Frightening Many with One Threat

Frightening Many with One Threat^[8]

So far we have assumed that a player threatens only one person or organization.

Sometimes, however, you need to threaten many. Interestingly, game theory demonstrates how many can be induced to comply even if you can carry out your threat against only one. To see this, assume that you have 10 employees.

Each employee gets the following payoff:

Employee’s Payoff
Work Hard	$5,000
Slack Off	$10,000
Get Fired	0

You want all your employees to work hard. If an employee knew that she wouldn’t get fired, however, she would prefer to slack off. An employee is worse off working hard than slacking off because of the mental anguish caused by effort. To get them to work hard, you need to threaten your employees with termination. An obvious solution to this game is for you to declare that an employee will get fired if she slacks off. If the employees all believe your threat, they will all work hard. Let’s make the game somewhat challenging by assuming that you can fire, at most, one employee, and your employees know this. Perhaps your company couldn’t survive if you fired more then one person.

If your strategy was to randomly choose one of the employees who slacked off and fire her then there would be two possible outcomes. In the good outcome, everyone would work hard. If all the other employees worked hard, each one would also want to, or else she would get fired. If the employees worked together, however, they should all agree to slack off. In this outcome you could fire only one employee. Therefore if an employee was lazy, then 90 percent of the time she would get a payoff of $10,000 and 10 percent of the time she would get 0. This outcome is probably better for an employee than always working hard and getting a payoff of $5,000.

How could you motivate all employees with just one threat? First, put the employees in some arbitrary but announced order. You tell employee 1 that if she doesn’t work hard, you will fire her. This would obviously cause employee 1 to work hard. Next, you tell employee 2 that if employee 1 does work hard, you will fire 2 if she doesn’t work hard. Since employee 2 expects employee 1 to work hard, this will cause employee 2 to work hard. Next, you tell employee 3 that if employees 1 and 2 work hard, you will fire 3 if she slacks off. Again, since employee 3 expects 1 and 2 to work hard, she will expect to get fired if she is lazy. You continue this process until all the employees have an incentive to work. Your employees can’t circumvent your system by colluding to be lazy because employee 1 would never agree to slack off since it would result in his termination.

Another application of this game would be if you had 10 errant suppliers, but your production demands meant that you could fire at most one. Even if the suppliers knew of your limitations, you could still effectively threaten them all by first threatening one, then the next, and so forth.

The key lesson from this game is that when you assign responsibility randomly, all might accept the chance of getting punished and choose not to work. It’s much better to have a clear chain of punishment. You choose one person most responsible and then go down the line. Unless your employees are crazy, your threat will induce them to comply with your demands.

^[8]Dixit and Nalebuff (1991), 16–19.