Game Theory: The Calculus of Strategy
If probability shows us what might happen, Game Theory simulates what actors will choose to make it happen. It is not about modeling the entire infinite chaotic set of possible events; instead, it mathematically winnows the universe down to only the most strategically viable scenarios.
Consider the famous Monty Hall problem. At first glance, a game with three doors appears to be a simple uniform distribution with 1/3 probability for each. But game theory and Bayesian updating reveal that the space is non-binary. When the host — an actor with knowledge of the system — opens an empty door, the probabilities do not reset to a 50/50 binary coin toss. The geometry of the probability space collapses, shifting a 2/3 probability mass to the remaining unopened door. Strategic viability dictates that changing your choice is mathematically superior.
Prior belief that the prize is behind door \(i\):
$$P(C_i)=1/3$$
After the host opens an empty door, switching wins with probability:
$$2/3$$
