Positional Games Research Articles

Howzat? The Financial Health of English Cricket: Not Out, Yet

In 1997 a review of the financial health of English county cricket highlighted strategic weaknesses within the professional game, principally an over-reliance by clubs on the annual grants provided to them by the England and Wales Cricket Board (ECB). Without such grants the teams, in general terms, would be insolvent. Using the financial statements of the First Class Cricket Counties, this paper explores how the financial position and performance of the county game has changed, 20 years on from the seminal study. A series of structural changes to the game had been made, yet financial problems are still evident. Counties are as reliant on central grant income as they were in 1997, although there are cases where clubs have made strategic enhancements and are becoming self-sustainable as going concerns. Rather than the ECB directly funding county revenue it should be working in collaboration with individual clubs to achieve developments in the game from the grassroots upwards, in order to help clubs grow their own revenue streams.

On two theorems of positional games

I give fully detailed proofs of two important theorems—the exact solution of the weak clique game and the compactness theorem—in the theory of positional games. Both results were published several years ago, including an outline of the proofs that explained the basic idea, but left some technical details to the reader. Unfortunately, in both cases these details turned out to be highly non-trivial. Several mathematicians asked me for help; asked me to clarify the missing details. This is why I felt obliged to write this paper.

Security analysis of Game Changer Password System

Abstract In the paper we present the results of security analysis of the Game Changer Password System, proposed by McLennan et al.—a mnemonic variant of password security that uses game positions as passwords. The idea is that several different games are graphically presented on the screen, allowing users to select the game first and then to enter his or her password in the game selected in the form of putting pieces on a board. For example, a user first selects chess, then puts four chess figures on the chessboard. A password is represented by the fact that chess was used and figures’ locations. The first issue with the proposed system is a small search space—the number of possible combinations of passwords, enabling relatively simple and quite inexpensive brute force attacks. The second issue is that users prefer specific locations and figures over others, further reducing the search space and thus enabling attackers to speed up the attacks with high probabilities of success. Based on the issue of non-uniformity of locations and figures attackers can build special dictionaries to launch dictionary-based attacks. We elaborate on the weaknesses and propose a solution that produces stronger passwords. However, the tradeoff between memorability and attack resilience must be taken into the account.

Aggressive Behavior in a Non-Antagonistic Positional Differential Game

This paper is a continuation of the research [3], in which a formalization of nonantagonistic positional differential two-player games (NPDGs) was offered for the case of different behavioral types of players (in short, the NPDGwBT) as follows. In addition to the usual (normal, nor) type of behavior oriented towards maximization of his/her own functional, each player can use other behavioral types described in [2, 7], namely, the altruistic (alt), aggressive (agg), and paradoxical (par) ones. By assumption, each player can switch from one behavioral type to another in the course of the game. Note that in [6, 7] such switching allowed to obtain new solutions in a repeated bimatrix 2 × 2 game. The formalization of players’ actions in the NPDGwBT presented in [3] relies on the formalization and results of the general theory of antagonistic positional differential games [4, 5]. By assumption, in the NPDGwBT each player chooses a closed-loop strategy simultaneously with his/her own indicator function that is defined on the whole game duration and takes values from the set {nor, alt, agg, par}. The indicator function shows the behavioral type dynamics of a given player. The strong and weak BT-solutions of the NPDGwBT were defined in [3]. Expectedly, in some cases the abnormal types of behavior (the ones differing from the normal type) may yield better outcomes for players in the NPDGwBT than in the NPDG. The main emphasis of the examples in [3] was stressed on the use of altruistic behavior by all players. This paper considers two examples of games with simple 2D motion dynamics and phase constraints in which each player can demonstrate altruism and also aggression towards a partner for certain time intervals, including the case of mutual aggression. In the first example, strong BT-solutions are constructed in which both players increase their payoffs in comparison with the game with the normal type of behavior. If the players are prohibited to choose the aggressive type of behavior, then the game has no BT-solutions. In the second example, strong BT-solutions are also constructed but, under the prohibition of aggression, they still exist and are induced by the altruistic type of behavior.

