Mechanisms for a regulated monopoly (with A. Mackenzie)
Suppose that a monopolist with a convex production technology must determine price and quantity by eliciting consumer demand. We know well how to proceed when the objective is to maximize profits. But how can we proceed if the objective is instead to maximize consumer surplus, subject to the monopolist not losing money? This can happen if the monopolist is regulated, is a cooperative, or if the buyers are organized as a cartel.
We characterize the class of mechanisms available to the regulated monopolist. Across those that moreover guarantee ex-post voluntary participation to both the consumers and the monopolist, we find that a novel class of mechanisms maximize consumer surplus. This latter class involves subsidies to consumers who purchase nothing.
Status: revising draft
Optimistic and pessimistic approaches in cooperative games (with A. Atay)
Cooperative game theory studies how to allocate the joint value generated by a set of players. These games are typically analyzed using the characteristic function form with transferable utility, which represents the value attainable by each coalition. In the presence of externalities, coalition values can be defined through various approaches, notably by trying to determine the best and worst-case scenarios. Typically, the optimistic and pessimistic perspectives offer valuable insights into strategic interactions. In many applications, these approaches correspond to the coalition either choosing first or choosing after the complement coalition. In a general framework in which the actions of a group affects the set of feasible actions for others, we explore this relationship and show that it always holds in the presence of negative externalities, but only partly with positive externalities. We then show that if choosing first/last corresponds to these extreme values, we also obtain a useful inclusion result: allocations that do not allocate more than the optimistic upper bounds also do not allocate less than the pessimistic lower bounds. Moreover, we show that when externalities are negative, it is always possible to guarantee the non-emptiness of these sets of allocations. Finally, we explore applications to illustrate how our findings provide new results and offer a means to derive results from the existing literature.
Status: Accepted, European Journal of Operational Research
On the (non-) coincidence of the Serial and Shapley solutions in multi-server waiting line problems (with S. Banerjee)
The existing literature on single-server waiting line problems—including sequencing, queueing, and scheduling—has traditionally designed equitable compensations by modeling each problem as a transferable utility (TU) game and assigning agents their respective Shapley payoffs. This paper demonstrates that, in more general settings, it is equally crucial to study the `serial' and `reverse serial' cost-sharing rules. Specifically, we show that the recommendation of the optimistic Shapley value aligns with the serial rule in the following cases: (1) multi-server queueing with divisible and indivisible jobs; and (2) multi-server scheduling with indivisible jobs. However, this coincidence fails in the case of multi-server scheduling with divisible jobs. Furthermore, we establish that the recommendation of the pessimistic Shapley value aligns with the reverse serial rule only for multi-server queueing with both divisible and indivisible jobs. This coincidence breaks down for multi-server scheduling with either divisible or indivisible jobs. Notably, in this latter scenario, the pessimistic Shapley value exhibits an undesirable property known as 'anti-ranking,' in contrast to the reverse serial rule, which satisfies 'ranking.' We also provide characterizations of the serial and reverse serial rules for the above mentioned classes of problems.
Status: submitted
Duality-driven insights about value capture (with M. Ryall)
The aim of this paper is to turn attention to a formalized treatment of duality in the context of value creation and capture under competition (Gans and Ryall, 2017). Just as informal duality reasoning reveals complementary perspectives and hidden relationships, mathematical duality provides a framework to systematically explore similarly interconnected viewpoints of the same underlying economic reality. The transition to a more technical perspective on duality allows us to use mathematical tools to generate additional, unifying insights across different perspectives on firm performance.
We examine how the same set of fundamental economic primitives can be analyzed through multiple lenses to draw clarifying, duality-based connections between the traditional, products-and-prices view (which, remains a central orientation in strategy) and the agents-and-profits view as elaborated in the value capture stream. Building on these connections, we develop additional duality results that show: i) how shadow prices in the value maximization-through-product-allocation problem identify market-clearing product prices in the products-and-prices setting; and ii) how shadow prices in value-maximization-through agent-allocation problem identify competitive value capture distributions in the agents-and profits setting.
