Statistical analysis of scores in Glines - a possible reflection of success and failure in life activities.#
Zbigniew Koziol, email@example.com
#First published in 2000. This version contains updated links.
There is an almost unexplored broad research field near the borders of sociology, psychology, physics, mathematics, where human behaviour could be studied in order to describe certain it's features in a quantitative way. As an example of that kind of studies let us remind an early publication on the Internet in 2000, by this author (1). The problem discussed there concerned statistical description of members activities on e-mail discussions lists. It has been shown than that a probability distribution function describing frequency of occuring of new messages (postings) to mailing lists resembles well Fermi-Dirac distribution of electron holes in a semiconductor. Or, within the accuracy of the available data, that distribution could be approximated well by a simpler function of the type P(s) = sa/(1+sa), as defined in the Abstract.
In this article, another aspect of human behaviour is researched and it is shown that it is very well described by similar statistical functions. This time, the success of playing a computer board game Glines is analyzed.
Glines (2) is the Gnome (open-source desktop for Linux operating system) port of the once popular Windows game called Color Lines (Fig. 1). The game's objective is to align as often as possible five balls or more of the same color causing them to disappear. The game ends when there is no space left on the board for new balls. More detailed information about playing Glines, and it's rules, can be found at https://help.gnome.org/users/glines/.
For the purpose of this article, it is sufficient to know that playing successfuly this game requires some concentration of the player and advance predicting of possible development of the game. However, playing Glines involves also a factor of chance, since the position of new objects on the board is random. Hence, from one side some thinking is required in order to achieve high score in the game, some basic at least experience with the game and developing certain rules of playing succesfuly. But in any case the factor of randomness may prevent a good player from reaching a high score. In practice, it looks like that a concentration and slow, wise decisions of the player help significantly to achieve high score. Hence, a success is largely determined by the condition of mind of the player, his or her patience, and concentration. In this meaning, the success and failure in this game reflect somehow success and failure of human beings in real life, where similarely both wise decisions of a person and chance play a significant role.
The game has been played more than 2000 times, over a few months period of time. A small change in it's source code before compiling it did allow to store all results in a log file. Every efford has been made in order to achieve accurate data: once the game has been started, it has always been played till the end, with an attempt to achive the highest possible score, regardless of the situation on the board. Otherwise, the results would be spoiled by arbitrary data.
Fig. 2 shows a simple presentation of the results: on a diagram with double-logarithmic axes numbers of scores from within certain range, is represented by vertical bars. For instance, the most left vertical bar shows that there was one only result observed falling into the range between 10 and 20 points earned.
Results on Fig. 2 do not help us much in finding out a good mathematical description of them. More useful is drawing the total number of results observed for scores, i.e. showing a probability distribution function. This is done in Fig. 3.
Blue points represent observed number of scores, normalized by the total number of results (see the note 3). The shape of the curve obtained clearly resembles for instance a Fermi-Dirac (F-D) probability distribution function. Since however there is no reason visible for using F-D function in this case, a simpler one is tryed:
where s=Score/k, and k is a certain normalization number while a is an exponent. The red curve on Fig. 3 has been drawn with the following parameters: k=398, a=3.6. The same parameters are used on all remaining figures, showing that these values and this function does indeed describe experimental results very well.
To the best of my knowledge, this is the first attempt to find a mathematical formula for probability distribution function describing scores in a computer game achieved by a player. It is likely however that the results have much in common with a broad range of human activities and as such these first studies might begin a development of a new descipline devoted to success and failure of humans in various life circumstances, a subject mathematicaly not explored, as far as the author is aware. This work and earlier studies (1) provide a likely evidence of the existence of a broad unexplored field for research.
There is a more practical aspect of this study, more directely exploitable. Glines is a one of plenty of games that could be played on computers for enjoyement. However, computer users more and more often start to spend money on games where real profits could be earned, like for instance by participating in online-gambling. The author believes that playing some of online casino games could bring the player more likely rewards once he or she understands better their chances. These chances could certainly be modeled by some mathematical functions and after that evaluated calmely before real money is thrown into a real game.
The k and a values of parameters used for fitting the data in Eq. 1 are certainly to some extend player-specific.
Acknowledgement:The author would like to thank to Shooby Ban, one of the authors of Glines, for discussions and help in preparing the source code of the game for logging of all results of the game. An acknowledgement deserves also the idea of using the type of function described in the text for fitting the results. That suggestion has been made first by Dr Mikolaj Sawicki of John A. Logan College in Illinois.