Anthropometric differences per playing position in Chilean professional footballers

Introduction: given the characteristics of soccer, the tendency in the selection of these athletes and their allocation in certain game positions has been conditioned by the morphological characteristics that they present, so that the knowledge of anthropometric particularities by playing position is a relevant antecedent for the technical bodies of professional football. The objective of the present investigation is to identify and compare the anthropometric characteristics by playing position of Chilean professional soccer players. Methods: a total of 390 professional male soccer players, from 15 Chilean professional clubs, were evaluated morphologically by means of the measurement of 25 anthropometric variables, with which the body composition, the somatotype and the Z-score of Phantom were estimated. A Chi-square test with significance level of p < 0.05 was used to compare the variables between groups using SPSS software others players (greater adipose and muscular mass). As for a sample of active subjects, soccer players have a significantly different body composition, similarly occurs when compared to Phantom, where height, weight and muscle mass are greater and fat mass is lower. Conclusion: professional footballers differ by game position compared to other non-sports players and Phantom.

Multi-Step Game of Reserves Management in the Attack-Defense Model

The authors describe a multi-step generalization of the “attack-defense” model, defined and studied by Germeier. It is a modification of the Gross’ model. The similar model was proposed by Gorelik for the gasoline production. In the military models the points are usually interpreted as directions and characterize the spatial distribution of defense resources across the width of the defense front. The dynamics of the average number of parties described by the “attack-defense” game can be described by finite-difference Osipov-Lanchester’ equations. Therefore, it would also be interesting to obtain a generalization of Germeyer’s classical model to the dynamic case when the “attack-defense” game is played many times. On this basis, in the present work, a dynamic expansion of the model is constructed in the form of a positional game with opposing interests of the distribution of parties’ reserves with complete information. The authors studied the simplest multi-step extension of the attack-defense model, which consists in the fact that the corresponding game is played repeatedly. Multi-step game with the complete information of the parties’ reserves management was built on this basis. It is assumed that the defense party makes the first move at each step and the attack party became aware about this move. The functional equation for the best guaranteed result of the defense, which is the value of the positional game due to the parties’ adopted sequence of moves was written out. Its analytical solution for a two-step game was obtained and it was shown that it is advantageous for an attack party to enter all reserves simultaneously, as in the classic attack-defense game.

Análise comparativa da potência aeróbia máxima em jogadoras de futebol de acordo com as posições do sistema de jogo

Introdução: O consumo máximo de oxigênio é uma variável que apresenta associação com a performance esportiva dos atletas. Seus valores podem ser diferentes de acordo com a posição tática. Objetivo: O objetivo do presente estudo foi comparar a potência aeróbia máxima de jogadoras de futebol de acordo com suas diferentes posições do sistema de jogo. Materiais e métodos: Participaram do estudo 18 atletas de futebol do sexo feminino (três zagueiras, quatro laterais, cinco meio-campistas e seis atacantes), com idade média de 19,1 (1,4) anos. Trata-se de um estudo de corte transversal. Foram realizadas as avaliações antropométricas (massa corporal e estatura) e da potência aeróbia máxima (Yo-yo test). Resultados: De acordo com a comparação realizada, não houve diferença estatisticamente significante sobre a potência aeróbia máxima das atletas. Conclusão: Não foram identificadas diferenças significativas quanto ao consumo máximo de oxigênio entre as jogadoras de futebol profissional, independentemente das posições de jogo.

On the optimality of the uniform random strategy

Biased Maker‐Breaker games, introduced by Chvátal and Erdős, are central to the field of positional games and have deep connections to the theory of random structures. The main questions are to determine the smallest bias needed by Breaker to ensure that Maker ends up with an independent set in a given hypergraph. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies certain “container‐type” regularity conditions. This will enable us to answer the main question for hypergraph generalizations of the H‐building games studied by Bednarska and Łuczak as well as a generalization of the van der Waerden games introduced by Beck. We find it remarkable that a purely game‐theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, while the analogous questions about sparse random discrete structures are usually quite challenging.

Cognitive Evoked Potentials (P300): Is the Decision To Press a Button Always Conscious?

