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
We study cooperative cost-sharing in scheduling problems where agents differ in job lengths but share identical waiting costs. While the optimistic and pessimistic cost functions are natural approaches to compute the waiting cost of a coalition, their symmetry—well established for queueing problems—breaks down for scheduling. In particular, the pessimistic Shapley value violates equity by rewarding longer jobs. To address this, we propose an alternative formulation, termed ``accountable cost" of serving a coalition. Under this, the coalitional members are served first, but they internalize the externality imposed on the non-coalitional members. We show that the accountable cost game is concave and the Shapley value of this game coincides with the decreasing serial rule, thereby restoring the lost symmetry. We also provide axiomatic characterizations of both the optimistic and accountable Shapley values, explore extensions to multi-server environments, and discuss connections with general sequencing problems. These results position the accountable function as a principled counterpart to the optimistic approach, unifying efficiency and fairness in the cooperative analysis of scheduling problems.
Status: Presenting results
Ratios, Symmetry, and Cores of TU Games (with E. Bahel and and S. Sarangi)
We provide a novel algorithm that, if the core is non-empty, provides the full list of its extreme points. The algorithm builds on two new insights. Firstly, if a game is symmetric (all coalitions of the same size generate the same value) then by generalizing the concept of lexicographic maximization we are able to provide the full list of extreme points. Secondly, suppose that we take a game and increase the value of a single coalition (not the grand coalition). We show that if we know the extreme points of the core of the original game, we can easily find the extreme points of the new game, by a simple procedure that adjusts, if needed, each of these extreme points. Combining these two insights, for an arbitrary game we first define its symmetric floor function (assigning to each cardinality the minimum value a coalition of that size takes), and using the first insight, define its extreme points. We then make a series of changes, one coalition at a time, to reach back our original game, adjusting the extreme points of the core at each step.
Status: Presenting results
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
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: revising draft
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
When do the serial and Shapley solutions coincide? (with S. Banerjee)
In a simple framework in which agents differ along a single dimension, we explore when the Shapley value and the serial rule coincide. We obtain a simple sufficient condition on marginal contributions that we test on various applications, from airport to queueing and joint production problems. The condition is satisfied either when the joint cost depends only on extreme values of the individual parameters or when the cost generated by an agent depends on its rank with respect to the individual parameters.
Status: writing initial draft
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