_INSURANCE AND GAMBLING._
1. If the reader will recall to mind the fundamental
postulate of the Science of Probability, established and explained
in the first few chapters, and so abundantly illustrated
since, he will readily recognize that the two opposite
characteristics of individual irregularity and average regularity
will naturally be differently estimated by different
minds. To some persons the elements of uncertainty may
be so painful, either in themselves or in their consequences,
that they are anxious to adopt some means of diminishing
them. To others the ultimate regularity of life, at any rate
within certain departments, its monotony as they consider it,
may be so wearisome that _they_ equally wish to effect some
alteration and improvement in its characteristics. We shall
discuss briefly these mental tendencies, and the most simple
and obvious modes of satisfying them.
To some persons, as we have said, the world is all too full
of change and irregularity and consequent uncertainty. Civilization
has done much to diminish these characteristics in
certain directions, but it has unquestionably aggravated them
in other directions, and it might not be very easy to say with
certainty in which of these respects its operation has been, at
present, on the whole most effective. The diminution of
irregularity is exemplified, amongst other things, in the case
of the staple products which supply our necessary food and
clothing. With respect to them, famine and scarcity are by
comparison almost unknown now, at any rate in tolerably
civilized communities. As a consequence of this, and of the
vast improvements in the means of transporting goods and
conveying intelligence, the fluctuations in the price of such
articles are much less than they once were. In other directions,
however, the reverse has been the case. Fashion, for
instance, now induces so many people in every large community
simultaneously to desire the same thing, that great
fluctuations in value may ensue. Moreover a whole group of
causes (to enter upon any discussion of which would be to
trench upon the ground of Political Economy) combine to
produce great and frequent variations in matters concerning
credit and the currency, which formerly had no existence.
Bankruptcy, for instance, is from the nature of the case,
almost wholly a creation of modern times. We will not attempt
to strike any balance between these opposite results
of modern civilization, beyond remarking that in matters of
prime importance the actual uncertainties have been probably
on the whole diminished, whereas in those which affect
the pocket rather than the life, they have been rather increased.
It might also be argued with some plausibility
that in cases where the actual uncertainties have not become
greater, they have for all practical purposes done so, by their
consequences frequently becoming more serious, or by our
estimate of these consequences becoming higher.
2. However the above question, as to the ultimate
balance of gain or loss, should be decided, there can be no
doubt that many persons find the present amount of uncertainty
in some of the affairs of life greater than suits their
taste. How are they to diminish it? Something of course
may be done, as regards the individual cases, by prudence
and foresight. Our houses may be built with a view not to
take fire so readily, or precautions may be taken that there
shall be fire-engines at hand. In the warding off of death
from disease and accident, something may be done by every
one who chooses to live prudently. Precautions of the above
kind, however, do not introduce any questions of Probability.
These latter considerations only come in when we begin to
invoke the regularity of the average to save us from the
irregularities of the details. We cannot, it is true, remove
the uncertainty in itself, but we can so act that the consequences
of that uncertainty shall be less to us, or to those in
whom we are interested. Take the case of Life Insurance.
A professional man who has nothing but the income he earns
to depend upon, knows that the whole of that income may
vanish in a moment by his death. This is a state of things
which he cannot prevent; and if he were the only one in
such a position, or were unable or unwilling to combine with
his fellow-men, there would be nothing more to be done in
the matter except to live within his income as much as possible,
and so leave a margin of savings.
3. There is however an easy mode of escape for him.
All that he has to do is to agree with a number of others,
who are in the same position as himself, to make up, so to
say, a common purse. They may resolve that those of their
number who live to work beyond the average length of life
shall contribute to support the families of those who die
earlier. If a few only concurred in such a resolution they
would not gain very much, for they would still be removed
by but a slight step from that uncertainty which they are
seeking to escape. What is essential is that a considerable
number should thus combine so as to get the benefit of that
comparative regularity which the average, as is well known,
almost always tends to exhibit.
4. The above simple considerations really contain the
essence of all insurance. Such points as the fact that the
agreement for indemnity extends only to a certain definite
sum of money; and that instead of calling for an occasional
general contribution at the time of the death of each member
they substitute a fixed annual premium, out of the proceeds
of which the payment is to be made, are merely accidents of
convenience and arrangement. Insurance is simply equivalent
to a mutual contract amongst those who dread the consequences
of the uncertainty of their life or employment,
that they will employ the aggregate regularity to neutralize
as far as possible the individual irregularity. They know
that for every one who gains by such a contract another will
lose as much; or if one gains a great deal many must have
lost a little. They know also that hardly any of their number
can expect to find the arrangement a 'fair' one, in the
sense that they just get back again what they have paid in
premiums, after deducting the necessary expenses of management;
but they deliberately prefer this state of things.
