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Shoot for the moon, and even if you miss, you’ll land among the stars, so the adage goes. But shooting straight for the stars might actually be the more effective option, according to mathematicians.

In life, people tend to strike a balance between being ambitious yet not overly so when it comes to pursuing their objectives, such as landing a better job, finding an appropriate partner or achieving political goals.

However, quantifying this balance has not been studied in detail, and much research has focused on when people stop looking too soon and aren’t ambitious enough, says at the University of Warwick, UK.

Now, using mathematical models, at the University of Wyoming and his colleagues have found that the best outcomes for uncertain scenarios typically come from aiming high, but not unrealistically high. “You can prove that the optimal ambition is strictly above average and strictly finite, meaning above average but you don’t shoot for the moon,” says Burgess.

Burgess and his team first came up with a statistical model for how a person might weigh up different outcomes, varying their willingness to settle for more or less ambitious result. From this, they derived a formula for the overall reward someone might receive according to their satisfaction threshold.

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Then they tested this model with random potential outcomes and varied how they might appear, such as how many outcomes a person has to choose between in a set chunk of time, how many bad outcomes compared to good outcomes there were, or how much time and effort it took to choose a particular outcome.

After running thousands of simulations and comparing the results to real-world datasets, such as university applications and US election polls, Burgess and his team found that the optimal outcomes indeed came when people aimed above  the average reward, but not near the maximum.

This wasn’t surprising given the common wisdom that people tend to follow, says Burgess, but the team was surprised to find that this picture changes when scenarios are biased towards one very bad or good outcome.

Typically, if most outcomes are mediocre but one is extremely bad, such as a recession once every 10 years, the common wisdom is to be cautious. But Burgess and his team found that the best approach is actually to be more ambitious than you would be if the rewards were more even. “We find, compared to the average, you want to be a little bit more ambitious [in these scenarios], because you don’t want to be thrown off by these bad years dragging the average down.”

Similarly, when one outcome is extremely good, such as a startup making a billion dollars or nothing, you should be a little less ambitious than average. “It’s actually initially so counterintuitive that when my colleagues showed me the result I thought that they had made a mistake,” says Burgess.

Hills points out that people might have different ideas on how they balance risk and reward. “Some people may prefer to have a stable income rather than an ‘optimal’ but potentially riskier income, for example,” says Hills. “Moreover, some environments are winner-take-all environments, where social comparisons are more important, and in those cases risk-seeking ambition may be more appropriate.”