### Risk Tolerance for Poker Players

Outcome is the result of either your decision or of the unrolling of a previously unrevealed state of nature. The value of the next card to be turned is an outcome, as is whether you win or lose a bet. The level of detail we use when describing outcomes will vary according to the situation. It might seem a little trite to go to the trouble to discuss the concept of an outcome but it’s important to keep in mind that we mean something specific while at the same time we aren’t always talking about the same level of specificity.

I’ll use the symbol X to denote outcome and the lower case x to denote a particular realization of the outcome. For example the result from flipping a coin is X and we have either x=tails or x=heads.

Outcomes don’t have to be expressed in numerical terms. For example, the outcomes might be the turn card is a spade. But we should be able to assign some sort of numerical value to an outcome.

Value of an outcome will be denoted by the functional notation V(x).

Probability is a way of expressing knowledge or belief that something will occur or has occurred. The study of probability began with Cardano in the 16th century and full development of the concept is attributed to Fermat and Pascal in the 17th century. The beginnings of probability theory is rooted in the analysis of gambling games. Cardano doesn’t often get full credit for his work on the fundamentals of probability theory because his work wasn’t published until after his death, and because it wasn’t translated from Latin into English until the early 1950’s.

In the 18th century Thomas Simpson applied concepts of probability to the study of errors in astronomical observations. But decision analysis tends to follow the branch of the development of probability theory that’s related to the analysis of gambling games.

The functional notation P(x) is used to denote the probability of x occurring.

Expected value is a technical concept that’s central to the application of decision theory or probability theory to poker. It’s simply the sum of all the outcome values multiplied by the probability of each outcome. We write E(x) = Sum[V(x)*P(x)].

A gamble is an uncertain event with two or more possible outcomes. We can often think of a coin flip where we either win or we lose, and there’s an associated value with each of the possible outcomes. In poker we tend to think of gambles as having discrete outcomes, although it can be more than 2 outcomes. For example the next card might complete our flush, corresponding to a large win, or it might pair our high card, corresponding to a moderate win, or it might miss us entirely, corresponding to a loss.

The certainty equivalent of a gamble is the fixed amount you would accept (or pay) in exchange for the gamble. This idea is where the concept of Risk Tolerance begins to take form. We’ll often think of the amount we would pay for a gamble as simply the expected value of a gamble. If we flip a fair coin and pay $1 for heads and get $1 for tails the expected value is E(coin flip) = .5*(1) + .5(-1) = 0. Flipping a biased coin might have an expected value of E(coin flip) = .6*(1) + .4(-1) = +.2

Notice that it’s important for Sum[P(x)] = 1.

It’s common to argue that we should be indifferent to a gamble that has a zero expected value and be willing to take any gamble that has a positive expected value. David Sklansky is well known for making this argument (see David Sklansky, Getting the Best of It, Two Plus Two Publishing, 1982). But that argument doesn’t really hold up to scrutiny. Just because I’m willing to take an even money coin flip for $1 should you actually expect me to be willing to make that same coin flip for $1,000?

I don’t think so.

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