A flow-based ascending auction to compute buyer-optimal Walrasian prices

TL;DR

This work tackles computing buyer-optimal Walrasian prices in multi-unit markets where buyers have truncated additive valuations. It introduces a flow-based ascending auction that identifies overdemanded sets via a left-most $s$-$t$ cut in an auxiliary network and raises prices accordingly until reaching buyer-optimal, market-clearing prices with a stable allocation. The method avoids submodular minimization, supports warm starts with flow updates, and establishes monotonicity of prices with respect to supply and demand changes. It also clarifies the relationship to classical ascending auctions and VCG prices, showing that buyer-optimal prices align with minimum competitive prices in this setting but per-object VCG pricing may fail for truncated additive valuations. The results provide practical algorithms and structural insights for auction design in multi-unit, additive-valuations markets, with potential extensions to broader valuation classes and reoptimization scenarios.

Abstract

We consider a market where a set of objects is sold to a set of buyers, each equipped with a valuation function for the objects. The goal of the auctioneer is to determine reasonable prices together with a stable allocation. One definition of "reasonable" and "stable" is a Walrasian equilibrium, which is a tuple consisting of a price vector together with an allocation satisfying the following desirable properties: (i) the allocation is market-clearing in the sense that as much as possible is sold, and (ii) the allocation is stable in the sense that every buyer ends up with an optimal set with respect to the given prices. Moreover, "buyer-optimal" means that the prices are smallest possible among all Walrasian prices. In this paper, we present a combinatorial network flow algorithm to compute buyer-optimal Walrasian prices in a multi-unit matching market with additive valuation functions and buyer demands. The algorithm can be seen as a generalization of the classical housing market auction and mimics the very natural procedure of an ascending auction. We use our structural insights to prove monotonicity of the buyer-optimal Walrasian prices with respect to changes in supply or demand.

Figures (2)

• Figure 1: The network $G(\boldsymbol{p})$ with two buyers $j_1, j_2$ and three objects $\alpha, \beta, \gamma$. We have valuations $v_{j_1} = (3, 2, 1)$ and $v_{j_2} = (0, 2, 0)$, demands $d_{j_1} = 4, d_{j_2} = 2$, supply $b_\alpha = b_\beta = 1, b_\gamma = 4$, and current prices $p(\alpha) = p(\beta) = p(\gamma) = 0$. In this example, the item sets at prices $\boldsymbol{p} = (0,0,0)$ are given by $\Omega_1' = \{\alpha, \beta \}$, $\Omega_1" = \{\gamma\}$, $\Omega_1"' = \varnothing$ and $\Omega_2' = \varnothing$, $\Omega_2" = \{\beta\}$, $\Omega_1"' = \{\alpha, \gamma\}$.
• Figure 2: Proof sketch. Illustration of a minimum $s$-$t$-cut $C$ and the induced sets $I$ (light blue) and $T$ (light green). The set $I^=$ is red, $I^<$ is orange, $T_1$ is yellow and $T_2$ is petrol.

Theorems & Definitions (45)

• Example 1
• Lemma 2
• proof
• Proposition 3
• Theorem 4
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• proof