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.

References

1. Riedl, Arno; Rohde, Ingrid M. T.; Strobel, Martin (April 2016). "Efficient Coordination in Weakest-Link Games" (PDF). The Review of Economic Studies. 83 (2): 737–767. doi:10.1093/restud/rdv040. ISSN   0034-6527.
2. Cartwright, Edward (9 June 2018). "The Optimal Strategy in the Minimum Effort Game" (PDF). Games. 9 (3). MDPI: 42. doi: 10.3390/g9030042 . ISSN   2073-4336.
3. Feri, Francesco; Gantner, Anita; Moffatt, Peter G.; Erharter, Dominik (13 October 2022). "Leading to efficient coordination: Individual traits, beliefs and choices in the minimum effort game". Games and Economic Behavior. 136: 403–427. doi: 10.1016/j.geb.2022.10.003 .