Winning with Losing
Games
An Examination of
Parrondo’s Paradox
A Fair Game
Start with a capital of $0.
Flip a fair coin.
If the coin lands on heads,
then your capital increases
by $1.
If the result is tails, then
your capital decreases by $1.
A Simple Game
As before, the starting capital
is $0.
Flip a biased coin–one that
will land on tails 50.5% of
the time.
Increase the capital by $1 if
the coin lands on heads and
decrease it by $1 if the coin
lands on tails.
Graphical Approach
$1
3
.495
$0
2
.505
-$1
Given the above
graph, one can form
an adjacency matrix
which will allow for
further analysis.
1
0
0 
 1
.505 0 .495


0
1 
 0
A Complicated Game
Start with a capital of $0.
If the capital is a multiple of 3,
then flip a coin that lands on
tails 90.5% of the time.
If the capital is not a multiple of
3, then flip a coin which lands
on heads 74.5% of the time.
As before, a flip of heads results
in gaining $1 while tails results
in losing $1.
Graphical Approach
$3
7
.745
$2
6
.745
.255
$1
5
.095
.255
$0
4
.745
.905
-$1
3
.745
.255
-$2
2
.255
-$3
1
Matrix Representation
0
0
0
0
0
0 
 1
.255 0 .745 0
0
0
0 


0
0 
 0 .255 0 .745 0
 0
0 .905 0 .095 0
0 


0
0 .255 0 .745 0 
 0


0
0
0
0
.
255
0
.
745


 0
0
0
0
0
0
1 
 1
.698

.592

.555
.179

.046
 0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 
.302
.408

.445 
.821

.954
1 
This matrix
is the above
matrix raised
to the 500th
power.
Introduction to the
Paradox
Coin A: Lands on heads 49.5% of the time
and lands on tails 50.5% of the
time. This coin is used when
playing the Simple Game.
Coin B: Lands on heads 9.5% of the time
and lands on tails 90.5% of the
time. This coin is used when on
playing the Complicated Game
and one’s capital is a multiple of
3.
Coin C: Lands on heads 74.5% of the time
and lands on tails 25.5% of the
time. This coin is used in the
Complicated Game when the
capital is not a multiple of 3.
Parrondo’s Paradox
Form a new game which is a
combination of the Simple
and Complicated games.
At each juncture, use a fair
coin to randomly choose
which game to play.
Randomly alternating between
the two games will yield a
winning result although both
are losing.
This is Parrondo’s Paradox.
Illustrations of
Parrondo’s Paradox
Chess
It is sometimes necessary to sacrifice
pieces in order to produce a winning
outcome.
Farming
It is known that both sparrows and
insects can eat all the crops. However,
by having a combination of sparrows
and insects, a healthy crop is
harvested.
Genetics
Some genes that are considered to be
detrimental can actually be beneficial
given the correct environmental
conditions.
A Brief Example
Game
Played
Coin
Flipped
Heads/
Capital
Tails
0
Complicated
Coin C
Tails
-1
Simple
Coin A
Tails
-2
Simple
Coin A
Heads
-1
Simple
Coin A
Heads
0
Simple
Coin A
Tails
-1
Complicated
Coin B
Heads
0
Simple
Coin A
Heads
1
Complicated
Coin C
Heads
2
Complicated
Coin C
Tails
1
Simple
Coin A
Heads
2
Example -- Graphically
$3
17
$2
16
15
14
$1
13
12
11
$0
10
9
8
-$1
7
6
-$2
4
-$3
3
5
2
Parrondo’s Graphical
Game
$3
17
$2
16
15
14
$1
13
12
11
$0
10
9
8
-$1
7
6
-$2
4
-$3
3
1
5
2
Simple
Complicated
Matrix Representation
 1
.505

 0
.255

 0

 0
 0

 0
 0

 0
 0

 0
 0

 0
 0

 0
 0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 .495 0
0
0
0
0
0
0
0
0
0
.5
0
.5 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 .745 0
0
0
0
0
0
0
0
0
0
0
0
0
0 .495 0
0
0
0
0
0
0
0 .5
0
.5 0
0
0
0
0
0
0
0
0
0
0 .745 0
0
0
0
0
0
0
0
0
0
0 .495 0
0
0
0
0 .5
0
.5 0
0
0
0
0
0
0
0 .095 0
0
0
0
0
0
0
0 .495 0
0 .5
0
.5 0
0
0
0 .505 0
0
0
0
0
0
0 .255 0
0
0
0
0
0 .505 0
0
0
0
0
0
0
0
0 .905 0
0
0
0
0
0
0
0
0 .505 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 .255 0
0
0
0
0
0
0
0
0
0
0
0 .505 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 .255 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 .745 0
0
0
0
0 .5
0
.5
0
0
0
0
0
0
0
0 
0 

0 
0 

0 

0 
0 

0 
0 

0 
0 

0 
0 

.495
0 

.745
1 
Matrix Powers
 1
.788

.735

.682
.605

.573
.540

.406
.473

.541

.283
.235

.187
.119

.089
.060

 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 
.212 
.265

.318
.395

.427 
.460 

.594 
.527 

.459 

.717 
.765

.813
.881

.910 
.940 

1 
The above matrix represents the combined
game after 500 coin flips. Notice, for
example, that the probability that you go
from Vertex 9 to Vertex 17 is .527. Thus,
one is more likely to progress up the graph.
Generalizations
Let the probabilities associated with
each coin be defined as follows:

Coin A



Coin B



P(H) = .5 – e
P(T) = .5 + e
P(H) = .1 – e
P(T) = .9 + e
Coin C


P(H) = .75 – e
P(T) = .25 + e
Let the probability of playing the
Simple game be p and the
probability of playing the
Complicated game be 1-p.
A Deeper Analysis
Given the generalizations, I
sought to determine the
widest p range that could be
used so that the Combined
Game was still winning.
Once this p range was
determined, I then
attempted to find the optimal
p which would allow for the
widest e range.
Conclusions
The p range, given that
e = .005 was calculated to be
(.08, .84).
The optimal p was found to
be p = .40. Given this p, the
e range was calculated to be
(0, .013717), accurate to the
millionth position.