A study of memory effects in a chess database

A series of recent works studying a database of chronologically sorted chess games --containing 1.4 million games played by humans between 1998 and 2007-- have shown that the popularity distribution of chess game-lines follows a Zipf's law, and that time series inferred from the sequences of those game-lines exhibit long-range memory effects. The presence of Zipf's law together with long-range memory effects was observed in several systems, however, the simultaneous emergence of these two phenomena were always studied separately up to now. In this work, by making use of a variant of the Yule-Simon preferential growth model, introduced by Cattuto et al., we provide an explanation for the simultaneous emergence of Zipf's law and long-range correlations memory effects in a chess database. We find that Cattuto's Model (CM) is able to reproduce both, Zipf's law and the long-range correlations, including size-dependent scaling of the Hurst exponent for the corresponding time series. CM allows an explanation for the simultaneous emergence of these two phenomena via a preferential growth dynamics, including a memory kernel, in the popularity distribution of chess game-lines. This mechanism results in an aging process in the chess game-line choice as the database grows. Moreover, we find burstiness in the activity of subsets of the most active players, although the aggregated activity of the pool of players displays inter-event times without burstiness. We show that CM is not able to produce time series with bursty behavior providing evidence that burstiness is not required for the explanation of the long-range correlation effects in the chess database.Comment: 18 pages, 7 figure

Memory and long-range correlations in chess games

In this paper we report the existence of long-range memory in the opening moves of a chronologically ordered set of chess games using an extensive chess database. We used two mapping rules to build discrete time series and analyzed them using two methods for detecting long-range correlations; rescaled range analysis and detrented fluctuation analysis. We found that long-range memory is related to the level of the players. When the database is filtered according to player levels we found differences in the persistence of the different subsets. For high level players, correlations are stronger at long time scales; whereas in intermediate and low level players they reach the maximum value at shorter time scales. This can be interpreted as a signature of the different strategies used by players with different levels of expertise. These results are robust against the assignation rules and the method employed in the analysis of the time series.Comment: 12 pages, 5 figures. Published in Physica

Innovation and Nested Preferential Growth in Chess Playing Behavior

Complexity develops via the incorporation of innovative properties. Chess is one of the most complex strategy games, where expert contenders exercise decision making by imitating old games or introducing innovations. In this work, we study innovation in chess by analyzing how different move sequences are played at the population level. It is found that the probability of exploring a new or innovative move decreases as a power law with the frequency of the preceding move sequence. Chess players also exploit already known move sequences according to their frequencies, following a preferential growth mechanism. Furthermore, innovation in chess exhibits Heaps' law suggesting similarities with the process of vocabulary growth. We propose a robust generative mechanism based on nested Yule-Simon preferential growth processes that reproduces the empirical observations. These results, supporting the self-similar nature of innovations in chess are important in the context of decision making in a competitive scenario, and extend the scope of relevant findings recently discovered regarding the emergence of Zipf's law in chess.Comment: 8 pages, 4 figures, accepted for publication in Europhysics Letters (EPL

Memory Kernel in the Expertise of Chess Players

In this work we investigate a mechanism for the emergence of long-range time correlations observed in a chronologically ordered database of chess games. We analyze a modified Yule-Simon preferential growth process proposed by Cattuto et al., which includes memory effects by means of a probabilistic kernel. According to the Hurst exponent of different constructed time series from the record of games, artificially generated databases from the model exhibit similar long-range correlations. In addition, the inter-event time frequency distribution is well reproduced by the model for realistic parameter values. In particular, we find the inter-event time distribution properties to be correlated with the expertise of the chess players through the memory kernel extension. Our work provides new information about the strategies implemented by players with different levels of expertise, showing an interesting example of how popularities and long-range correlations build together during a collective learning process

