Fig. 2.

Sharp transitions in optimal number of sampled alternatives at low capacity, and power law behavior at large capacity. (A) Average reward (points, simulations; lines, theoretical expressions; Eq. 4) as a function of the number of sampled alternatives $M$ for three different capacities ($C=4,10,100$; light, intermediate, and dark lines, respectively) for the flat environment (uniform prior). The maximum occurs at the large extreme for low capacity but at a relatively low value for large capacity. Note log horizontal scale. (B) Optimal number of sampled alternatives as a function of capacity. When capacity is smaller than around 5, a linear trend of unit slope is observed (dashed green line), but when capacity is above 7, a sublinear behavior is observed (dashed red line corresponds to the best power law fit, with exponent close to 1/2). The black line corresponds to analytical predictions. The jagged nature of this prediction and simulation lines in this and other panels is due to the discrete values that the optimal $M$ can only take, not due to numerical undersampling. (C) The sharp transition is clearer when plotting the optimal number of sampled alternatives to capacity ratio as a function of capacity. For low capacity, the ratio is 1, but for large capacity the ratio decreases very rapidly. The last point below which the optimal ratio is always 1 (critical capacity) corresponds to capacity equal to 5 (indicated with a vertical red line). (D) Number of sampled alternatives to capacity ratios for different prior distributions ($\alpha =\beta =3,$ bell-shaped, green line; $\alpha =3,\beta =1$, negatively skewed prior modeling a “rich” environment, brown line; $\alpha =1,\beta =3$, positively skewed distribution modeling looking for a “needle in a haystack,” that is, a “poor” environment, blue line). Lines correspond to analytical predictions from SI Appendix, Eq. 9; points correspond to numerical simulations; error bars are smaller than data points in all panels.