Status: writing initial draft
Suppose that a group of agents are connected to a source via a pipeline. The flow in the pipeline creates a negative local externality, that can be interpreted as pollution or risk of a spill. We first examine the problem of determining the optimal flow in the pipeline, balancing the consumption gains with the negative externalities. We then examine how to share the benefits generated by the pipeline. To do so, we define an optimistic and a pessimistic version of the problem (depending if a coalition consumes before or after the other agents). We explore the resulting cores. Given that the optimal use of the pipeline typically implies that some agents will restrict their consumption for the benefits of the group, we consider providing them with compensation for this sacrifice.
Status: submitted
In Alcalde-Ulzu et al. (Journal of Economic Theory, 2022), a novel model to measure the value of a language is developed, and a family of value functions is axiomatically characterized. These functions are reminiscent of the Shapley valye. We argue that a monotonicity axiom used to axiomatize the family favors majority languages. We show that by modifying this axiom we can obtain drastically different families of values and of policy implications. We examine a family that simply defines the value of a language by the number of its speakers, while another starts by defining the value at the individual level before aggregating. By examining their policy implications in a variety of applications, we show how the families differ and argue that each might be tailored for different uses.
Status: submitted
Preconvex games (with E. Bahel and H. Wang)
In a convex game, the marginal contribution of a player is an increasing (with respect to set inclusion) function of the coalition joined by that player. We define below a new family of TU games for which the players’ marginal contributions do not necessarily satisfy this property. In a k-semiconvex TU game, the marginal contribution of a player i to a coalition T containing at least n−k+1 players is weakly higher than i’s marginal contribution to a subset of T and the reverse inequality must hold for coalitions T that contain at most n−k players (and their nonempty subsets S). We show a necessary condition for such games to have a non-empty core, by defining a core allocation that generalizes the Shapley value. Multiple applications are provided.
Status: Revised and resubmitted.
Queueing problems when users can move (with A. Atay)
We study the situation where we have a finite set of machines that can process jobs, and users that are assigned to their local machine. However, if they are impatient and the queue at their local machine, they can travel to a neighboring machine, at a cost. Using a cooperative game theory framework, we study how this option of moving affects stability and cost sharing. We consider optimistic (where a coalition of agents could use all machines without the presene of others) and pessimistic (where others have already arrived at the machines) approaches to determine bounds, and compare the two approaches. We show that the optimistic approach is more demanding, but it is always possible to find allocations in which no coalition does better than the best case scenario of the optimistic approach.
Status: gathering results
Scheduling with divisible jobs (with S. Banerjee)
Traditionally, scheduling problems suppose that agents have jobs of various lengths to be processed on a machine. When we have more than one machine, it matters whether we suppose that the jobs are divisible (we can process them simultaneously on multiple machines) or not. The divisible case, unexplored so far, is the subject of the paper. We define optimistic (coalitions have priority access to the machines) and pessimistic (coalitions have access to the machine after the complement set of agents) games, and study the optimistic anticore and the pessimistic core. We study their Shapley values, providing closed-form expressions. We also study the serial allocation rule, and show that it is in the pessimistic core.
Status: writing initial draft, presenting results
Core constraints when scheduling with deadlines (with L. Streekstra)
We consider a version of the classic job scheduling problem in which all machines are identical, but in which jobs must be processed before a deadline. We study this problem using cooperative game theory, focusing on the question of how to divide the total cost between the agents. We consider both a simple version of the problem, in which all jobs have the same length, and a more general version where jobs of different lengths either need to be processed continuously on one machine or may be interrupted and processed on several machines at the same time. In all these versions the core may be empty. We therefore focus on stability and fairness and define two allocation mechanisms that satisfy the core constraints and either the most favorable scenario lower bounds or the egalitarian fair lower bounds. For the simple version we also provide necessary and sucient conditions for the core to be non-empty and for the general version we provide sufficient conditions.
Status: revising draft