The Pythagorean Expected Wins Percentage Model was developed by Bill James to estimate a baseball team expected wins percentage over the course of a season. As such, the model can be used to assess how lucky or unfortunate a team was over the course of a season. From a sports analytics perspective, such information is valuable in that it is important to understand how reproducible a given result may be in the next time period. In contest theoretic (game theoretic) parlance, the original model represents a (restricted) Tullock contest success function (CSF). We transform, estimate, and compare the original model and two alternative models from contest theory, the serial and difference form CSFs, using MLB team win data (2003 to 2015) and perform a cross-validation exercise to test the accuracy of the alternative models. The serial CSF estimator dramatically improves wins estimation (reduces root mean squared error) compared to the original model, an optimized version of the model, or an optimized difference form model. We conclude that the serial CSF model of wins estimation substantially improves estimates of team quality, on average. The work provides a real world test of alternat
Ultimate Tic-Tac-Toe is a variant of the popular Tic-Tac-Toe game. Two players compete to win three aligned "fields," with each field constituting its own miniature tic-tac-toe game. Each move determines which field the next player must play in. Prior studies have shown that there exists a forced winning strategy for the first player, whereby they can win in at least 29 moves and at most 43 moves. This paper proposes a practical solution to the forced-win problem discovered by Bertholon et al., by putting forth a simple method for randomizing the first set of moves that are played. This method uses 5 randomly generated digits between 0 and 8 to arbitrarily place the first 4 moves of the game, and helps players avoid forced wins without having to change other rules of the game. This paper also investigates the probability that a random placement of the first 4 moves will lead to an easily calculable forced win, and shows that this probability is precisely 56/59049, or 0.0948%.
Finkelstein-Schoenfeld, Buyse, Pocock, and other authors have developed generalizations of the Mann-Whitney test that allow for pairwise patient comparisons to include a hierarchy of measurements. Various authors present either asymptotic or randomized methods for analyzing the wins. We use graph theory concepts to derive exact means and variances for the number of wins, as a replacement for approximate values obtained from bootstrap analysis or random sampling from the permutation distribution. The time complexity of our algorithm is $O(N^2)$, where $N$ is the total number of patients. In any situation where the mean and variance of a bootstrap sample are used to draw conclusions, our methodology will be faster and more accurate than the randomized bootstrap or permutation test.
We consider two games between two players Ann and Ben who build a word together by adding alternatively a letter at the end of the shared word. In the nonrepetitive game, Ben wins the game if he can create a square of length at least $4$, and Ann wins if she can build an arbitrarily long word before that. In the erase-repetition game, whenever a square occurs the second part of the square is erased and the goal of Ann is still to build an arbitrarily large word (Ben simply wants to limit the size of the word in this game). Grytczuk, Kozik, and Micek showed that Ann has a winning strategy for the nonrepetitive game if the alphabet is of size at least $6$ and for the erase-repetition game is the alphabet is of size at least $8$. In this article, we lower these bounds to respectively $4$ and $6$. The bound obtain by Grytczuk et al. relied on the so-called entropy compression and the previous bound by Pegden relied on some particular version of the Lovász Local Lemma. We recently introduced a counting argument that can be applied to the same set of problems as entropy compression or the Lovász Local Lemma and we use our method here. For these two games, we know that Ben has a winning s
Using mostly elementary considerations, we find out who wins the game of Domineering on all rectangular boards of width 2, 3, 5, and 7. We obtain bounds on other boards as well, and prove the existence of polynomial-time strategies for playing on all boards of width 2, 3, 4, 5, 7, 9, and 11. We also comment briefly on toroidal and cylindrical boards.
