Averaging-type PBSL

A specific form of averaging-type PBSL was first proposed McElreath et al. [120], which is why we call it PBSL McElreath for short. This strategy works differently than scoring-type PBSL. It takes the average generated by options A and B as observed in the sample and then chooses the option with the higher performance. A greater sample size might seem favorable to this approach, as a small sample could be more subject to randomly assigning the higher average payoff to the worse option. In the original work, only sample size 3 was tested, so we were left wondering whether higher sample sizes increase performance. To check this, we simulated PBSL McElreath for sample sizes ranging from 2 to 7. The resulting performance using our default parameters is shown in figure 2.10. In contrast to the expectation, higher sample sizes do not lead to higher performance. Instead, the best performance by a fair margin is reached for a sample size of 3, the sample sizes studied in the original work. Here, as was the case for scoring-type PBSL with weights [1/0], the principle seems to be: less is more.

Figure 2.11:Mean performance of PBSL McElreath as a function of the mean success rate of the environment (left panel) or as a function of the step sizek_incr (right panel). Shown is the excess performance over chance level (50%) depending on sample size, each stacked on top of one another. Numbers in the bars indicate the best performance achieved for each sample size, bars with thick borders indicate the best performance achieved for each mean parameter value. Left: A sample size of 2 leads to a rather stable but low performance (≈60−61%) over meanp. For a sample size of 3, very good performance (≈74−75.5%) is attained for medium to high meanp (0.45 to 0.65). The reverse trend in performance is found for higher sample sizes; for instance, the best overall performance (77.38%) is found for sample size 6 and meanpof 0.25. Right: Sample size 3 is favored for lowkincr, while sample sizes of 4 and higher are favored for higherkincr.

2.3.2.2 Mean success rate of the environment

It is crucial to study how strategies perform depending on the mean success rate provided by the environment. The left panel of figure 2.11 shows the performance of PBSL McElreath as a function of sample size and meanp_A and p_B. To emphasize the differences, we only show performance above the chance level of 50%. For each mean p, we stack the performance for all sample sizes on top of each other. For low or high mean p, higher sample sizes provide the best performance. For example, for a mean p of 0.25, sample sizes of 6 yield the best results; and for a mean p of 0.75, a sample size of 4 yields the best performance. A sample size of 2 generally leads to low but stable performance. A sample size of 3 leads to good performance in the p range of 0.45 to 0.65 but for low mean p also leads to the lowest observed performance, hardly exceeding chance level.

2.3.2.3 Reversion factor

We simulated the influence of the the reversion factorron performance. For PBSL McElreath, we found that performance decreases with increasing r, regardless of the sample size. For all r except for r = 0, sample size of 3 delivered the best performance; forr = 0, performance peaked at a sample

size of 4. The advantage over a sample size of 3 was, however, only 2.4 percentage points forr = 0 and was on average 5.7 percentage points lower for all other r. It is thus safe to state that a sample size of 3 results in the highest performance and that this is robust with regard to the parameterr.

2.3.2.4 Speed of environmental change

We varied p_incr from 0.1 to 1 in steps of 0.1 and found no large differences to the previous results. Sample size of 3 remains the best choice by far over the whole parameter range. Generally, when changes become slower (lower p_incr), performance is stable or increases. The only important change is that for environments that change very slowly (pincr ≤ 0.5), PBSL McElreath with sample size of 2 see a performance boost and becomes the second best choice. PBSL McElreath with sample size 3 is the only strategy to outper-form scoring-type PBSL that ignores failures (i.e. with weights [1/0] and equivalent); slightly for high p_incr and more so for lowp_incr.

2.3.2.5 Step size

We varied the step size k_incr, which reflects the absolute change inp_A and p_B each time they change. The results are shown in the right panel of figure 2.11. A sample size of 3 leads to the best performance as long as kincr is small (k_incr ≤ 0.03). For higher values, 0.04 ≤ k_incr ≤0.08, a sample size of 4 is best. For even higher values, 0.09 ≤ k_incr ≤ 0.1, a sample size of 5 is best. It thus seems that the higher kincr, the higher the optimal sample size.

2.3.2.6 Three choice options

We tested the performance of PBSL McElreath for three instead of two choice options. The results are shown in figure 2.12. Overall, PBSL McElreath, regardless of its sample size, improves its performance when there are three options. Strategies with sample size 4 profit most, showing an increase of 6.8 percentage points. However, sample size 3 still remains the best choice, delivering the best performance for both two and three choice options.

Of the environmental parameters, we found ∆p, which determines the mean value of p_A, p_B, and p_C, to be the most important. Therefore, we re-simulated performance for different values of ∆p, this time with three choice options. The results were almost identical to those found for two choice options. Interesting to note, over the whole parameter range of ∆p, performance improved for each studied sample size when there were three instead of two choice options; with only 3.3 percentage points on average, these improvements were rather small, though, and they never exceeded 9 percentage points.

0.3 0.4 0.5 0.6 0.7

Figure 2.13: Mean performance of PBSL with a payoff-conformism trade-off as a function of the mean success rate of the environment and sample size. Shown is the excess performance over chance level (50%), with sample sizes stacked on top of one another. Numbers in the bars indicate the best performance for each sample size, bars with thick borders indicate the best performance for each mean p. For low to moderately high meanpA and pB, a sample size of 6 yields the best result, for high meanpA andpB, a sample size of 5 is better, and for very high mean pA

and pB, a sample size of 4 is best. Sample size of 3, which was dominant without the trade-off, is never the best.

performance. The result is shown in figure 2.13. Previously, we found that PBSL McElreath with a sample size of 3 produced the best performance, except for low or very high mean p_A and p_B. After including the payoff-conformism trade-off, a sample size of 3 never leads to the best performance.

Instead, a sample size of 6 is the best choice for mean values of pA and pB

between 0.25 and 0.55. For higher values, smaller sample sizes are better. In summary, introducing the trade-off increases performance considerably and shifts the best sample size upwards.

We furthermore analyzed the modified social learning strategy under the condition that there are three choice options. The results were very similar to those found for two choice options. Increasing the number of options does therefore not affect performance in an important way.