They consist of a body of persons who think it decidedly
better to leave behind them a comparatively fixed fortune,
rather than one which is extremely uncertain in amount;
although they are perfectly aware that, owing to the unavoidable
expenses of managing the affairs of such a society,
the comparatively fixed sum, so to be left, will be a trifle less
than the average fortunes which would have been left had
no such system of insurance been adopted.
As this is not a regular treatise upon Insurance no more
need be said upon the exact nature of such societies, beyond
pointing out that they are of various different kinds. Sometimes
they really are what we have compared them with,
viz. mutual agreements amongst a group of persons to make
up each other's losses to a certain extent. Into this category
fall the Mutual Insurance Societies, Benefit Societies, Trades
Unions (in respect of some of their functions), together with
innumerable other societies which go by various names.
Sometimes they are companies worked by proprietors or
shareholders for a profit, like any other industrial enterprise.
This is the case, I believe, with the majority of the
ordinary Life Insurance Societies. Sometimes, again, it is
the State which undertakes the management, as in the case
of our Post Office Insurance business.
5. It is clear that there is no necessary limit to the
range of application of this principle.[1] It is quite conceivable
that the majority of the inhabitants of some nation
might be so enamoured of security that they should devise
a grand insurance society to cover almost every concern in
life. They could not indeed abolish uncertainty, for the
conditions of life are very far from permitting this, but they
could without much difficulty get rid of the worst of the
_consequences_ of it. They might determine to insure not
merely their lives, houses, ships, and other things in respect
of which sudden and total loss is possible, but also
to insure their business; in the sense of avoiding not only
bankruptcy, but even casual bad years, on the same principle
of commutation. Unfamiliar as such an aim may appear
when introduced in this language, it is nevertheless one
which under a name of suspicious import to the conservative
classes has had a good deal of attention directed to it. It is
really scarcely anything else than Communism, which might
indeed be defined as a universal and compulsory[2] insurance
society which is to take account of all departments of business,
and, in some at least of its forms, to invade the province
of social and domestic life as well.
Although nothing so comprehensive as this is likely to
be practically carried out on any very large scale, it deserves
notice that the principle itself is steadily spreading in every
direction in matters of detail. It is, for instance, the great
complaint against Trades Unions that they too often seek to
secure these results in respect of the equalization of the
workmen's wages, thus insuring to some degree against incompetence,
as they rightly and wisely do against illness and
loss of work. Again, there is the Tradesman's Mutual Protection
Society, which insures against the occasional loss
entailed by the necessity of having to conduct prosecutions
at law. There are societies in many towns for the prosecution
of petty thefts, with the object of escaping the same
uncertain and perhaps serious loss. Amongst instances of
insurance _for_ the people rather than _by_ them, there is of
course the giant example of the English Poor Law, in
which the resemblance to an initial Communistic system
becomes very marked. The poor are insured against loss
of work arising not only from illness and old age, but from
any cause except wilful idleness. They do not, it is true,
pay the whole premium, but since they mostly bear some
portion of the burden of municipal and county taxation
they must certainly be considered as paying a part of
the premium. In some branches also of the public and
private services the system is adopted of deducting a percentage
from the wage or salary, for the purpose of a
semi-compulsory insurance against death, illness or superannuation.
6. Closely connected with Insurance, as an application
of Probability, though of course by contrast, stands Gambling.
Though we cannot, in strictness, term either of these practices
the converse of the other, it seems nevertheless correct
to say that they spring from opposite mental tendencies.
Some persons, as has been said, find life too monotonous for
their taste, or rather the region of what can be predicted
with certainty is too large and predominant in their estimation.
They can easily adopt two courses for securing the
changes they desire. They may, for one thing, aggravate
and intensify the results of events which are comparatively
incapable of prevision, these events not being in themselves
of sufficient importance to excite any strong emotions. The
most obvious way of doing this is by betting upon them.
