Pascal’s wager 3/4: Mathematical model and rebuttal

It is legitimate to put Pascal's wager on the school curriculum. But it happens that some teachers, with little respect for secularism, develop this theme beyond what is required by the culture to make it a missionary tool, the aim being to prepare the pupils to welcome the faith1. When ideology prevails over the critical sense, the pupil must perceive it clearly. Reason then requires a counterweight to be opposed to it.

Pascal's wager

«But your bliss? Let us weigh the winning and the loss, betting that God is. Let us consider these two cases: if you win, you win everything; if you lose, you lose nothing. Wager, then, that He is, without hesitation.» Blaise Pascal, Thoughts, 1670

The reasoning behind Pascal's wager is circular

Let us temporarily assume the value of one chance in two for the probability that God exists. If this is the case, one gains eternal life in Paradise, and the gain is infinite. If not, one loses nothing. The choice seems easy to make.

However, one must be wary of hidden assumptions. First of all, in the object of the wager, there is not only the existence of God, but also that the Catholic religion would be true and that religious practice would lead to Paradise. Secondly, it is prudent to examine what is covered by the term "infinite".

In mathematics, infinity appears as the limit of sequences. Consider for example the following suggested sequence:

• in a game with a zero bet, every time you try, you win a thousand euros randomly every other time;
• in a game with a zero bet, every time you try, you win a million euros randomly every other time;
• in a game with a zero bet, every time you try, you win a billion euros randomly every other time,
• and "so on".

However, the earth's resources are limited. To pronounce the "so on", one must admit that the supernatural exists. In other words, Pascal implicitly assumes the existence of God, which constitutes a vicious circle, a circular reasoning.

Generalised formulation of Pascal's wager

Initially, Pascal's wager was supposed to support the Catholic faith. But its central element - the possibility of a gigantic gain - is not specifically Christian and can be adapted to any doctrine that promises much. Its versatility even allows its principle to be exploited far beyond the religious realm. Its general formulation is: "The more wonderful the promise, the more justified it is to bet on it".

Variations on Pascal's wager

An advertisement is displayed: "If you buy this product, you will be happier. If you give it up, you are depriving yourself of a great service. Weigh the pros and cons, and don't hesitate to buy it!".

A speech by a politician: "I'm going to improve the future of society, and you will be able to enjoy it at your leisure. It's worth betting on me: I'm counting on your vote!".

A healer who asks to have faith in his powers: "If you trust me, your illness will disappear and you will be able to live a long time. Why not try, since there is so much to be gained?"

The Christian priest who speaks in the name of Jesus: "If you follow me, you will be rewarded with eternal happiness. Become my disciple, and your gain will be infinite!"

Beyond charlatanism

An unverified hypothesis remains a hypothesis whose confirmation or rebuttal is postponed to the future. On the other hand, an "unverifiable hypothesis" loses its status as a hypothesis to become a fable or an ideology.

The principle of Pascal's wager puts the gullible to sleep by the immediate comfort provided by the hope of a miraculous payoff. The huckster is indifferent to true and false, for he is concerned only with pleasing, to his greatest advantage. While the promises of charlatans can be invalidated by the absence of expected results, those of religious propagandists, being absolutely unverifiable, go further than charlatanism.

For lovers of mathematical expectation

In the context of Pascal's wager, the bet, which is the Christian commitment, is fixed, or at least capped. In what follows, we assume it to be constant. Two variables remain: the winning and the probability of winning. In all games of chance, the more you aim for a high winning, the lower the probability of winning. For example, if you bet 1 euro, it is a fair game to be able to win 1000 euros with a probability of 1/1000; in another game, if you bet 1 euro, it is a fair game to be able to win 1,000,000 euros with a probability of 1/1,000,000. In this context, we can affirm that, when the winning tends towards infinity, the probability of winning tends towards 0.

What happens if the mathematical expectation of the net winning E of the game is non-zero? The formula to be considered is as follows:

\[ p = \frac{E+bet}{winning} \]

While the players to whom the wager is addressed expect a net winning expectation close to zero, i.e. a game that is not too biased, believers imagine an immense net winning expectation. But this doesn't change anything: even if E is worth a billion, when the winning tends towards infinity, the probability of winning tends towards 0.

If the probability of winning is positive, to make the winning tend towards infinity is tantamount to admitting the supernatural. But this cannot be hypothesised, since that is precisely what we want to prove. In the context of games of chance, the two assertions "the winning is infinite" and "the probability of winning is a real positive" are incompatible.

The above principle can now be corrected: «The more wonderful the promise, the less likely it is. And, in the end, it is implausible.»

To reinforce by another argument that "the probability of obtaining an infinite winning is null", we can refer to the document On the likelihood that a given religion is true [see Pascal's wager 2], which brings us to the following situation:

\[ E = -bet + \underbrace{\overbrace{winning}^{\to \infty} \cdot \overbrace{p}^{\to 0}}_{\text{indefinite}} \]

where (winning) tends towards infinity and p tends towards 0.

We are faced with an indetermination of the type "infinite times zero". Thus the mathematical reasoning comes to an impasse, and the conclusions drawn by Pascal are unfounded.

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.

Christiaan Huygens

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

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

The mathematical expectation of the net winning is

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

The formula for mathematical expectationThis 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.

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, \( \quad | \text{mathematical expectation} | \leq E \)

In the end, Pascal's wager is unfounded.

Discussion

Question or objection

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

Answer

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.

Question or objection

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

Answer

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 [see Pascal's wager 4: Reversal of the wager].

Question or objection

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"?

Answer

    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  \( w_\epsilon = (E+b)/ \epsilon \). As the mathematical model produces a sequence of probabilities that tends towards zero, the consequence is that all winnings that are greater than \( w_\epsilon \) 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 \( E = -b + w \cdot p \) , 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.

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