# Wankelmut: A Simple Benchmark for the Evolvability of Behavioral Complexity
## 5 Quantitative Analysis
### Evolving Solutions with Non-Pre-Structured Neural Networks in Various Scenarios
#### 5.2.1 Single Environment vs. Double Environment and Mean Fitness vs. Minimum Fitness  
At first, we evaluated the performance of evolution in a fixed environment as is depicted in Figure 1. We found that evolution could quickly converge to a sort of pre-programmed trajectory (see Figure 6c,e) that achieved very high fitness (see Figure 6a,b), but was non-reactive as it did not consider sensor inputs in the wanted way. A post-hoc test of the best evolved genomes in a flipped environment, where the gradient pointed to the other side, failed for both evolved controller types, clearly indicating that no reactive Wankelmut behavior had evolved (see Figure 6d,f). The trajectories in both post-hoc runs show some difference from the runs in the environment that was used in evolution, and this indicates that some sensor input was affecting the behavior, but not in the desired way: the agents still started off in the wrong direction, and also, the oscillations ceased after some time. \begin{table}
\begin{tabular}{l c l c} \hline
**Parameter** & **ANN** & **Parameter** & **CTRNN** \\ \hline Population size & 150 & Population size & 150 \\ Number of genes & 41 & Number of genes (weights) & 121 \\  & & Number of genes (\(\theta\)) & 11 \\  & & Number of genes (\(\tau\)) & 11 \\ Init. range (weight) & \((-0.5,0.5)\) & Init. range (weight) & \((-0.5,0.5)\) \\  & & Init. range (\(\tau\)) & (0.9,5.9) \\  & & Init. range (\(\theta\)) & as in [31] \\ Mutation rate & 0.3 & Mutation rate (weights) & 0.1 \\  & & Mutation rate (\(\theta\), \(\tau\)) & randomly, one at a time \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters used for the evolutionary algorithm. CTRNN, Continuous-Time Recurrent Neural Network. As this approach did not yield the wanted reactive Wankelmut behavior, we decided to evaluate each individual genome twice: One time, the gradient pointed uphill to the left, and one time, it pointed uphill to the right side. The fitness function in all three scenarios used the _Switch_ fitness regime, which is described as fitness \(=F(1,0,0)\). We evaluated two methods of associating a fitness value with each genome from these two evaluations: first, we calculated the arithmetic mean value of both runs and, as a second variant, we took the minimum value of both runs. As a consequence, a good genome had to perform well in both environments; in the \"minimum\"-fitness function, the selection was even harsher than in the \"mean\" variant. We found that the \"minimum\" variant of selection showed a higher selection pressure on evolving the reactive control (compare Figures 7 and 8): with the minimum fitness regime, the controllers found it harder to gather high fitness values, as they had to perform well in both environments, preventing evolution from \"just considering one of the two environments\". We hoped that this would  
Figure 6: Single Environment, _Switchfitness_\(=F(1,0,0)\). Top row: fitness dynamics of the ANNs (left) and CTRNNs (right). The horizontal orange line indicates the maximum fitness achievable by a reactive Wankelmut controller, as it was achieved by the simple hand-coded controller. Bottom row: trajectory of the best performing genome of each controller type in the environment used in evolution versus a run in a flipped environment as a post-hoc evaluation (only the initial 56 time steps of 250 time steps are shown). support the evolution of reactive control. As Figure 8c-f shows, it still did not produce the desired reactive Wankelmut behavior in the long run. Based on these findings, we used the double environment evaluation with a minimum-of-both-runs fitness function for all further quantitative analysis in this study, in order to maximize the selection pressure towards the reactivity of the evolved controller. Figure 7: Evolutionary results of the double environment setting, fitness regime _Switch_: fitness \(=F(1,0,0)\). The arithmetic mean of both evaluations was used as the fitness value for a genome. Top row: fitness dynamics of the ANNs (left) and CTRNNs (right).