Delusions of Grandeur and Their Benefits (and Hazards)

Abstract: We study a population-wide tournament in which agents, who care both about their absolute and relative wealth, experiment by searching over correlated objects. We explore the role of the agents' beliefs about the environment; namely, the stochastic processes corresponding to their experimentation. We find that although optimism leads to higher output, it also produces greater inequality. We connect these observations with empirical evidence suggesting a positive relationship between inequality and entrepreneurship.

• The paper develops a novel equilibrium model that integrates overconfidence and social status in the decision-making process of entrepreneurial search.
• It finds that higher optimism drives prolonged search behavior, yielding greater innovation outcomes alongside rising output inequality.
• Empirical evidence confirms that belief distortions, rather than risk tolerance, critically shape entrepreneurial persistence and economic disparity.

Delusions of Grandeur in Entrepreneurial Experimentation: Benefits and Hazards

Model Overview

This paper develops an equilibrium model of entrepreneurial experimentation under delusional optimism. Agents face a costly sequential search problem: each runs an independent driftless Brownian motion representing the stochastic value of their projects, with incremental exploration costly at a constant rate. Agents may stop search at any time to implement the best value observed to date. Crucially, agents care both about their absolute gain (the quality of the best innovation found) and their rank relative to peers. Thus, the model integrates private returns and social status in entrepreneurial incentives.

The novel mechanism centers on belief distortion. Although all agents are ex-ante identical and face the same underlying process, each agent holds an optimistic, misspecified model—believing their own search process has a strictly positive drift, despite the true process being driftless. There is 'common knowledge' that every agent is overconfident, yet each believes only their confidence is justified. This misspecification parametrized by $\mu$ (the perceived drift) systematically induces overconfidence-driven persistence.

The agents’ problem is then a dynamic optimal stopping problem with both idiosyncratic and population-level externalities, with subjective expectations running under $\mathbb{P}^\mu$ even though realized outcomes are governed by $\mathbb{P}^0$.

Equilibrium Characterization

The central technical contribution is a full equilibrium characterization leveraging the structure of the experimenters' stopping rules. Equilibrium is formulated in terms of drawdown boundaries: an agent continues to search until her current value falls a specified gap below her all-time best. The paper derives, via a free boundary ODE, that the equilibrium stopping threshold depends both on the agent’s optimism parameter and on the shape of the social ranking function.

By a change of variables to hazard-time quantiles, the equilibrium reduces to a one-dimensional nonlinear ODE for the tolerated drawdown as a function of population rank. This ODE admits a unique bounded solution $\delta_\mu(t)$ characterized by terminal value at the single-agent limit. The equilibrium quantile function $m_\mu(t)$ is then recovered as $m_\mu'(t)=\delta_\mu(t)$, providing a full mapping from hazard 'time' (which corresponds to the exponential rank distribution in the population) to realized outcomes.

This reduction enables explicit analysis of how belief distortion shifts the entire equilibrium distribution.

Comparative Statics

A core set of results establishes how increases in overconfidence (i.e., higher $\mu$) alter aggregate and distributional outcomes, evaluated under the true law.

• Output and Discovery: Optimistic agents search longer and tolerate larger setbacks before stopping, resulting in strictly higher realized maxima in distribution (FOSD improvement). This is a pure behavioral effect: persistence, not an actual improvement in the search technology, boosts aggregate discoveries.
• Inequality: The gains from extra search are convex; improvements are greatest for those at the right tail of the distribution. Thus, higher optimism increases the variance of realized outcomes. The model yields the strong claim that overconfidence simultaneously raises average output and increases output inequality.
• Search Duration: The expected real-time experimentation increases monotonically with optimism, with a closed-form solution for average search time as a mixture over squared drawdown tolerances.
• Distributional Tail Effects: The mapping from optimism to absolute discoveries is strictly increasing in quantile rank, leading to right-tail stretching of the output distribution without parallel shifts.

Empirical Analysis

The theoretical predictions are confronted with multi-country data. Using cross-sectional country-level measures of entrepreneurship (startup rates, self-employment shares) and inequality (Gini, 90/10 ratio), the empirical analysis shows:

• Positive Correlation: Economies with higher measured entrepreneurship exhibit systematically higher income inequality, controlling for GDP per capita and population.
• Risk Aversion Exclusion: Alternative metrics of risk appetite (willingness to take risks, uncertainty avoidance, equity investment shares, lottery participation) do not explain entrepreneurship rates or attenuate the entrepreneurship-inequality relationship.
• Belief-Driven Entrepreneurship: Survey-based measures of individual self-belief regarding entrepreneurial ability are strongly correlated with entrepreneurship at the country level, dominating risk preference variables.

These findings are consistent exclusively with the model’s mechanism: overconfidence, not risk tolerance per se, is the primary driver of entrepreneurial persistence and associated distributional effects.

Theoretical and Practical Implications

The model provides a rigorous micro-foundation for belief-driven experimentation and the resulting trade-off between aggregate technological advance and economic inequality. Unlike classical models where risk preference shapes entry, this framework implies that the behavioral channel operates through changed stopping thresholds, not through who chooses to enter search.

Practically, this suggests that policies or environments fostering overconfident expectations could catalyze more innovation and higher social output at the cost of increased dispersion in success. The inequality is not attributed to ex-ante ability heterogeneity, credit constraints, or institutional factors, but emerges purely from belief-driven experimentation in a symmetric population. Welfare consequences depend critically on the social valuation of output versus equality, as the gains accrue disproportionately to rare winners.

Theoretically, this work bridges literatures on optimal stopping, search and innovation tournaments, and the psychology of entrepreneurial behavior. It provides a tractable, equilibrium-based tool for further analysis of belief heterogeneity and dynamic experimentation in economic agent populations.

Future Research Directions

Several avenues present themselves for extension:

• Generalization of Search Process: Introducing heterogeneity in search technologies, drift, or initial conditions could interact with belief distortions in nontrivial ways.
• Dynamics of Belief Updating: Allowing for Bayesian or adaptive correction of overconfident beliefs based on observed failures and successes.
• Institutional Interventions: Modeling the impact of policy instruments on belief formation or search costs to analyze trade-offs between innovation incentives and equality.
• Multistage or Recursive Feedback: Incorporating market entry and exit, wealth feedback loops, or model uncertainty at the population level could further detail the macroeconomic implications.

Conclusion

The paper provides a formal theory of how overconfidence among experimenting agents governs both the average level and the dispersion of entrepreneurial outcomes. Optimistic belief distortions act as self-imposed subsidies for continued search, fueling higher discovery but also amplifying inequality. Empirical evidence supports the mechanism by directly linking self-belief measures to entrepreneurship and, in turn, to inequality, independent of risk tolerance controls. The equilibrium approach and analytical tractability set the stage for significant further advancement in the modeling of innovation, belief-driven behavior, and economic disparity.

Reference: "Delusions of Grandeur and Their Benefits (and Hazards)" (2606.02306)