**We know that our decisions are imperfect, we give into temptation, we shy away from risk and we are inconsistent both in our preferences and our planning.**

### Becoming better decision-makers

We don’t need science to tell us we aren’t rational decision-makers, at least in the way assumed by traditional economic models. We only need a moment of introspection and will recognize her own limitations.

But we do need science to help us understand those limitations and we do need science to help us identify ways of overcoming those limitations in becoming better decision-makers.

### The mathematics of probability

Let’s begin our exploration of the science by going back to Renaissance Italy the town of Milan in the middle of the 16th century. In Milan lived a physician named Girolamo Cardano. Cardano was not just any physician, he was one of the most famous physicians of his time.

Royal families and even the Pope would consult with him about their ailments, but Cardano is most famous today not for his medical virtues which were considerable but for one of his vices: he was a gambler. Today we call him a pathological gambler.

Cardano’s masterwork “A Book on Games of Chance” introduces probability as a way of thinking about gambling. If a gambler throws a six sided die, the probability of rolling a five is one in six. Probability is thus defined as the number of desired outcomes divided by the number of possible outcomes.

This definition can be applied to sets of outcomes as well. There are four distinct ways of getting a total of five when rolling to dice out of 36 possible combinations, that means that the probability of rolling 5 on two dice is 4 out of 36 or 1 in 9.

Knowing such probabilities gave Cardano and other informed gamblers a mathematical cheat sheet that help them in their betting. At that time the odds associated with different games of chance were not as standardized as they were now, so savvy gamblers could identify bets that were in their favor.

### The expected value

By multiplying the amount that could be one by the probability of winning the gambler could calculate what came to be called expected value of a risky choice. Expected value is one of the landmark concepts of all of decision science as well as in statistics and economics more broadly.

Expected value provides a simple rule for decision-making: choose what ever option maximizes expected value, but expected value has its limitations. Let’s take the simple sort of gamble: betting on the outcome of a single flip of a coin.

Suppose that I pull out a coin and offer you a bent: if it lands heads I’ll pay you $10 if it lands tails you will pay me $5. Now Cardano would’ve leapt across the table to make this bed, the expected values clearly in the batter’s favor.

A $10 gain is twice as much as a five dollar loss and indeed most people are willing to take this bet. But what happens if I increase the stakes? Suppose that if the coin lands heads I’ll pay you $10,000, but if the coin lands tails will pay me $5000.

Do you take that bet? Expected value argues that you should. It’s still very much in your favor, but many people would decline the spent even though it’s a better investment than anything that your eye will see in our lifetime.

### The expected utility

The mathematician and physicist Daniel Bernoulli writing in the middle of the 18th century used problems like this to argue that expected value did not account for people’s real decisions for two basic reasons:

- first, people placed different value on a good like money depending on their desires and circumstances and
- second a given amount of money was more important to people who had very little then the people who had very much.

So the chance of a large gain might not be attractive if we risk a large loss

Bernoulli argued that expected value needed to be replaced by a new concept which we now call expected utility.

Economics defines utility as a subjective value that is placed on some good or action. Over the next couple of centuries economists developed models that assume that people act to maximize their expected utility.

Two people might make different choices because they prefer different goods, one prefers apples and the other ice cream or one might be risk-averse while the other is willing to tolerate risk but each choice is assumed to maximize their own utility.

These are the so-called rational choice models.