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UNO is a popular multiplayer card game. In each turn, a player has to play a card in their hand having the same number or color as the most recently played card. When having few people, adding virtual players to play the game can easily be done in UNO video games. However, this is a challenging task for physical UNO without computers. In this paper, we propose an unconventional protocol that can simulate virtual players using nothing but physical UNO cards. In particular, our protocol can uniformly select a valid card to play from each virtual player's hand at random, or report that none exists, without revealing the rest of its hand. The protocol can also be applied to simulate virtual players in other turn-based card or tile games where each player has to select a valid card or tile to play in each turn.

This study proposes a simple method for multi-object tracking (MOT) of players in a badminton court. We leverage two off-the-shelf cameras, one on the top of the court and the other on the side of the court. The one on the top is to track players' trajectories, while the one on the side is to analyze the pixel features of players. By computing the correlations between adjacent frames and engaging the information of the two cameras, MOT of badminton players is obtained. This two-camera approach addresses the challenge of player occlusion and overlapping in a badminton court, providing player trajectory tracking and multi-angle analysis. The presented system offers insights into the positions and movements of badminton players, thus serving as a coaching or self-training tool for badminton players to improve their gaming strategies.

In most sports, especially football, most coaches and analysts search for key performance indicators using notational analysis. This method utilizes a statistical summary of events based on video footage and numerical records of goal scores. Unfortunately, this approach is now obsolete owing to the continuous evolutionary increase in technology that simplifies the analysis of more complex process variables through machine learning (ML). Machine learning, a form of artificial intelligence (AI), uses algorithms to detect meaningful patterns and define a structure based on positional data. This research investigates a new method to evaluate the value of current football players, based on establishing the machine learning models to investigate the relations among the various features of players, the salary of players, and the market value of players. The data of the football players used for this project is from several football websites. The data on the salary of football players will be the proxy for evaluating the value of players, and other features will be used to establish and train the ML model for predicting the suitable salary for the players. The motivation is to explore what

A commonly held opinion is that left-handed tennis players are overrepresented compared to the percentage of left-handers within the general population. This study provides the domain insights supported by data analysis that could help inform the decision of parents and coaches considering whether a child should start playing tennis as left- or right-handed when there is no strong arm-handed dominance. Compared to the commonly cited figure of about 10% of left-handed male population, data analysis from the official ATP web site for the top 100 ranked tennis players over the past decades (1985-2016) shows evidence of overrepresentation of left-handed elite tennis players (about 15%). The insights and data analysis can inform the handedness decision, advance coaching and strategic game concepts, enhance media coverage/analytics, left-handed facts and statistics, and inform tennis equipment manufacturing.

In this paper we introduce the $Γ$ value, a new value for cooperative games with transferable utility. We also provide an axiomatic characterization of the $Γ$ value based on a property concerning the so-called necessary players. A necessary players of a game is one without which the characteristic function is zero. We illustrate the performance of the $Γ$ value in a particular cost allocation problem that arises when the owners of the apartments in a building plan to install an elevator and share its installation cost; in the resulting example we compare the proposals of the $Γ$ value, the equal division value and the Shapley value in two different scenarios. In addition, we propose an extension of the $Γ$ value for cooperative games with transferable utility and with a coalition structure. Finally, we provide axiomatic characterizations of the coalitional $Γ$ value and of the Owen and Banzhaf-Owen values using alternative properties concerning necessary players.

We propose a game-theoretic model of the reliability of decentralised systems based on Varian's model of system reliability, to which we add a new normalised total effort case that models \textit{decentralisation conscious players} who prioritise decentralisation. We derive the Nash equilibria in the normalised total effort game. In these equilibria, either one or two values are played by players that do not free ride. The speed at which players can adjust their contributions can determine how an equilibrium is reached and equilibrium values. The behaviour of decentralisation conscious players is robust to deviations by other players. Our results highlight the role that decentralisation conscious players can play in maintaining decentralisation. They also highlight, however, that by supporting an equilibrium that requires an important contribution they cannot be expected to increase decentralisation as contributing the equilibrium value may still imply a loss for many players. We also discuss practical constraints on decentralisation in the context of our model.

We investigate multi-round team competitions between two teams, where each team selects one of its players simultaneously in each round and each player can play at most once. The competition defines an extensive-form game with perfect recall and can be solved efficiently by standard methods. We are interested in the properties of the subgame perfect equilibria of this game. We first show that uniformly random strategy is a subgame perfect equilibrium strategy for both teams when there are no redundant players (i.e., the number of players in each team equals to the number of rounds of the competition). Secondly, a team can safely abandon its weak players if it has redundant players and the strength of players is transitive. We then focus on the more interesting case where there are redundant players and the strength of players is not transitive. In this case, we obtain several counterintuitive results. First of all, a player might help improve the payoff of its team, even if it is dominated by the entire other team. We give a necessary condition for a dominated player to be useful. We also study the extent to which the dominated players can increase the payoff. These results bring i

It is challenging to get access to datasets related to the physical performance of soccer players. The teams consider such information highly confidential, especially if it covers in-game performance.Hence, most of the analysis and evaluation of the players' performance do not contain much information on the physical aspect of the game, creating a blindspot in performance analysis. We propose a novel method to solve this issue by deriving movement characteristics of soccer players. We use event-based datasets from data provider companies covering 50+ soccer leagues allowing us to analyze the movement profiles of potentially tens of thousands of players without any major investment. Our methodology does not require expensive, dedicated player tracking system deployed in the stadium. We also compute the similarity of the players based on their movement characteristics and as such identify potential candidates who may be able to replace a given player. Finally, we quantify the uniqueness and consistency of players in terms of their in-game movements. Our study is the first of its kind that focuses on the movements of soccer players at scale, while it derives novel, actionable insights