Or again, they may invent games or other pursuits, the
individual contingencies of which are entirely removed from
all possible human prevision, and then make heavy money
consequences depend upon these contingencies. This is
gambling proper, carried on mostly by means of cards and
dice and the roulette.
The gambling spirit, as we have said, seeks for the excitement
of uncertainty and variety. When therefore people
make a long continued practice of playing, especially if the
stakes for which they play are moderate in comparison with
their fortune, this uncertainty from the nature of the case
begins to diminish. The thoroughly practised gambler, if
he possesses more than usual skill (in games where skill
counts for something), must be regarded as a man following
a profession, though a profession for the most part of a risky
and exciting kind, to say nothing of its ignoble and often
dishonest character. If, on the other hand, his skill is below
the average, or the game is one in which skill does not tell
and the odds are slightly in favour of his antagonist, as in
the gaming tables, one light in which he can be regarded
is that of a man who is following a favourite amusement;
if this amusement involves a constant annual outlay on his
part, that is nothing more than what has to be said of most
other amusements.
7. We cannot, of course, give such a rational explanation
as the above in every case. There are plenty of
novices, and plenty of fanatics, who go on steadily losing in
the full conviction that they will eventually come out winners.
But it is hard to believe that such ignorance, or such
intellectual twist, can really be so widely prevalent as would
be requisite to constitute them the rule rather than the exception.
There must surely be some very general impulse
which is gratified by such resources, and it is not easy to see
what else this can be than a love of that variety and consequent
excitement which can only be found in perfection
where exact prevision is impossible.
It is of course very difficult to make any generalization
here as to the comparative prevalence of various motives
amongst mankind; but when one considers what is the
difference which most quiet ordinary whist players feel
between a game for 'love' and one in which there is a
small stake, one cannot but assign a high value to the
influence of a wish to emphasize the excitement of loss
and gain.
I would not for a moment underrate the practical dangers
which are found to attend the practice of gambling. It
is remarked that the gambler, if he continues to play for a
long time, is under an almost irresistible impulse to increase
his stakes, and so re-introduce the element of uncertainty.
It is in fact this tendency to be thus led on, which makes
the principal danger and mischief of the practice. Risk and
uncertainty are still such normal characteristics of even civilized
life, that the mere extension of such tendencies into
new fields does not in itself offer any very alarming prospect.
It is only to be deprecated in so far as there is a
danger, which experience shows to be no trifling one, that
the fascination found in the pursuit should lead men into
following it up into excessive lengths.[3]
8. The above general treatment of Gambling and
Insurance seems to me the only rational and sound principle
of division;--namely, that on which the different practices
which, under various names, are known as gambling
or insurance, are arranged in accordance with the spirit of
which they are the outcome, and therefore of the results
which they are designed to secure. If we were to attempt
to judge and arrange them according to the names which
they currently bear, we should find ourselves led to no
kind of systematic division whatever; the fact being that
since they all alike involve, as their essential characteristic,
payments and receipts, one or both of which are necessarily
uncertain in their date or amount, the names may often be
interchanged.
For instance, a lottery and an ordinary insurance society
against accident, if we merely look to the processes performed
in them, are to all intents and purposes identical.
In each alike there is a small payment which is certain in
amount, and a great receipt which is uncertain in amount.
A great many persons pay the small premium, whereas a
few only of their number obtain a prize, the rest getting
no return whatever for their outlay. In each case alike,
also, the aggregate receipts and losses are intended to
balance each other, after allowing for the profits of those
who carry on the undertaking. But of course when we
take into account the occasions upon which the insurers
get their prizes, we see that there is all the difference in the
world between receiving them at haphazard, as in a lottery,
and receiving them as a partial set-off to a broken limb or
injured constitution, as in the insurance society.
Again, the language of betting may be easily made to
cover almost every kind of insurance. Indeed De Morgan
has described life insurance as a bet which the individual
makes with the company, that he will not live beyond a
certain age. If he dies young, he is pecuniarily a gainer, if
he dies late he is a loser.[4] Here, too, though the expression
is technically quite correct (since any such deliberate risk of
money, upon an unproductive venture, may fall under the
definition of a bet), there is the broadest distinction between
betting with no other view whatever than that of risking
money, and betting with the view of diminishing risk and
loss as much as possible. In fact, if the language of sporting
life is to be introduced into the matter, we ought, I presume,
to speak of the insurer as 'hedging' against his death.
