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 GameAs 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

Given the above graph, one can form an adjacency matrix which will allow for further analysis.

1

3

2

.495

.505

100

495.0505.

001

$0

-$1

$1

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

1

2

7

6

5

4

3

.745

.745

.095

.745

.745

.255

.255

.255

.255

.905

$0

$1

$2

$3

-$1

-$2

-$3

Matrix Representation

1 0 0 0 0 0 0

255 0 745 0 0 0 0

0 255 0 745 0 0 0

0 0 905 0 095 0 0

0 0 0 255 0 745 0

0 0 0 0 255 0 745

0 0 0 0 0 0 1

. .

. .

. .

. .

. .

1000000

954.00000046.

821.00000179.

00000

408.00000592.

302.00000698.

0000001This matrix is the above matrix raised to the 500th power.

.555 .445

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/Tails

Capital0

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

15

12

9

6

14

11

8

5

2

16

13

10

7

4

$0

$2

$1

-$1

-$2

-$3

$3

Parrondo’s Graphical Game

1

3

17

15

12

9

6

14

11

8

5

2

16

13

10

7

4

Simple

Complicated

$0

$2

$1

-$2

-$3

$3

-$1

Matrix Representation

10000000000000000

745.0000255.00000000000

05.05.0000000000000

495.0000505.00000000000

00745.00000255.00000000

00005.05.0000000000

00495.00000505.00000000

00000095.00000905.00000

00000005.05.0000000

00000495.00000505.00000

00000000745.00000255.00

00000000005.05.0000

00000000495.00000505.00

00000000000745.0000255.

00000000000005.05.0

00000000000495.0000505.

00000000000000001

Matrix Powers

10000000000000001

940.000000000000000060.

910.000000000000000089.

881.000000000000000119.

813.000000000000000187.

765.000000000000000235.

717.000000000000000283.

459.000000000000000541.

000000000000000

594.000000000000000406.

460.000000000000000540.

427.000000000000000573.

395.000000000000000605.

318.000000000000000682.

265.000000000000000735.

212.000000000000000788.

00000000000000001

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.

.527.473

Generalizations

Let the probabilities associated with each coin be defined as follows: Coin A

P(H) = .5 – P(T) = .5

Coin B P(H) = .1 – P(T) = .9 +

Coin C P(H) = .75 – P(T) = .25 +

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 range.

Conclusions

The p range, given that

was calculated to be (.08, .84).

The optimal p was found to be p = .40. Given this p, the range was calculated to be (0, .013717), accurate to the millionth position.