-$k$22 algorithm runs in polynomial time and computes a revenue-maximizing envy-free pricing; by contrast, for arbitrary demands the envy-free optimization problem is NP-hard (Chen et al., 2012).

The competitive-equilibrium algorithm has two stages. Stage 1 finds a “candidate winner set” $k$23 by partitioning buyers by distinct $k$24-levels $k$25, highest values first, and maintaining the residual supply $k$26. If $k$27, the algorithm checks whether there is a subset $k$28 with $k$29; if so, it moves to Stage 2, and otherwise declares that no equilibrium exists. If $k$30, it adds $k$31 to $k$32 and decreases $k$33 by $k$34. The claim is that if any competitive equilibrium exists, this procedure finds $k$35 equal to the set of winners in some competitive equilibrium (Chen et al., 2012).

Stage 2 fixes the unique “greedy” allocation $k$36 that assigns the top-quality items to buyers in $k$37 in descending order of $k$38. Prices are then obtained from an exponential-size LP maximizing revenue subject to nonnegativity, zero price on unallocated items, winner envy-freeness constraints

$k$39

and loser constraints

$k$40

Because separation for loser constraints takes $k$41 by sorting $k$42, the ellipsoid method solves the LP in polynomial time. Stage 1 runs in $k$43, and Stage 2 is polynomial in $k$44 (Chen et al., 2012).

For envy-free pricing, the algorithm specializes the general bounded-demand method to $k$45. Candidate winner sets $k$46 satisfy $k$47 and the monotonicity condition that no buyer with higher $k$48 can become a loser and still be envy-free. In the pair-demand case, the total number of candidate sets is $k$49, and they can be generated by a simple DP in time $k$50. For each candidate set, the method either enumerates small subsets $k$51 of items when $k$52, or fixes the top-4 items and uses a dynamic program Solve-DLP to choose the remaining $k$53 items. The subroutine MaxRevenue$k$54 is a tiny LP, constant-size when $k$55, and decides in $k$56 time whether prices exist for the fixed allocation and, if so, returns revenue-maximizing prices (Chen et al., 2012).

Several simplifications are specific to pair demand: every winner blocks exactly $k$57 units of supply; subset-sum checks reduce to checking whether $k$58 is even and at most $k$59; the loser checks involve only subsets of size at most $k$60; and Solve-DLP has only two phases. These observations explain why the bounded-demand framework becomes particularly explicit in the $k$61 case (Chen et al., 2012).

6. Bayesian multi-unit pricing with two possible demands

In the unlimited-supply single-buyer model with type $k$62 and $k$63, the central regularity notion is decreasing marginal revenue (DMR). For a distribution with CDF $k$64 and density $k$65, DMR means that

$k$66

is concave in $k$67, equivalently

$k$68

or, equivalently, $k$69 is non-decreasing in $k$70. In the two-demand setting, both conditional distributions $k$71 and $k$72 are assumed to satisfy DMR (Devanur et al., 2017).

Under DMR, the optimal menu is deterministic. The seller posts two prices $k$73, offering $k$74 units for price $k$75 and $k$76 units for price $k$77. A buyer of type $k$78 chooses whichever bundle maximizes

$k$79

Expected revenue is

$k$80

For a $k$81-buyer, the relevant thresholds are

$k$82

which determine whether the buyer purchases the $k$83-bundle, the $k$84-bundle, or nothing. Writing these regions out yields a closed-form expression for $k$85, and the key proposition is that under DMR the map $k$86 is jointly concave on the convex domain $k$87 (Devanur et al., 2017).

Concavity gives both structure and computation. The proof partitions the $k$88-plane into regions where the threshold ordering is fixed, shows concavity of each piece via DMR, and then checks that both the values and gradients agree on boundaries, yielding global concavity. As a consequence, the optimal prices are determined by either an interior optimum or one of two boundary faces (Devanur et al., 2017).

The closed-form solution uses the monopoly-reserve points

$k$89

and $k$90. Three cases arise. In Case A, if $k$91, the separate-reserves solution $k$92 is feasible and optimal. In Case B, if $k$93, the optimum lies on the lower boundary $k$94, so $k$95 and $k$96. In Case C, if $k$97, the optimum lies on the upper boundary $k$98, where $k$99 maximizes

$k$00

In all cases, the problem reduces to maximizing a concave function on a convex set, so standard convex-optimization methods yield a polynomial-time algorithm to any desired precision (Devanur et al., 2017).

Taken together, these results show that the significance of pair-demand depends on market architecture. In combinatorial exchange, it is the first setting in which complementarities decisively disrupt Walrasian existence and tractability. In dynamic pricing for bi-demand buyers, LP duality and tight-edge structure are still strong enough to support welfare-optimal posted prices. In revenue maximization, the same cardinality restriction can either permit polynomial-time envy-free and competitive-equilibrium pricing under correlated values, or admit a deterministic and concave optimal price curve under DMR in the single-buyer unlimited-supply model.