Related papers: Generalized Kings and Single-Elimination Winners i…

Consider an election between k candidates in which each voter votes randomly (but not necessarily independently) and suppose that there is a single candidate that every voter prefers (in the sense that each voter is more likely to vote for…

Probabilistic properties of tennis scoring systems are examined and compared with best-of-K systems. A model, where each player has his/her own probability of winning his/her service point and which remains invariant for the duration of the…

The determinant of a tournament $T$ is defined as the determinant of the skew-adjacency matrix of $T$. For a positive odd integer $k$, let $\mathcal{D}_k$ be the set of tournaments whose all subtournaments have determinant at most $k^2$.…

With the aid of mathematical modelling (basic tool is the random walk with absorbing barriers) we derive subsequent formulas to study the effect of different versions of possible rules. For different rules the probability of winning a game,…

A vertex $x$ in a tournament $T$ is called a king if for every vertex $y$ of $T$ there is a directed path from $x$ to $y$ of length at most 2. It is not hard to show that every vertex of maximum out-degree in a tournament is a king.…

For a one-locus haploid infinite population with discrete generations, the celebrated Kingman's model describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability.…

Weak selection, which means a phenotype is slightly advantageous over another, is an important limiting case in evolutionary biology. Recently it has been introduced into evolutionary game theory. In evolutionary game dynamics, the…

Simple stochastic games are turn-based 2.5-player games with a reachability objective. The basic question asks whether one player can ensure reaching a given target with at least a given probability. A natural extension is games with a…

A discrete-time stochastic process derived from a model of basketball is used to generalize any discrete distribution. The generalized distributions can have one or two more parameters than the parent distribution. Those derived from…

We study tournaments where winning a rank-dependent prize requires passing a minimum performance standard. We show that, for any prize allocation, the optimal standard is always at a mode of performance that is weakly higher than the global…

We give elementary examples within a framework for studying decisions under uncertainty where probabilities are only roughly known. The framework, in gambling terms, is that the size of a bet is proportional to the gambler's perceived…

Determining how close a winner of an election is to becoming a loser, or distinguishing between different possible winners of an election, are major problems in computational social choice. We tackle these problems for so-called weighted…

Games on graphs provide a natural and powerful model for reactive systems. In this paper, we consider generalized reachability objectives, defined as conjunctions of reachability objectives. We first prove that deciding the winner in such…

A tournament is a complete directed graph. A king in a tournament is a vertex v such that every other vertex is reachable from v via a path of length at most 2. It is well known that every tournament has at least one king, one of which is a…

A $k$-coloring of a tournament is a partition of its vertices into $k$ acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a…

We present an extensive statistical analysis of the results of all sports competitions in five major sports leagues in England and the United States. We characterize the parity among teams by the variance in the winning fraction from…

In J. Schwenk.(2018) ['What is the Correct Way to Seed a Knockout Tournament?' Retrieved from The American Mathematical Monthly], Schwenk identified a surprising weakness in the standard method of seeding a single elimination (or knockout)…

The $k$-majority game is played with $n$ numbered balls, each coloured with one of two colours. It is given that there are at least $k$ balls of the majority colour, where $k$ is a fixed integer greater than $n/2$. On each turn the player…

A generalized tournament matrix $M$ is a nonnegative matrix that satisfies $M+M^{t}=J-I$, where $J$ is the all ones matrix and $I$ is the identity matrix. In this paper, a characterization of generalized tournament matrices with the same…

According to recent empirical studies, the group draw of major sports tournaments can imply a high level of uncertainty, and some lucky teams enjoy an unfair advantage over the other teams. We propose a novel technique to quantify this draw…