Related papers: But some are more equal than others
We study the problem of fair allocation of a set of indivisible goods among $n$ agents with $k$ distinct additive valuations, with the goal of achieving approximate envy-freeness up to any good ($\alpha-\mathrm{EFX}$). It is known that EFX…
We point out that the Nobel prize production of the USA, the UK, Germany and France has been in numbers that are large enough to allow for a reliable analysis or the long-term historical developments. Nobel prizes are often split, such that…
A famous result in game theory known as Zermelo's theorem says that "in chess either White can force a win, or Black can force a win, or both sides can force at least a draw". The present paper extends this result to the class of all…
Group fairness is achieved by equalising prediction distributions between protected sub-populations; individual fairness requires treating similar individuals alike. These two objectives, however, are incompatible when a scoring model is…
Let f be a polynomial with irrational leading coefficient. We obtain inequalities for the distance from the nearest integer of f(p) that hold for infinitely many primes p. These results improve work of Harman in 1981 and 1983 and Wong in…
In this note by using elementary considerations, we settle Fr\"oberg's conjecture for a large number of cases, when all generators of ideals have the same degree.
In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the ground of geometric probability theory by defining an enlightening problem: What is the probability that a needle thrown randomly on a ground made of equispaced parallel…
The problem of fairly allocating a set of indivisible items is a well-known challenge in the field of (computational) social choice. In this scenario, there is a fundamental incompatibility between notions of fairness (such as envy-freeness…
We consider the problem of fair allocation of indivisible items among $n$ agents with additive valuations, when agents have equal entitlements to the goods, and there are no transfers. Best-of-Both-Worlds (BoBW) fairness mechanisms aim to…
New fairness notions aligned with the merit principle are proposed for designing exchange rules. We show that for an obviously strategy-proof, efficient and individually rational rule, (i) an agent receives her favorite object when others…
We prove that there exists a constant $\gamma_{\mathrm{crit}}\approx .17566$ such that if $G\sim \mathbb{G}(n,1/2)$ then for any $\varepsilon > 0$ with high probability $G$ has a equipartition such that each vertex has…
Einstein's happiest thought was his leap from the observation that a falling person feels no gravity to the realization that gravity might be equivalent to acceleration. It affects all bodies in the same way because it is a property of…
Apportionment is the act of distributing the seats of a legislature among political parties (or states) in proportion to their vote shares (or populations). A famous impossibility by Balinski and Young (2001) shows that no apportionment…
We are interested in the following version of Jeffreys's law: if two predictors are predicting the same sequence of events and either is doing a satisfactory job, they will make similar predictions in the long run. We give a classification…
It has been argued by Shepard that there is a robust psychological law that relates the distance between a pair of items in psychological space and the probability that they will be confused with each other. Specifically, the probability of…
In the year 1876 the mathematician Charles Dodgson, who wrote fiction under the now more famous name of Lewis Carroll, devised a beautiful voting system that has long fascinated political scientists. However, determining the winner of a…
The property of exponential dichotomy can be seen as a generalization of the hyperbolicity condition for non autonomous linear finite dimensional systems of ordinary differential equations. In 1978 W.A. Coppel proved that the exponential…
If the continued fractions of two irrational numbers have a common complete quotient, then these two numbers are in the same orbit under the action of $\mathrm{PGL}(2,\mathbb{Z})$. The converse is Serret's well-known theorem, but we give a…
Consider $2k-1$ voters, each of which has a preference ranking between $n$ given alternatives. An alternative $A$ is called a Condorcet winner, if it wins against every other alternative $B$ in majority voting (meaning that for every other…
In the prophet inequality problem, a gambler faces a sequence of items arriving online with values drawn independently from known distributions. On seeing an item, the gambler must choose whether to accept its value as her reward and quit…