Evoked cognitive potentials are a promising experimental technique that can be useful in diagnosis of various cognitive disorders, especially connected to the various memory impairments. Of particular interest to researchers is the study of the cognitive evoked potentials’ latency when recognition of various stimuli isinvolved. To study the physiological nature and functional role of evoked cognitive potentials observed during the reaction to stimuli, and to determine the degree of conscious control involved in this reaction, the experimental data were statistically analyzed to determine the numerical correlation between the working memory capacity and the cognitive potential’s latency. The cognitive EPs were measured for 166 healthy subjects and 63 others that had some pathological brain condition (discirculatory encephalopathy and other brain discirculatory disorders), varied by the age and gender. Measurements were done with the Neurosoft Neuro-MEP 4 computerized encephalography system, which implemented the standard method of large latency neural response measurement for P300 waves. Measurements were done in the setup where the subject had to recognize the significant stimuli and react to them by pressing a button. The significant stimulus was represented by a 2 kHz tone, while the insignificant stimuli were the frequently repeated 1 kHz tones.The brainwaves were registered in Cz-M1 and Cz-M2 outputs. This method allowed detection and latency measurements of the main response components N2, P3a P3ab and N3. Working memory capacity was determined by the number of words from a 10–15 word sets correctly reproduced shortly after exposures, as outlined by Luria. A statistically significant nonlinear correlation was observed between the memory capacity and the EP’s latency, which was fitted by a modified hyperbolic function. Additionally, greater latencies were observed for subjects suffering from the memory disorders, and much shorter latencies were in general noted for the male subjects as opposed to the female ones. One more interesting observation was that in manycases the absolute reaction time was less than the cognitive potential peak taken as a conscious decision, suggesting that it is in many cases an automatic reaction. Cognitive potentials measurements, being an instrumental method, allow a clinical psychologist much greater flexibility in his diagnostic repertory, as it can be used even in the cases where word tests cannot. Also, the reaction times shorter than the latency of conscious reaction to stimuli adds weight to the simultaneous codes theory of automatic reactions.&#x0D; Keywords: working memory capacity (WMC), regression line, nonlinear correlation, test less WMC estimates, cognitive processes, working memory, cognitive evoked potentials simultaneous codes, positional game, conscious cognitive comparison, P300

Evaluation of Wheelchair Basketball Skill Performance of Wheelchair Basketball Players in Different Game Positions

In wheelchair basketball the evaluation of the quality of disabled players’ movement, motion and action variety, sensomotoric, intelectual, social interaction skills, coordinative and physical abilities besides the natural tests (Brasile, 1986), quantitative and qualitative play results (technical actions carried out by a player during a game and during a minute of a game) are considered to be objective and informative. The aim of the research was to state and evaluate the playing skills of wheelchair basketball players in different game positions (a playmaker, a forward, a center) in official competitions. The play of 32 baketball players was under research. Aiming to analyse playing results of wheelchair basketball players in different game positions 20 games were video recorded by camera ”Panasonic NV-GS27“. The following actions of the players (playmakers, forwards, centers) with different functions in the team were recorded: playing time, passing, dribbling, shooting and its efficiency, rebounding and other important technical actions characterizing the activity of the players during the game. The integral preparation, ability to play, the playing quality were evaluated according to the methods of D. Byrnes and B. Hendrick (1994), giving high or low marks for each action. Results showed that centers were the most active ones, carrying out on average 2.5 actions per minute (forwards — 2.1, playmakers — 1.2), shooting 16 times per game (forwards — 7.7, playmakers — 5). The most universal players were also the centers, their passing and dribbling actions making up 25%, shooting actions — 20% of all the movements. The most accurate main technical actions were carried out by the centers: their shooting accuracy was 36% (that of wing players and playmakers being 30%). Centers’ integral qualification and playing skills (according to the methods used) were the highest rated + 42 points. Quantitative and qualitative playing results and the results of integral preparation of wheelchair basketball players in center position were significantly better than those of the playmakers and forwards.Keywords: wheelchair basketball, quantitative and qualitative playing results, integral qualification, player’s game positions.

Dynamic Games Methods in Synthesis of Safe Ship Control Algorithms

The paper presents application of dynamic games methods, multistage positional and multistep matrix games, to automate the process control of moving objects, on the example of safe control of own ship in collision situations when passing many ships encountered. Taking into consideration two types of ships cooperation, for each of the two types of games, positional and matrix, four control algorithms for determining a safe ship trajectory supporting the navigator’s maneuvering decision in a collision situation are presented. The considerations are illustrated by examples of computer simulation in Matlab/Simulink software of safe trajectories of a ship in a real situation at sea. Taking into account the smallest final deviation game trajectory from the reference trajectory of movement, in good visibility at sea, the best is trajectory for cooperative matrix game, but in restricted visibility at sea, the best is trajectory for cooperative positional game.

