## Introduction

## Studying intuition with chess

## Complexity in chess

## Aim of the present study

## Method

### Participants

^{1}the participants were assigned to one of four levels of expertise: candidate masters (1999 < Elo < 2200), masters (2199 < Elo < 2400), international players (2399 < Elo < 2600) and World-class players (Elo > 2599) players. Eight players had more than 2700 Elo, including one player with a rating above 2800 Elo. At the time of data collection, only 39 players worldwide had 2700 Elo or more. The four groups had significantly different levels of skill, F(3,59) = 250.86, p < 0.001: candidate masters (M = 2106.43 Elo, SD = 66.67 Elo, n = 14), masters (M = 2314.13 Elo, SD = 56.27 Elo, n = 15), international players (M = 2485.71 Elo, SD = 64.78 Elo, n = 17), and World-class players (M = 2690.94 Elo, SD = 58.82 Elo, n = 17). The level of expertise was independent of age, which did not vary between groups, F(3,59) = 0.01, p = 0.99. The sample size (N = 63) was deemed appropriate because effects sizes tend to be strong and results replicable in research on chess players’ expertise (e.g., Gobet et al., 2004). Ethics approval was granted by Brunel University, UK.

### Task and material

Chess symbol | Meaning |
---|---|

+– | White has a decisive advantage |

± | White has a clear advantage |

\(\underline{\underline{ + }}\) | White has a small advantage |

= | The position is equal |

\(\overline{\overline{ + }}\) | Black has a small advantage |

\(\mp\) | Black has a clear advantage |

−+ | Black has a decisive advantage |

### Procedure

### Preprocessing of response times and evaluation errors

### Statistical analyses

^{®}.

## Results

### Analysis of response times

Complexity | Balance | Skill | |||
---|---|---|---|---|---|

Candidate | Masters | International | World class | ||

Simple | −+ | 2517.15 (195.98) | 2504.6 (148.34) | 2507.67 (231.16) | 2218.14 (216.45) |

−/+ | 2672.97 (244.94) | 2828.28 (227.79) | 2617.65 (221.33) | 2653.68 (312.8) | |

=+ | 2710.81 (206.35) | 2540.24 (174.02) | 2764.62 (252.90) | 2508.41 (174.81) | |

= | 2532.77 (297.27) | 2526.61 (180.02) | 2444.84 (219.26) | 2445.52 (229.15) | |

+= | 2631.66 (198.95) | 2845.74 (189.22) | 2652.66 (169.88) | 2899.61 (315.81) | |

± | 2527.49 (205.71) | 2466.84 (168.94) | 2701.37 (251.69) | 2267.22 (241.89) | |

+− | 2679.1 (252.61) | 2787.68 (202.26) | 2672.85 (211.80) | 2300.24 (215.98) | |

Complex | −+ | 2472.38 (202.61) | 2677.86 (164.38) | 2820.36 (216.17) | 2580.37 (298.21) |

−/+ | 2769.18 (202.15) | 2993.75 (217.12) | 2718.23 (216.55) | 2886.22 (272.51) | |

=+ | 2856.89 (246.34) | 2660.89 (198.56) | 2682.24 (189.15) | 2654.79 (295.84) | |

= | 2693.56 (245.93) | 2722.24 (161.7) | 2362.98 (140.38) | 2478.56 (252.97) | |

+= | 2587.91 (259.60) | 2526.01 (115.67) | 2442.46 (215.06) | 2456.19 (256.27) | |

± | 2695.44 (245.76) | 2383.12 (154.62) | 2512.73 (206.99) | 1993.15 (150.59) | |

+− | 2703.62 (232.49) | 2502.03 (162.39) | 2242.72 (115.02) | 2539.44 (231.06) |

Factor | Degrees of freedom | F | MSE | η ^{2} | |
---|---|---|---|---|---|

Numerator | Denominator | ||||

Balance | 6 | 354 | 5.763** | 0.007 | 0.089 |

Complexity | 1 | 59 | 0.501 | 0.009 | 0.008 |

Balance × complexity | 6 | 354 | 3.425* | 0.006 | 0.055 |

Skill | 3 | 59 | 0.713 | 0.185 | 0.035 |

Balance × skill | 18 | 354 | 1.324 | 0.007 | 0.063 |

Complexity × skill | 3 | 59 | 0.447 | 0.009 | 0.022 |

balance × complexity × skill | 18 | 354 | 0.893 | 0.006 | 0.043 |

### Analysis of evaluation errors

Factor | Degrees of freedom | F | MSE | η ^{2} | |
---|---|---|---|---|---|

Numerator | Denominator | ||||

Balance | 1 | 59 | 43.138** | 2.190 | 0.422 |

Complexity | 1 | 59 | 34.271** | 0.245 | 0.367 |

Balance × complexity | 1 | 59 | 14.283** | 1.737 | 0.195 |

Skill | 3 | 59 | 14.558** | 0.489 | 0.425 |

Balance × skill | 18 | 354 | 1.247 | 2.190 | 0.060 |

Complexity × skill | 3 | 59 | 3.256* | 0.245 | 0.142 |

Balance × complexity × skill | 3 | 59 | 2.100 | 1.737 | 0.096 |

^{2}= 0.14; balance × skill, F(3,59) = 1.25, p = 0.30, MSE = 2.19, lower bound corrected; and complexity × balance, F(6,354) = 14.28, p < 0.01, MSE = 0.29, η

^{2}= 0.19 with a linear trend, F(1,59) = 65.90, p < 0.001. MSE = 0.27.

^{2}= 0.10.

### Predicting evaluation error with Elo rating

Model summary | Parameter estimates | ||||||
---|---|---|---|---|---|---|---|

Bin | R square | F | df1 | df2 | Sign | Constant | Coefficient |

0–1 s | 0.151 | 4.631 | 1 | 26 | 0.041 | 5.760 | − 0.002 |

1–2 s | 0.189 | 14.210 | 1 | 61 | 0.001 | 2.901 | − 0.001 |

2–3 s | 0.171 | 12.358 | 1 | 60 | 0.001 | 2.968 | − 0.001 |

3–4 s | 0.153 | 10.148 | 1 | 56 | 0.002 | 3.945 | − 0.001 |

4–5 s | 0.005 | 0.242 | 1 | 46 | 0.625 | 2.100 | 0.000 |