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This note briefly summarizes the consequences of adding correlated individual differences to the best baseline model in the Games competition, I-SAW. I find evidence that the traits of an individual are correlated, but refining I-SAW to capture these correlations does not significantly improve the model's accuracy when predicting average behavior.

Are individual differences correlated and can modeling them as such increase the accuracy of a model's predictions? Correlations between individual differences was one of the features of the model I entered in the

One of the regularities observed in the results of the CPC estimation experiment and previous studies is that individuals differ in their behavior [

In the standard version of I-SAW, it is assumed that the values an individual's trait parameters take are independent. This study is motivated by the conjecture that traits are correlated. For example, one might expect that exploration (ε_{i}), the tendency to choose at random, is negatively correlated with inertia (π_{i})_{i}) is positively correlated with inertia (π_{i}).

Data from the CPC experiments were used to investigate correlations between trait parameters. First, the individual decisions made by the 120 participants in the estimation experiment were used to calculate maximum likelihood estimates of the five trait parameters for each participant. These estimates provide evidence that traits are correlated and not independent. Second, the estimation experiment was simulated with a refined I-SAW model. Parameters were constrained to have the same uniform distributions as in the baseline I-SAW model. Selected correlations between trait parameters were introduced and estimated using a grid search. Interestingly, adding these correlations between trait parameters while holding the model and distribution of parameters constant had only a very small effect on how accurately the model predicts average behavior in the estimation and competition experiments.

Entry decisions are stochastic in the I-SAW model. In the first trial, each player enters with a fixed probability. In subsequent trials, the probability player

Each of the 120 participants in the CPC estimation experiment played 10 games, _{i}_{i}_{i}_{i}_{i}_{i}_{g}_{g}

For exploration and inertia, the exact probability of entry was calculated directly. For exploitation, to accommodate the internal stochastic component of I-SAW (drawing a small sample of previous outcomes of the current game), it was calculated as follows

The vector of parameters _{i}_{i} have a lower tendency to exhibit inertia π_{i}. They also give less weight to average payoffs over all previous trials w_{i} and more weight to a small sample of previous trials (1 – _{i}_{i}, in contrast, give more weight to average payoffs w_{i}. Finally, in sampling during exploitation trials, participants with a greater tendency to bias their sample by selecting the most recent trial ρ_{i}, give less weight to average payoffs w_{i}.

The only constraints on the parameter estimates reported in the previous section were the upper and lower bounds. To refine the I-SAW model to accommodate correlated traits, two additional constraints were imposed: traits were assumed to be uniformly distributed between the upper and lower bounds, and the correlations between traits were assumed to have a specific structure as described below.

The following procedure was used to generate _{i}_{ij}_{θ,j}_{1}, …,x_{5}) of independent standard normal values was generated. This was used to generate a vector of correlated standard normal values
_{i}_{θj}

Notice that when _{i}_{i}

The matrix

On the estimation set, using these estimates gives a score of 1.34 compared to 1.37 when there are no correlations. The respective figures for the competition set are 1.17 and 1.19. In both cases, introducing correlations leads to a slight increase in the accuracy of the model's predictions.

This study found that while correlations between traits matter for individual behavior, refining I-SAW to capture these correlations does not significantly improve the prediction of averages such as average entry rates, efficiency, and alternation rates. Hence, when the goal is prediction of average rather than individual behavior, assuming individual trait parameters are independently distributed appears to be a sound simplifying assumption. A natural question for future research is why the refinement of I-SAW did not produce better predictions. One possibility is that since participants interacted in groups, there may be group effects that carry over to the parameters obtained by fitting the model separately to each individual. Another direction for future research is testing models with fewer trait parameters. If, as this study suggests, correlations between trait parameters only have a small effect on the accuracy of the model's prediction of average behavior, it may be possible to achieve the same degree of predictive accuracy with a model that is simplified by combining some of the trait parameters.

The five trait parameters.

ε_{i} ∼ U[0,0.24] |
Probability of exploration in trials after the first one. |

π_{i} ∼ U[0,0.6] |
Tendency for inertia. |

μ_{i} = {1, 2, or 3} |
Number of samples taken in exploitation trials. |

ρ_{i} ∼ U[0,0.2] |
Probability a sample draw is biased. If the draw is biased, the most recent trial is selected. If it is unbiased, a previous trial is selected at random. |

w_{i} ∼ U[0,0.8] |
In exploitation trials, the sample mean is given weight (1 – w) and the mean of all previous trials weight w. |

Correlation coefficients and summary statistics for estimated trait parameters.

_{i} |
_{i} |
_{i} |
_{i} |
_{i} | |
---|---|---|---|---|---|

ε_{i} |
1.00 | ||||

π_{i} |
–0.38 |
1.00 | |||

μ_{i} |
–0.03 | 0.14 | 1.00 | ||

ρ_{i} |
–0.22 |
–0.14 | –0.14 | 1.00 | |

w_{i} |
–0.31 |
0.24 |
0.05 | –0.27 |
1.00 |

mean | 0.12 | 0.31 | 1.87 | 0.12 | 0.29 |

variance | 0.01 | 0.04 | 0.70 | 0.01 | 0.05 |

max | 0.24 | 0.60 | 3.00 | 0.20 | 0.80 |

min | 0.00 | 0.00 | 1.00 | 0.00 | 0.00 |

:p < .05,

:p < 0.01,

:p < 0.001

The sum of independent normally distributed random variables has the following property: if
^{2}) and ^{2}^{2}). Let _{1},_{n}_{i}a_{i}x_{i}