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We consider a multi-player non-zero-sum turn-based game (abbreviated as multi-player game) on a finite directed graph. A secure equilibrium (SE) is a strategy profile in which no player has the incentive to deviate from the strategy because no player can increase her own payoff or lower the payoff of another player. SE is a promising refinement of Nash equilibrium in which a player does not care the payoff of another player. In this paper, we discuss the decidability and complexity of the problem of deciding whether a secure equilibrium with constraints (a payoff profile specifying which players must win) exists for a given multi-player game.

The availability of tracking data in football presents unique opportunities for analyzing team shape and player roles, but leveraging it effectively remains challenging. This difficulty arises from the significant overlap in player positions, which complicates the identification of distinct roles and team formations. In this work, we propose a novel model that incorporates a hidden permutation matrix to simultaneously estimate team formations and assign roles to players at the frame level. To address the cardinality of permutation sets, we develop a statistical procedure to parsimoniously select relevant matrices prior to parameter estimation. Additionally, to capture formation changes during a match, we introduce a latent regime variable, enabling the modeling of dynamic tactical adjustments. This framework disentangles player locations from role-specific positions, providing a clear representation of team structure. We demonstrate the applicability of our approach using player tracking data, showcasing its potential for detailed team and player analysis.

In a previous note, a two-player game between a random player and a deterministic player was introduced, and it was shown analytically that the winning probability of the deterministic player is the derangement probability dn/n!. The natural question left open was to explain this coincidence combinatorially. This paper gives such an explanation and extends it to a larger family of games. In the generalized game, the deterministic player removes q elements per turn, or all remaining elements if fewer than q remain. We couple the game exactly to the cycle decomposition of a uniformly random permutation. Under this coupling, the random player wins precisely when the first cycle of length at most q, read in canonical cycle order, is a fixed point. The case q = 1 recovers derangements, while the general case is governed by the first short cycle of the permutation.

Consider a game involving a team with $n$ players, $k$ of which wear shirts marked with a letter $A$, while the others with a letter $B$, and such that only $s$ people play, while the remaining $n-s$ wait outside the court. At certain times the players rotate, and one player enters the court while another one leaves, each player keeping the same neighbors at all times. The house rules want at least $t$ players wearing a shirt with the letter $A$ in the court at each rotation. We show that this can be achieved if and only if $nt\leq ks$. We use classical work on balanced words dating back to Christoffel and Smith.

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

Cooperative game theory deals with systems where players want to cooperate to improve their payoffs. But players may choose coalitions in a non-cooperative manner, leading to a coalition-formation game. We consider such a game with several players (willing to cooperate) and an adamant player (unwilling to cooperate) involved in resource-sharing. Here, the strategy of a player is the set of players with whom it wants to form a coalition. Given a strategy profile, an appropriate partition of coalitions is formed; players in each coalition maximize their collective utilities leading to a non-cooperative resource-sharing game among the coalitions, the utilities at the resulting equilibrium are shared via Shapley-value; these shares define the utilities of players for the given strategy profile in coalition-formation game. We also consider the utilitarian solution to derive the price of anarchy (PoA). We considered a case with symmetric players and an adamant player; wherein we observed that players prefer to stay alone at Nash equilibrium when the number of players (n) is more than 4. In contrast, in the majority of the cases, the utilitarian partition is grand coalition. Interestingly

We describe our system for the SoccerNet 2026 Player-Centric Ball-Action Spotting Challenge, which requires predicting who performs which action and when, across eight classes in broadcast soccer. Building on the three FOOTPASS baselines [1] (TAAD, TAAD+GNN, and TAAD+DST), we contribute four extensions: (1) gradient check pointing to enable full-backbone fine-tuning on a single GPU; (2) fusion of GNN logits into the DST encoder, combining graph-based tactical context with per-player visual features; (3) square-root frequency class weighting to address the 213:1 pass-to-tackle imbalance in the training data; and (4) a post processing pipeline comprising per-class logit gating, temporal frame refinement, jersey re-assignment, and a two-model ensemble. Our system achieves 0.548 Macro F1 on the test set and 0.446 on the challenge set (server evaluation).

In an interactive belief model, are the players "commonly meta-certain" of the model itself? This paper formalizes such implicit "common meta-certainty" assumption. To that end, the paper expands the objects of players' beliefs from events to functions defined on the underlying states. Then, the paper defines a player's belief-generating map: it associates, with each state, whether a player believes each event at that state. The paper formalizes what it means by: "a player is (meta-)certain of her own belief-generating map" or "the players are (meta-)certain of the profile of belief-generating maps (i.e., the model)." The paper shows: a player is (meta-)certain of her own belief-generating map if and only if her beliefs are introspective. The players are commonly (meta-)certain of the model if and only if, for any event which some player i believes at some state, it is common belief at the state that player i believes the event. This paper then asks whether the "common meta-certainty" assumption is needed for an epistemic characterization of game-theoretic solution concepts. The paper shows: if each player is logical and (meta-)certain of her own strategy and belief-generating map,

Characterizing playing style is important for football clubs on scouting, monitoring and match preparation. Previous studies considered a player's style as a combination of technical performances, failing to consider the spatial information. Therefore, this study aimed to characterize the playing styles of each playing position in the Chinese Football Super League (CSL) matches, integrating a recently adopted Player Vectors framework. Data of 960 matches from 2016-2019 CSL were used. Match ratings, and ten types of match events with the corresponding coordinates for all the lineup players whose on-pitch time exceeded 45 minutes were extracted. Players were first clustered into 8 positions. A player vector was constructed for each player in each match based on the Player Vectors using Nonnegative Matrix Factorization (NMF). Another NMF process was run on the player vectors to extract different types of playing styles. The resulting player vectors discovered 18 different playing styles in the CSL. Six performance indicators of each style were investigated to observe their contributions. In general, the playing styles of forwards and midfielders are in line with football performance e

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

This work focuses on studying players behaviour in interactive narratives with the aim to simulate their choices. Besides sub-optimal player behaviour due to limited knowledge about the environment, the difference in each player's style and preferences represents a challenge when trying to make an intelligent system mimic their actions. Based on observations from players interactions with an extract from the interactive fiction Anchorhead, we created a player profile to guide the behaviour of a generic player model based on the BDI (Belief-Desire-Intention) model of agency. We evaluated our approach using qualitative and quantitative methods and found that the player profile can improve the performance of the BDI player model. However, we found that players self-assessment did not yield accurate data to populate their player profile under our current approach.

