# Search on an NK Landscape with Swarm Intelligence: Limitations and Future Research Opportunities
## 3 Swarm Intelligence Algorithms
### Mathematical Development of Swarm
We conjecture that the sub-group arguments made in the literature also make no substantial differences. Instead, one must include other considerations, such as the differential costs of various choices, to alter the results. We summarize the results and discuss the limitations and opportunities for further development in the following two sections. (A) Pure Exploration

Discussion of Simulation Results and Identification of Limitations of the NK Landscape for Swarm-Based Search

### Discussion of Simulation Results

We run multiple simulations comparing exploration and exploitation search strategies on smooth and rugged landscapes and with a small and large number of firms. We also examine performance when firms are restricted to one-mutant search strategies. We label following the global best as \"exploitation\" and randomly exploring on its own as \"exploration\". Exploration tends to outperform exploitation if gauged by the probability of a focal firm reaching the highest peak or the number of firms reaching the highest peak. If we take a snapshot at some intermediate point in the simulation (say \(t=10\)), then exploitation typically creates the highest value for firms on average, as firms quickly converge to the best-known location (in terms of payoff), which is unlikely to be the global maximum but may be higher than a random location on the hypercube. Overall, the results may not be that surprising. Early in the simulation run, firms gain more information

Figure 6.5 \(N=15\) and \(K=14\). Each simulated firm is represented by a different color. by copying other firms (maximizing over N other locations) as compared to exploring randomly, so exploitation performs better but will likely result in the firm finding a local rather than global optima. While we can technically discuss which search strategy performs best under a given set of criteria, it is unclear if such an assessment has practical meaning to the study of firm search due to the limitations the NK landscape places on our swarm model, specifically the use of only one information input from firms rather than a more sophisticated set of inputs (as given in Equations (6) and (7)). The one information input we used was the global best position from the entire firm population. While we could choose some other formulation (a different reference set, etc. ), the main limitation is that we can only use one factor rather than multiple factors. The swarm model that we present in Equations (6) and (7) provides a rich way to capture a variety of influences from the environment. These equations can be modified in a variety of ways to capture realistic behaviors. Using the NK landscape, the way swarm can model influences from other firms is too simplistic. ### Further Discussion of NK Limitations for Swarm-Based Search Studies

Prior literature has made extensive use of the NK landscape to represent a combination of interdependent decisions. By modeling each decision as a binary choice that can have interdependencies with other (binary) choices, modelers using NK have produced numerous insights about firms' or individuals' search behaviors, which have greatly benefited the strategy and organization literature. While the simplicity of the NK landscape--the hypercube of binary coordinates--is arguably one of its strengths, it also creates several limitations. First, the NK landscape seems more ideally suited for examining the performance feedback-driven search of one agent rather than how multiple agents influence each other's search through multiple variables, which is the main application and advantage of swarm in the search literature. NK forces the agent to move along the (0/1) nodes of the hypercube rather than in a real-dimensional space. Therefore, models that take multiple variables (like in Equation (7)) and inform the agent's search through a weighted average (like in Equation (6)) cannot be fully utilized and instead collapse to one variable input. The ability to draw nuanced inferences about competitive interactions (e.g., imitation, differentiation, etc.)