If you’ve been reading this blog for a while you know I like to play poker. I have read numerous poker books and articles over the years, and the concept of “expected value” is at the core of every one. Expected value theory helps skilled players calculate whether a particular play will, in the long run, be profitable.
Here’s a simple example. Suppose I’m playing Texas Hold ‘em and my hole cards are the king of hearts and the three of hearts. The flop comes and the community cards are the ace of hearts, the four of clubs and the nine of hearts. My opponent is in front of me and bets out for $10 into a pot of $25. Everyone else folds. What should I do? This is a very complex question with numerous variables, but for my purposes here I want to isolate the variable of “mathematical expectation” with respect to the current pot (which, for the initiated, means I am setting aside issues of semi-bluffing, implied odds, stack size, opponent profile, fold equity, etc.).
If I narrow the question to mathematical expected value, the answer is simple. First, I missed the flop and have nothing but a king high hand. I assume my opponent either hit the flop (probably with a pair of aces) or had a pocket pair to begin with. As things stand now, I am almost certainly beat. Should I just fold and move on to the next hand? No. A call here has a positive expected value. What? How can calling a bet when you are beat have a positive expected value? The answer lies in the probability of certain future events. While I currently have nothing, if another heart comes I will have an ace-high flush, which will almost certainly win the pot. With two cards coming I have approximately a 35% chance of completing my flush by the end of the hand.
Let’s do the expected value math: The pot currently contains $35 (the original $25 plus my opponent’s bet). I have a 35% chance of winning by the end of the hand. $35 X 0.35 = $12.25. This means that even though I will lose 65% of the time and win only 35% of the time, over the long run I should expect to receive a net average of $12.25 if I call. I only have to invest $10 to obtain an expectation of $12.25, which gives me an expected profit of $2.25. I should call.
Blaise Pascal and Pierre de Fermat jointly worked out the mathematical underpinnings of expected value theory in 1654. This is interesting, because Pascal’s famous “Wager” is at bottom simply a special case of expected value theory. Let’s see how.
Pascal describes the wager as follows:
“God is, or He is not.” But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? According to reason, you can do neither the one thing nor the other; according to reason, you can defend neither of the propositions.
Do not, then, reprove for error those who have made a choice; for you know nothing about it. “No, but I blame them for having made, not this choice, but a choice; for again both he who chooses heads and he who chooses tails are equally at fault, they are both in the wrong. The true course is not to wager at all.”
Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.
“That is very fine. Yes, I must wager; but I may perhaps wager too much.” Let us see. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to chance your life to gain three at a game where there is an equal risk of loss and gain. But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity of chances there is one for you, if there were an infinity of an infinitely happy life to gain. But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite.
Pensée #272 (emphasis added).
Note that Pascal’s Wager is NOT a proof of the existence of God. Indeed, the wager assumes for the sake of argument that God’s existence cannot be definitely proved. Nevertheless, the wager demonstrates that even in a state of uncertainty (even extreme uncertainty) the rational man chooses to live his life as though God exists.
Very importantly, you cannot avoid the bet. God exists or he does not. Every person must choose which side of this question he comes down on. Attempting to ignore the choice (i.e., choosing not to choose) is futile. You cannot avoid the Wager. Choosing not to choose does not mean you have not chosen. It means you have wagered on “does not exist.”
If we must bet, then how should we bet? The Wager demonstrates that only a fool would bet that God does not exist even if he is all but certain he is right. What? Why should a person bet on God’s existence even if he is all but certain that God does not exist?
The proposition can be expressed mathematically as an expected value analysis. No honest person can ever say he is 100% absolutely positive God does not exist. Even arch-atheist Richard Dawkins expresses the matter as “God probably does not exist.” So let’s assume for the sake of argument that there is a 99.999999% chance that God does not exist. What is the expected value of betting on “God does not exist” with respect to the afterlife? Here there are actually two bets. The bettor bets that God does not exist. He also bets against the proposition that he does exist. To find the expected value of these two bets we find their independent expected values and add them together.
Bet 1: God does not exist. The bettor is wagering that God does not exist and that when he dies he simply ceases to be. The value of ceasing to exist is zero.
EV = 99.999999% X 0 = 0. In other words, if the bettor wins his bet he gains nothing. Therefore, the expected value of the bet is zero.
Bet 2: God exists. The bettor is wagering that God does not exist and that the penalty of eternal suffering in hell will not be applied to him. We will assign negative ∞ to the outcome “eternal suffering.”
EV = .000001 X –∞ = –∞
So the expected value of the wager “God does not exist” is the sum of these two bets: 0+-∞ = –∞
Now let us calculate the expected value of the bet “God exists.”
Bet 1: God does exist. The bettor is wagering that God does exist and that when he dies he will have an eternal reward in heaven. We will assign ∞ to the outcome “eternal reward in heaven.”
EV = .000001 X ∞ = ∞. If the bettor wins his bet he gains eternal bliss. Therefore, the expected value of the bet is ∞.
Bet 2: God does not exist. The bettor is wagering that God exists. If he loses and the atheist is correct he simply ceases to be at death and loses nothing.
EV = 99.999999% X 0 = 0
So the expected value of the wager “God does exist” is the sum of these two bets: EV = 0 + ∞ = ∞.
In summary, the rational man chooses to bet “God exists” even if he believes the probability of that proposition being true is very low. Keep in mind that I have assumed that the probability of God existing is very low ONLY for the sake of the argument. In fact, given the various proofs of God’s existence that have been adduced over the centuries, I believe that it is far more likely than not that God does in fact exist. This only adds force to the conclusions to be drawn from the Wager.