Chain Stability in Trading Networks

Abstract

We show that in general trading networks with bilateral contracts, a suitably adapted chain stability concept (Ostrovsky, 2008) is equivalent to stability (Hatfield and Kominers, 2012;Hatfield et al., 2013) if all agents' preferences are jointly fully substitutable and satisfy the Laws of Aggregate Supply and Demand. We also show that in the special case of trading networks with transferable utility, an outcome is consistent with competitive equilibrium if and only if it is not blocked by any chain of contracts. Moreover, from a computational perspective, checking whether an outcome is chain stable is substantially easier than directly checking whether an outcome is stable.

The Structure of Equilibria in Trading Networks with Frictions

“…Our results can be summarized as follow: For a model of trading networks with frictions (Fleiner et al, 2019) and under the assumptions of full substitutability (Sun and Yang, 2006;Ostrovsky, 2008;Hatfield et al, 2013) and the laws of aggregate demand and supply (Hatfield and Kominers, 2012;Hatfield et al, 2021), we show that -the set of competitive equilibria is a sublattice of the price space (first part of Theorem 1), -a generalized "rural hospitals theorem" holds: the difference between the number of signed downstream and the number of signed upstream contracts is the same for each firm in each equilibrium (second part of Theorem 1), -assuming additionally "bounded willingness to pay" (Fleiner et al, 2019), 2 there is an equilibrium that is most preferred by terminal sellers and an equilibrium that is most preferred by terminal buyers (Theorem 2), -a mechanism that selects buyer-optimal equilibria is group-strategy-proof for terminal buyers on the domain of unit-demand utility functions and similarly a mechanism that selects seller-optimal equilibria is group-strategy-proof for terminal sellers on the domain of unit-supply utility functions (Theorem 3).…”

“…The model follows Hatfield et al (2013), and the extensions of Fleiner et al (2019) and Hatfield et al (2021). We consider a finite set of firms F and a finite set of trades Ω.…”

“…Expansion Law of Aggregate Supply (LAS): For p, p ∈ R Ω f and each Ψ ∈ Hatfield et al (2021) introduce the notion of Monotone Substitutability which requires that FS, LAD and LAS hold jointly for the same bundles of trades. Hatfield et al (2021) do not assume continuity of utility functions for their main result, and in that case, monotone substitutability is generally a stronger property than the combination of FS, LAD and LAS. With continuity, however, monotone substitutability is equivalent to the combination of FS, LAD and LAS.…”

“…Even if wealth effects are absent, transaction frictions such as those induced by taxation, subsidies, or transaction costs, make a transferable utility model inapplicable. This has motivated researchers to explore how results for matching markets with transfers (Demange and Gale, 1985;Legros and Newman, 2007;Nöldeke and Samuelson, 2018;Galichon et al, 2019) and for trading networks (Fleiner et al, 2019;Hatfield et al, 2021) can be generalized beyond transferable utility.…”

“…Related issues occur inHatfield et al (2021) where the stronger Monotone Substitutability property is needed that restricts the choice in circumstances where the choice correspondence is multi-valued.…”