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.

Trading Networks With General Preferences

“…Thus they reserve the term full substitutability for the weaker notion and all of their results hold for the weaker notion of full substitutability. Our use of the term full substitutability is consistent with the use of the term inHatfield et al (2018). They establish the equivalence of chain stability and stability in trading networks for general utility functions under the assumption of the stronger version of full substitutability that we also use.…”

“…Our results can be summarized as follow: Even without quasi-linearity, we show that the set of competitive equilibria has a lattice structure, provided that full substitutability (Sun and Yang, 2006;Ostrovsky, 2008;Hatfield et al, 2013) and the laws of aggregate demand and supply (Ostrovsky, 2008;Hatfield et al, 2018) hold. Moreover, under the same assumptions, we show that the difference between the number of signed downstream and the number of signed upstream contracts is the same for each firm in each equilibrium (this is a generalization of the rural hospitals theorem from matching theory).…”

Trading networks with bilateral contracts

“…Our work complements a recent paper by Hatfield et al (2015) on the properties of set-stable outcomes in general contract networks. They show that in general contract networks, under certain conditions, setstable outcomes coincide with (what we call) strongly trail-stable outcomes i.e.…”

“…There is no trail T , such that T ∩ A = ∅ and T is (A, f )-rational for all f ∈ F (T ). Hatfield et al (2015) showed that in general contract networks set-stable outcomes are equivalent to strongly trail-stable outcomes whenever choice functions satisfy full substitutability and Laws of Aggregate Demand and Supply. 9 However, Fleiner (2009) and Hatfield and Kominers (2012) showed that a set-stable outcomes may not exist in general contract networks (see Example 1 below).…”

A Compact, Logical Approach to Large-Market Analysis

“…Thus our exercise here is in some sense similar to that of Blume and Zame (1994), who unified our understanding of perfect and sequential equilibria by way of the Tarski-Seidenberg Theorem. But at the same time, we use our methods to prove completely new results: our results for trading networks-all of which are novel to this work-generalize the existence result of Hatfield et al (2013) to infinite trading networks (see also Hatfield et al, 2019Hatfield et al, , 2018Fleiner et al, forthcoming). Meanwhile, we show a way that Logical Compactness can be used to convert a dynamic game with a finite start time into an "ongoing" dynamic game with neither start nor end-connecting our work to the broad literature on infinite-horizon games (see, e.g., Fudenberg and Levine, 2009) .…”