History of the problem of points.

1. Ryan Wang
STAT267
5/30/2012
Term Paper
Various Solutions to the Problem of Points
The problem of points is a classical problem in the history of statistics, and indeed all of
mathematics. It provided the motivation for a correspondence between Blaise Pascal and
Pierre de Fermat in 1654, the results of which became the starting point for probability
theory. Landmark works such as Huygens’s De ratiociniis in aleae ludo (1657), Montmort’s
Essai danalyse sur les jeux de hazards (1708) and De Moivre’s De mensura sortis (1712)
discuss solutions to the problem in increasing generality. Laplace’s landmark article on
inverse probability, “Memoire sur la probabilite des causes par les evenemens” (1774),
includes a Bayesian analysis of the problem. A problem which must have started out as
a gambler’s curiosity ultimately attracted the attention of various giants from the early
development of probability theory. Also known as the problem of division or the problem
of the unfinished game, the problem of points concerns a hypothetical gambling game which
has been prematurely ended. At the outset, players contribute equal stakes and agree to
play until a predetermined number of points have been achieved. Play proceeds in rounds,
with the winner of each round receiving a point. If the game is interrupted, how should
the total stake be divided amongst the players? In this paper, I trace the history of and
some solutions to this problem of points.
Early Proposals: Pacioli, Cardano and Tartaglia
The problem of points has a long history, with special cases appearing in Italian
manuscripts as early as 1380 (Ore, 1960). The earliest known printed version appears
in a mathematical textbook, Summa de arithmetica, geometria, proportioni et proportion-
alita (1494), written by Luca Pacioli. There, the problem does not concern a gambling
1

