Should you invest in umbrellas or ice-cream? How should you choose between the two?
Some simply to go with their gut feeling. That may work sometimes, but it turns out that gut feeling is often wrong. Being led by your stomack or other bodily parts often lead to systematic mistakes and bad outcomes. To improve the way we make decisions, in life and in business, it is useful to explore both when gut feelings tend to lead us astray as well as what we should do to make better descisions.
How should one make the best possible decisions? Intuitively one should weight the costs and benefits of the two actions. The question is how one can do this in an uncertain world. Should you bring an umbrella or not when going out of the house? The decision depends on your beliefs about the weather, the hassle of carrying an umbrella and how bad you believe it is to be outside in the rain without an umbrella. All of this has to be estimated and somehow integrated into a single value in order to determine the best action.
Continue with the example and imagine that your subjective probability for rainy weather is p. To calculate the expected consequences of bringing an umbrella, you consider the two possible outcomes. If it rains you will avoid getting wet, if it does not rain you will look like a fool with an umbrelly on a sunny day. The higher the probability of rain, the more likely it is that you will experience the first outcome. In this way the probability of the outcome can serve as a weight. The more probable an outcome is, the more weight we should attach to it when calculating the expected consequence. So to make a rational decision we list all the possible outcomes from an action and weight the outcomes according to how probable they are. In short:
Expected consequence of an action = Sum of (probability * Value of outcome) for all outcomes of that action
St. Petersburg Paradox
This may sound obvious, but there are always surprises hidden in the details. To illustrate a potential problem, suppose you are offered a chance to play a game in which a coin is thrown until it lands heads up. If it lands heads up after the first throw, you get two dollars. If it takes two throws to get heads, you get four dollars. Three throws to gives you eight dollars and so on. In short, the money doubles for every time tails comes up and as soon as the coin comes heads up, you get that sum of money. Now, the question is how much you are willing to pay in order to be allowed to play this game.
Being rational, you start to calculate the expected value of playing the game. The outcomes are easy, you can end up winning 2, 4, 8, 16 dollars and so on. The probabilities are also quite easy. The probability of getting heads on the first throw is 1/2. To win four dollars, the coin has to land with tails up on the first throw (p=1/2) and heads on the second throw (p=1/2). Each event has probability a half, so the probability for both to happen is 1/2*1/2 = 1/4. next, the third outcome - winning eight dollars - implies tails, tails, head or a probability of 1/2 * 1/2 * 1/2 = 1/8. If we go on we have:
Outcomes (2 , 4 , 8 , 16 , 32 , ...)
Probabilities (1/2, 1/4, 1/8, 1/16, 1/32, ...)
Multiplying each outcome with the probability that it will happen, we then have (2 * 1/2) + (4*1/4) + 8 * 1/8 and so on:
Expected value of the gamble = 1 + 1 + 1 + 1 ...
Retunr to the question of how much you would be willing to pay to play the game. Since the expected value is infinite (1 + 1 + 1 forever), one might think that we should be willing to pay whatever we have available to play the game. In reality this is not what happens. Most people are only wiling to pay a dollar or two in order to play the game. Either the theory about how to make a good decision is wrong, or there is something wrong with the example!
As promised the interesting and surprising devil is in the details. The relevant outcome is not that you get a certain amount of money, but how much utility you get from that money. You do not care about the money in itself, only the utility it gives you. This small distinction is vitally important and it was first pointed out by the brilliant Bernoulli several hundred years ago: What we want to maximize when making a decision is not expected value, but expected utility. Each outcome should be transformed into the utility we get from that outcome before we multiply it with its probability:
Expected value = Sum of (probability * outcome) for all outcomes of that action
Expected value = Sum of (probability * utility of outcome) for all outcomes of that action
In the case of the coin game, we then have
Expected utility of coin game = 1/2 * utility of (2 dollars) + 1/4 * utility (4 dollars) + 1/8 * utility (8 dollars)
How does this solve the problem? The basic problem when using the expected value, is that the sum became infinite because the growth of the money balanced the decrease in probability so we got an infinite sum of 1 + 1 + 1 + ... When using utilities, this is no longer the case. 4 dollars is twice as much money as 2, but the utility we get from four dollars is probably not twice as high as the utility we get from two. A billionaire is not a thousand times happier than a millionaire. This means thet the value of the outcome will not increase as much when we use utilities as outcome: the probabilities will decrease faster than the increase in the value of the outcomes and the sum no longer becomes infinite.
A rational decision
The lesson? Being rational implies, first of all, to consider all the possible actions, outcomes and their respective probabilities. The best decision is then the one that maximizes expected utility, defined as the sum of the utility of the outcomes associated with the action weighted by the probabilities of these outcomes.
This is, at least, the starting point!
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