This paper presents discrete convex analysis as a tool for use in economics and game theory. Discrete convex analysis is a new framework of discrete mathematics and optimization, developed during the last two decades. Recently, it has been recognized as a powerful tool for analyzing economic or game models with indivisibilities. The main feature of discrete convex analysis is the distinction of two convexity concepts, M-convexity and L-convexity, for functions in integer or binary variables, together with their conjugacy relationship. The crucial fact is that M-concavity in its variant is equivalent to the gross substitutes property in economics. Fundamental theorems in discrete convex analysis such as the M-L conjugacy theorems, discrete separation theorems and discrete fixed point theorems yield structural results in economics such as the existence of equilibria and the lattice structure of equilibrium price vectors. Algorithms in discrete convex analysis provide iterative auction algorithms for finding equilibria.

http://www.mechanism-design.org/arch/v001-1/p_05.pdf

Discrete convex analysis: A tool for economics and game theory

Conventional algorithmic fairness is West-centric, as seen in its subgroups, values, and methods. In this paper, we de-center algorithmic fairness and analyse AI power in India. Based on 36 qualitative interviews and a discourse analysis of algorithmic deployments in India, we find that several assumptions of algorithmic fairness are challenged. We find that in India, data is not always reliable due to socio-economic factors, ML makers appear to follow double standards, and AI evokes unquestioning aspiration. We contend that localising model fairness alone can be window dressing in India, where the distance between models and oppressed communities is large. Instead, we re-imagine algorithmic fairness in India and provide a roadmap to re-contextualise data and models, empower oppressed communities, and enable Fair-ML ecosystems.

Re-imagining Algorithmic Fairness in India and Beyond

Stability concepts in matching under distributional constraints

Designing matching mechanisms under constraints: An approach from discrete convex analysis

School choice programs are implemented to give students/parents an opportunity to choose the public school the students attend. Controlled school choice programs need to provide choices for students/parents while maintaining distributional constraints on the balance on the composition of students, typically in terms of so-cioeconomic status. Previous works show that setting soft-bounds, which flexibly change the priorities of students based on their types, is more appropriate than setting hard-bounds, which strictly limit the number of accepted students for each type. We consider a case where soft-bounds are imposed and one student can belong to multiple types, e.g., \"financially-distressed\" and \"minority\" types. We first show that when we apply a model that is a straightforward extension of an existing model for disjoint types, there is a chance that no stable matching exists. Thus, we propose an alternative model and an alternative stability definition, where a school has reserved seats for each type. We show that a stable matching is guaranteed to exist in this model, and develop a mechanism called Deferred Acceptance for Overlapping Types (DA-OT). The DA-OT mechanism is strategy-proof and obtains the student-optimal matching within all stable matchings. Computer simulation results illustrate that the DA-OT outperforms an artificial cap mechanism, where the number of seats for each type is fixed.

/pdf/controlled-school-choice-with-soft-bounds-and-overlapping-2oeztafsst.pdf

Controlled School Choice with Soft Bounds and Overlapping Types

Strategyproof matching with regional minimum and maximum quotas

To handle various applications, we study matching under constraints. The only requirement on the constraints is heredity; given a feasible matching, any matching with fewer students at each school is also feasible. Heredity subsumes existing constraints such as regional maximum quotas and diversity constraints. With constraints, there may not exist a matching that satisfies fairness and nonwastefulness (i.e., stability). We demonstrate our new mechanism, the Adaptive Deferred Acceptance mechanism (ADA), satisfies strategy-proofness for students, nonwastefulness, and a weaker fairness property. We also offer a technique to apply ADA even if heredity is violated (e.g., minimum quotas).

/pdf/designing-matching-mechanisms-under-general-distributional-4g57ukqakc.pdf

Designing Matching Mechanisms under General Distributional Constraints

Strategy-proof school choice mechanisms with minimum quotas and initial endowments

School choice programs are implemented to give students/parents an opportunity to choose the public school the students attend. Controlled school choice programs need to provide choices for students/parents while maintaining distributional constraints on the balance on the composition of students, typically in terms of so-cioeconomic status. Previous works show that setting soft-bounds, which flexibly change the priorities of students based on their types, is more appropriate than setting hard-bounds, which strictly limit the number of accepted students for each type. We consider a case where soft-bounds are imposed and one student can belong to multiple types, e.g., "financially-distressed" and "minority" types. We first show that when we apply a model that is a straightforward extension of an existing model for disjoint types, there is a chance that no stable matching exists. Thus, we propose an alternative model and an alternative stability definition, where a school has reserved seats for each type. We show that a stable matching is guaranteed to exist in this model, and develop a mechanism called Deferred Acceptance for Overlapping Types (DA-OT). The DA-OT mechanism is strategy-proof and obtains the student-optimal matching within all stable matchings. Computer simulation results illustrate that the DA-OT outperforms an artificial cap mechanism, where the number of seats for each type is fixed.

Ryoji Kurata

Papers

Controlled School Choice with Soft Bounds and Overlapping Types

Strategyproof matching with regional minimum and maximum quotas

Designing Matching Mechanisms under General Distributional Constraints

Strategy-proof school choice mechanisms with minimum quotas and initial endowments

Controlled school choice with soft bounds and overlapping types