2. game but an interrupted ball game for which an appropriate division of prize money is
required. Suppose players A and B have a and b points, respectively. Pacioli’s proposal is
essentially to divide the stake according to each player’s relative number of points, so that
A receives a
a+b
parts of the stake and B receives b
a+b
parts (Pacioli, 1494).
Gerolamo Cardano, another Italian mathematician, discussed Pacioli’s errors “that even
a child should recognize” in his textbook, Practica arithmetice et mensurandi singularis.
If only one round were played, Pacioli’s proposal would call for the entire stake to be
given to the player with one point regardless of how many points were required to win.
Moreover, it is possible for a player to receive a smaller portion despite getting closer to the
required number of points. For example, suppose that A and B play to 13 points. When
A has 3 points and B has 1 point, A would receive 3/4 of the stake. However if the game
were to continue such that A has 12 points and B has 9 points, then A would receive 4/7
of the stake. In this case A receives a smaller portion than before, despite progressing to
within one point of the objective. Cardano suggests that the number of points still required
to win should determine the division. If A has a points, B has b points and n points are
required to win, then the stakes should be divided according to the arithmetic progressions,
p(a, n) = 1 + ... + (n − a) and p(b, n) = 1 + ... + (n − b). In particular A’s stake should be
proportional to p(b,n)
p(a,n)+p(b,n)
(Cardano, 1539).
Niccolo Tartaglia, Cardano’s great rival, provided yet another proposal in his textbook,
La prima parte del general trattato di numeri et misure (1556). Tartaglia prefaced his
proposal with the qualification that “the resolution of such a question is judicial rather
than mathematical, so that in whatever way the division is made there will be found
arguing.” He suggested to divide the stake according to the difference in the two players’
points. That is, if A has more points than B, then A should receive his own original stake
as well as a part of B’s original stake proportional to the difference in their points. In
particular A’s stake should be proportional to n+a−b
2n
. If each player were to stake S at the
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3. outset, so that the total stake were 2S, then A should receive S + a−b
n
S.
We see that the problem of points was known in Europe well before the Chevalier de
Mere posed it to Blaise Pascal in 1654. Although early proposals appeared in mathematical
texts, they lacked grounding in mathematical principle. Note that they deal only with the
case of two players and assume that each player has an equal chance of winning any given
point. Interestingly, the method of exhaustive enumeration of possibilities was known to
Cardano and can lead to a correct solution (Edwards, 1982). Indeed Cardano had applied
such a method to study various games of dice and cards. Nevertheless, a satisfactory
solution remained elusive.
Pascal and Fermat’s 1654 Correspondence
Pascal and Fermat’s famous correspondence in the summer of 1654 resulted in not one,
but three methods for correctly solving the problem of points for players of equal skill. Their
formulation of the problem involved a dice game, but this game is equivalent to flipping a
two-sided coin. An expression for the general solution was not given in the correspondence,
buts methods to do so were clearly laid out. Note that Pascal did give an expression in
his Traite du triangle arithmetique (Edwards, 1982), and proved that if A and B require a
and b points, respectively, than A’s stake should be proportional to
a−1
k=0
a + b − 1
k
/2a+b−1
.
The first method of solution proceeds via an enumeration of all possible sequences of
heads and tails, and a subsequent tabulation of the winner in each case. This required first
finding the maximum number of rounds remaining. In Pascal’s letter dated July 29th, he
wrote, “It is necessary to say in the first place, if one has one game of 5, for example, and
that thus he lacks 4, the game will be infallibly decided in 8, which is double of 4.” More
generally if A and B require a and b points, respectively, then the game must be completed
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4. in at most a + b − 1 more rounds. Thus, there are 2a+b−1
equally probable sequences
for which the winner can be determined, although some sequences include rounds played
after the completion of the original game. In another letter, Pascal responded to M. de
Roberval’s objections to this aspect. Note that both Pascal and Fermat produced similar
such solutions (Edwards, 1982).
The second method, found in Pascal’s letter dated July 29th, involves recursively ap-
plying the idea of expectation. His method is effectively to work backwards from the end
of game and apply the principle that if gains of X and Y are equally probable, then the
expected gain is worth X+Y
2
. For example, consider the case where the players play to 3
games, with 64 pistoles at stake, and A and B having 2 points and 1 point, respectively.
Pascal imagines, “Now the play one game... if the first wins it, he wins all the money which
is in the game, namely 64 pistoles; if the other wins it, they are two games to two games,
and consequently, if they wish to separate themselves, it is necessary that they each take
back their stake, namely 32 pistoles each.” Pascal next reasons according to the idea of
expectation, “Therefore, if they do not wish to risk this game and to separate themselves
without playing it, the first must say: I am certain to have 32 pistoles, because even the
loss gives them to me; but for the other 32, perhaps I will have them, perhaps you will
have them, the chance is equal. Therefore we divide these 32 pistoles in half and give me,
beyond those, my 32 which are certain for me.” Pascal does not fully generalize his method
in this correspondence, but he does address the cases where A requires 1 point and B any,
using induction, and where A and B are separated by only one point, using his method of
combinations. The third method, due to Fermat, involves enumerating the possibilities by
working forwards along an event tree but does not introduce any new principle (Edwards,
1982).
Extensions to Multiple Players of Unequal Skill
Following Pascal and Fermat’s breakthrough, various solutions to extensions of the
4

