Mathematical aspects of Pascal's wager

Pascal's wager draws its arguments from the framework of games of chance.

The mathematical model of game theory

Many contemporary commentators formalise Pascal's wager with game theory, the foundations of which were described in the 1920s by Ernst Zermelo and developed by Oskar Morgenstern and John von Neumann in 1944. As Pascal died in 1662, it is an anachronism to interpret Pascal's wager by means of game theory, and there is a great risk of betraying his thought.

Moreover, infinity is treated as an entity, which poses problems of realism that we will discuss later.

Huygens' mathematical model

The first person to successfully pursue Pascal's work on games of chance was the Dutch mathematician and physicist Christiaan Huygens. In the period 1655 - 1657, while Pascal was still alive, he generalised Pascal's method to the case where the transition probabilities are unevenly distributed. He was also the first to use the term expectation (Hoffnung). It is this historical way of formalising Pascal's wager that seems relevant to me and that I have retained.

As far as infinity is concerned, it will not be treated as an entity, but as a limit.

The example of roulette wheel: bet on a single number

The play mat has 37 squares numbered from 0 to 36. Playing "single" consists in placing the bet, noted b, on a single square. If the chosen number comes up, the player wins 36 times the bet, which is the gross winnings from which the bet must be deducted to obtain the net winning. In our model, we do not take into account what the player usually leaves for the casino staff. The random variable of the game is

\[ \begin{equation*} \left\{ \begin{array}{ccc} −b + 36b = 35b & \text{with a probability of } & 1/37,\\ -b & \text{ with a probability of } & 36/37. \end{array} \right. \end{equation*} \]

The mathematical expectation of the net winning is

\[ \begin{equation*} \begin{aligned} E &= 35 b \cdot \frac{1}{37} + (−b) \cdot \frac{36}{37}\\ &= (-\frac{1}{37}) \cdot b \end{aligned} \end{equation*} \]

This means that, over a large number of games, the player loses on average 1/37 of his bets to the casino. It is a game with negative mathematical expectation.

The formula for mathematical expectation

To generalize, let us consider a game of chance in which, for a bet b, you can get the winning w with a probability p. The random variable is

\[ \begin{equation*} \left\{ \begin{array}{ccc} -b + w & \text{with a probability of } & p,\\ -b & \text{ with a probability of } & 1-p. \end{array} \right. \end{equation*} \]

The mathematical expectation of the net winning is

\[ E = (-b+w) \cdot p + (−b) \cdot (1-p) = -b + w \cdot p \]

Remember

\[ E = -b + w \cdot p \]

From the latter formula is derived the expression of the probability :

\[ p = \frac{E+b}{w} \qquad \text{ where } w>0 \]

Conditions 0 ≤ p ≤ 1 result in 0  ≤ (E+b) ≤ w

The case of fair games

If the mathematical expectation of the net winning is zero, the game is said to be fair. The probability of winning is then p = b/w. For example, by betting 1 €, it is a fair game to be able to win 1000 € with a probability of 1/1000; in another game, by betting 1 €, it is fair to be able to win 1,000,000 € with a probability of 1/1,000,000. When the winning is huge, the probability of winning is tiny. With a constant bet, if the winning tends towards infinity, the probability of winning tends towards 0:

\[ p = \lim_{w\to\infty} \frac{b}{w} = 0\]

Case of games with high mathematical expectation

If the mathematical expectation of the net winning is positive, a generous sponsor is needed to contribute to the financing of the winning. While the players to whom the wager is addressed wait for a mathematical expectation close to zero, i.e. a game not too biased, believers imagine an immense mathematical expectation. Suppose for example that E is worth a billion times the bet. Since (E+b) is constant, the limit probability remains zero:

\[ p = \lim_{w\to\infty} \frac{E+b}{w} = 0\]

i.e. with a constant bet, however great the mathematical expectation, when the winning tends towards infinity, the probability of winning tends towards 0.

To be convinced of this, consider the following sequence of winnings: 10(E+b), 100(E+b), 1000(E+b), 10000(E+b), and so on. The corresponding probabilities will have the values :

w	p
10(E+b)	0.1
100(E+b)	0.01
1000(E+b)	0.001
10000(E+b)	0.0001
...	...
∞	0

To obtain this result, it is not necessary for the mathematical expectation to be constant, but only for its absolute value to be capped by an upper bound, i.e. there is a number E such that, for all winnings,
|mathematical expectation|⩽E.

