Daniela Saban

The Complexity of Computing the Random Priority Allocation Matrix

The random priority (RP) mechanism is a popular way to allocate n objects to n agents with strict ordinal preferences over the objects. In the RP mechanism, an ordering over the agents is selected uniformly at random; the first agent is then allocated his most-preferred object, the second agent is allocated his most-preferred object among the remaining ones, and so on. The outcome of the mechanism is a bi-stochastic matrix in which entry ( i , a ) represents the probability that agent i is given object a . It is shown that the problem of computing the RP allocation matrix is #P-complete. Furthermore, it is NP-complete to decide if a given agent i receives a given object a with positive probability under the RP mechanism, whereas it is possible to decide in polynomial time whether or not agent i receives object a with probability 1. The implications of these results for approximating the RP allocation matrix as well as on finding constrained Pareto optimal matchings are discussed.

The competitive facility location problem in a duopoly: connections to the 1-median problem

We consider a competitive facility location problem on a network, in which consumers are located on the vertices and wish to connect to the nearest facility. Knowing this, competitive players locate their facilities on vertices that capture the largest possible market share. In 1991, Eiselt and Laporte established the first relation between Nash equilibria of a facility location game in a duopoly and the solutions to the 1-median problem. They showed that an equilibrium always exists in a tree because a location profile is at equilibrium if and only if both players select a 1-median of that tree [4]. In this work, we further explore the relations between the solutions to the 1-median problem and the equilibrium profiles. We show that if an equilibrium in a cycle exists, both players must choose a solution to the 1-median problem. We also obtain the same property for some other classes of graphs such as quasi-median graphs, median graphs, Helly graphs, and strongly-chordal graphs. Finally, we prove the converse for the latter class, establishing that, as for trees, any median of a strongly-chordal graph is a winning strategy that leads to an equilibrium.

A polyhedral study of the maximum edge subgraph problem

The study of cohesive subgroups is an important aspect of social network analysis. Cohesive subgroups are studied using different relaxations of the notion of clique in a graph. For instance, given a graph and an integer k, the maximum edge subgraph problem consists of finding a k-vertex subset such that the number of edges within the subset is maximum. This work proposes a polyhedral approach for this NP-hard problem. We study the polytope associated to an integer programming formulation of the problem, present several families of facet-inducing valid inequalities, and discuss the separation problem associated to these families. Finally, we implement a branch and cut algorithm for this problem. This computational study illustrates the effectiveness of the classes of inequalities presented in this work.

Optimal Commissions and Subscriptions in Networked Markets

Platforms facilitating the exchange of goods and services between individuals are prevalent: one can purchase goods from others on eBay, arrange accommodation through Airbnb, and find temporary projects/workers on online labor markets such as Upwork. The majority of these markets exhibit three key features. First, the platforms do not dictate the transaction prices, i.e., buyers and sellers determine at which price the goods/services will be exchanged. Second, not all buyers or sellers on a platform are compatible. This may be due to taste differences (a buyer may be interested only in the types of goods/services a subset of the sellers offer), geographical or import/export restrictions (e.g., being able to provide services only regionally), or other sources of mismatch (e.g., a mismatch in the desired and available skills in online labor markets). Finally, buyers/sellers are heterogeneous in their valuations for goods or services they receive/provide. In exchange for facilitating trade, these platforms commonly obtain a commission from each transaction and/or charge subscription fees to sellers and buyers who access the platform. As such, their revenues depend both on the chosen commission/subscription fees and on the prices at which buyers/sellers transact. In settings that exhibit the aforementioned features, how should a platform design commission/subscription fees with the objective of maximizing its revenues? Is it sufficient to charge these to only one side of the market or is it necessary to charge them to both sides? What is the role of the underlying compatibility structure, and which structures are more conducive to higher revenues? In order to answer these questions, we consider a model where potential buyers and sellers are divided into types. Not all buyer and seller types are compatible with each other, and we represent the compatibility across these types using a bipartite network: nodes on one side correspond to buyer types, nodes on the other side correspond to seller types, and edges capture compatibility between different types of buyers and sellers. Agents within each type differ in their valuation for the goods they buy/sell. The platform chooses subscription fees, which must be paid by all agents who participate in the market. In addition, it chooses commissions, and a buyer/seller who exchanges goods/services pays a fraction of the transaction price to the platform as indicated by these commissions. The value distributions, subscription fees, and commissions are all possibly type-specific. Given the commissions/subscriptions chosen by the platform, the transaction prices and equilibrium trades are formed endogenously at a competitive equilibrium. We establish that, in order to maximize its revenues, the platform may need to charge different commissions/subscriptions to different types, depending on their network position. In fact, we show that if the same commissions/subscriptions are employed for all agents on the same side, the revenue loss can be unbounded. We complement this worst-case result by providing a bound on the revenue loss in terms of the supply/demand imbalance across the network under homogeneous value distributions. Surprisingly, we also show that, in general, charging commissions/subscriptions to only one side of the market (i.e., only to buyers or only to sellers) leads to lower revenues than optimal, even when different types on the same side can be charged different fees. Furthermore, we characterize the impact of the network structure on the revenues of the platform. Finally, we investigate how the commissions/subscriptions chosen by the platform impact social welfare. We establish that, under mild convexity assumptions on the value distributions, the revenue-maximizing commissions/subscriptions induce at least 2/3 of the maximum achievable social welfare.