5. problem of points began to appear. Huygens’s textbook De ratiociniis in aleae ludo (1657)
discussed the problem in the case of any number of players of equal skill. Huygens did
not give a general expression but, as with Pascal and Fermat, the method of solution is
clearly laid out in the text. Later, Pascal’s general solution was included as a formula
in the first edition of Pierre Montmort’s textbook Essai d’analyse sur les jeux de hazards
(1708). John Bernoulli communicated a formula for the case of players of unequal skill,
which Montmort published in the second edition of his book in 1713 (Edwards, 1982). This
formula is later cited by Laplace in his Bayesian treatment of the problem. Montmort’s
treatment begins with a reprinting of a letter from Pascal to Fermat dated August 24,
1654 before publishing a general expression for their solution. Montmort considers a game
between Pierre and Paul, the characters from Pascal and Fermat’s correspondence, “Let p
be the number of points which are lacking to Pierre, q the number of which points which
are lacking to Paul. We demand a formula which expresses the lot of the Players.” Letting
p + q − 1 = m, Montmort gives Pascal’s formula for Pierre’s stake. Next, he considers the
case when Pierre and Paul are not on equal footing, “supppose that the number of chances
that Pierre has to win each point, or if we wish that his lot be to that of Paul as a to b”
(Montmort, 1713). Expressed as a series, Pierre’s stake is thus proportional to
am
b0
+ mam−1
b +
m · (m − 1)
1 · 2
am−2
b2
+ &c.
where the number of terms to include is q and the denominator is (a + b)m
rather than 2m
in the case of equal skill.
Laplace’s Bayesian Solution
In 1774, the great French mathematician Pierre Simon Laplace published his “Memoir
sur la probabilite des causes par les evenemens,” a vastly influential article on inverse
probability. Section IV of this article gives what we would today call a Bayesian analysis of
the problem of points. Far from the focus of the work, the problem is presented as a simple
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6. example. Interestingly, it also appears as an example in Laplace’s work on the theory of
recurrent series, “Recherches, sur l’integration des equations differentielles aux differences
finies, & sur leur usage dans la theorie des hasards,” read to the Paris Academy of Sciences
in 1773 and published in 1776. Laplace’s approach using inverse probability is quite familiar
to the modern reader. He considers the posterior distribution of the unknown probability
that A wins a point by applying a principle equivalent to Bayes’s Theorem. To derive the
appropriate division, he effectively calculates the expectation of the predictive distribution
under binomial sampling and a uniform prior. Extending the previous treatments, Laplace’s
solution treats the respective skills of the players as unknown, though it only holds for two
players.
Laplace begins by establishing a principle which will form the foundation for the re-
mainder of the article. He states, “If an event can be produced by a number n of different
causes, the probabilities of these causes given the event are to each other as the probabilities
of the event given the causes, and the probability of the existence of each of these is equal
to the probability of the event given that cause, divided by the sum of all the probabilities
of the event given each of these causes” (Laplace, 1774). This translates to
P(E|Ci)
P(E|Cj)
=
P(Ci|E)
P(Cj|E)
and
P(Ci|E) =
P(E|Ci)
n
j=1 P(E|Cj)
where E denotes an event and Ci denote causes for i = 1, ..., n. This is equivalent to
Bayes’s Theorem when all the causes are a priori equally likely, that is when P(C1) =
... = P(Cn). Stigler (1986) argues that Laplace’s principle was derived independently of
Bayes’s Theorem, and indeed that it was Laplace’s article and not Bayes’s 1764 article
that influenced the spread of inverse probability prior to the twentieth century. Regardless
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7. of priority, Laplace’s work is a great achievement. Among other things, he evaluates the
integral of the normal density, introduces techniques for asymptotic analysis of posterior
distributions, and characterizes the posterior median as the estimator that minimizes the
absolute posterior expected loss.
The problem of points appears as Problem II in Laplace’s article. Laplace phrases the
problem in a familiar way, “Two players A and B, whose respective skills are unknown, play
some game, for example piquet, where the first player to win a number n points receives
a sum S (originally a) deposited at the beginning of play. I suppose that the two players
are forced to abandon play with player A lacking f points and player B lacking h points.”
Unlike previously, the respective skills of the players are unknown. By skill, Laplace means
the probability of winning a given round. If these probabilities were known to be x and
1 − x for A and B, respectively, then B should receive a stake of
S · (1 − x)f+h−1
·



1 +
x
1 − x
· (f + h − 1)
+
x2
(1 − x)2
·
(f + h − 1) · (f + h − 2)
1 · 2
+ ...
+
xf−1
(1 − x)f−1
·
(f + h − 1) · · · (h + 1)
1 · 2 · · · (f − 1)
citing the formula given in the second edition of Montmort’s textbook. Denote this stake
by φ(x|S, f, h), since x will be allowed to vary in what follows.
Now take the respective skills, x and 1 − x, to be unknown. Since A lacks f points and
B lacks h points, they have accumulated n − f and n − h points, respectively. Laplace
implicitly assumes a binomial model. Thus the probability of observing n − f points to A
and n − h points to B after 2n − f − h total points, an event which we will denote by E,
given x is
P(E|x) = xn−f
(1 − x)n−h
.
7

