麻豆传媒

Dreams and careers rest on the decision. Do you go all out in the semi-final or rest your stars so they are fresh for the final?

of Pukyong National University in Busan, South Korea, and his colleagues have applied game theory to this 鈥渟emi-finalists鈥� dilemma鈥�, and the results could apply in the boardroom as much as in basketball or football.

They constructed a simple mathematical model in which winning depends on stamina, which is a finite quantity that each team must choose how to distribute over its final two matches of a tournament.

In a winner-takes-all format, the best strategy is to copy what everyone else does, Baek鈥檚 team says. If your opponents rest their stars, you ought to as well. Otherwise, you risk being more tired than the opponent in the final. This sort of scenario in which no player has an incentive to go off-strategy is what game theorists call a Nash equilibrium.

Chasing bronze

But it鈥檚 different when there鈥檚 the chance of redemption in a third-place play off. You need to introduce an element of chance to catch your opponents off guard, says Baek鈥檚 team. Either go all out in the semis, or save everything for the next match. A team may be crushed in the semi-final, but saving energy increases the odds of taking home a prize: the bronze.

The work could help explain why animals sometimes seem to cooperate in nature, even when competing for a mate, says co-author of Sejong University in Seoul, South Korea. A seemingly cooperative pair of males may not be altruistic 鈥� they may have simply converged on saving their stamina for future rounds with other opponents.

The results could also guide companies in setting incentive schemes. 鈥淭he pay-off system is effectively sending a signal to people on how to behave,鈥� says Baek. For instance, a company wanting to cultivate specialists on its management team may wish to avoid paying the CEO far more than others, because that might push people to develop the same skill set to compete for the top job.

Group theory

鈥淵ou could see that career structure as a bunch of subsequent stages where some proceed to the next stage and other people don鈥檛,鈥� says , an evolutionary game theorist at the University of Amsterdam, the Netherlands. He says the work could also be extended into exploring the game theory of an entire World Cup tournament, group stage and all: 鈥淭his shows the principle.鈥�

He says the Dutch side has reached the World Cup semis five times without solving the dilemma.

Jeong says his nation鈥檚 Olympic squad is one example of how a different pay-off structure might alter a team鈥檚 strategy. South Korean male athletes are exempted from their two years of mandatory military service if they bring back an Olympic medal, giving them an incentive to ensure they get a bronze.

It might be coincidence, but in the football at the 2012 London Olympics, South Korea suffered a 3鈥�0 thrashing from Brazil in the semi-final, but went on to overcome arch-rivals Japan 2鈥�0 and snatch bronze.

Journal reference: Physical Review E; journals.aps.org/pre/accepted/7807dR6fFd419718908d1c99d99366454f9e6b9a5