We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

/pdf/the-theory-of-industrial-organization-19bejph0ej.pdf

The Theory of Industrial Organization

In this paper, I construct an equilibrium for markets with frictions, which is competitive in the sense that all agents are price takers and maximize utility subject to a set of market parameters. I show that the equilibrium allocation is socially optimal. I also show how the competitive search equilibrium can be achieved if employers with vacancies can advertise publicly the wages they pay.

Competitive Search Equilibrium

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Perturbation Analysis Of Optimization Problems

Ingredients of Locational Analysis (J. Krarup & P. Pruzan). The p-Median Problem and Generalizations (P. Mirchandani). The Uncapacitated Facility Location Problem (G. Cornuejols, et al.). Multiperiod Capacitated Location Models (S. Jacobsen). Decomposition Methods for Facility Location Problems (T. Magnanti & R. Wong). Covering Problems (A. Kolen & A. Tamir). p-Center Problems (G. Handler). Duality: Covering and Constraining p-Center Problems on Trees (B. Tansel, et al.). Locations with Spatial Interactions: The Quadratic Assignment Problem (R. Burkard). Locations with Spatial Interactions: Competitive Locations and Games (S. Hakimi). Equilibrium Analysis for Voting and Competitive Location Problems (P. Hansen, et al.). Location of Mobile Units in a Stochastic Environment (O. Berman, et al.). Index.

https://link.springer.com/content/pdf/10.1057%2Fjors.1991.208.pdf

Discrete location theory

Motivated by the prevalence of ride-sharing platforms, in “Spatial Pricing in Ride-Sharing Networks,” Bimpikis, Candogan, and Saban explore the impact of the demand pattern for rides across a netwo...

Spatial Pricing in Ride-Sharing Networks

We explore spatial price discrimination in the context of a ride-sharing platform that serves a network of locations. Riders at different locations are heterogeneous in terms of their destination preferences, as captured by the demand pattern of the underlying network. Drivers decide whether, when, and where to provide service so as to maximize their expected earnings given the platform's pricing policy. Our findings highlight the impact of the demand pattern of the underlying network on the platform's optimal profits and aggregate consumer surplus. In particular, we establish that both profits and consumer surplus are maximized when the demand pattern is "balanced" across the network's locations. In addition, we show that profits and consumer surplus are monotonic with the balancedness of the demand pattern (as formalized by the pattern's structural properties). Furthermore, we explore the widely adopted compensation scheme that allocates a constant fraction of the fare to drivers and identify a class of networks for which it can implement the optimal equilibrium outcome. However, we also showcase that generally this scheme leads to significantly lower profits for the platform than the optimal pricing policy especially in the presence of heterogeneity among the demand patterns in different locations. Together, these results illustrate the value of accounting for the demand pattern across a network's locations when designing the platform's pricing policy, and complement the existing focus on the benefits of dynamic (surge) pricing to deal with demand fluctuations over time.

We explore spatial price discrimination in the context of a ride-sharing platform that serves a network of locations. Riders at different locations are heterogeneous in terms of their destination preferences, as captured by the demand pattern of the underlying network. Drivers decide whether, when, and where to provide service so as to maximize their expected earnings given the platform’s pricing policy. Our findings highlight the impact of the demand pattern of the underlying network on the platform’s optimal profits and aggregate consumer surplus. In particular, we establish that both profits and consumer surplus are maximized when the demand pattern is “balanced” across the network’s locations. In addition, we show that profits and consumer surplus are monotonic with the balancedness of the demand pattern (as formalized by the pattern’s structural properties). Furthermore, we explore the widely adopted compensation scheme that allocates a constant fraction of the fare to drivers and identify a class of networks for which it can implement the optimal equilibrium outcome. However, we also showcase that generally this scheme leads to significantly lower profits for the platform than the optimal pricing policy especially in the presence of heterogeneity among the demand patterns in different locations. Together, these results illustrate the value of accounting for the demand pattern across a network’s locations when designing the platform’s pricing policy, and complement the existing focus on the benefits of dynamic (surge) pricing to deal with demand fluctuations over time.

http://www.columbia.edu/~dhs2131/allocation_of_divisible_goods.pdf

A note on object allocation under lexicographic preferences

We consider the problem of reallocating indivisible objects amongst a set of agents when the preference ordering of each agent may contain indifferences. The same model, but with strict preferences, goes back to the seminal work of Shapley and Scarf in 1974. When preferences are strict, we now know that the Top-Trading Cycles (TTC) mechanism invented by Gale is Pareto efficient, strategy-proof, and finds a core allocation, and that it is the only mechanism satisfying these properties. In the extensive literature on this problem since then, the TTC mechanism has been characterized in multiple ways, establishing its central role within the class of all allocation mechanisms. The question motivating our work is the extent to which these results can be generalized to the setting with indifferences. Our main contribution is a general framework to design strategyproof mechanisms that find a Pareto optimal allocation in the weak-core. Along the way, we establish a sufficient condition for a mechanism (within a broad class of mechanisms) to be strategyproof and use this condition to design fast algorithms for finding a "good" reallocation. Our results generalize and unify two (different) mechanisms for the reallocation problem derived, independently of each other, by Manjunath and Jaramillo, and Alcalde-Unzu and Molis.

/pdf/house-allocation-with-indifferences-a-generalization-and-a-4ii0lybwzx.pdf

Daniela Saban

Papers

Spatial Pricing in Ride-Sharing Networks

Spatial Pricing in Ride-Sharing Networks

Spatial Pricing in Ride-Sharing Networks

A note on object allocation under lexicographic preferences

House allocation with indifferences: a generalization and a unified view