Player modeling attempts to create a computational model which accurately approximates a player's behavior in a game. Most player modeling techniques rely on domain knowledge and are not transferable across games. Additionally, player models do not currently yield any explanatory insight about a player's cognitive processes, such as the creation and refinement of mental models. In this paper, we present our findings with using action model learning (AML), in which an action model is learned given data in the form of a play trace, to learn a player model in a domain-agnostic manner. We demonstrate the utility of this model by introducing a technique to quantitatively estimate how well a player understands the mechanics of a game. We evaluate an existing AML algorithm (FAMA) for player modeling and develop a novel algorithm called Blackout that is inspired by player cognition. We compare Blackout with FAMA using the puzzle game Sokoban and show that Blackout generates better player models.

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

Tracking and identifying players is a fundamental step in computer vision-based ice hockey analytics. The data generated by tracking is used in many other downstream tasks, such as game event detection and game strategy analysis. Player tracking and identification is a challenging problem since the motion of players in hockey is fast-paced and non-linear when compared to pedestrians. There is also significant camera panning and zooming in hockey broadcast video. Identifying players in ice hockey is challenging since the players of the same team look almost identical, with the jersey number the only discriminating factor between players. In this paper, an automated system to track and identify players in broadcast NHL hockey videos is introduced. The system is composed of three components (1) Player tracking, (2) Team identification and (3) Player identification. Due to the absence of publicly available datasets, the datasets used to train the three components are annotated manually. Player tracking is performed with the help of a state of the art tracking algorithm obtaining a Multi-Object Tracking Accuracy (MOTA) score of 94.5%. For team identification, the away-team jerseys are g

We analyze the dynamics of competitions with a large number of players. In our model, n players compete against each other and the winner is decided based on the standings: in each competition, the mth ranked player wins. We solve for the long time limit of the distribution of the number of wins for all n and m and find three different scenarios. When the best player wins, the standings are most competitive as there is one-tier with a clear differentiation between strong and weak players. When an intermediate player wins, the standings are two-tier with equally-strong players in the top tier and clearly-separated players in the lower tier. When the worst player wins, the standings are least competitive as there is one tier in which all of the players are equal. This behavior is understood via scaling analysis of the nonlinear evolution equations.

We investigate a stochastic differential game in which a major player has a private information (the knowledge of a random variable), which she discloses through her control to a population of small players playing in a Nash Mean Field Game equilibrium. The major player's cost depends on the distribution of the population, while the cost of the population depends on the random variable known by the major player. We show that the game has a relaxed solution and that the optimal control of the major player is approximatively optimal in games with a large but finite number of small players.

Video Games are boring when they are too easy, and frustrating when they are too hard. In terms of providing game experience such as enjoyment to the player by match players with different levels of ability to player ability, We assume that implementing DDA for providing matches between player ability and overall game difficulty to the game, especially the modern game, has limitations in terms of increasing computational cost and complexities in the design of modeling the difficulty in modern games. To overcome limitations underlying the method of providing static difficulty changes to player, and DDA, we proposed a novel idea, Player Domination adjustment (PDA). The proposed idea is that to control the AI's actions based on the player's inputs so as to adjust the player's dominant power (e.g. the AI recognizes the player's attack actions but defends it in a wrong side to let the player incur damage to itself), which was proved as it leads to promotion of game-related self-efficacy in our work. Several pieces of research on were conducted on a social deduction game and a fighting game respectively, show our proposed idea has its potential of promoting User Experience(UX). As in an

In this paper we introduce novel algorithmic strategies for effciently playing two-player games in which the players have different or identical player roles. In the case of identical roles, the players compete for the same objective (that of winning the game). The case with different player roles assumes that one of the players asks questions in order to identify a secret pattern and the other one answers them. The purpose of the first player is to ask as few questions as possible (or that the questions and their number satisfy some previously known constraints) and the purpose of the secret player is to answer the questions in a way that will maximize the number of questions asked by the first player (or in a way which forces the first player to break the constraints of the game). We consider both previously known games (or extensions of theirs) and new types of games, introduced in this paper.

Successful analysis of player skills in video games has important impacts on the process of enhancing player experience without undermining their continuous skill development. Moreover, player skill analysis becomes more intriguing in team-based video games because such form of study can help discover useful factors in effective team formation. In this paper, we consider the problem of skill decomposition in MOBA (MultiPlayer Online Battle Arena) games, with the goal to understand what player skill factors are essential for the outcome of a game match. To understand the construct of MOBA player skills, we utilize various skill-based predictive models to decompose player skills into interpretative parts, the impact of which are assessed in statistical terms. We apply this analysis approach on two widely known MOBAs, namely League of Legends (LoL) and Defense of the Ancients 2 (DOTA2). The finding is that base skills of in-game avatars, base skills of players, and players' champion-specific skills are three prominent skill components influencing LoL's match outcomes, while those of DOTA2 are mainly impacted by in-game avatars' base skills but not much by the other two.