Convergence of the Core in Assignment Markets

We consider a two-sided assignment market with agent types and a stochastic structure, similar to models used in empirical studies. We characterize the size of the core in such markets. Each agent has a randomly drawn productivity with respect to each type of agent on the other side. The value generated from a match between a pair of agents is the sum of the two productivity terms, each of which depends only on the type (but not the identity) of one of the agents, and a third deterministic term driven by the pair of types. We prove, under reasonable assumptions, that when the number of agent types is kept fixed, the relative size of the core vanishes rapidly as the number of agents grows. Numerical experiments confirm that the core is typically small. Our results provide justification for the typical assumption of a unique core outcome in such markets, which is close to a limit point. Further, our results suggest that, given the market composition, wages are almost uniquely determined in equilibrium. The ...

Technical Note—The Competitive Facility Location Problem in a Duopoly: Advances Beyond Trees

We consider a competitive facility location problem on a network where consumers located on vertices wish to connect to the nearest facility. Knowing this, each competitor locates a facility on a vertex, trying to maximize market share. We focus on the two-player case and study conditions that guarantee the existence of a pure-strategy Nash equilibrium for progressively more complicated classes of networks. For general graphs, we show that attention can be restricted to a subset of vertices referred to as the central block. By constructing trees of maximal bi-connected components, we obtain sufficient conditions for equilibrium existence. Moreover, when the central block is a vertex or a cycle (for example, in cactus graphs), this provides a complete and efficient characterization of equilibria. In that case, we show that both competitors locate their facilities in a solution to the 1-median problem, generalizing a well-known insight arising from Hotelling’s model. We further show that an equilibrium must...

Facilitating the Search for Partners on Matching Platforms: Restricting Agent Actions

Two-sided matching platforms, such as those for labor, accommodation, dating, and taxi hailing, can control and optimize over many aspects of the search for partners. To understand how the search for partners should be designed, we consider a dynamic model of search by strategic agents with costly discovery of pair-specific match value. We find that in many settings, the platform can mitigate wasteful competition in partner search via restricting what agents can see/do. For medium-sized screening costs (relative to idiosyncratic variation in utilities), the platform should prevent one side of the market from exercising choice (similar to Instant Book on Airbnb), whereas for large screening costs, the platform should centrally determine matches (similar to taxi hailing marketplaces). Surprisingly, simple restrictions can improve social welfare even when screening costs are small, and agents on each side are ex-ante homogeneous. In asymmetric markets where agents on one side have a tendency to be more selective (due to smaller screening costs or greater market power), the platform should force the more selective side of the market to reach out first, by explicitly disallowing the less selective side from doing so. This allows the agents on the less selective side to exercise more choice in equilibrium.When agents are vertically differentiated, forcing one side of the market to propose results in a significant increase in welfare even in the limit of vanishing screening costs. Furthermore, a Pareto improvement in welfare is possible in this limit: the weakest agents can be helped without hurting other agents. In addition, in this setting the platform can further boost welfare by hiding quality information.

Facilitating the Search for Partners on Matching Platforms

Two-sided matching platforms can control and optimize over many aspects of the search for partners. To understand how matching platforms should be designed, we introduce a dynamic two-sided search model with strategic agents who must bear a cost to discover their value for each potential partner, and can do so non-simultaneously. We characterize equilibria and find that, in many settings, the platform can mitigate wasted search effort by imposing suitable restrictions on agents. In unbalanced markets, the platform should force the short side of the market to initiate contact with potential partners, by disallowing the long side from doing so. This allows the agents on the long side to exercise more choice in equilibrium. When agents are vertically differentiated, the platform can significantly improve welfare even in the limit of vanishing screening costs by forcing the shorter side of the market to propose and by hiding information about the quality of potential partners. Furthermore, a Pareto improvement in welfare is possible in this limit.

Combinatorial properties and further facets of maximum edge subgraph polytopes

Abstract Given a graph G and an integer k, the maximum edge subgraph problem consists in finding a k-vertex subset of G such that the number of edges within the subset is maximum. This NP-hard problem arises in the analysis of cohesive subgroups in social networks. In this work we study the polytope P ( G , k ) associated with a straightforward integer programming formulation of the maximum edge subgraph problem. We characterize the graph generated by P ( G , k ) and give a tight bound on its diameter. We give a complete description of P ( K 1 n , k ) , where K 1 n is the star on n + 1 vertices, and we conjecture a complete description of P ( m K 2 , k ) , where m K 2 is the graph composed by m disjoint edges. Finally, we introduce three families of facet-inducing inequalities for P ( G , k ) , which generalize known families of valid inequalities for this polytope.