This is a fascinating post, thanks to all the contributors. Robert, this is a very clear explanation of this Parrondo paradox. I wasn't familiar with it (though I'd heard of it), but I'm immediately drawn to it! Like Alex, I also value the verbal descriptions of what's going on.

First question: point of detail. Refer to Robert's post with subject "0.495" above. There it says "Now if you play a turn of game A, even though that is unfavourable in itself, it decreases the probability that your fortune will be 0 mod 3 in the next turns." How does playing game A make it less likely that fortune will be a multiple of 3? Sorry if it's obvious.

And second question: general point. Is it a correct description of the Parrondo game structure to make the following generalization?

Let x,y,z denote state variables (i.e. they evolve over time according to the outcome of the stochastic games). Let x denote wealth and (for simplicity) let y,z denote binary, indicator variables equal to either 1 or 0. Let E(x) denote the expected value of wealth x in a particular subgame.

I write the architecture of the Parrondo game as follows:

**if z=0, then ****[game A] **

** E(x)<0**

**if z=1, then ****[game B]**

** if y=1, then **** ****[subgame B1]**

E(x)>0

** if y=0, then **** ****[subgame B2] **

** E(x)<0**

Above, the indicator variable z is "time" (turn number), which evolves deterministically and independently, while the indicator variable y is "the property that x=0 mod 3", which evolves stochastically and also depends on z.

Both game A and game B are losing games when played independently of each other. However, game B contains a winning subgame (subgame B1). The Parondo paradox arises whenever the outcome of game A can bias game B in favor of the winning subgame B1. What turns a combination of losing games into a winning one is that the outcome of game A influences the value of the indicator/state variable y in favor of the winning subgame B1.

Is this a fair description of the general mechanism at work in the Parrondo paradox? And if so, doesn't the paradox arise from the "framing" of game B as one single losing game, rather than framing it as two separate (albeit interdependent) games B1 and B2?

By "framing" I just mean the way the Parrondo game is presented, where bundling subgames B1 and B2 into one single game B obscures the existence of a winning subgame that it is actually possible to play more often than it would first seem possible. No?