Minimum effort game explained

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.
• 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^[2]

If there are

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

There are

(K(K -1 ))/(2) non pure Nash equilibria, given as follows: each player chooses two effort levels k < l and puts in k units of effort with probability \left ( (c)/(b) \right ) ^{(1)/(n -1)} and l units of effort with probability 1- \left ( (c)/(b) \right ) ^{(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 . Efficient Coordination in Weakest-Link Games . The Review of Economic Studies . 83 . 2 . April 2016 . 0034-6527 . 10.1093/restud/rdv040 . 737–767.
2. Cartwright . Edward . The Optimal Strategy in the Minimum Effort Game . Games . . 9 . 3 . 9 June 2018 . 2073-4336 . 10.3390/g9030042 . free . 42.
3. Feri . Francesco . Gantner . Anita . Moffatt . Peter G. . Erharter . Dominik . Leading to efficient coordination: Individual traits, beliefs and choices in the minimum effort game . Games and Economic Behavior . 136 . 13 October 2022 . 10.1016/j.geb.2022.10.003 . 403–427.