Minimum effort game

In game theory, the minimum effort game or weakest link game is a game in which each person decides how much effort to put in and is rewarded based on the least amount of effort anyone puts in.^[1] It is assumed that the reward per unit of effort is greater than the cost per unit effort, otherwise there would be no reason to put in effort.

Examples

• On an island, each person tries to build barriers to protect an island from flooding. Because even a single failed barriers causes the whole island to flood, the flood protection is determined by the weakest barrier. ^[1]
• An airport ground crew must complete all their tasks before an airplane can take off. As a result, the time spent is based on the slowest member of the ground crew.

Nash equilibria

Source:^[2]

If there are ${\displaystyle n}$ players, the set of effort levels is ${\displaystyle A=\{1,...,K\}}$, it costs each player ${\displaystyle c}$ dollars to put in one unit of effort, and each player is rewarded ${\displaystyle b}$ dollars for each unit of effort the laziest person puts in, then there are ${\displaystyle K}$ pure-strategy Nash equilibria, one for each ${\displaystyle k\in A}$, with each player putting in the same amount of effort ${\displaystyle k}$, because putting more effort costs more money without extra reward, and because putting less effort reduces the reward earned.

There are ${\displaystyle {\frac {K(K-1)}{2}}}$ non pure Nash equilibria, given as follows: each player chooses two effort levels ${\displaystyle k<l}$ and puts in ${\displaystyle k}$ units of effort with probability ${\displaystyle \left({\frac {c}{b}}\right)^{\frac {1}{n-1}}}$ and ${\displaystyle l}$ units of effort with probability ${\displaystyle 1-\left({\frac {c}{b}}\right)^{\frac {1}{n-1}}}$.

In practice

The amount of effort players put in depends on the amount of effort they think other players will put in.^[3] In addition, some players will put more effort than expected in an attempt to get others to put in more effort.