## The Cave of Surrender

Remember the Dunning-Kruger (DK) effect? It is a cognitive bias first identified in a 1999 study published by social psychologists David Dunning and Justin Kruger. Simply put, the DK effect is the non-monotonic relationship between an individual's competence/knowledge and her/his self-confidence/self-assessment regarding a particular task. At the earliest stages of performing a task, an individual has little knowledge but very high confidence. Since knowledge is limited, success rate is low. The failures lead the individual to reassess his/her confidence, resulting in a decrease in confidence. However, at the later stages, the individual continues to accumulate knowledge even if confidence is at relatively low levels. Increasing knowledge in turn leads to higher success rates, and this causes confidence to increase again.

Clearly, there is a missing dimension here; intuition suggests that a person's effort must be endogenous to his/her confidence and knowledge. Put differently, confidence and knowledge interact with effort. So, the question is how we can understand such an interaction. An interesting but still very simple view is summarized below:

This assumes that the individual's effort at time t is (positively) associated with how confident she/he is and how much she/he knows at time t. The outcome is stochastic; with varying degrees of success or failure. The crude assumption here is that failures decrease confidence and successes increase it. But effort always increases knowledge.

Imagine now a stochastic dynamical system that represents these ideas in concrete mathematical form. With simple functional forms and under any arbitrary parameterization, we can run such a dynamical system for a large number of individuals. Here is what we get for parameter values and initial conditions that imply multiple equilibria:

For 1,000 individuals, all starting with high confidence and little knowledge, a run for 300 periods allows us to discover the Cave of Surrender. Some people survive the Valley of Despair when confidence is low, eventually gaining confidence and converging to the high confidence-high knowledge equilibrium. Others are facing more failures, on average, and they eventually enter the Cave of Surrender. They (optimally) decrease their effort given the confidence erosion faced in earlier periods. But decreasing effort increases the probability of failure. Eventually, knowledge depreciation dominates knowledge expansion due to effort. The surrenders, of course, converge to the no confidence-no knowledge equilibrium.

So, don't give up, people!

Not just yet!