What is the difference between and a game theory model and reality?
Game Theory is an exercise in modeling strategic interaction between intentions.
Strategic interaction is familiar to transactional lawyers, who must make contingency plans based upon what they think the other parties might do, and implement those plans into contractual language.
Judging what might happen is different from calculating with a model what might happen. The type of person making the judgment or calculation is also important.
Trying to judge what sort of person you are responding to is no simple task, as an analysis of the Monty Hall problem shows.
The Monty Hall problem reveals the difference in strategic reasoning between layman, experts, and players.
Strategic reasoning is not the same as probabilistic reasoning -it may use some concepts from probability, but we don't end our thought process with counting the number of possible states.
The Monty Hall problem is easy to state. There are three doors, behind one of the doors is $100 and the other two doors have nothing. The prize is distributed at random, equally behind the doors.
You get to pick a door, say Door 1. But before opening it, Monty Hall offers to show you what is behind another door, say Door 2. He opens Door 2, which is empty. Next, Monty Hall allows you the choice of switching your choice between Door 1 and Door 3. Should you switch? (By the rules of the Game, Monty Hall, cannot open the door which shows the prize.)
Here is a Monty Hall game you can play online.
The layman will reason this way. First, he might say, is Monty Hall trying to trick me into taking a choice I might regret? Well, there are only three possibilities: the prize is behind Door 1, Door 2 or Door 3. If I picked Door 1, and it contained the prize, then Monty Hall could always open Door 2 or Door 3. But if Door 1 did not contain the prize, Monty Hall could also open one of Door 2 or Door 3, to show me an empty door. It doesn't appear that Monty Hall is giving me any new information that would make it reasonable for me to switch.
The expert, who loves models, draws up the situation differently. He draws a pay-off table like the one below. He first realizes that there are not three possibilities, but four. This gives him great confidence that he has discovered something important. He also likes to talk about states of nature, which he denotes by Si because this too sounds important. Again, assume that Door 1 has been chosen.
S1 is the state of nature where Door 1 has the prize and Monty Hall opens Door 2
S2 is the state of nature where Door 1 has the prize and Monty Hall opens Door 3
S3 is the state of nature where Door 2 has the prize and Monty Hall opens Door 3
S4 is the state of nature where Door 3 has the prize and Monty Hall opens Door 2
Our expert realizes two things. The probabilities of S3 and S4 are the same, 1/3. The probabilities of S1 + S2 also equal 1/3, and the probability of S1 = S2, i.e. S1=S2=1/6.
He comes to the startling conclusion, when Monty Hall reveals Door 2, eliminating S1 and S4, then changing from S2 to S3 dramatically improves your chances of winning the prize, since S3 occurs twice as many times as S2. Thus, the expert recommends to switch, always. Switching always will produce a higher expected monetary return.
The player dismisses the layman, and sees the expert's result for what it is: an inherently incomplete model, which can never be made complete. The player wonders about whether Monty Hall really has to open a Door with no prize, whether the prizes can be moved once the choice is revealed, whether the $100 is real or not.
The player also notes a possible flaw in the expert's scheme: if the expert commits to playing switch all the time, and will pay to switch, can Monty Hall take advantage of this?
What happens if Monty Hall's strategy is to always open Door 2, when the prize is behind Door 1? (Assume that the expert doesn't know this.)
This commitment defeats the expert's analysis: if S2 is eliminated by Monty Hall's play -he never opens Door 3, then all the states are equal and it turns out that the layman's analysis is correct.
Can the player get the expert to take the bet? Once the expert finds out he has been had, will he demand a replay which allows him to change or switch each round? Or will the expert stick with his model? Can the player make enough money from the expert before this happens?
If you don't know whether you are a layman, expert or player -you are a layman. If you think that you are expert, you probably are. John Farina, whose game I referenced above is a good expert.
But, players are only players if they are making money from experts and layman.