9. Again, in Tontines we have a system of what is
often called Insurance, and in certain points rightly so,
but which is to all intents and purposes simply and absolutely
a gambling transaction. They have been entirely
abandoned, I believe, for some time, but were once rather
popular, especially in France. On this plan the State, or
whatever society manages the business, does not gain anything
until the last member of the Tontine is dead. As the
number of the survivors diminishes, the same sum-total of
annuities still continues to be paid amongst them, as long as
any are left alive, so that each receives a gradually increasing
sum. Hence those who die early, instead of receiving the
most, as on the ordinary plan, receive the least; for at the
death of each member the annuity ceases absolutely, so far
as he and his relations are concerned. The whole affair
therefore is to all intents and purposes a gigantic system
of betting, to see which can live the longest; the State
being the common stake-holder, and receiving a heavy commission
for its superintendence, this commission being naturally
its sole motive for encouraging such a transaction. It is
recorded of one of the French Tontines[5] that a widow of 97
was left, as the last survivor, to receive an annuity of 73,500
livres during the rest of the life which she could manage to
drag on after that age;--she having originally subscribed a
single sum of 300 livres only. It is obvious that such a system
as this, though it may sometimes go by the name of insurance,
is utterly opposed to the spirit of true insurance,
since it tends to aggravate existing inequalities of fortune
instead of to mitigate them. The insurer here bets that
he will die old; in ordinary insurance he bets that he will
die young.
Again, to take one final instance, common opinion often
regards the bank or company which keeps a _rouge et noir_
table, and the individuals who risk their money at it, as being
both alike engaged in gambling. So they may be, technically,
but for all practical purposes such a bank is as sure
and safe a business as that of any ordinary insurance society,
and probably far steadier in its receipts than the majority of
ordinary trades in a manufacturing or commercial city. The
bank goes in for many and small transactions, in proportion
to its capital; their customers, very often, in proportion to
their incomes go in for very heavy transactions. That the
former comes out a gainer year after year depends, of course,
upon the fact that the tables are notoriously slightly in their
favour. But the _steadiness_ of these gains when compared
with the unsteadiness of the individual losses depends simply
upon,--in fact, is merely an illustration of,--the one great
permanent contrast which lies at the basis of all reasoning
in Probability.
10. We have so far regarded Insurance and Gambling
as being each the product of a natural impulse, and as having
each, if we look merely to experience, a great mass of human
judgment in its favour. The popular moral judgment, however,
which applauds the one and condemns the other rests in great
part upon an assumption, which has doubtless much truth
in it, but which is often interpreted with an absoluteness which
leads to error in each direction;--the duty of insurance being
too peremptorily urged upon every one, and the practice of
gambling too universally regarded as involving a sacrifice of
real self-interest, as being in fact little better than a persistent
blunder. The assumption in question seems to be
extracted from the acknowledged advantages of insurance,
and then invoked to condemn the practice of gambling. But
in so doing the fact does not seem to be sufficiently recognized
that the latter practice, if we merely look to the extent
and antiquity of the tacit vote of mankind in its favour,
might surely claim to carry the day.
It is of course obvious that in all cases with which we
are concerned, the aggregate wealth is unaltered; money
being merely transferred from one person to another. The
loss of one is precisely equivalent to the gain of another.
At least this is the approximation to the truth with which we
find it convenient to start.[6] Now if the happiness which is
yielded by wealth were always in direct proportion to its
amount, it is not easy to see why insurance should be advocated
or gambling condemned. In the case of the latter this
is obvious enough. I have lost £50, say, but others (one or
more as the case may be) have gained it, and the increase
of their happiness would exactly balance the diminution of
mine. In the case of Insurance there is a slight complication,
arising from the fact that the falling in of the policy
does not happen at random (otherwise, as already pointed
out, it would be simply a lottery), but is made contingent
upon some kind of loss, which it is intended as far as possible
to balance. I insure myself on a railway journey, break my
leg in an accident, and, having paid threepence for my
ticket, receive say £200 compensation from the insurance
company. The same remarks, however, apply here; the
happiness I acquire by this £200 would only just balance the
aggregate loss of the 16,000 who have paid their threepences
and received no return for them, were happiness always
directly proportional to wealth.
