Christiaan Huygens is well-known for his use of symmetry arguments in mechanics. But he also used symmetry arguments when he set out the foundations of the modern theory of probability, in delightfully easy form. That is my reading, I’ll explain it here.
Note: I rely on Freudenthal’s translation from the Dutch where possible, my own otherwise, and will list the English translation in the Bibliography,for comparison.
1. What is a symmetry argument?
An object or situation is symmetric in a certain respect if relevant sorts of changes leave it just the same in that respect. For example if you hang a square painting upside down it is still square, though in other respects it is not the same. If you say it is symmetric, you are referring to certain characteristics and ignoring certain differences
A symmetry argument exploits the differences that remain after the selection of what counts as relevant. Suppose we want to solve a problem that pertains to a given situation S. We state the same problem for a different situation S*, mutatis mutandis, and solve it there. Then we argue that S* and S are essentially the same, that is, the same in all respects relevant to the problem. On this basis we take the solution for this problem for S to be the same as the solution we found for S*.
Whatever be the problem at issue, it is not well posed except in the presence of a prior selection of which aspects will count as the relevant respects. So that prior selection must be understood as a given in each case.
2. Huygens’ fundamental postulate
Postulate. For a decision to be made under uncertainty, in a given situation S, there is an equitable (just, fair, “rechtmatig”) game of chance S* that is in all relevant respects the same as S.
More specifically, if I am offered an opportunity for an investment, what that opportunity is worth equals what it would be worth for me to enter the corresponding equitable game.
A game is equitable if no player is at a disadvantage compared to the others (“daer in niemandt verlies geboden is”). That is a symmetry: the players all have the same role. If roles are reversed the game is still the same. For example if a Bookie offers a bet to a Player, the bet is an equitable bet if the chances and amounts of gain and loss would remain the same for each if their roles were reversed.
The Netherlands was a mercantile nation, financiers would get together to outfit ships for trade i the Orient. When this sort of situations, with their opportunities for investment, the merchant must determine what those opportunities are worth. The relevant respects are just the chances the players have of gaining the various possible gains or losses, and the amounts of those gains or losses. Note well that the selection of the respects which alone count as relevant is a matter of choice, of the participants. Given that Huygens addresses the case in which the relevant respects are the gains and chances alone, the Postulate is not a substantive assertion, nor an empirical claim.
Given this postulate and the concept of an equitable game, everything is in place for a typical symmetry argument.
3. A chancy set-up with two outcomes with equal chances
Proposition I. If I have the same chance to get a or b it is worth as much to me as (a + b)/2.
The situation might be the offer of an investment opportunity. What is the corresponding equitable game?
In an equitable game each player places the same stake, and the winner will get the total stake, but my have side-contracts with other players (as long as the relevant symmetries are not broken).
I play against another person, we each put up the same stake x (what we take the game to be worth for us to play), and we agree that whoever wins will give a to the other. The event (e.g. a coin toss) has two possible outcomes, with equal chance. What must be the stake x so that the game situation is perfectly symmetric for us, that is, for each of us to have equal chances to receive either a or b?
We have equal chances to be winner or loser. The winner gets the total stake, which is 2x, but gives a to the loser. The result must be that the winner ends up with b. So b = (2x – a). So solving for x, we arrive at the solution that the stake is x = (a + b)/2.
That is straightforward, and all the relevant details are explicit. It is not so straightforward when the proposition is generalized to a number of possible outcomes.
4. A chancy set-up with three outcomes with equal chances
Proposition II. If I have the same chance to get a or b or c it is worth as much to me as (a + b + c)/3.
Huygens’ argument here makes the corresponding equitable game to be one with three players. Each will enter with the same stake x. Let me be player P1. With player P2 I agree that each of us, if winning, will give the other b. With player P3 I agree that each of us, if winning, will give the other c.
The event is one with three outcomes, with equal chances, and the winner being P1 on the first outcome, P2 on the second, and P3 on the third. If P2 wins, I get b. If P3 wins, I get c. What must the stake x be to complete the picture, with me getting a if I win?
If I win I get the total stake 3x, but pay out b to P2 and c to P3. So what is required is that a = (3x – b – c), or equivalently, that the stake x = (a + b + c)/3.
At first sight this is again straightforward. It answers what my stake should be.
But what if the other players didn’t want to put up that stake? Is player P2 in the same position in this game as P3, given that they have made different contracts with me?
What was not explicit in Huygens’ proof is that players P2 and P3 must do something ‘just like’ what I did by entering those special contracts.
Suppose all players do place stake x = (a + b + c)/3. Player P3 will get a if he wins, and will get c if I win. To complete the picture he must get b if P2 wins. And similarly, P2 needs to get c if P3 wins. So P2 and P3 would have to make an un-symmetric contract.
