With all of our work on the probability of cascades forming, review of the central limit theorem, and applying power laws to the rich get richer phenomenon, I started to get really interested in how statistics can be used to explain real world events. This article by John Tierney of The New York Times does just that. There’s even a fun game you can play that illustrates some of the probability theory introduced into class.

Tierney first discusses how a dominant strategy forms in one of America’s most beloved old game shows, “Let’s Make a Deal.” Now, I know that most of us wouldn’t know about this show unless we have seen it on Game show network, but here’s the general gist of the game. You are told to pick among one of three doors. One door has a car behind it, and two of the doors have a donkey behind them. After you make your first choice (let’s assume this is door number 1), Monty Hall will then opens one of the doors you didn’t choose to reveal a donkey behind it. You are then given the choice to either stick with your original choice of door number 1 or switch to the other door. The question is: Which door should you choose? Originally, I thought that you would get the same payoff from staying with your original choice or switching. This is because I thought that there is a 50% chance that either of the remaining doors have the donkey behind it. However, it is actually a dominant strategy to switch to the second door. Why? Well the article explains that with your original choice, your probability of being right is 1/3. By switching, your probability of being right is 2/3. If you play the game, you find out that you will win more by switching doors once the donkey is originally revealed. The reason for the higher probability for correctness by choosing to switch doors links back to our discussion of Baye’s rule and signals. Opening the chosen door (let’s say door number 3) provides a signal that changes the posterior probability that door number 2 is actually the winning door. The concept is a little confusing to me, but it closely follows the taxi cab example in class. Before the signal, the probability that a yellow cab was implicated in the accident was only 20%, but after the signal, that probability increased to 50%. The same logic can be applied here: before the signal, the probability that door number 2 is correct is 1/3, but after the signal and after you have chosen door number 1, the probability of door number 2 being correct is 2/3.

The same statistical calculations were used by Dr. Chen of Yale to explain the Monkey M&M study pictured above. Here’s the general gist of the study: a monkey chooses between either a blue or red M&M. When the monkey chooses the red candy, it is then given the choice between a blue and green M&M. Most of the time, the monkey chooses the green candy. Findings like these have been used to support the theory of cognitive dissonance. This theory states that when we are given a choice between two objects, we later rationalize our choice by convincing ourselves that we really did not like the object we rejected. However, application of Bayes rule and some simple probability theory shows that this behavior may largely be described through statistics. As shown by the diagram above, if you give the monkey a choice between all three candies given that we know the monkey prefers red to blue, we see that the monkey will choose green over blue 2/3 of the time. This is exactly the ratio of monkeys choosing green over blue that the researchers found. Again, we can go back to our discussion of Bayes rule and conditional probability to explain this behavior. Choosing red over blue provides a signal that blue is not the monkeys preferred candy color. Thus conditioning on choosing red over blue, the probability that the monkey likes blue compared to some other color is drastically different than if we did not condition on choosing red. Now, Dr. Chen does not completely reject the notion of cognitive dissonance on this finding but it does add an interesting twist to traditional psychological theory. This is just another example of how network analysis can describe a large amount of the social phenomena we see today. It also shows how knowing your p’s and q’s can be quite lucrative. If I ever am in a Monty Hall position, I’m definitely pursuing the dominant strategy of switching doors.

http://www.nytimes.com/2008/04/08/science/08tier.html?_r=1&emc=eta1&oref=slogin