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Risk, Uncertainty and Monty Hall

I find Gerd Gigerenzer’s grouchy attacks on other scholars entertaining and so I enjoyed his new book, Risk Savvy. In this he discusses risk, uncertainty and Monty Hall.

Risk, Uncertainty and Monty Hall

In Risk Savvy Gigerenzer brings a new perspective to the Monty Hall problem. Monty was a gameshow host famous for having a great prize behind one of three doors and undesirable prizes (goats) behind the other two doors. In the traditional telling a contestant must choose a door. Monty then reveals the prize behind a door that the contestant didn’t choose. The contestant is then offered a chance to change their choice. People often refuse to believe that changing their choice is the right thing to do. The standard, faulty, logic is that there are now only two doors, the good prize is equally likely behind either door so why change?

But in the standard telling changing your door choice doubles your chance of winning. To see why imagine three different people choosing. Let’s assume the great prize is behind door 2.

The first contestant picks door 1. Monty’s only option is to open door 3 [so as not to reveal the great prize behind door 2]…. Switching to door 2 wins. The second contestant picks door 3. This time, Monty has to open door 1, and switching to door 2 again wins. Only the third contestant who picks door 2 will lose when switching.

Gigerenzer, 2014, page 127

Switching your choice of door doubles your chance of winning the great prize. The trick is that Monty knows where the prize is and by observing his actions you gain information.

The Monty Hall Problem

Monty Didn’t Always Offer A Choice

Gigerenzer then adds a detail I was unaware of. Monty didn’t always offer the choice to switch. The problem is, therefore, not a problem of risk but one of uncertainty. A risk problem is one where we know the possibilities and how likely they are. With uncertainty, on the other hand, we can’t simply calculate the best choice because we don’t know how likely each outcome is or even what possible outcomes there are. Monty’s unpredictable behavior creates uncertainty. For example, Monty may only offer the chance to switch when the contestant has chosen the correct door. In such a case switching doesn’t make sense. Unfortunately we are uncertain whether that is how Monty decides when to make the offer to switch. Monty may offer the choice randomly in which case switching makes sense.

Gigerenzer suggests the world has more uncertainty than we allow for. We usually don’t know the possible outcomes never mind how likely each outcome is. To him ours is a world of uncertainty not just risk.

For more on decision making see here.

Read: Gerd Gigerenzer (2014) Risk Savvy: How To Make Good Decisions, Viking, New York, NY

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