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The Evolution of Probability: Decision Theory

The evolution of probability began in the 17th Century with the great minds of the Enlightenment. Pascal and Fermat solved the Problem of Points. They and other academics of the time confronted the inherent vagaries of belief. The Port-Royal Logic and other contemporaneous publications formalized logic and established the need for evidence-based inquiry. The next big moment in the evolution of probability came with Decision Theory.

Decision Theory

Decision Theory is the study of the reasons an actor makes choices. In other words, decision theory studies what we decide to do when faced with uncertainty. This is closely linked with a concept known as expected value. Expected value represents the sum of every possible value of a variable multiplied by its probability of occurring. It is relatively easy to find the expected values of measured events. One example of this is the rolling of a six-sided die. Here is how the math breaks down for a six-sided die:

1(1⁄6) + 2(1⁄6) + 3(1⁄6) + 4(1⁄6) + 5(1⁄6) + 6(1⁄6) ≈ 3.5

With mathematical probabilities like this and many others, an actor can rationally predict outcomes. The highest possible expected value can easily be found. In this case, that number is 3.5. The benefit of this example is not immediately obvious with one die. Add another die to the situation, however, and you have the probability set that produced the game of craps. This basic math explains why the number “7” is the linchpin of that game and others.

But dice are mathematical. The outcome of a dice roll is easily measured. Decision theory is predicated on achieving the highest possible expected value, but what if our variables aren’t numbers? Blaise Pascal asked this question as well. When he began developing decision theory, he did so with this question in mind.

Pascal’s Wager

One of Pascal’s most infamous contributions to philosophy is his famous wager. Despite being a relatively well known thought experiment, the implications of Pascal’s Wager are not always fully explored. On the surface, Pascal’s Wager famously deals with the great unanswerable question: Does God exist? But it is not inherently a religious thought experiment. Rather, this argument shows that one can still find the highest expected value, even when dealing with unknown probabilities. Pascal argued that belief in God is not as relevant as the expected value of the decision to believe. The belief in God must be seen through the lens of cost and benefit. The chart below explains:

pascalswager

Image found in “The Emergence of Probability”, 2nd Ed., by Ian Hacking

Pascal rationalized that belief in God carries fewer risks and better rewards than unbelief. If you believe in God, risk is mitigated, and the expected value of that belief is higher. Pascal’s Wager lacks quantifiability, yet it amply explains how to arrive at an optimal decision anyway. Not only did Pascal develop decision theory, but via his Wager, he also contributed to an understanding of two different kinds of decision theory. Next week we will discuss these two kinds of decision theory in greater detail.

This is Part Four of a series on probability. Click here for Part One, Part Two, Part Three, and Part Five.