11. The practice of Insurance does not, I think, give
rise to many questions of theoretic interest, and need not
therefore detain us longer. The fact is that it has hardly
yet been applied sufficiently long and widely, or to matters
which admit of sufficiently accurate statistical treatment,
except in one department. This, of course, is Life Insurance;
but the subject is one which requires constant attention to
details of statistics, and is (rightly) mainly carried out in
strict accordance with routine. As an illustration of this
we need merely refer to the works of De Morgan,--a professional
actuary as well as a writer on the theory of Probability,--who
has found but little opportunity to aid his speculative
treatment of Probability by examples drawn from this class
of considerations.
With Gambling it is otherwise. Not only have a variety
of interesting single problems been discussed (of which the
Petersburg problem is the best known) but several speculative
questions of considerable importance have been raised.
One of these concerns the disadvantages of the practice of
gambling. There have been a number of writers who, not
content with dwelling upon the obvious moral and indirect
mischief which results, in the shape of over-excitement,
consequent greed, withdrawal from the steady business
habits which alone insure prosperity in the long run, diversion
of wealth into dishonest hands, &c., have endeavoured
to demonstrate the necessary loss caused by the practice.
12. These attempts may be divided into two classes.
There are (1) those which appeal to merely numerical considerations,
and (2) those which introduce what is called the
'moral' as distinguished from the mathematical value of a
future contingency.
(1) For instance, an ingenious attempt has been made
by Mr Whitworth to prove that gambling is necessarily
disadvantageous on purely mathematical grounds.
When two persons play against each other one of the two
must be ruined sooner or later, even though the game be a
fair one, supposing that they go on playing long enough; the
one with the smaller income having of course the worst
chance of being the lucky survivor. If one of them has a
finite, and the other an infinite income, it must clearly be
the former who will be the ultimate sufferer if they go on
long enough. It is then maintained that this is in fact every
individual gambler's position, "no one is restricted to gambling
with one single opponent; the speculator deals with the
public at large, with a world whose resources are practically
unlimited. There is a prospect that his operations may
terminate to his own disadvantage, through his having nothing
more to stake; but there is no prospect that it will
terminate to his advantage through the exhaustion of the
resources of the world. Every one who gambles is carrying
on an unequal warfare: he is ranged with a restricted capital
against an adversary whose means are infinite."[7]
In the above argument it is surely overlooked that the
adversaries against whom he plays are not one body with a
common purse, like the bank in a gambling establishment.
Each of these adversaries is in exactly the same position as
he himself is, and a precisely similar proof might be employed
to show that each of them must be eventually ruined
which is of course a reduction to absurdity. Gambling can
only transfer money from one player to another, and therefore
none of it can be actually lost.
13. What really becomes of the money, when they
play to extremity, is not difficult to see. First suppose a
limited number of players. If they go on long enough, the
money will at last all find its way into the pocket of some
one of their number. If their fortunes were originally equal,
each stands the same chance of being the lucky survivor;
in which case we cannot assert, on any numerical grounds,
that the prospect of the play is disadvantageous to any one
of them. If their fortunes were unequal, the one who had
the largest sum to begin with can be shown to have the
best chance, according to some assignable law, of being left
the final winner; in which case it must be just as advantageous
for him, as it was disadvantageous for his less wealthy
competitors.
When, instead of a limited number of players, we suppose
an unlimited number, each as he is ruined retiring
from the table and letting another come in, the results are
more complicated, but their general tendency can be readily
distinguished. If we supposed that no one retired except
when he was ruined, we should have a state of things in
which all the old players were growing gradually richer. In
this case the prospect before the new comers would steadily
grow worse and worse, for their chance of winning against
such rich opponents would be exceedingly small. But as
this is an unreasonable supposition, we ought rather to
assume that not only do the ruined victims retire, but also
that those who have gained fortunes of a certain amount
retire also, so that the aggregate and average wealth of the
gambling body remains pretty steady. What chance any
given player has of being ruined, and how long he may
expect to hold out before being ruined, will depend of course
upon the initial incomes of the players, the rules of the
game, the stakes for which they play, and other considerations.
But it is clear that for all that is lost by one, a
precisely equal sum must be gained by others, and that
therefore any particular gambler can only be cautioned beforehand
that his conduct is not to be recommended, by
appealing to some such suppositions as those already mentioned
in a former section.