Points out the errors in the paper "J.G. Chen, S.D. Kominers, and R.W. Sinnott. Walk versus wait: The lazy mathematician wins. arXiv.org Mathematics 2008. arXiv:0801.0297"
The angel-devil game is played on an infinite two-dimensional ``chessboard''. The squares of the board are all white at the beginning. The players called angel and devil take turns in their steps. When it is the devil's turn, he can turn a square black. The angel always stays on a white square, and when it is her turn she can fly at a distance of at most J steps (each of which can be horizontal, vertical or diagonal) to a new white square. Here J is a constant. The devil wins if the angel does not find any more white squares to land on. The result of the paper is that if J is sufficiently large then the angel has a strategy such that the devil will never capture her. This deceptively easy-sounding result has been a conjecture, surprisingly, for about thirty years. Several other independent solutions have appeared simultaneously, some of them prove that J=2 is sufficient (see the Wikipedia on the angel problem). Still, it is hoped that the hierarchical solution presented here may prove useful for some generalizations.
Maker-Breaker subgraph games are among the most famous combinatorial games. For $n,q\in\mathbb{N}$ and a fixed subgraph $C$ of the complete graph $K_n$, the two players, called Maker and Breaker, alternately claim edges of $K_n$. Maker claims one unclaimed edge per round and Breaker may claim up to $q$ edges per round. If Maker is able to claim all edges of a copy of $C$, he wins the game. Otherwise Breaker wins. Bednarska and Łuczak (2000) determined in a landmark work the asymptotics of the treshold bias as $Θ(n^{1/m(C)})$ where $m(C)$ is the 2-density of $C$, analysing random strategies. Since then it has been a major open problem to determine the treshhold bias, if it exists, with corresponding strategies, leading to sharp constants in the $Θ$-notion. A famous case is the triangle game ($C=C_3$), studied by Chvatal and Erd"os (1978), who showed Maker wins if $q\le \sqrt{2n}$ and Breaker wins if $q\ge2\sqrt{n}$. Glazik and Srivastav (2022) improved this via a potential method, showing Breaker wins already for $q\ge\sqrt{8/3}\sqrt{n}$. Spencer (2019) conjectured generalizability to arbitrary subgraphs $C$. We confirm this conjecture, presenting a general winning strategy for Brea
Consider a collection of m competing machine learning algorithms. Given their performance on a benchmark of datasets, we would like to identify the best performing algorithm. Specifically, which algorithm is most likely to ``win'' (rank highest) on a future, unseen dataset. The standard maximum likelihood approach suggests counting the number of wins per each algorithm. In this work, we argue that there is much more information in the complete rankings. That is, the number of times that each algorithm finished second, third and so forth. Yet, it is not entirely clear how to effectively utilize this information for our purpose. In this work we introduce a novel conceptual framework for estimating the win probability for each of the m algorithms, given their complete rankings over a benchmark of datasets. Our proposed framework significantly improves upon currently known methods in synthetic and real-world examples.
The win ratio is increasingly used in randomized trials due to its intuitive clinical interpretation, ability to incorporate the relative importance of composite endpoints, and its capacity for combining different types of outcomes (e.g. time-to-event, binary, counts, etc.) to be combined. There are open questions, however, about how to implement adaptive design approaches when the primary endpoint is a win ratio, including in group sequential designs. A key requirement allowing for straightforward application of classical group sequential methods is the independence of incremental interim test statistics. This paper derives the covariance structure of incremental U-statistics that evaluate the win ratio under its asymptotic distribution. The derived covariance shows that the independent increments assumption holds for the asymptotic distribution of U-statistics that test the win ratio. Simulations confirm that traditional $α$-spending preserves Type I error across interim looks. A retrospective look at the IN.PACT SFA clinical trial data illustrates the potential for stopping early in a group sequential design using the win ratio. We have demonstrated that straightforward use of L
The game of cops and robbers is played on a fixed (finite or infinite) graph $G$. The cop chooses his starting position, then the robber chooses his. After that, they take turns and move to adjacent vertices, or stay at their current vertex, with the cop moving first. The game finishes if the cop lands on the robber's vertex. In that case we say that the cop wins, while if the robber is never caught then we say that the robber wins. The graph $G$ is called cop-win if the cop has a winning strategy. In this paper we construct an infinite cop-win graph in which, for any two given starting positions of the cop and the robber, we can name in advance a finite time in which the cop can capture the robber, but these finite times are not bounded above. This shows that this graph has maximum capture time (CR-ordinal) $ω$, disproving a conjecture of Bonato, Gordinowicz and Hahn that no such graph should exist.