The pairing strategies of the 9-in-a-row game

One of the most useful strategies for proving Breaker’s win in Maker-Breaker Positional Games is to find a pairing strategy. In some cases there are no pairing strategies at all, in some cases there are unique or almost unique strategies. For the k -in-a-row game, the case k = 9 is the smallest (sharp) for which there exists a Breaker winning pairing (paving) strategy. One pairing strategy for this game was given by Hales and Jewett. In this paper we show that there are other winning pairings for the 9 -in-a-row game, all have a very symmetric torus structure. While describing these symmetries we prove that there are only a finite number of non-isomorphic pairings for the game (around 200 thousand), which can be also listed up by a computer program. In addition, we prove that there are no “irregular”, non-symmetric pairings. At the end of the paper we also show a pairing strategy for a variant of the 3 -dimensional k -in-a-row game.

Differential Games on Minmax of the Positional Quality Index

The paper goes back to the research of N. N. Krasovskii devoted to two-person zero-sum positional differential games on minmax of non-terminal quality indices, which evaluate a set of system’s states realized at given times. The first part of the paper gives a survey of the results concerning existence of the value and saddle point in such differential games. A special attention is paid to the case when the quality index has a certain positional structure. The second part of the paper overviews a method for constructing the value and optimal strategies in the case when the dynamical system is linear in the state vector, and the quality index has the appropriate convexity properties. The method is based on the recurrent procedure of constructing the upper convex hulls of certain auxiliary functions. To illustrate that this method can be numerically realized on modern computers, a model example is considered.

Backward induction in presence of cycles

For the classical backward induction algorithm, the input is an arbitrary $n$-person positional game with perfect information modeled by a finite acyclic directed graph (digraph) and the output is a profile $(x_1, \ldots, x_n)$ of pure positional strategies that form some special subgame perfect Nash equilibrium. We extend this algorithm to work with digraphs that may have directed cycles. Each digraph admits a unique partition into strongly connected components, which will be treated as the outcomes of the game. Such a game will be called a {\em deterministic graphical multistage}(DGMS) game. If we identify the outcomes corresponding to all strongly connected components, except terminal positions, we obtain the so-called {\em deterministic graphical}(DG) games, which are frequent in the literature. The outcomes of a DG game are all terminal positions and one special outcome $c$ that is assigned to all infinite plays. We modify the backward induction procedure to adapt it for the DGMS games. However, by doing so, we lose two important properties: the modified algorithm always outputs a {\em Nash equilibrium} (NE) only when $n = 2$ and, even in this case, this NE may be not {\em subgame perfect}. (Yet, in the zero-sum case it is.) The lack of these two properties is not a fault of the algorithm, just (subgame perfect) Nash equilibria in pure positional strategies may fail to exist in the considered game. {\bf Keywords:} deterministic graphical (multistage) game, game in normal and in positional form, saddle point, Nash equilibrium, Nash-solvability, game form, positional structure, directed graph, digraph, directed cycle, acyclic digraph.

Prevalence and severity of all head impacts among semi-professional soccer players over one full season, according to their game position

Soccer players are frequently subject to head injuries, from impacts with another player's head or body part and impacts with the ground. Moreover, game-play in soccer involving intentional and repeated head impacts through heading the ball. These impacts can sometimes cause concussion and soccer players might have a risk of developing neurodegenerative disorders on the long term by cumulative effect. However, the prevalence of head impacts when playing soccer is relatively unknown, because until now this was evaluated most of the time by only self-assessment questionnaires. The objective of this study was to quantify players’ exposure to all various head impacts during one soccer season. We have followed a group of 51 semi-professional soccer players, aged from 16- to 35-years old, during one full season. With video analysis of the games, the number of head impacts per player, including heading and concussion, has been counted. The average of heading per hour during a game was from 0 to 9 depending on the player position, with a median of 4.61 for the forwards, 5.84 for the centre backs, 2.85 for the fullbacks, and 2.48 for the midfielders ( P = 0.0001.) We recorded three concussions due to a direct head to head trauma, including two on the same player. The number of head trauma recorded during practices was lower than during games. We have evaluated the exact prevalence of heading and all other head impacts during a full soccer season, and their direct consequences during the games (stoppage time, medical stoppage, exit game). Forwards and centre backs are mostly exposed. Next step should focus on improving the evaluation of the consequences of these repeated head impacts on brain structure and function, in order to consider possible prevention and protection tools for soccer players.