Memory and long-range correlations in chess games

In this paper we report the existence of long-range memory in the opening moves of a chronologically ordered set of chess games using an extensive chess database. We used two mapping rules to build discrete time series and analyzed them using two methods for detecting long-range correlations; rescaled range analysis and detrended fluctuation analysis. We found that long-range memory is related to the level of the players. When the database is filtered according to player levels we found differences in the persistence of the different subsets. For high level players, correlations are stronger at long time scales; whereas in intermediate and low level players they reach the maximum value at shorter time scales. This can be interpreted as a signature of the different strategies used by players with different levels of expertise. These results are robust against the assignation rules and the method employed in the analysis of the time series.submittedVersionFil: Schaigorodsky, Ana L. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Billoni, Orlando V. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Billoni, Orlando V. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Física Enrique Gaviola; Argentina.Fil: Billoni, Orlando V. Universidad Nacional de Córdoba. Instituto de Física Enrique Gaviola; Argentina.Fil: Perotti, Juan I. Aalto University. Department of Biomedical Engineering and Computational Science, Aalto University; Finland.Otras Ciencias Física

Structure constrained by metadata in networks of chess players

Abstract Chess is an emblematic sport that stands out because of its age, popularity and complexity. It has served to study human behavior from the perspective of a wide number of disciplines, from cognitive skills such as memory and learning, to aspects like innovation and decision-making. Given that an extensive documentation of chess games played throughout history is available, it is possible to perform detailed and statistically significant studies about this sport. Here we use one of the most extensive chess databases in the world to construct two networks of chess players. One of the networks includes games that were played over-the-board and the other contains games played on the Internet. We study the main topological characteristics of the networks, such as degree distribution and correlations, transitivity and community structure. We complement the structural analysis by incorporating players’ level of play as node metadata. Although both networks are topologically different, we show that in both cases players gather in communities according to their expertise and that an emergent rich-club structure, composed by the top-rated players, is also present

A Study of Memory Effects in a Chess Database - Fig 1

<p>(a) Chess tree corresponding to the main opening-lines up to depth <i>d</i> = 4. The size of the nodes is proportional to their popularity. Here only the main lines are shown. (b) Distribution of popularities of the nodes at depth <i>d</i> = 1, 2, 3 and 4; these distributions are well fitted by power laws <i>P</i>(<i>k</i>) ∝ <i>k</i>^−<i>α</i> with <i>α</i> = 1.10 ± 0.05, 1.29 ± 0.03, 1.47 ± 0.02 and 1.59 ± 0.02 (<i>R</i>² = 0.972, 0.993, 0.996 and 0.997), respectively. Errors estimated by the fitting.</p

(a) Burstiness parameter <i>B</i> and (b) burstiness exponent <i>β</i> as a function of the length of the time series for the database growing by Player Aggregation (black triangles) and by chronological game aggregation (gray diamonds).

<p>The error bars are not shown because they are of the same size as the points.</p

Cumulative distribution of inter-event times of the most popular opening-line measured in the database (black triangles) and generated with CM for <i>p</i> = 0.005 and <i>τ</i>_<i>c</i> = 96 (green diamonds).

<p>The lines are fits according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0168213#pone.0168213.e020" target="_blank">Eq (7)</a> with <i>β</i> = 0.927 ± 0.003 (<i>R</i>² = 0.999) for the database (black dashed line) and for CM with <i>β</i> = 1.035 ± 0.003 (<i>R</i>² = 0.999) (green full line). Inset: cumulative frequency distribution of inter-event times measured in the database (black triangles), YSM for <i>p</i> = 0.1 (magenta circles) and <i>p</i> = 0.005 (cyan squares).</p

Hurst exponent obtained by the DFA method as a function of the length of the time series in the database (dotted black line with triangles), and generated with CM, <i>p</i> = 0.005 and <i>τ</i>_<i>c</i> = 96 (full green line with diamonds): (a) using the PAR; (b) using the GAR; and (c) using the UAR.

<p>Hurst exponent obtained by the DFA method as a function of the length of the time series in the database (dotted black line with triangles), and generated with CM, <i>p</i> = 0.005 and <i>τ</i>_<i>c</i> = 96 (full green line with diamonds): (a) using the PAR; (b) using the GAR; and (c) using the UAR.</p