Decision-making by Monty Hall strategy

The Monty Hall strategy is a way to make decisions when only one party is making a random selection. Let's look at how the probabilities of picking the winning option dramatically changes when the counterparty is making selections in a non-random manner.

Background

Monty Hall was the host of the gameshow Let's Make a Deal. In the game, a contestant would start by selecting one door from amongst a number of doors that s/he believes has a car. After selecting the door, Monty would systematically eliminate one of the doors remaining. After eliminating the door, he'd ask the contestant if s/he wanted to keep his/her selection, or switch to one of the remaining doors. After the contestant makes a decision, Monty eliminates the next door, and asks the question again. This goes on until two doors remain - the one the contestant has chosen and one remaining. Monty asks one last time if the contestant wants to switch, and after s/he makes his/her final decision, Monty reveals the contents behind each door.

Uniqueness of the Monty Hall strategy: p(non-random selection)

What the contestant does not know is that Monty is fully aware of the contents of each and every door. He purposely....in a non-random manner...eliminates each door to keep the game going. The contestant, on the other hand, starts the game making a random selection. It is this difference - the difference in the circumstances by which the contestant and Monty make their selections - that forms the foundation of the Monty Hall strategy. Let's see how this plays out.

Some analogies before we begin

Here are the appropriate analogies to make the Monty Hall scenario pertinent to your entrepreneurial case:

1. Game: the entrepreneurial scenario that you encounter

2. Contestant: you, the entrepreneur

3. Monty Hall: the counterparty: a competitor or the marketplace

4. Selections or doors: the options available to you

5. Win: an optimal, NPV > 0 choice

How the p(random selection) is affected by p(non-random selection) - hint: it isn't

The keys to the Monty Hall strategy are:

1. identifying whether the counterparty making selections is doing so in a non-random manner (in other words, in a manner opposite of you), and

2. understanding how a non-random selection affects:

   1. the probability of your selection being the winner and

   2. the probability of the remaining choices being the winner

In the Monty Hall scenario, your initial selection, and all subsequent selections (if you decide to switch) are random. Therefore, your probabilities are random probabilities. Monty Hall, however, makes non-random selections. His probability is a non-random probability. How does Monty's strategy affect your random probability?

• A random probability of Monty's selection will affect the random probability of your selection

• A non-random probability of Monty's selection will not affect the random probability of your selection

Let's walk through an example in which there are five (5) options at the start.

The start of the game

There are five (5) options available to you and you select one. Because you have no ex ante information about any of the options, your selection is random. Therefore, your probability of selecting the winning option is 0.2 (20%; 1 out of 5). Since your the probability of your selection producing a winning result is 0.2, the probability of your selection NOT producing a winning result is 0.8 (1-0.2). 

The 80% probability of your selection not producing a winning result means that the sum probability of the *remaining* options producing the winning result is 0.8. Since you selected 1 of the 5 options, there are 4 options remaining: the 0.8 probability of winning must be divided amongst those 4 remaining options. Therefore, each option that you *did not* select has a 0.2 probability of yielding a winning result.

Study the table above. It displays the logic that we just walked through. A key point in this table is the last column: it indicates the marginal increase in your probability of winning if you make a switch. At the start of the game and after you make your first selection, there is no point in immediately switching because every option amongst the set of 5 options carries the same probability of winning: 0.20.

Monty's turn

Monty will select from one of the remaining 4 options. Let's compare the changes in probabilty (if any) when Monty makes a random or non-random selection.

Random selection

If Monty makes a random selection, he will remove any one of the 4 options from the game. Now, a total of 4 options remain. Because Monty made a random selection, all random probabilities (like yours) will change. Instead of having a 1 in 5 chance of picking the winning option (0.2), you now have a 1 in 4 chance (0.25). As a result of Monty's random choice, there is a 0.75 chance that one of the remaining 3 options is the winning choice - in other words, the probability of each remaining choice being the winner is 0.75/3 = 0.25.

Look a the last column. The additional probability that you would earn by switching is 0.00. When Monty makes random choices, you don't earn additional probability by switching.

Non-random selection

If Monty makes a non-random selection - in other words, Monty knows what's behind every door - his/her selection will not affect the probabilities of any randomly selected option. Since you made a random selection, its probability will not change after Monty makes his non-random selection.
After making his/her selection, four (4) options remain. Recall you picked one of those selections with a probability of winning = 0.2 - a probability that remains 0.2. Since your probability remains 0.2 after Monty's selection, the probability that one of the remaining options is the winning one is 0.8 (1-0.2). This time, however, the probability of 0.8 is divided across 3 remaining options (instead of the 4 remaining options at the start of the game). 0.8 divided by 3  = 0.27.

Take a look at the last column. If you were to switch after Monty's selection, you'd enjoy an increase in probability (of winning) by 7%.  Far different than if Monty had made a random selection.

Iterating the game until 2 options remain

If we continue with the game, the following probabilities (of winning) are realized based on the type of selection Monty makes: random or non-random.

Each time Monty makes a random selection, your probability of winning increases as much as the probabilities of the remaining options. As a result, there really is no mathematical reason for you to switch.

Each time Monty makes a non-random selection, your probability of winning remains unchanged, and the probabilities of the remaining options increase. The more rounds you go with Monty making non-random selections, the greater the marginal increase in your probability if you switch.

The key is not to determine your random probability, but rather if the counterparty is making a non-random decision. Some entrepreneurs believe that the marketplace, with its collective wisdom, experience, and knowledge, makes decisions in a non-random manner. In such a case, take advantage of those non-random decisions and increase your marginal probability of winning.