We propose a generalized win fraction regression framework for prioritized composite survival outcomes. The framework models the conditional win fraction through a chosen link function (including identity, logit, or probit), thereby accommodating multi-component time-to-event endpoints within a unified regression structure. To handle right censoring, we construct inverse-probability-of-censoring-weighted estimating equations that target the win fraction as if censoring were absent. Under the identity link, regression parameters characterize covariate associations on the natural win fraction scale. Under the logit link, they characterize the log odds of winning -- a new and complementary effect measure that treats ties as failures to win, imposing a more conservative standard than the win ratio or win odds. When there are no ties, the logit win fraction model reduces to proportional win fraction regression; moreover, the unweighted version of our estimating equations numerically coincides with the proportional win fraction point estimator regardless of ties. We establish large-sample properties of the proposed estimators and derive a consistent sandwich variance estimator that accou
We prove that for a Manneville--Pomeau type interval map, the set of points whose orbit closures miss a prescribed countable set is absolute winning in the sense of McMullen. The proof has three parts. First we directly prove that the exceptional set for the distinguished endpoint of the induced first-return map is absolute winning. Then we use the finite-branch winning theorem of Hu--Li--Yu, together with the one-dimensional implication from $1/2$-strong winning to absolute winning, to obtain absolute winning for all countable induced targets. Finally, a quasisymmetric pullback argument transfers these induced results back to the original map.
Treatment effect estimands based on win statistics, including the win ratio, win odds, and win difference are increasingly popular targets for summarizing endpoints in clinical trials. Such win estimands offer an intuitive approach for prioritizing outcomes by clinical importance. The implementation and interpretation of win estimands is complicated in cluster randomized trials (CRTs), where researchers can target fundamentally different estimands on the individual-level or cluster-level. We numerically demonstrate that individual-pair and cluster-pair win estimands can substantially differ when cluster size is informative: where outcomes and/or treatment effects depend on cluster size. With such informative cluster sizes, individual-pair and cluster-pair win estimands can even yield opposite conclusions regarding treatment benefit. We describe consistent estimators for individual-pair and cluster-pair win estimands and propose a leave-one-cluster-out jackknife variance estimator for inference. Despite being consistent, our simulations highlight that some caution is needed when implementing individual-pair win estimators due to finite-sample bias. In contrast, cluster-pair win esti
This paper proposes an analytical framework for the orbital Target-Attacker-Defender game with a non-maneuvering target along elliptic orbits. Focusing on the linear quadratic game, we derive an analytical solution to the matrix Riccati equation, which yields analytical Nash-equilibrium strategies for the game. Based on the analytical strategies, we derive the analytical form of the necessary and sufficient winning conditions for the attacker. The simulation results show good consistency between the analytical and numerical methods, exhibiting 0.004$\%$ relative error in the cost function. The analytical method achieves over 99.9$\%$ reduction in CPU time compared to the conventional numerical method, strengthening the advantage of developing the analytical strategies. Furthermore, we verify the proposed winning conditions and investigate the effects of eccentricity on the game outcomes. Our analysis reveals that for games with hovering initial states, the initial position of the defender should be constrained inside a mathematically definable set to ensure that the attacker wins the game. This constrained set further permits geometric interpretation through our proposed method. Th
