Transitive Avoidance Games on Boards of Odd Size

Avoidance (or misère) games are a type of positional games. Two players alternately claim points of a set $N$ (the `board' of the game). The game is determined by a family $L$ of subsets of $N$ and the following rule: The first player who claims every point of some set in $L$ loses the avoidance game. The game is called transitive if the group of all permutations of $N$ leaving $L$ invariant acts transitively on $N$. Johnson, Leader and Walters show that for a board size which is neither a prime number nor a power of two there are transitive avoidance games where the first player can force his win. For primes of size at least $17$, the corresponding problem remained open. We are going to close this gap and prove that for all primes $n$ of size at least $17$ there are also transitive avoidance games with board size $n$ where the first player can force his win.

Multi-Criteria Optimisation of Multi-Stage Positional Game of Vessels

The paper presents a mathematical model of a positional game of the safe control of a vessel in collision situations at sea, containing a description of control, state variables and state constraints as well as sets of acceptable ship strategies, as a multi-criteria optimisation task. The three possible tasks of multi-criteria optimisation were formulated in the form of non-cooperative and cooperative multi-stage positional games as well as optimal non-game controls. The multi-criteria control algorithms corresponding to these tasks were subjected to computer simulation in Matlab/Simulink software based on the example of the real navigational situation of the passing of one’s own vessel with eighteen objects encountered in the North Sea.

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.

Transitive Avoidance Games

Positional games are a well-studied class of combinatorial games. In their usual form, two players take turns to play moves in a set ('the board'), and certain subsets are designated as 'winning': the first person to occupy such a set wins the game. For these games, it is well known that (with correct play) the game cannot be a second-player win.In the avoidance (or misère) form, the first person to occupy such a set loses the game. Here it would be natural to expect that the game cannot be a first-player win, at least if the game is transitive, meaning that all points of the board look the same. Our main result is that, contrary to this expectation, there are transitive games that are first-player wins, for all board sizes which are not prime or a power of 2.Further, we show that such games can have additional properties such as stronger transitivity conditions, fast winning times, and 'small' winning sets.