14. As an additional justification of this view the
reader may observe that the state of things in the last
example is one which, expressed in somewhat different
language and with a slight alteration of circumstances, is
being incessantly carried on upon a gigantic scale upon
every side of us. Call it the competition of merchants and
traders in a commercial country, and the general results are
familiar enough. It is true that in so far as skill comes into
the question, they are not properly gamblers; but in so far
as chance and risk do, they may be fairly so termed, and in
many branches of business this must necessarily be the case
to a very considerable extent. Whenever business is carried
on in a reckless way, the comparison is on general grounds
fair enough. In each case alike we find some retiring ruined,
and some making their fortunes; and in each case alike
also the chances, _coeteris paribus_, lie with those who have
the largest fortunes. Every one is, in a sense, struggling
against the collective commercial world, but since each of his
competitors is doing the same, we clearly could not caution
any of them (except indeed the poorer ones) that their efforts
must finally end in disadvantage.
15. If we wish to see this result displayed in its
most decisive form we may find a good analogy in a very
different class of events, viz. in the fate of _surnames_. We
are all gamblers in this respect, and the game is carried
out to the last farthing with a rigour unknown at Newmarket
or Monte Carlo. In its complete treatment the
subject is a very intricate one,[8] but a simple example will
serve to display the general tendency. Suppose a colony
comprising 1000 couples of different surnames, and suppose
that each of these has four children who grow up to marry.
Approximately, one in 16 of these families will consist of
girls only; and therefore, under ordinary conventions, about
62 of the names will have disappeared for ever after the
next generation. Four again out of 16 will have but one
boy, each of whom will of course be in the same position
as his father, viz. the sole representative of his name.
Accordingly in the next generation one in 16 of these names
will again drop out, and so the process continues. The
number which disappears in each successive generation becomes
smaller, as the stability of the survivors becomes
greater owing to their larger numbers. But there is no
check to the process.
16. The analogy here is a very close one, the names
which thus disappear corresponding to the gamblers who
retire ruined and those which increase in number corresponding
to the lucky winners. The ultimate goal in each
case alike,--of course an exceedingly remote one,--is the
exclusive survival of one at the expense of all the others.
That one surname does thus drop out after another must
have struck every one who has made any enquiry into family
genealogy, and various fanciful accounts have been given
by those unfamiliar with the theory of probability. What
is often apt to be overlooked is the extreme slightness of
what may be termed the "turn of the tables" in favour
of the survival at each generation. In the above numerical
example we have made an extravagantly favourable supposition,
by assuming that the population doubles at every
generation. In an old and thickly populated country where
the numbers increase very slowly, we should be much nearer
the mark in assuming that the average effective family,--that
is, the average number of children who live to marry,--was
only _two_. In this case every family which was represented
at any time by but a single male would have but
three chances in four of surviving extinction, and of course
the process of thinning out would be a more rapid one.
17. The most interesting class of attempts to prove
the disadvantages of gambling appeal to what is technically
called 'moral expectation' as distinguished from
'mathematical expectation.' The latter may be defined
simply as the average money value of the venture in
question; that is, it is the product of the amount to be
gained (or lost) and the chance of gaining (or losing) it.
For instance, if I bet four to one in sovereigns against the
occurrence of ace with a single die there would be, on the
average of many throws, a loss of four pounds against a gain
of five pounds on each set of six occurrences; i.e. there
would be an average gain of three shillings and fourpence
on each throw. This is called the true or mathematical
expectation. The so-called 'moral expectation', on the other
hand, is the subjective value of this mathematical expectation.
That is, instead of reckoning a money fortune in
the ordinary way, as what it _is_, the attempt is made to
reckon it at what it is _felt_ to be. The elements of computation
therefore become, not pounds and shillings, but sums
of pleasure enjoyed actually or in prospect. Accordingly
when reckoning the present value of a future gain, we must
now multiply, not the objective but the subjective value,
by the chance we have of securing that gain.
With regard to the exact relation of this moral fortune
to the physical various more or less arbitrary assumptions
have been made. One writer (Buffon) considers that the
moral value of any given sum varies inversely with the
total wealth of the person who gains it. Another (D. Bernoulli)
starting from a different assumption, which we shall
presently have to notice more particularly, makes the moral
value of a fortune vary as the logarithm of its actual amount.[9]
A third (Cramer) makes it vary with the square root of the
amount.
