Goldbach's Weak Conjecture

The famous problem about to be solved? ... Terence Tao makes headway.

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Smart Lotteries in School Choice Ex-ante Pareto-Improvement with Ex-post Stability

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Working Paper: https://cgi.cse.unsw.edu.au/~haziz/PIRMES.pdf  Abstract: In a typical school choice application, the students have strict preferences over the schools while the schools have coarse priorities over the students based on their distance and their enrolled siblings. The outcome of a centralized admission mechanism is then usually obtained by the Deferred Acceptance (DA) algorithm with random tie-breaking. Therefore, every possible outcome of this mechanism is a stable solution for the coarse priorities that will arise with certain probability. This implies a probabilistic assignment, where the admission probability for each student-school pair is specified. In this paper, we propose a new efficiency-improving stable `smart lottery' mechanism. We aim to improve the probabilistic assignment ex-ante in a stochastic dominance sense, while ensuring that the improved random matching is still ex-post stable, meaning that it can be decomposed into stable matchings regarding the ...
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Fair budgets from ranked ballots: PB-EAR

How do you split a shared budget fairly when people rank projects that cost very different amounts? PB-EAR, from our AAAI 2021 paper  Proportionally Representative Participatory Budgeting with Ordinal Preferences  ( AAAI version ), gives every voter an equal slice of the budget and lets groups pool their slices to fund the projects they rank highest, looking further down the ballots only when nothing is yet affordable. Every cohesive group is guaranteed its proportional share, and the rule extends our  Expanding Approvals Rule  for elections to budgets with project costs, ties and unequal voter weights.  That makes it a natural fit for DAO grant rounds such as   TheDAO Security Fund 's ETHSecurity Initiatives, where badge holders will rank the security initiatives listed on   the app  to allocate the fund's budget. The two-minute video below walks through a worked example. Here's an educational tool someone built to show how PB-EAR works. 
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Proportionally Representative Clustering

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H. Aziz, B. Lee, S. Morota Chu, and J. Vollen. Proportionally Representative Clustering.  WINE 2024 .  [  Pdf  ]  In recent years, there has been a surge in effort to formalize notions of fairness in machine learning. We focus on centroid clustering— one of the fundamental tasks in unsupervised machine learning. We pro- pose a new axiom “proportionally representative fairness” (PRF) that is designed for clustering problems where the selection of centroids re- flects the distribution of data points and how tightly they are clustered together. Our fairness concept is not satisfied by existing fair cluster- ing algorithms. We design efficient algorithms to achieve PRF both for unconstrained and discrete clustering problems. Our algorithm for the unconstrained setting is also the first known polynomial-time approxi- mation algorithm for the well-studied Proportional Fairness (PF) axiom [ 13 ]. Our algorithm for the discrete setting also matches the best kn...
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