Covariate adjustment can enhance precision and power in clinical trials, yet its application to the win odds remains unclear. The win odds is an extension of the win ratio that counts ties as half a win for the treatment and the control group, respectively. In their original form, both the win ratio and the win odds rely on comparing each individual from the treatment group to each individual from the control group in a pairwise manner, and count the number of wins, losses, and ties from these pairwise comparisons. A priori, it is not clear how covariate adjustment can be implemented for the win odds. To address this, we establish a connection between the win odds and the marginal probabilistic index, a measure for which covariate adjustment theory is well-developed. Using this connection, we show how covariate adjustment for the win odds is possible, leading to potentially more precise estimators and larger power as compared to the unadjusted win odds. We present the underlying theory for covariate adjustment for the win odds in an accessible way and apply the method on synthetic data based on the CANTOS trial (ClinicalTrials.gov identifier: NCT01327846) characteristics, on a subs
The Strong Ramsey game $\mathcal{R}(B,G)$ is a two player game with players $P_1$ and $P_2$, where $B$ and $G$ are $k$-uniform hypergraphs for some $k \geq 2$. $G$ is always finite, while $B$ may be infinite. $P_1$ and $P_2$ alternately color uncolored edges $e \in B$ in their respective color and $P_1$ begins. Whoever completes a monochromatic copy of $G$ in their own color first, wins the game. If no one claims a monochromatic copy of $G$ in a finite number of moves, the game is declared a draw. For a $t \in \mathbb{N}$, let $\hat{K}_{2,t}$ denote the $K_{2,t}$ together with the edge connecting the two vertices in the partition class of size 2. The purpose of this paper is to give a winning strategy for $P_1$ in the game $\mathcal{R}(K_{\aleph_0}, \hat{K}_{2,3})$.
In Major League Baseball, strategy and planning are major factors in determining the outcome of a game. Previous studies have aided this by building machine learning models for predicting the winning team of any given game. We extend this work by training a comprehensive set of machine learning models using a common dataset. In addition, we relate the win probabilities produced by these models to win strength as measured by score differential. In doing so we show that the most common machine learning models do indeed demonstrate a relationship between predicted win probability and the strength of the win. Finally, we analyze the results of using predicted win probabilities as a decision making mechanism on run-line betting. We demonstrate positive returns when utilizing appropriate betting strategies, and show that naive use of machine learning models for betting lead to significant loses.
Maker-Breaker subgraph games are among the most famous combinatorial games. For given $n,q \in \mathbb{N}$ and a subgraph $C$ of the complete graph $K_n$, the two players, called Maker and Breaker, alternately claim edges of $K_n$. In each round of the game Maker claims one edge and Breaker is allowed to claim up to $q$ edges. If Maker is able to claim all edges of a copy of $C$, he wins the game. Otherwise Breaker wins. In this work we introduce the first constructive strategy for Maker for the $C_4$-Maker-Breaker game and show that he can win the game if $q < 0.16 n^{2/3}$. According to the theorem of Bednarska and Luczak (2000) $n^{2/3}$ is asymptotically optimal for this game, but the constant given there for a random Maker strategy is magnitudes apart from our constant 0.16.
The Win Ratio has gained significant traction in cardiovascular trials as a novel method for analyzing composite endpoints (Pocock and others, 2012). Compared with conventional approaches based on time to the first event, the Win Ratio accommodates the varying priorities and types of outcomes among components, potentially offering greater statistical power by fully utilizing the information contained within each outcome. However, studies using Win Ratio have largely been confined to fixed design, limiting flexibility for early decisions, such as stopping for futility or efficacy. Our study proposes a sequential design framework incorporating multiple interim analyses based on Win Ratio or Net Benefit statistics. Moreover, we provide rigorous proof of the canonical joint distribution for sequential Win Ratio and Net Benefit statistics, and an algorithm for sample size determination is developed. We also provide results from a finite sample simulation study, which show that our proposed method controls Type I error maintains power level, and has a smaller average sample size than the fixed design. A real study of cardiovascular study is applied to illustrate the proposed method.