18. Historically, these proposals have sprung from
the wish to reconcile the conclusions of the Petersburg
problem with the dictates of practical common sense; for,
by substituting the moral for the physical estimate the
total value of the expectation could be reduced to a finite
sum. On this ground therefore such proposals have no great
interest, for, as we have seen, there is no serious difficulty in
the problem when rightly understood.
These same proposals however have been employed in
order to prove that gambling is necessarily disadvantageous,
and this to both parties. Take, for instance, Bernoulli's supposition.
It can be readily shown that if two persons each
with a sum of £50 to start with choose to risk, say, £10 upon
an even wager there will be a loss of happiness as a result;
for the pleasure gained by the possessor of £60 will not
be equal to that which is lost by the man who leaves off
with £40.[10]
19. This is the form of argument commonly adopted;
but, as it stands, it does not seem conclusive. It may surely
be replied that all which is thus proved is that _inequality_ is
bad, on the ground that two fortunes of £50 are better than
one of £60 and one of £40. Conceive for instance that the
original fortunes had been £60 and £40 respectively, the
event may result in an increase of happiness; for this will
certainly be the case if the richer man loses and the fortunes
are thus equalized. This is quite true; and we are therefore
obliged to show,--what can be very easily shown,--that if
the other alternative had taken place and the two fortunes
had been made still more unequal (viz. £65 and £35 respectively)
the happiness thus lost would more than balance
what would have been gained by the equalization. And
since these two suppositions are equally likely there will be a
loss in the long run.
The consideration just adduced seems however to show
that the common way of stating the conclusion is rather
misleading; and that, on the assumption in question as
to the law of dependence of happiness on wealth, it really is
the case that the effective element in rendering gambling
disadvantageous is its tendency to the increase of the inequality
in the distribution of wealth.
20. This raises two questions, one of some speculative
interest in connection with our subject, and the other of
supreme importance in the conduct of life. The first is this:
quite apart from any particular assumption which we make
about moral fortunes or laws of variation of happiness, is
it the fact that gambling tends to increase the existing
inequalities of wealth? Theoretically there is no doubt that
this is so. Take the simplest case and suppose two people
tossing for a pound. If their fortunes were equal to begin
with there must be resultant inequality. If they were
unequal there is an even chance of the inequality being
increased or diminished; but since the increase is proportionally
greater than the decrease, the final result remains of
the same kind as when the fortunes were equal.[11] Taking a
more general view the same conclusion underlies all our
reasoning as to the averages of large numbers, viz. that the
resultant divergencies increase absolutely (however they
diminish relatively) as the numbers become greater. And of
course we refer to these absolute divergencies when we
are talking of the distribution of wealth.
21. This is the theoretic conclusion. How far the
actual practice of gambling introduces counteracting agencies
must be left to the determination of those who are competent
to pronounce. So far as outsiders are authorised
to judge from what they read in the newspapers and other
public sources of information, it would appear that these
counteracting agencies are very considerable, and that in
consequence it is a rather insecure argument to advance
against gambling. Many a large fortune has notoriously
been squandered on the race-course or in gambling saloons,
and most certainly a large portion, if not the major part,
has gone to swell the incomes of many who were by comparison
poor. But the solution of this question must clearly be
left to those who have better opportunities of knowing
the facts than is to be expected on the part of writers on
Probability.
22. The general conclusion to be drawn is that those
who invoked this principle of moral fortune as an argument
against gambling were really raising a much more intricate
and far-reaching problem than they were aware of. What
they were at work upon was the question, What is the
distribution of wealth which tends to secure the maximum
of happiness? Is this best secured by equality or inequality?
Had they really followed out the doctrine on which their
denunciation of gambling was founded they ought to have
adopted the Socialist's ideal as being distinctly that which
tends to increase happiness. And they ought to have
brought under the same disapprobation which they expressed
against gambling all those tendencies of modern
civilized life which work in the same direction. For instance;
keen competition, speculative operations, extended
facilities of credit, mechanical inventions, enlargement of
business operations into vast firms:--all these, and other
similar tendencies too numerous to mention here, have had
some influence in the way of adding to existing inequalities.
They are, or have been, in consequence denounced by
socialists: are we honestly to bring them to this test in
order to ascertain whether or not they are to be condemned?
The reader who wishes to see what sort of problems this
assumption of 'moral fortune' ought to introduce may be
recommended to read Mr F. Y. Edgeworth's _Mathematical
Psychics_, the only work with which I am acquainted which
treats of these questions.
