Optimal ambition in business, politics and life

TL;DR

The paper addresses how ambitious decision-makers should be under uncertainty in a finite horizon. It develops a simple yet general AR(1) search model with a satisfaction threshold $T$ and proves that the optimal threshold is finite, strictly above the mean $bc$, and increases with longer search horizons, landscape ruggedness, and left-skewness while decreasing with search costs and adverse social comparison. The results elucidate when being moderately ambitious outperforms aiming for the mean or the moon, reveal a nuanced distinction between ambition and risk-taking, and connect theory to real-world contexts such as entrepreneurship, policy, and campaigns. The work offers testable hypotheses, qualitative guidance for calibrating ambition across domains, and a platform for future extensions that incorporate smarter search and richer reward landscapes.

Abstract

In business, politics and life, folk wisdom encourages people to aim for above-average results, but to not let the perfect be the enemy of the good. Here, we mathematically formalize and extend this folk wisdom. We model a time-limited search for strategies having uncertain rewards. At each time step, the searcher either is satisfied with their current reward or continues searching. We prove that the optimal satisfaction threshold is both finite and strictly larger than the mean of available rewards -- matching the folk wisdom. This result is robust to search costs, unless they are high enough to prohibit all search. We show that being too ambitious has a higher expected cost than being too cautious. We show that the optimal satisfaction threshold increases if the search time is longer, or if the reward distribution is rugged (i.e., has low autocorrelation) or left-skewed. The skewness result reveals counterintuitive contrasts between optimal ambition and optimal risk taking. We show that using upward social comparison to assess the reward landscape substantially harms expected performance. We show how these insights can be applied qualitatively to real-world settings, using examples from entrepreneurship, economic policy, political campaigns, online dating and college admissions. We discuss implications of several possible extensions of our model, including intelligent search, reward landscape uncertainty and risk aversion.

Figures (11)

• Figure 1: Reward landscapes. Stylized representations of reward landscape ruggedness (left) and skewness (right). Ruggedness determines how different successive rewards can be from each other. Skewness determines the relative abundance of low and high rewards, among landscapes that share the same mean and variance.
• Figure 2: Expected rewards vs. satisfaction thresholds, and empirical examples. Panel a: analytical and numerical calculations of the expected reward distribution, as a function of the target threshold, on a maximally rugged landscape. The expected reward distribution is unimodal and negatively skewed, with an optimal threshold above zero (the landscape mean). The analytical expression given by equation \ref{['eq:expected-reward']} (black, dashed) matches the simulation results (red). Parameter values are: $\epsilon_t \sim \mathcal{N}(0,1)$, $\varphi=0$, and $t_{\text{max}}=1000$. Results shown are averaged over $10^4$ simulations. We note that many real-world applications occur on shorter time scales ($t_{\text{max}}\ll1000$), and consequently have lower optimal thresholds (e.g., optimal $T\approx 1.6$ with $t_{\text{max}}=100$). Panels b and c: examples of real-world search strategies. Panel b: When online dating, heterosexual men (blue) and women (red) are most likely to message potential partners who are slightly more desirable than they are (averaged over the four cities shown in fig. 2 of ref. bruch2018aspirational). Panel c: When applying for college, members of the 2008 U.S. high-school graduating class concentrated their applications on schools with median SAT scores similar to, or slightly below, their own score (data from ref. hoxby2012missing). This suggests students were either being sub-optimally ambitious, or they faced other constraints (e.g., on income or geography), as ref. hoxby2012missing suggests was the case.
• Figure 3: Effects of smoothness and skewness. Smoother reward landscapes yield lower cumulative rewards than the maximally rugged, non-skewed landscapes (a). Left-skewed reward landscapes yield higher cumulative rewards (b). Smooth landscapes are generated analytically. For the skewed landscape simulations, $\epsilon_t \sim \mathcal{SN}(0, 1,\alpha)$, $\varphi=0$, $t_{\text{max}}=1000$, where the parameter $\alpha$ is varied to result in skew values from $-0.8$ to $0.8$, each averaged over $10^4$ simulations.
• Figure 4: Social comparison is costly. Mean rewards for cohort comparisons and upward social comparisons are shown on a rough, non-skewed landscape. Cohort comparison (yellow curve) lowers the optimal satisfaction threshold and cumulative reward. Upward social comparison (purple curve) further lowers the optimal threshold and substantially hinders performance. Both the expected reward and the optimal satisfaction threshold are lower compared to the reference landscape (black, dashed curve), which assumes no social comparison. Parameter values are: $\epsilon_t \sim \mathcal{N}(0,1)$, $\varphi=0$, and $t_{\text{max}}=1000$. Cohorts include $100$ agents, with thresholds uniformly sampled from $[-3,6]$. Results shown are averaged over $10^4$ simulations.
• Figure 5: Effects of search costs. Expected rewards for search with a per-timestep search cost on a rough, non-skewed landscape. The cost values are: (a) $c=2$, (b) $c=10$, (c) $c=395$, (d) $c=800$. Introducing a search cost (colored curves) lowers the optimal satisfaction threshold and can produce negative expected rewards when either the cost or the threshold are too high. In all cases, both the expected reward and the optimal satisfaction threshold are lower than those of the reference landscape (black dashed curve). The sums are computed numerically based on equation \ref{['eq:expected-reward-cost']}. Parameter values are: $\epsilon_t \sim \mathcal{N}(0,1)$, $\varphi=0$, and $t_{\text{max}}=1000$.