Mathematician’s Advice: Aim High, But Don’t Shoot for the Moon

There’s a popular saying: if you aim for the moon and miss, you’ll land among the stars. However, mathematicians assert that targeting the stars directly might yield better results.

In the pursuit of aspirations—whether securing a new job, finding a partner, or achieving political ambitions—individuals often strike a balance between ambition and realism.

Yet, the intricate dynamics of this balance remain underexplored, with many studies highlighting instances where individuals cease their pursuit too hastily due to insufficient ambition, as highlighted by Thomas Hills from the University of Warwick, UK.

Through a novel mathematical model, Matt Burgess and his colleagues at the University of Wyoming discovered that optimal outcomes arise from aiming slightly above average, but not to the extremes. “We validated that optimal ambition lies strictly above average and within reasonable limits: aim high, but not unreasonably so,” Burgess stated.

Initially, the team developed a statistical framework illustrating how individuals assess different potential outcomes based on their willingness to compromise based on varying ambitions. This led them to create a formula outlining overall compensation based on satisfaction criteria.

The researchers tested this model by simulating random potential outcomes, adjusting parameters like the number of choices available over time, the ratio of positive to negative outcomes, and the time and effort expended in making decisions.

After conducting thousands of simulations and aligning the findings with real-world datasets (e.g., college applications and U.S. election polls), Burgess’s team found that achieving optimal results typically requires aiming for compensation that exceeds the average but is modestly close to the maximum.

While this aligns with prevailing wisdom, Burgess noted an interesting twist: scenarios biased towards extreme outcomes—be it exceptionally good or bad—shifted the approach required for the best results.

In most cases, outcomes tend to cluster around the average; however, when faced with a rare but severe downturn, like a major recession, a cautious approach becomes intuitive. Yet, Burgess and his team contended that a bolder approach is advisable even in challenging scenarios. “In these instances, aiming slightly higher than average is beneficial,” Burgess explained, emphasizing the risk of having a poor outcome skew your overall average.

Conversely, in situations where the potential for exceptionally positive results exists—like a startup achieving a billion-dollar valuation—aiming slightly higher than average is advisable. “Initially, I found this counterintuitive; my first thought was that my colleagues may have miscalculated,” Burgess remarked.

Hills, who did not participate in this study, noted that individuals might perceive the balance between risk and reward differently. “Some may prefer a stable income, while others chase the ‘optimal’ but riskier paths,” he mentioned. “Problematic environments, like winner-takes-all scenarios, may warrant a more ambitious approach based on social comparisons.”