Parrondo's Paradox

I think what you say might be true. I once found a dumbed down version which goes like this: (A simplified version of Mia [0])

On your turn, you secretly roll a die and then make a statement about what number you rolled. The next player may then call your bluff. If you lied, you lose, otherwise they lose. Or they may roll again, but must claim to have rolled something higher than whatever you rolled. If you don't lose, you win.

Suppose you have to explain your (mixed, i.e. involving randomness) strategy to other players before playing. This is a simulation of the realistic situation of playing many times and other players learning your strategy.

Let us examine the situation where you are given a 4 from the previous player and you decided to roll. (You would have to prove that this can be a correct move, but what follows is in principle true in any situation where you roll.)

If you roll a 5 or 6, you say so. But if you roll a lower number, you have to claim a 5 or 6 regardless. If you announce a 6, you lose instantly, because the next player has no chance of rolling higher, so they must call your bluff. Thus in a single game the correct move is to announce a 5.

But since the other players know your strategy, you can increase you chances of them believing you have a 5 if you sometimes claim a 6 and thereby forefeit the game. It is an easy calculation to see that this strategy yields more wins than always claiming a 5 in this situation.

I think this is analogue to the poker example: Play a clearly bad move and thus gain an advantage by being unpredictable. In my example it is not so hard to see that the advantage can in principle outweigh the cost of playing bad moves.

[0] https://en.wikipedia.org/wiki/Mia_(game)

Later on it's revealed. The whole thing is indeed pointlessly daft:

> In summary, Parrondo's paradox is an example of how dependence can wreak havoc with probabilistic computations made under a naive assumption of independence.

In other words, this is a load of time-wasting BS, based on doing a stupid thing at the outset anyone thinking clearly sees right away.

I also thought of poker, but I'm not sure exactly if there's a connection here:

https://en.wikipedia.org/wiki/Morton%27s_theorem

I'm somewhat skeptical any time anyone references "the fundamental theorem of poker" (like that wiki page), given that Sklansky definitely doesn't understand game theory and the "fundamental theorem" is just wrong. This Morton's theorem shows one example of that.

For people who don't know the so-called fundamental theorem of poker[1] was proposed by David Sklansky who was definitely more mathematically-minded than most poker players of his generation but was not really a mathematician. It's gained a lot of mindshare even though it's clearly at odds with game theory. Probably this is because it's very simple to state and seems intuitive. It is usually stated as something like "Every time you play a hand differently than you would have if you could see all your opponent's cards you lose and they gain and vice versa, and conversely every time they play a hand differently from how they would if they could see all your cards you gain and they lose and vice versa".

This is one of those things that sounds sort of obviously correct and so is very seductive as an idea, but it completely fails if you just think for a second about the fact that you can't see their cards and they can't see your cards. In that world (actual poker), we know that the optimal strategy is a Nash equilibrium in mixed strategies(we know this is true for all games of hidden information). By definition that means you are sometimes doing something different from "what you would do if you could see their cards", because you have a mixed strategy (ie you don't do exactly the same thing every time but choose options with some randomness). The same is true for the opponent. Therefore we can see that it follows trivially that the "fundamental theorem" can't be correct.

[1] https://en.wikipedia.org/wiki/Fundamental_theorem_of_poker

Ok I misunderstood what it means to be a paradox, this does seem to apply. The point I was intending to make is that the composition of 2 games entails a new game which creates an entirely new ruleset and therefore new potential outcomes. On the spectrum of paradoxes this one doesn’t feel particularly profound, but perhaps it’s due to the examples being especially contrived.