3.2.1
Overview
We consider agents searching a space of possible N-digit solutions by modifying dig- its in their current solution in order to improve their performance. Examples include trying to identify an organizational configuration that maximizes performance or modi- fying components of existing technologies to improve them. These problems share the characteristic of being path dependent and interdependent in nature; getting to a solution usually involves modifying previous solutions and the success of changing some feature depends on the other features within the structure. The extent to which these character- istics dominate the problem space can be straightforwardly captured by tunably rugged fitness landscapes (see Environment). Agents by default engage in social learning (ex- ploitation) and switch to innovation (exploration) if the former does not prove successful (see Agents and Strategies). Thus a strategy that often proves useful will engage in more exploitation and less exploration. We measured the performance of different strategies by the average payoff achieved relative to other strategies and to the highest obtainable performance level in the environment.
3.2.2
Environment
The environment is characterised by a tunably rugged NK landscape, where N denotes the number of components of the system and K represents the number of interdependen-
cies between the components. A value of K= 0 produces single peak environment, while
3.2. Model 37 solutions, making local exploration ineffective. Intermediate values of K produce land- scapes with both local and global peaks, with some correlation between nearby solutions. Each solution in the environment is an N-length vector composed of binary strings: that is,
each element of the vector can take on two values, 0 and 1, leading to a total of 2N possible
solutions in the problem space. Each solution has a payoff that is calculated as the average of the payoff contributions of each element and the other elements with which they are interdependent. The payoff contribution of each element is a random number drawn from
a uniform distribution between 0 and 1. In the case of K= 0, a simple average of the N
elements is taken: 1/N ∗∑Ni=1Ni, whereas with K> 1, individual payoff contributions are
determined by values of the other K− 1 elements, that is , f (Ni| Ni,Ni+1,...,NK), where
f() is the payoff function and the total payoff is 1/N ∗ ∑Ni=1f(Ni | Ni,Ni+1,...,NK). In
other words, when K= 0, changing any single element of the solution will affect only
the contribution of that element, whereas when K > 0, changing a single element will
change the payoff contribution of the K− 1 other elements. As mentioned above, when
K= 0, exploration of solutions through the modification of single components can prove
effective, but as K increases, local exploration becomes less and less effective (Levinthal,
1997). In all results reported we let N= 15 and K = 0 or K = 7 to create two different
environments that we call simple and complex, respectively. Figure 3.1 displays a graphi- cal illustration of the environments studied. Panel A shows the simple environment where only one unique peak exists and it is possible to reach this peak by gradually modifying digits in one’s solution. In contrast, Panel B shows a multi-peaked environment, which mean that agents can get stuck in a local peaks and be unable to reach higher payoffs via local search.
Figure 3.1: Illustration of the two environments studied. A: Simple environment with a single peak. B: Complex environment with multiple peaks. In the simple environment solutions one-digit apart from each other have very similar payoffs, therefore, modifying single digits in a solution will eventually lead to the global peak. In the complex environ- ment payoffs of nearby solutions can be very different, therefore, search by single digit modification can lead to local peaks from which it is impossible to improve and, as a result, to find the global peak.
Our choices for values of N and K are representative of the literature, and sensitivity analyses revealed that changing these values would not affect our results. The simple
environment (N= 15, K = 0) would correspond to the environment studied in the previous
chapter since it is characterized by a single peak and all other solutions can be ranked by their payoffs.
Following several authors, we normalized the payoffs of different solutions by di-
viding them by the maximum obtainable payoff on a landscape PNorm= Pi/max(P) (Lazer
& Friedman, 2007; Siggelkow & Rivkin, 2005). The distribution of normalized payoffs tends to follow a normal distribution with decreasing variance as K increases. This implies that most solutions tend to cluster around very similar payoff values. Following Lazer and
3.2. Model 39 making most solutions "mediocre" and only a few solutions "very good". Note that this assumption does not change any of the results.
Since our main focus is on the properties of different strategies, the majority of our
simulations is based on a fully connected network with N= 100 nodes. We compare the
fully connected network with a locally connected network with 100 nodes and a degree
of d = 4 (see Section 3.3.2 for network versus strategy efficiency and Figure 2.1 for
an illustration of the networks) but omit the small-world network since we saw in the previous chapter that it lies in-between the two more extreme networks and, therefore, we do not expect anything unique to happen there. This allows us to compare more and less efficient networks as in previous studies (Lazer & Friedman, 2007; Mason & Watts, 2012).
3.2.3
Agents and Strategies
We simulated N= 100 agents. On each time step agents simultaneously interacted with
the environment in order to avoid possible sequence effects (Bikhchandani, Hirshleifer, & Welch, 1992).
On the first time step agents started out with a randomly assigned solution of N digits and on each subsequent time step went through the following steps:
(1) Implement social learning by following the steps specified by the building blocks:
(i) Search rule: search randomly among the population
(ii) Stopping rule: stop searching after looking up the solutions of s2 other in-
dividuals. We focused on two sample sizes, a relatively smaller (s=3) and a
relatively larger (s=9) sample size3.
(iii) Decision rule: select the best performing agent (best member); select the
most frequent solution (conformity)4; select a random agent (random copying).
(2) Observe whether the solution identified via social learning produces a higher fitness score than the current solution. If yes, switch to the alternative solution; otherwise go to Step 3.
(3) Engage in exploration by modifying a single digit in the current solution and observe whether it produces a higher payoff than the current solution. If yes, switch to the alternative solution; otherwise keep the current solution.
This procedure is repeated for t= 200 time steps and the average payoff in the population
is recorded for different strategies and environments. Results reported are averaged across 1000 different NK environments.