The "No Justice in the Universe" phenomenon: why honesty of effort may not be rewarded in tournaments

In 2000 Allen Schwenk, using a well-known mathematical model of matchplay tournaments in which the probability of one player beating another in a single match is fixed for each pair of players, showed that the classical single-elimination, seeded format can be "unfair" in the sense that situations can arise where an indisputibly better (and thus higher seeded) player may have a smaller probability of winning the tournament than a worse one. This in turn implies that, if the players are able to influence their seeding in some preliminary competition, situations can arise where it is in a player's interest to behave "dishonestly", by deliberately trying to lose a match. This motivated us to ask whether it is possible for a tournament to be both honest, meaning that it is impossible for a situation to arise where a rational player throws a match, and "symmetric" - meaning basically that the rules treat everyone the same - yet unfair, in the sense that an objectively better player has a smaller probability of winning than a worse one. After rigorously defining our terms, our main result is that such tournaments exist and we construct explicit examples for any number n >= 3 of players. For n=3, we show (Theorem 3.6) that the collection of win-probability vectors for such tournaments form a 5-vertex convex polygon in R^3, minus some boundary points. We conjecture a similar result for any n >= 4 and prove some partial results towards it.Comment: 26 pages, 2 figure

Combinatorial Games on Graphs

Combinatorial games are intriguing and have a tendency to engross students and lead them into a serious study of mathematics. The engaging nature of games is the basis for this thesis. Two combinatorial games along with some educational tools were developed in the pursuit of the solution of these games. The game of Nim is at least centuries old, possibly originating in China, but noted in the 16th century in European countries. It consists of several stacks of tokens, and two players alternate taking one or more tokens from one of the stacks, and the player who cannot make a move loses. The formal and intense study of Nim culminated in the celebrated Sprague-Grundy Theorem, which is now one of the centerpieces in the theory of impartial combinatorial games. We study a variation on Nim, played on a graph. Graph Nim, for which the theory of Sprague-Grundy does not provide a clear strategy, was originally developed at the University of Colorado Denver. Graph Nim was first played on graphs of three vertices. The winning strategy, and losing position, of three vertex Graph Nim has been discovered, but we will expand the game to four vertices and develop the winning strategies for four vertex Graph Nim. Graph Theory is a markedly visual field of mathematics. It is extremely useful for graph theorists and students to visualize the graphs they are studying. There exists software to visualize and analyze graphs, such as SAGE, but it is often extremely difficult to learn how use such programs. The tools in GeoGebra make pretty graphs, but there is no automated way to make a graph or analyze a graph that has been built. Fortunately GeoGebra allows the use of JavaScript in the creation of buttons which allow us to build useful Graph Theory tools in GeoGebra. We will discuss two applets we have created that can be used to help students learn some of the basics of Graph Theory. The game of thrones is a two-player impartial combinatorial game played on an oriented complete graph (or tournament) named after the popular fantasy book and TV series. The game of thrones relies on a special type of vertex called a king. A king is a vertex, k, in a tournament, T, which for all x in T either k beats x or there exists a vertex y such that k beats y and y beats x. Players take turns removing vertices from a given tournament until there is only one king left in the resulting tournament. The winning player is the one which makes the final move. We develop a winning position and classify those tournaments that are optimal for the first or second-moving player

Single-Elimination Brackets Fail to Approximate Copeland Winner

Single-elimination (SE) brackets appear commonly in both sports tournaments and the voting theory literature. In certain tournament models, they have been shown to select the unambiguously-strongest competitor with optimum probability. By contrast, we reevaluate SE brackets through the lens of approximation, where the goal is to select a winner who would beat the most other competitors in a round robin (i.e., maximize the Copeland score), and find them lacking. Our primary result establishes the approximation ratio of a randomly-seeded SE bracket is 2^{- Theta(sqrt{log n})}; this is underwhelming considering a 1/2 ratio is achieved by choosing a winner uniformly at random. We also establish that a generalized version of the SE bracket performs nearly as poorly, with an approximation ratio of 2^{- Omega(sqrt[4]{log n})}, addressing a decade-old open question in the voting tree literature

Promoting golf in the Golden Age : a frame analysis of the writings of O.B. Keelor and Grantland Rice

This study uses a frame analysis to show how O.B. Keelor (1882-1950) of the Atlanta Journal and Grantland Rice (1880-1954) of the New York Herald Tribune influenced the growth of golf in the United States with their newspaper writings in the 1920s. How media frame an issue influences how the public perceives it, and news slant or angle influences public opinion. Many things, including urbanization, industrialization and the end of World War I, contributed to the growing popularity of golf during the 1920s. The media, specifically newspapers, played an important role in golf\u27s development. The number of golf courses registered with the United States Golf Association tripled during the 1920s, and the number of rounds played also increased. Keelor is partly responsible for creating the legendary status attained by Bobby Jones (1902-1971). Jones won 13 major championships during his career including the Grand Slam in the 1930. This research identifies the frames Keelor used in the Atlanta Journal to promote Jones. Through the use of descriptive language Keelor was able to develop the mythical excellence of Jones. Rice\u27s two columns for the New York Herald Tribune were Sportlight and Tales of a Wayside Tee. Rice promoted golf in both columns as well as contributed to the mythmaking of Jones. This research examines and identifies the specific language both writers used to promote Jones and the image of golf. An index of articles Keelor wrote during major championships in which Jones played is provided

