1. Winning with Losing Games An Examination of Parrondo’s Paradox

2. 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.

3. 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.

4. Graphical Approach Given the above graph, one can form an adjacency matrix which will allow for further analysis. 1 3 2.495.505 $0 -$1 $1

5. 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.

6. Graphical Approach 1 2 7 6 5 4 3.745.095.745.255.905 $0 $1 $2 $3 -$1 -$2 -$3

7. Matrix Representation This matrix is the above matrix raised to the 500 th power..555.445

8. 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.

9. 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.

10. 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.

11. A Brief Example Game Played Coin Flipped Heads/ Tails Capital 0 ComplicatedCoin CTails SimpleCoin ATails-2 SimpleCoin AHeads SimpleCoin AHeads0 SimpleCoin ATails ComplicatedCoin BHeads0 SimpleCoin AHeads1 ComplicatedCoin CHeads2 ComplicatedCoin CTails1 SimpleCoin AHeads2

12. 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

13. 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

14. Matrix Representation

15. Matrix Powers 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

16. 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.

17. 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.

18. 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.