Dynamic Matching Market Design

We introduce a simple benchmark model of dynamic matching in networked markets, where agents arrive and depart stochastically and the network of acceptable transactions among agents forms a random graph. We analyze our model from three perspectives: waiting, optimization, and information. The main insight of our analysis is that waiting to thicken the market can be substantially more important than increasing the speed of transactions, and this is quite robust to the presence of waiting costs. From an optimization perspective, naive local algorithms, that choose the right time to match agents but do not exploit global network structure, can perform very close to optimal algorithms. From an information perspective, algorithms that employ even partial information on agents' departure times perform substantially better than those that lack such information. To elicit agents' departure times, we design an incentive-compatible continuous-time dynamic mechanism without transfers

Fully Dynamic Matching in Bipartite Graphs

Maximum cardinality matching in bipartite graphs is an important and well-studied problem. The fully dynamic version, in which edges are inserted and deleted over time has also been the subject of much attention. Existing algorithms for dynamic matching (in general graphs) seem to fall into two groups: there are fast (mostly randomized) algorithms that do not achieve a better than 2-approximation, and there slow algorithms with \O(\sqrt{m}) update time that achieve a better-than-2 approximation. Thus the obvious question is whether we can design an algorithm -- deterministic or randomized -- that achieves a tradeoff between these two: a $o(\sqrt{m})$ approximation and a better-than-2 approximation simultaneously. We answer this question in the affirmative for bipartite graphs. Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give stronger results for graphs whose arboricity is at most \al, achieving a (1+ \eps) approximation in worst-case time O(\al (\al + \log n)) for constant \eps. When the arboricity is constant, this bound is $O(\log n)$ and when the arboricity is polylogarithmic the update time is also polylogarithmic. The most important technical developement is the use of an intermediate graph we call an edge degree constrained subgraph (EDCS). This graph places constraints on the sum of the degrees of the endpoints of each edge: upper bounds for matched edges and lower bounds for unmatched edges. The main technical content of our paper involves showing both how to maintain an EDCS dynamically and that and EDCS always contains a sufficiently large matching. We also make use of graph orientations to help bound the amount of work done during each update.Comment: Longer version of paper that appears in ICALP 201

Markov Equilibria in Dynamic Matching and Bargaining Games

Rubinstein and Wolinsky (1990) show that a simple homogeneous market with exogenous matching has continuum of (non-competitive) perfect equilibria, but the unique Markov perfect equilibrium is competitive. By contrast, in the more general case of heterogeneous markets, we show there exists a continuum of (non-competitive) Markov perfect equilibria. However, a refinement of the Markov property, which we call monotonicity, does suffice to guarantee perfectly competitive equilibria, if, and only if, it is monotonic. The monotonicity property is closely related to the concept of Nash equilibrium with complexity costs

Dynamic Matching and Bargaining: The Role of Deadlines

We consider a dynamic model where traders in each period are matched randomly into pairs who then bargain about the division of a fixed surplus. When agreement is reached the traders leave the market. Traders who do not come to an agreement return next period in which they will be matched again, as long as their deadline has not expired yet. New traders enter exogenously in each period. We assume that traders within a pair know each other's deadline. We define and characterize the stationary equilibrium configurations. Traders with longer deadlines fare better than traders with short deadlines. It is shown that the heterogeneity of deadlines may cause delay. It is then shown that a centralized mechanism that controls the matching protocol, but does not interfere with the bargaining, eliminates all delay. Even though this efficient centralized mechanism is not as good for traders with long deadlines, it is shown that in a model where all traders can choose which mechanism toBargaining, Deadlines, Markets

Dynamic matching and bargaining games: A general approach

This paper presents a new characterization result for competitive allocations in quasilinear economies. This result is informed by the analysis of non-cooperative dynamic search and bargaining games. Such games provide models of decentralized markets with trading frictions. A central objective of this literature is to investigate how equilibrium outcomes depend on the level of the frictions. In particular, does the trading outcome become Walrasian when frictions become small? Existing specifications of such games provide divergent answers. The characterization result is used to investigate what causes these differences and to generalize insights from the analysis of specific search and bargaining games.Dynamic Matching and Bargaining, Decentralized Markets, Non-cooperative Foundations of Competitive Equilibrium, Search Theory

Online Algorithms for Dynamic Matching Markets in Power Distribution Systems

This paper proposes online algorithms for dynamic matching markets in power distribution systems, which at any real-time operation instance decides about matching -- or delaying the supply of -- flexible loads with available renewable generation with the objective of maximizing the social welfare of the exchange in the system. More specifically, two online matching algorithms are proposed for the following generation-load scenarios: (i) when the mean of renewable generation is greater than the mean of the flexible load, and (ii) when the condition (i) is reversed. With the intuition that the performance of such algorithms degrades with increasing randomness of the supply and demand, two properties are proposed for assessing the performance of the algorithms. First property is convergence to optimality (CO) as the underlying randomness of renewable generation and customer loads goes to zero. The second property is deviation from optimality, is measured as a function of the standard deviation of the underlying randomness of renewable generation and customer loads. The algorithm proposed for the first scenario is shown to satisfy CO and a deviation from optimal that varies linearly with the variation in the standard deviation. But the same algorithm is shown to not satisfy CO for the second scenario. We then show that the algorithm proposed for the second scenario satisfies CO and a deviation from optimal that varies linearly with the variation in standard deviation plus an offset

Essay on Dynamic Matching

abstract: In the first chapter, I study the two-sided, dynamic matching problem that occurs in the United States (US) foster care system. In this market, foster parents and foster children can form reversible foster matches, which may disrupt, continue in a reversible state, or transition into permanency via adoption. I first present an empirical analysis that yields four new stylized facts related to match transitions of children in foster care and their exit through adoption. Thereafter, I develop a two-sided dynamic matching model with five key features: (a) children are heterogeneous (with and without a disability), (b) children must be foster matched before being adopted, (c) children search for parents while foster matched to another parent, (d) parents receive a smaller per-period payoff when adopting than fostering (capturing the presence of a financial penalty on adoption), and (e) matches differ in their quality. I use the model to derive conditions for the stylized facts to arise in equilibrium and carry out predictions regarding match quality. The main insight is that the intrinsic disadvantage (being less preferred by foster parents) faced by children with a disability exacerbates due to the penalty. Moreover, I show that foster parents in high-quality matches (relative to foster parents in low-quality matches) might have fewer incentives to adopt. In the second chapter, I study the Minnesota's 2015 Northstar Care Program which eliminated the adoption penalty (i.e., the decrease in fostering-based financial transfers associated with adoption) for children aged six and older, while maintaining it for children under age six. Using a differences-in-differences estimation strategy that controls for a rich set of covariates, I find that parents were responsive to the change in direct financial payments; the annual adoption rate of older foster children (aged six to eleven) increased by approximately 8 percentage points (24% at the mean) as a result of the program. I additionally find evidence of strategic adoption behavior as the adoption rate of younger children temporarily increased by 9 percentage points (23% at the mean) while the adoption rate of the oldest children (aged fifteen) temporarily decreased by 9 percentage points (65% at the mean) in the year prior to the program's implementation.Dissertation/ThesisDoctoral Dissertation Economics 201

Unraveling in a dynamic matching market with Nash bargaining

Equilibrium sorting in a finite-horizon, two-sided matching market with heterogeneous agents is considered. It is shown that, if the match production function is additively separable in agent-types and if the division of match output is determined by the Nash bargaining solution, then an unraveling of the market obtains as the unique equilibrium in which all matches are formed in the first period.