In the end, Pascal's wager is unfounded.

Discussion

I still have a doubt. For me, the probability that God exists may be small, but positive.

Let's take a specific Church that offers you salvation on the condition that you pay it, for example, €100 per month. The probability that this is true is small, but one can have a doubt and judge that this probability is not nil. If you do not make the payments, it is because you do not support to the end the idea of taking into account events of low probability. What is the reason for this? Presumably because it is impossible to take into account everything that might possibly be possible. You have to decide what is serious and credible, and reject everything else.

Personally, I don't have the kind of doubt that your question evokes, because I firmly believe that I am not endowed with immortality. So Pascal's wager is pointless.

Could it be envisaged that, with w tending towards infinity, E also tending towards infinity?

1. We would end up with an indeterminacy of the "infinity over infinity" type; the limit probability would be undetermined, and we would have failed to show that the limit probability is positive.
2. Pascal concedes that the probability of winning could be 1/2 and decrees that the bet is zero. Thus, for him, the formula to consider is E = w/2. For example,
   • if a game allows you to win 1000 €, you would win an average of 500 € each time you try it with a zero bet;
   • if a game allows you to win 1,000,000 €, you would win an average of 500,000 € each time you try it with a zero bet;
   • if a game allows you to win 1,000,000,000 €, you would win an average of 500,000,000 € each time you try it with a zero bet;
   • By prolonging this family of fairy tale games to infinity, we obviously obtain a miracle, in this case Pascal's wager.
   Unfortunately, as natural resources are finite, to go to the limit, it is necessary to assume that the supernatural exists. But this approach consists in assuming that God exists in order to prove that God exists. It is a vicious circle. We can conclude that, if the probability is fixed, the winning cannot be stretched to infinity.
3. If the aim is to convince sceptical players, it is unconvincing to call for an act of faith that requires accepting a priori that the game is miraculous, as this is a characteristic of scams. Since you have to be a believer for the wager to be convincing, the wager loses much of its substance: it is not intended to incite non-believers to become believers, but only believers to become practitioners.
4. One would have accepted as a hypothesis that "when w tends towards infinity, the mathematical expectation E also tends towards infinity", which is an avatar of Pascal's Wager as described in point 2 above. Now, in a reasoning, admitting what one wants to demonstrate as a hypothesis is called a vicious circle.
5. By making a promise - paradise - which commits a third party over whom he has no control - God - the supporter of Pascal's wager implements a process similar to that of a swindler. On this subject, read the fourth objection "Reversal of the wager ... ".

What to answer to "The probability of obtaining an infinite winning may be close to 0, but it does not tend towards 0! It is a real positive fixed." ?

1. The approach consists in situating Pascal's wager among the games of chance whose winnings are gigantic, close to infinity. The expression "when the winning tends towards ..." simply means that a comparison is made with neighbouring games whose winnings are gigantic, close to infinity.
2. One should be able to approach the infinite winning through a sequence of increasing winnings and observe the impact this has on the probability of winning. Let us name ε the "fixed positive real". We can calculate the winning w = (E+b)/p which corresponds to p=ε : it is w_ε = (E+b)/ε. As the mathematical model produces a sequence of probabilities that tends towards zero, the consequence is that all winnings that are greater than w_ε correspond to probabilities of winning that are less than ε :

   w	p
   ...	...
   wε	ε
   10⋅wε	ε/10
   100⋅wε	ε/100
   ...	...
   ∞	ε ou 0 ?

   Uneasiness.
3. The limit is the continuous extension of the mathematical law of the game. When the so-called "fixed positive real" differs from the limit, it means that we are in the presence of a jump, a discontinuity, and that the mathematical law of the game is not respected to the end. In a game of chance, the two assertions "the winning is infinite" and "the probability of winning is a positive real" are incompatible. Pascal's wager is not in the line of games of chance, but in a break with them. Pascal's reasoning goes beyond the framework in which he placed himself. If it is a kind of miracle, it will have to be explained, preferably by reason rather than by faith.
4. Moreover, by substituting the assertions "the winning is infinite" and "the probability of winning is a real positive" in the formula E = -b + w⋅p, we obtain an infinite mathematical expectation, which can be approached by "if the promise of winning is gigantic, then one is almost certain to become immensely rich". This is an assertion that the victims of charlatans wrongly feed on.