Tournament Directed Graphs

Paired comparison is the process of comparing objects two at a time. A tournament in Graph Theory is a representation of such paired comparison data. Formally, an n-tournament is an oriented complete graph on n vertices; that is, it is the representation of a paired comparison, where the winner of the comparison between objects x and y (x and y are called vertices) is depicted with an arrow or arc from the winner to the other. In this thesis, we shall prove several results on tournaments. In Chapter 2, we will prove that the maximum number of vertices that can beat exactly m other vertices in an n-tournament is min{2m + 1,2n - 2m - 1}. The remainder of this thesis will deal with tournaments whose arcs have been colored. In Chapter 3 we will define what it means for a k-coloring of a tournament to be k-primitive. We will prove that the maximum k such that some strong n-tournament can be k-colored to be k-primitive lies in the interval [(n-12), (n2) - [n/4]). In Chapter 4, we shall prove special cases of the following 1982 conjecture of Sands, Sauer, and Woodrow from [14]: Let T be a 3-arc-colored tournament containing no 3-cycle C such that each arc in C is a different color. Then T contains a vertex v such that for any other vertex x, x has a monochromatic path to v

An algorithmic analysis of deliberation and representation in collective behaviour

The selection of a nominee by a group of players in the process of selecting a winner is present in many contexts. In sports, it is a major strategic problem to select the best team members. Crucially, in politics, this problem is essential for the process of primaries. There, factions decide which of their candidates should take part in the elections. We study the strategic behaviour of coalitions from the game-theoretic perspective. More precisely, we analyse the existence of a pure Nash equilibrium in the games capturing the strategic nomination problem. First, we adapt the well-known Hotelling-Downs model, capturing the strategic behaviour of political parties in primaries. Subsequently, we explore this problem for tournament-based rules. There, winners are chosen based on the pairwise comparisons between candidates. First, we study the setting of knockout-tournaments. Next, we investigate tournaments, in which participants do not compete in rounds. For each of these mechanisms, we analyse the computational complexity of checking the existence of a pure Nash equilibrium. Nominee selection can also be influenced by the deliberation between the voters. To account for that, we investigate the complexity of checking the convergence of a synchronous, threshold-based protocol. There, in every time step all agents update their opinion if the strict majority of their influencers disagrees with them. Furthermore, we explore computational aspects of majority illusion. This phenomenon occurs when a large number of agents in a network perceives the opinion, which is a minority view, as the one which is held by the majority of agents. We study the problem of checking the possibility of assigning opinions to agents, so that it holds for a large fraction of them. We further address the complexity of checking the possibility of eliminating the majority illusion by changing a small number of edges in a social network

Gender, competition and performance: evidence from chess players

This paper studies gender differences in performance in a male‐dominated competitive environment chess tournaments. We find that the gender composition of chess games affects the behaviors of both men and women in ways that worsen the outcomes for women. Using a unique measure of within‐game quality of play, we show that women make more mistakes when playing against men. Men, however, play equally well against male and female opponents. We also find that men persist longer before losing to women. Our results shed some light on the behavioral changes that lead to differential outcomes when the gender composition of competitions varies

Approximately Strategyproof Tournament Rules: On Large Manipulating Sets and Cover-Consistence

We consider the manipulability of tournament rules, in which n teams play a round robin tournament and a winner is (possibly randomly) selected based on the outcome of all binom{n}{2} matches. Prior work defines a tournament rule to be k-SNM-? if no set of ? k teams can fix the ? binom{k}{2} matches among them to increase their probability of winning by >? and asks: for each k, what is the minimum ?(k) such that a Condorcet-consistent (i.e. always selects a Condorcet winner when one exists) k-SNM-?(k) tournament rule exists? A simple example witnesses that ?(k) ? (k-1)/(2k-1) for all k, and [Jon Schneider et al., 2017] conjectures that this is tight (and prove it is tight for k=2). Our first result refutes this conjecture: there exists a sufficiently large k such that no Condorcet-consistent tournament rule is k-SNM-1/2. Our second result leverages similar machinery to design a new tournament rule which is k-SNM-2/3 for all k (and this is the first tournament rule which is k-SNM-(<1) for all k). Our final result extends prior work, which proves that single-elimination bracket with random seeding is 2-SNM-1/3 [Jon Schneider et al., 2017], in a different direction by seeking a stronger notion of fairness than Condorcet-consistence. We design a new tournament rule, which we call Randomized-King-of-the-Hill, which is 2-SNM-1/3 and cover-consistent (the winner is an uncovered team with probability 1)