216 gambler’s ruin

P

(1857–1936), much of whose work was greatly influ-

enced by Galton’s studies, was once director of the

laboratory.

Galton died in Surrey, England, on January 17,

1911.

See also

HISTORY OF PROBABILITY AND STATISTICS

(essay).

gambler’s ruin A class of problems in

PROBABILITY

theory, all known as problems of gambler’s ruin, con-

tains questions of the following ilk:

A gambler with $5 repeatedly bets $1 on a

coin toss. Each time he plays he has a 50 per-

cent chance of winning a dollar and a 50 per-

cent chance of losing a dollar. The gambler will

play until he either has $10 or has gone broke.

What is the probability that he will succeed in

doubling his money?

Problems like these are typically solved as follows: let

P(k) be the probability of reaching the $10 goal when

the gambler currently has kdollars in hand. We wish to

compute P(5). Note three things:

1. P(0) = 0. (With zero dollars, the gambler has lost

and will not win.)

2. P(10) = 1. (With $10 in hand, the gambler has won.)

3.

(With kdollars in hand, there is a 50 percent chance

the gambler will lose a dollar and will have to try to

win with k–1 dollars in hand, and a 50 percent

chance that he will win a dollar and will continue

play with k+ 1 dollars in hand.)

Thus we have eleven values, P(0),P(1),…,P(10), with

the first value equal to zero, the last equal to 1, and all

intermediate values equal to the average of the two

values around it. One can check that there is only one

sequence of numbers satisfying these conditions,

namely: 0, , ,…, = 1. This shows that P(1) = ,

P(2) = and, in particular, that P(5) = = .

Notice that with one dollar in hand there is only a

1/10 chance that the gambler will win $10 before going

broke. One can similarly show that there is only a one

in 100 chance that a gambler will win $100, and a one

in 1,000 chance that he will win $1000. Casinos are

well aware that gamblers are reluctant to stop playing

with small profits—especially if a player has had a

string of losses and is down to his or her last dollar.

The gambler’s ruin shows that in all likelihood, gam-

blers will end up broke before receiving profits they are

content with.

See also

HARMONIC FUNCTION

;

RANDOM WALK

.

game theory Game theory is the branch of mathe-

matics that attempts to analyze situations involving

parties with conflicting interests for which the outcome

of the situation depends on the choices made by those

parties. Situations of conflict arise in nearly every real-

world problem that involves decision making, and

game theory has consequently found profound applica-

tions to the study of business competition, economics,

politics, military operations, property division, and

even the study of personal relationships. Although it is

difficult to provide a complete analysis of all the types

of games that arise, complete solutions exist for simple

“matrix games” (which we describe below) involving a

small number of players. These simple games can be

used to model more-complicated multiplayer situations.

Game theory was first studied in 1921 by French

mathematician Félix Edouard Émile Borel (1871–1956),

but the importance of the topic was not properly

acknowledged until the 1944 release of the monumental

work Theory of Games and Economic Behavior by

J

OHN VON

N

EUMANN

(1903–57) and Oskar Morgen-

stern (1902–77). These two scholars completely ana-

lyzed situations of conflict satisfying the following basic

conditions:

i. There are a finite number of players.

ii. Each player can select one of a finite number of

possible actions. The choice of actions available to

each player need not be the same.

iii. All players are aware of the actions, and the conse-

quences, others may choose to take.

iv. Each player is a rational thinker and will select the

action that best suits his or her interests.

v. At the play of the game, no participant knows what

actions will be taken by the other players.

vi. The outcome of the game can be modeled as a set

of payments (positive, zero, or negative) to each of

the players.

1

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2

5

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10

2

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5

1

––

10

10

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10

2

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10

1

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10