After worrying it for a week, I finally got a Markov state representation of Parrondo's game to behave the way Parrondo, Harmer, and Abbott said it would, using an Excel spreadsheet.

The spreadsheet uses a 40x60 array of cells. Columns represent the levels of capital. Successive rows are the outcomes of successive plays. The number in the cell is the probability being in the corresponding Markov state.

I have placed a copy of the Excel worksheet and a pdf file print of its result at: http://www.frontiernet.net/~jmb184/parrondo/ Feel free to download either or both for your own persual.

Parameters which finally worked were:
Game A
P(winning)= 0.49

Game B
capital mod 3=0
P(winning)= 0.09

capital mod 3 ~-0
P(winning)= 0.749

AABB(0) 39
AABB(20) 39.13565 (capital from a sequence of games A, A, B, B, after 20 plays)
AABB(30) 39.17738
AABB(37) 39.27911

I was put off by my expectation that an epsilon offset of 0.05 would lead to a marked effect. At least in this Markov representation, much smaller offsets (biases toward loosing) were needed to obtain a win with the mixed sequence while loosing with either pure game. Even then the gain after 40 rounds is small. Much smaller, I note than the gains shown on Parrondo's page. Is this difference just due to technique--simulation vs Markov state probabilities? A Markov state model for 40 plays is the equivalent of 1,000,000,000,000 runs vs 50,000 that might be typically used in a simulation.

One last thing: A non-probabilistic model of the effect can be obtained with the games
A being if capital is odd, gain 1, else lose 2
B being if capital is even, gain 1, else loose 2

These play out as follows,
AAAAA plays as 7,8,6,4,2,0
AAA plays as 6,4,2,0
BBBB plays as 7,5,3,1,0
BBBBB plays as 6,7,5,3,1,0
ABABAB... plays as 7,8,9,10,...
ABABAB... plays as 6,4,5,6,...
BABABA... plays as 7,5,6,7,8
BABABA... plays as 6,,7,8,9...

Now turn this into a probabilistic model, by making the selection of game A vs B a random event. Another toy to play with.

Part of what is driving me to take this very seriously, is the deep parallels to the Black-Scholes model for pricing derivatives. A Black-Scholes price for an equity option is based on continuous hedging of the equity and a risk free bond corresponding roughly to games A and B. See http://www.ggw.org/donorware/options/

Thanks for letting me rant.

John Bailey
February 4, 2000