The minimum effort (or weakest link) game is fascinating - simple, yet capable of yielding profound insight. The basic idea is that there is a group of people who individually have to decide how much effort to put into a group task. Effort is costly for the individual but beneficial for the group. Crucially, group output is determined by the minimum effort that any one group member puts in to the task. A classic example is an airline flight: If any person involved in the flight - pilot, fuel attendant, mechanic, luggage handler etc. - gets delayed, then the flight is delayed, no matter how hard others try.
In experiments the minimum effort game is usually reduced to the matrix in Figure 1. Here subjects are asked to choose a number between 1 and 7 with the interpretation that a higher number is higher effort. Someone choosing effort 1 is guaranteed a payoff of 70. Someone who chooses 2 gets 80 if everyone else chooses 2 or more, but only gets 60 if someone in the group chooses 1. Someone who chooses 7 can get from 10 to 130 depending on the choice of others.
Suppose the minimum choice of others is 5. What should you choose? You do best to choose 5 and get a payoff of 110. What if the minimum of others is 6? You do best to choose 6 and get payoff 120. Following this logic we can see that 'everyone choose the same number' is a Nash equilibrium. Now here are the key points: Everyone choose 1 is the worst equilibrium while everyone choose 7 is the best equilibrium; but choosing 7 is very risky because others might let you down. To return to the airline example. It is no use the pilot racing around to get the flight ready to go if the fuel attendant is having an after-lunch snooze.
In experiments we typically observe that effort converges over time to 1 - the worst equilibrium. Most people start by choosing high effort. But it takes only one bad egg to ruin the team and that drags average effort down. This is a bad outcome! The airline is not going to be on time. So how to fix things?
An obvious answer seemed to be leadership. So, together with Mark van Vugt and Joris Gillet we ran some experiments on leadership. The basic idea was that one person chooses first and then others follow. We reasoned that if the leader chose 7 this would signal to the others in the group to also choose 7. Problem solved!
Things did not work out quite as well as expected. The figure below summarizes average effort over the 10 rounds subjects played the game. In a 4 player simultaneous choice version (Sim4) effort fell over time. In the leadership treatments, with either an endogenously or exogenously chosen leader (End and Exo), effort was higher but not by much. Moreover, it was no higher than we got if we just took one person out of the group (Sim3). This is not a ringing endorsement of leadership. We found that leadership failed to increase efficiency as much as expected because leaders were not bold enough. If a leader chose high effort followers responded but leaders were reluctant to choose high effort.
A recent study by Selhan Garip Sahin, Catherine Eckel and Mana Komai adds a slightly different twist. The figure below shows the average contributions they observed in a 6 player version (where effort could go up to 9). They looked at leadership by example (Exemplar) and leadership by 'communication' (manager). There overall results are very similar to ours. Again, leadership merely seems to stabilize effort and stop it falling. Their results, though, suggested more blame should be placed on followers. Specifically, leader effort increased over the rounds while follower effort fell.
There is still a lot we can learn about leadership in the minimum effort game. The failure of leadership to push effort up to the efficient level does though clearly illustrate the difficulty of getting groups to coordinate. And note that this is not because of some social dilemma like incentive to free-ride. There is no way to free-ride in this game. The problem is one of group members trusting that others will put in high effort. Trust, it seems, does not come easy.