In this paper, I introduce RisingBALLER, the first publicly available approach that leverages a transformer model trained on football match data to learn match-specific player representations. Drawing inspiration from advances in language modeling, RisingBALLER treats each football match as a unique sequence in which players serve as tokens, with their embeddings shaped by the specific context of the match. Through the use of masked player prediction (MPP) as a pre-training task, RisingBALLER learns foundational features for football player representations, similar to how language models learn semantic features for text representations. As a downstream task, I introduce next match statistics prediction (NMSP) to showcase the effectiveness of the learned player embeddings. The NMSP model surpasses a strong baseline commonly used for performance forecasting within the community. Furthermore, I conduct an in-depth analysis to demonstrate how the learned embeddings by RisingBALLER can be used in various football analytics tasks, such as producing meaningful positional features that capture the essence and variety of player roles beyond rigid x,y coordinates, team cohesion estimation, a

We formulate and study a decentralized multi-armed bandit (MAB) problem. There are M distributed players competing for N independent arms. Each arm, when played, offers i.i.d. reward according to a distribution with an unknown parameter. At each time, each player chooses one arm to play without exchanging observations or any information with other players. Players choosing the same arm collide, and, depending on the collision model, either no one receives reward or the colliding players share the reward in an arbitrary way. We show that the minimum system regret of the decentralized MAB grows with time at the same logarithmic order as in the centralized counterpart where players act collectively as a single entity by exchanging observations and making decisions jointly. A decentralized policy is constructed to achieve this optimal order while ensuring fairness among players and without assuming any pre-agreement or information exchange among players. Based on a Time Division Fair Sharing (TDFS) of the M best arms, the proposed policy is constructed and its order optimality is proven under a general reward model. Furthermore, the basic structure of the TDFS policy can be used with a

Positions of chess players in intransitive (rock-paper-scissors) relations are considered. Namely, position A of White is preferable (it should be chosen if choice is possible) to position B of Black, position B of Black is preferable to position C of White, position C of White is preferable to position D of Black, but position D of Black is preferable to position A of White. Intransitivity of winningness of positions of chess players is considered to be a consequence of complexity of the chess environment -- in contrast with simpler games with transitive positions only. The space of relations between winningness of positions of chess players is non-Euclidean. The Zermelo-von Neumann theorem is complemented by statements about possibility vs. impossibility of building pure winning strategies based on the assumption of transitivity of positions of chess players. Questions about the possibility of intransitive positions of players in other positional games are raised.

We consider a coalitional game with the same payoff for all players. To maximize the payoff, the players need to use one collective strategy, if all players are in certain states, and the other strategy otherwise. The current state of each player changes according to external conditions and is not known to the other players. In one example of such a game, quantum entanglement between players results in the optimal payoff thrice the maximal payoff for unentangled players.

There has been a consistent criticism over the past decade of the NFL franchise tag's monetary limitations due to its biased institutions in favor of the team rather than the player. But the question whether the NFL's franchise tag is fair or unfair to players has never been systematically studied. In this paper, I investigate the effects of NFL players' contract extensions when on a franchise tag compared to when they are not and analyze them through statistical and economic lens. Through my research, I find that indeed the current franchise tag designation is unfair to players when it comes to contract extension. I then propose a solution to remedy this unfairness, that is, removing the opportunity to franchise tag players for multiple years, and adding an option for the player to either test free agency but receive zero pay until they settle on a contract (the team can also match the offer) or sign the franchise tag, to provide more flexibility for the player and the team.

We study a routing game in which one of the players unilaterally acts altruistically by taking into consideration the latency cost of other players as well as his own. By not playing selfishly, a player can not only improve the other players' equilibrium utility but also improve his own equilibrium utility. To quantify the effect, we define a metric called the Value of Unilateral Altruism (VoU) to be the ratio of the equilibrium utility of the altruistic user to the equilibrium utility he would have received in Nash equilibrium if he were selfish. We show by example that the VoU, in a game with nonlinear latency functions and atomic players, can be arbitrarily large. Since the Nash equilibrium social welfare of this example is arbitrarily far from social optimum, this example also has a Price of Anarchy (PoA) that is unbounded. The example is driven by there being a small number of players since the same example with non-atomic players yields a Nash equilibrium that is fully efficient.

We study adaptive learning in a typical p-player game. The payoffs of the games are randomly generated and then held fixed. The strategies of the players evolve through time as the players learn. The trajectories in the strategy space display a range of qualitatively different behaviors, with attractors that include unique fixed points, multiple fixed points, limit cycles and chaos. In the limit where the game is complicated, in the sense that the players can take many possible actions, we use a generating-functional approach to establish the parameter range in which learning dynamics converge to a stable fixed point. The size of this region goes to zero as the number of players goes to infinity, suggesting that complex non-equilibrium behavior, exemplified by chaos, may be the norm for complicated games with many players.

In increasingly different contexts, it happens that a human player has to interact with artificial players who make decisions following decision-making algorithms. How should the human player play against these algorithms to maximize his utility? Does anything change if he faces one or more artificial players? The main goal of the paper is to answer these two questions. Consider n-player games in normal form repeated over time, where we call the human player optimizer, and the (n -- 1) artificial players, learners. We assume that learners play no-regret algorithms, a class of algorithms widely used in online learning and decision-making. In these games, we consider the concept of Stackelberg equilibrium. In a recent paper, Deng, Schneider, and Sivan have shown that in a 2-player game the optimizer can always guarantee an expected cumulative utility of at least the Stackelberg value per round. In our first result, we show, with counterexamples, that this result is no longer true if the optimizer has to face more than one player. Therefore, we generalize the definition of Stackelberg equilibrium introducing the concept of correlated Stackelberg equilibrium. Finally, in the main resul