Is that not to the disadvantage of one, depending on which of b or c is the greater? No, for whoever wins will get a, and if they do not win the have an equal chance of getting b or c.
| WHAT THEY RECEIVE | |||
| If the winner is: | P1 | P2 | P3 |
| P1 | 3x – b- c | b | c |
| P2 | b | 3x-b-c (c to P3) | c |
| P3 | c | b (from P3) | 3x – b – c (b to P2) |
In the description of the game, the roles I gave myself and the roles the others play are not the same. But in fact the game is equitable, because the different roles are the same in the relevant respects. The difference does not affect the chances that each of us have to get any of the three amounts a, b, c, which are the same for all of us. Therefore if we reverse roles, if P3 and I change chairs, so to speak, our chances for receiving any of those good outcomes are not changed. Think of a game of cards in which one of the players holds the bank. If the game is equitable, it does not matter who holds the bank, and makes no difference if the banker changes roles with one of the others.
What matters once again is the prior selection of what will count as relevant.
As Huygens points out, this argument is easily extended to 4, 5, … different outcomes with equal chances.
5. A chancy set-up with unequal chances
So far we have looked at games in which the deciding event is something like a toss with a fair coin or with a fair die, or with several of them, or some other such device. What about games in which the decision about who wins is made with a biased coin or biased die? What if the possible gains are a and b but their chances are different?
The words “chance” and “chances”. There is some ambiguity in how we use the word “chance” which appears saliently in Proposition III. If I buy a ticket in a 1000-ticket lottery, the chance I have of winning is 1/1000. What if I buy five tickets? Then I have five chances of winning! Or, we also calculate, then the chance of winning I have is 5/1000.
In the vernacular these are two ways of saying the same thing, but these two ways of speaking do not go together. Putting them together we get nonsense: If I say that I have five chances of winning, and that the chance of winning is 5/1000, can I then ask which of those five chances is the chance of winning?
Proposition III. If the number of chances I have for a is p, and the number of chances I have for b is q, then assuming that every chance can happen as easily, it is worth as much to me as (pa + qb)/(p + q).
Huygens uses the first way of speaking, and we can understand as follows, with an example. For each game with a biased coin or die there is an equivalent game with cards. Consider a game in which the deciding event is the toss which a biased coin, so that the chance of Heads is 2/3 and chance of Tails is 1/3. This game is equivalent to a 3-card game, with equal chances, in which two cards are labeled H and one is labeled T, and the prize is a if a card with H is drawn and is b if a card with T is drawn.
This assertion of equivalence is an appeal to symmetry. The two games are the same in all relevant respects, thereforereasoning about the one is equally relevant reasoning about the other.
Just by choosing the first way of speaking, Huygens has reduced the problem of the biased coin or die to the previous case. The game with unequal chances is equivalent to a game with equal chances, and this Proposition III is the straightforward generalization of Proposition II to arbitrarily large cases.
NOTES
Expectation value. What Huygens calls ‘what it is worth for me” (e.g. “dit is mij zoveel weerdt als …” in Proposition I) matches what we now call the expectation value. We nevertheless read Huygens’ monograph as a treatise on probability, for the two notions are interdefinable. For example the probability of proposition A is the expectation value of a gamble on A with outcome 1 if true and 0 if false.
Finitude. Huygens arguments go only as far as cases with a finite number of outcomes, with probabilities that are all rational numbers.
Translation. Christiaan Huygens’ little 1657 monograph was the first modern book on probability theory. It was first written in Dutch with the title Van Rekeningh in Spelen van Geluck. We note in passing that the Dutch word (“geluk”, in modern spelling) means “chance” in this context but it is the same word for fortune and for happiness. (Does that reveal something about the Dutch character?) Van Schooten translated this into Latin and published it as part of a larger workd. Hans Freudenthal (1980) offered a number of criticisms of how the text is understood (the French translation was excellent, he says, and the German abysmal) so provided his own English translation of various parts. The English translation of 1714 is precious for its poetic character, and still easily available.
The original Dutch version and information on the translations is available online from the University of Leiden:
Text: https://web.universiteitleiden.nl/fsw/verduin/stathist/huygens/1660.pdf
Information: https://web.universiteitleiden.nl/fsw/verduin/stathist/huygens/factfigs.htm
BIBLIOGRAPHY
Freudenthal, Hans (1980) “Huygens’ Foundations of Probability. Historia Mathematica 7: 113-117. Available online: https://core.ac.uk/download/pdf/81947283.pdf
Huygens, Christiaan and W. Kleijne (1998) Van Rekeningh in Spelen van Geluck. Epsilon Uitgaven.
Huygens, Christian. The Value of Chances in Games of Fortune. English translation 1714. https://math.dartmouth.edu/~doyle/docs/huygens/huygens.pdf
Shafer, Glenn (2019) Pascal’s and Huygens’s game-theoretic foundations for probability. Sartoniana 32 117-145. 2019
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