8. Applying Laplace’s principles gives the posterior distribution of x as
P(x|E) =
xn−f
(1 − x)n−h
1
0
xn−f (1 − x)n−hdx
.
Thus B’s stake should be given by the posterior expectation of φ(x|S, f, h),
Ex|E[φ(x|S, f, h)] =
1
0
φ(x|S, f, h) · p(x|E)dx
= S
1
0
(1 − x)f+h−1
xn−f
(1 − x)n−h
·
1 + x
1−x
· (f + h − 1) + ... + xf−1
(1−x)f−1 · (f+h−1)···(h+1)
1·2···(f−1)
1
0
xn−f (1 − x)n−hdx
.
In modern notation, the expression involves a number of Beta integrals e.g.
1
0
xn−f
(1 − x)n−h
dx = Beta(n − f + 1, n − h + 1)
=
(n − f)!(n − h)!
(2n − f − h + 1)!
=
1 · 2 · · · (n − h)
(n − f + 1) · · · (2n − f − h + 1)
which can easily be calculated using integration by parts.
I believe that there may be a typo in the manuscript that I have consulted (Laplace,
1774), which gives
1
0
xn−f
dx · (1 − x)f+h−1
=
1 · 2 · 3 · · · (f + h − 1)
(n − f + 1) · · · 2n
=
(n − f)!(f + h − 1)!
(2n)!
but should actually be
1 · 2 · 3 · · · (f + h − 1)
(n − f + 1) · · · (n + h)
=
(n − f)!(f + h − 1)!
(n + h)!
= Beta(n − f + 1, f + h).
8

9. for two reasons. The first expression does not actually appear to coincide with any Beta
integral. The source of the error seems to be a missing factor of (1−x)n−h
in the expression
given for “the sum that should really be returned to player B.” Since (1 − x)n−f
· (1 −
x)f+h−1
= (1 − x)n+h−1
, the integral of interest becomes
1
0
xn−f
dx · (1 − x)f+n−1
=
1 · 2 · 3 · · · (f + n − 1)
(n − f + 1) · · · 2n
in which case the formula two formulas are the same after replacing f +h−1 with f +n−1.
However, it seems strange to have a term only involving f and not h, so perhaps I have
miscalculated, since this also leads to a different expression for the final sum that should
be returned to B compared to what is given in the manuscript. The final sum is given to
be
S · (n−h+1)···(h+f−1)
(2n−f−h+2)···2n
·



1 +
(f + h − 1)
1
·
n − f + 1
f + h − 1
+
(f + h − 1) · (f + h − 2)
1 · 2
·
(n − f + 1) · (n − f + 2)
(f + h − 1) · (f + h − 2)
+ ...
+
(f + h − 1) · · · (h + 1)
1 · 2 · · · (f − 1)
·
(n − f + 1) · · · (n − 1)
(f + h − 1) · · · (h + 1)
.
Conclusion
The problem of points has a rich history and has drawn the attention of great thinkers from
Pascal and Fermat to Huygens to Laplace. Its solution by Pascal and Fermat introduced key
principles which would provide the foundation for probability theory. It was also present
in Laplace’s landmark treatise on inverse probability, and his Bayesian treatment extended
the problem to the case of players of unknown skill. A study of the various solutions to
this problem serves as an enriching introduction to some of the important techniques in
the early history of statistics.
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10. Sources
Cardano, Gerolamo. 1539. Practica arithmetice et mensurandi singularis. English transla-
tion by Richard Pulskamp.
Edwards, A.W.F. 1982. “Pascal and the Problem of Points,” International Statistical
Review, 50(3): 259-266.
Laplace, Pierre Simon. 1774. “Memoire sur la probabilite des causes par les evenemens.”
English translation by Stephen M. Stigler, appearing in Statistical Science, 1(3): 364-378.
Montmort, Pierre. 1713. Essai d’analyse sur les jeux de hazard. 2nd Edition. English
translation by Richard Pulskamp.
Ore, Oystein. 1960. “Pascal and the Invention of Probability Theory,” American Mathe-
matical Monthly, 67(5): 409-419.
Pacioli, Fra Luca. 1494. Summa de arithmetica, geometria, proportioni et proportionalita.
At http://echo.mpiwg-berlin.mpg.de/content. English translation by Richard Pulskamp.
Pascal, Blaise and Pierre Fermat. 1654. “Correspondence on the problem of points”
published in Oeuvres de Fermat 2: 299-314. English translation by Richard Pulskamp.
Stigler, Stephen M. 1986. “Laplace’s 1774 Memoir on Inverse Probability,” Statistical
Science, 1(3): 359-363.
Tartaglia, Niccolo. 1556. La prima parte del general trattato di numeri et misure. English
translation by Richard Pulskamp.
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