1. The question of the advisability of inoculation against the
small-pox, which gave rise to much discussion amongst the writers on
Probability during the last century, is a case in point of the same
principles applied to a very different kind of instance. The loss
against which the insurance was directed was death by small-pox, the
premium paid was the illness and other inconvenience, and the very
small risk of death, from the inoculation. The disputes which thence
arose amongst writers on the subject involved the same difficulties
as to the balance between certain moderate loss and contingent great
loss. In the seventeenth century it seems to have been an occasional
practice, before a journey into the Mediterranean, to insure against
capture by Moorish pirates, with a view to secure having the ransom
paid. (See, for an account of some extraordinary developments of the
insurance principle, Walford's _Insurance Guide and Handbook_. It is
not written in a very scientific spirit, but it contains much
information on all matters connected with insurance.)
2. All that is meant by the above comparison is that the ideal aimed
at by Communism is similar to that of Insurance. If we look at the
processes by which it would be carried out, and the means for
enforcing it, the matter would of course assume a very different
aspect. Similarly with the action of Trades Unionism referred to in
the next paragraph.
3. One of the best discussions that I have recently seen on these
subjects, by a writer at once thoroughly competent and well
informed, is in Mr Proctor's _Chance and Luck_. It appears to me
however that he runs into an extreme in his denunciation not of the
folly but of the dishonesty of all gambling. Surely also it is a
strained use of language to speak of all lotteries as 'unfair' and
even 'swindling' on the ground that the sum-total of what they
distribute in prizes is less than that of what they receive in
payments. The difference, in respect of information deliberately
withheld and false reports wilfully spread, between most of the
lotteries that have been supported, and the bubble companies which
justly deserve the name of swindles, ought to prevent the same name
being applied to both.
4. "A fire insurance is a simple bet between the office and the party,
and a life insurance is a collection of wagers. There is something
of the principle of a wager in every transaction in which the
results of a future event are to bring gain or loss." _Penny
Cyclopædia_, under the head of _Wager_.
5. _Encyclopédie Methodique_, under the head of _Tontines_.
6. Of course, if we introduce considerations of Political Economy,
corrections will have to be made. For one thing, every Insurance
Office is, as De Morgan repeatedly insists, a _Savings Bank_ as well
as an Insurance Office. The Office invests the premiums, and can
therefore afford to pay a larger sum than would otherwise be the
case. Again, in the case of gambling, a large loss of capital by any
one will almost necessarily involve an actual destruction of wealth;
to say nothing of the fact that, practically, gambling often causes
a constant transfer of wealth from productive to unproductive
purposes.
7. _Choice and Chance_, Ed. II. p. 208.
8. It was, I believe, first treated as a serious problem by Mr Galton.
(See the _Journal Anthrop. Inst._ Vol. IV. 1875, where a complete
mathematical solution is indicated by Mr H. W. Watson.)
9. Bernoulli himself does not seem to have based his conclusions upon
actual experience. But it is a noteworthy fact that the assumption
with which he starts, viz. that the subjective value of any small
increment (dx) is inversely proportional to the sum then possessed
(x), and which leads at once to the logarithmic law above mentioned,
is identical with one which is now familiar enough to every
psychologist. It is what is commonly called Fechner's Law, which he
has established by aid of an enormous amount of careful experiment
in the case of a number of our simple sensations. But I do not
believe that he has made any claim that such a law holds good in the
far more intricate dependence of happiness upon wealth.
10. The formula expressive of this moral happiness is c log x/a; where
x stands for the physical fortune possessed at the time, and a for
that small value of it at which happiness is supposed to disappear:
c being an arbitrary constant. Let two persons, whose fortune is x,
risk y on an even bet. Then the balance, as regards happiness, must
be drawn between
c log x/a and 1/2c log (x + y)/a + 1/2c log (x - y)/a,
or log x^{2} and log(x + y)(x - y),
or x^{2} and x^{2} - y^{2},
the former of which is necessarily the greater.
11. This may be seen more clearly as follows. Suppose two pair of
gamblers, each pair consisting of men possessing £50 and £30
respectively. Now if we suppose the richer man to win in one case
and the poorer in the other these two results will be a fair
representation of the average; for there are only two alternatives
and these will be equally frequent in the long run. It is obvious
that we have had two fortunes of £50 and two of £30 converted into
one of £20, two of £40, and one of £60. And this is clearly an
increase of inequality.