Approximate optimality with bounded regret in dynamic matching models

Abstract

We consider a discrete-time bipartite matching model with random arrivals of units of supply and demand that can wait in queues located at the nodes in the network. A control policy determines which are matched at each time. The focus is on the infinite-horizon average-cost optimal control problem. A relaxation of the stochastic control problem is proposed, which is found to be a special case of an inventory model, as treated in the classical theory of Clark and Scarf. The optimal policy for the relaxation admits a closed-form expression. Based on the policy for this relaxation, a new matching policy is proposed. For a parameterized family of models in which the network load approaches capacity, this policy is shown to be approximately optimal, with bounded regret, even though the average cost grows without bound.

A stochastic matching model on hypergraphs

“…This completes the proof of (11) in view of (13). To conclude, as B r ∈ I(H) and in view of (10), we have that μ(B r ) ≤ H∈H |H ∩ B r | min k∈H μ(k), which implies, by an immediate induction using (11), that μ(B) ≤ H∈H |H ∩ B| min k∈H μ(k). This completes the proof.…”

“…hence (10). Now fix B, a subset of V that is not an independent set of H. Then we construct by induction the family of sets B := B 0 ⊃ B 1 ⊃ B 2 ⊃ ... ⊃ B r , where r is properly defined below, as follows: for any i ≥ 0, if B i is not an independent set of I(H), then we take an arbitrary hyperedge H j i ∈ H such that H j i ⊂ B i , and set B i+1 = B i \ {k i }, for an arbitrary k i ∈ L μ (H j i ).…”

“…Other references address specific models for designated applications: [5] on kidney transplants, [22] on housing allocations systems, and [21] on ride-sharing models. In another line of research, such stochastic matching architectures are addressed from the point of view of stochastic optimization in [10], [13], and [20], among others.…”

Asymptotically Optimal Control of a Centralized Dynamic Matching Market with General Utilities

“…They show that the optimal policy is of threshold form under vertical and unidirectionally horizontal differentiated types. Ding et al (2021) allow the matching utilities to depend on the class of buyer and seller and perform a fluid analysis of a greedy policy, and Bušić and Meyn (2015) minimize linear holding costs in a system without classdependent matching utilities or abandonment but also find that matches are not made until there is a sufficient number of agents in the market (see Moyal and Perry 2017, where these systems are referred to as matching queues, for other references to these types of models). Gurvich and Ward (2014) and Nazari and Stolyar (2019) study a control problem in a more general setting than the aforementioned studies, where arriving customers wait to be matched to agents of other classes.…”

A product form for the general stochastic matching model

“…Various aspects and properties of the BM model have been studied in the recent literature, along with various applications. The max-weight policy is shown to be asymptotically optimal for models with relaxations in [10]. A stabilizing policy and fluid/diffusion approximations are given respectively in [8] and [9] for a model in which the compatibility graph is bipartite-complete, but the nominal matchings between items are themselves random.…”