Excessive punishments and game theory

How much punishment is enough?

This is an interesting question for game theory. One of the reference models is the repeated Prisoner’s dilemma where players have to use the threat of punishment as a means to achieve cooperation.

Sometimes harsh punishment is the right move. I discussed this before in the context of the grim trigger strategy used in estate planning, specifically the Crummey trust.

But often excessive punishment is the wrong move. Ashish asks me the following question:

My economics professor talked about why Draconian laws don’t work in real life (if you steal an apple, we will chop your hands off)…Can you tell me how that works from a game theoretic standpoint? Why do cops need speeders? Why aren’t speeding tickets draconian – we take your car if you speed beyond 90 mph?

This is a very interesting topic, so let me give a few thoughts.

The short answer

One issue is that excessive punishment can lead to more dangerous behavior. To see this, suppose anyone who speeds over 90 mph will be executed, and consider a speeding driver being chased by the cop. How will this turn out?

The speeder faces death if he gets caught. So in a sense, he has nothing to lose from trying to evade the cop. He will drive faster and faster. He might think, hey why not speed and try to escape? I am going to executed if I am caught. I will resist with full force.

This example might seem trivial, but it is illustrative. The same type of logic happens with enemy soldiers who face excessive punishment and interrogation, as discussed briefly in Zachary Shore’s nice book Blunder: Why Smart People Make Bad Decisions.

Another cost of excessive punishment

The topic of setting punishment is also discussed in the fun game theory book Thinking Strategically.

Here is an excerpt from a chapter regarding the Prisoner’s dilemma that discusses another issue with excessive punishment:

Next we ask how severe a punishment should be. Most people’s instinctive feeling is that it should “fit the crime.” But that may not be big enough to deter cheating. The surest way to deter cheating is to make the punishment as big as possible. Since the punishment threat succeeds in sustaining cooperation, it should not matter how dire it is. The fear keeps everyone from defecting, hence the breakdown never actually occurs and its cost is irrelevant.

The problem with this approach is that it ignores the risk of mistakes. The detection process may go wrong…If punishment is as big as possible, then mistakes will be very costly. To reduce the cost of mistakes, the punishment should be the smallest size that suffices to deter cheating. Minimal deterrence accomplishes its purpose without imposing any extra costs when the inevitable mistakes occur. (105-106)

Mistakes in detection and miscommunications are frequent in business and personal relationships, and excessive punishment raises the cost of these errors.

Summary

It seems strange, but economics and game theory show why zero crime or full compliance may be unrealistic goals. There is usually an optimal level of punishment that can both deter crime at a cost effective level with low costs of mistakes.