Weak Invariance of a Cylindrical Set with Smooth Boundary with Respect to a Control System

We consider the problem of constructing resolving sets for a differential game or an optimal control problem based on information on the dynamics of the system, control resources, and boundary conditions. The construction of largest possible sets with such properties (the maximal stable bridge in a differential game or the controllability set in a control problem) is a nontrivial problem due to their complicated geometry; in particular, the boundaries may be nonconvex and nonsmooth. In practical engineering tasks, which permit some tolerance and deviations, it is often admissible to construct a resolving set that is not maximal. The constructed set may possess certain characteristics that would make the formation of control actions easier. For example, the set may have convex sections or a smooth boundary. In this context, we study the property of stability (weak invariance) for one class of sets in the space of positions of a differential game. Using the notion of stability defect of a set introduced by V.N. Ushakov, we derive a criterion of weak invariance with respect to a conflict control dynamic system for cylindrical sets. In a particular case of a linear control system, we obtain easily verified sufficient conditions of weak invariance for cylindrical sets with ellipsoidal sections. The proof of the conditions is based on constructions and facts of subdifferential calculation. An illustrating example is given.

Perfil antropométrico e composição corporal de jovens futebolistas de nível regional

O estudo objetivou avaliar o perfil antropométrico e a composição corporal de jovens futebolistas de nível regional. A amostra foi composta por 55 jogadores das categorias sub 13, sub 15 e sub 17 classificados também por posição de jogo. Para comparação foram adotados os testes ANOVA One-Way e post-hoc de Bonferroni. A categoria sub 17 apresentou maiores valores (p&lt;0,05) nas variáveis relacionadas ao crescimento e desenvolvimento físico. O percentual de gordura corporal permaneceu estável entre as categorias e posições. A homogeneidade verificada sugere que a manutenção destes índices é resultante dos similares níveis de esforço exigidos na modalidade.

Futsal World Cup: Differences Created by Winning, Losing and Drawing Variables in Scored Goals and Offensive Variations

Futsal, which can be defined as a new sports branch compared to football, has some shortcomings in comparison with football as well according to the studies carried out. Studies have rather focused on the movement characteristics of futsal players during the game and the physiological effects of games. On the other hand, there are only a limited number of studies conducted on game analysis. Therefore, as the primary goal of the present study, offensive variations that ended up as goals in all matches played in Futsal World Cup were analyzed. As the secondary goal, these parameters were studied in terms of the teams’ winning-drawing-losing states during games. In the present study, a total of 52 matches of the teams playing in 2016 FIFA Futsal World Championship were analyzed. The analyses of the games were performed using the hand notation method. Differences between offensive variations (Diagonal, parallel, central and transition game forms, set games etc.) and the goals scored (shots taken, shots at goal, the area where the goals were scored, the way the goals were scored, the area where the foot hit the ball, block shoots etc.) were determined with One-way Anova and Kruskal-Wallis tests on SPSS version 20 in terms of 3 variables (win-loss-draw). The findings show that the winning teams had significantly more shots than the losing teams (10.80, 95%CI (from 3, 73 to 17, 87) and that the number of shots at goal was also higher (5.82, 95%CI (from 2, 43 to 9, 91) in terms of the teams’ winning-losing-drawing states. However, no statistically significant difference was found between the areas where the shots were taken and the points they ended up at the goal. As for position specific games, on the other hand, statistically significant differences were found in terms of winning, losing and drawing states during the game (central= χ2(2) = 10.500, p = .005, diagonal= χ2(2) = 17.142, p = .000, parallel= χ2(2) = 10.931, p = .004, transition= χ2(2) = 28,587 p = .000). No statistically significant difference was found between set games ending up with goals and the teams’ game score variables. The results obtained concerning offensive variations in a superior futsal championship reveal that positional games of dynamic style could be much more effective compared to set games of static starting positions in winning the game. Moreover, producing shooting positions and their accuracy levels seem to be an important factor affecting success. According to these results, it can be recommended that the trainings to be held should be designed to improve positional games that these should be planned as practices that result in shots.

Long Cycles in Locally Expanding Graphs, with Applications

We provide sufficient conditions for the existence of long cycles in locally expanding graphs, and present applications of our conditions and techniques to Ramsey theory, random graphs and positional games.

Numerical Construction of Stackelberg Solutions in a Linear Positional Differential Game Based on the Method of Polyhedra

We consider the problem of constructing approximate Stackelberg solutions in a linear non-zero-sum positional differential game of two players with terminal payoffs and player controls chosen on convex polyhedra. A formalization of player strategies and motions generated by them is based on the formalization and results of the theory of zero-sum positional differential games developed by N.N. Krasovskii and his scientific school. The problem of finding a Stackelberg solution reduces to solving nonstandard optimal control problems. We propose an approach based on operations with convex polyhedra.