Parrondo's Paradox

If you take two mutually independent events and somehow introduce a dependence between the two, then the events are no longer mutually independent and therefore by law of conditional probability, you can influence the probability of the outcome of the combination.<p>Corollary: If you take two mutually dependent events and somehow break the dependence between the two, then the events are no longer dependent and therefore by law of conditional probability and law of independent events, you can influence the probability of the outcome of the combination.<p>A practical example of the corollary would be password re-entry user interfaces during account registration.<p>The reason while re-entering the password the second time, we are not shown what we typed earlier is to make the reentry independent of the previous entry. Otherwise, we <i>may</i> look at what was typed earlier and subconsciously type the same thing -- which would be bad while we are doing account registration.

This seems like such a pointless semantic flex to me...<p>In this case has the game not become Game A + Game B ?<p>It&#x27;s just a larger game with a distinct winning strategy because the ruleset is expanded right?<p>What&#x27;s the significance?

I don&#x27;t understand why it is obvious that for any games A and B, composition C has any relation to A or B.

I often wondered whether some old-school poker players are actually unintentionally doing a variant of this. If you analyze the play of someone like Daniel Negranu from a game theoretic perspective it&#x27;s clear that lots of the things he does are just bad (blind limps, check in the dark on the flop etc)[1] but taken together over more than one hand they create situations which are positive ev by widening the ranges they might have in a particular spot and therefore increasing the advantage of information asymmetry (they know exactly what they have whereas the opponent only knows a now very wide range they could have).<p>[1] By which I mean these are strategies that are strictly dominated in the game-theory sense.

Started reading the examples and my eyes glazed over. Someone have a better example?

Recalls a vulgar job interview joke. The candidate claims that he can detect prostate cancer with his finger for, say, $1k.<p>The interviewer (who knows he has a positive diagnosis), sensing easy money, plays along, and an exam ensues on the spot.<p>The candidate pronounces that the interviewer is clear. The interviewer produces the diagnosis, and demands to be paid.<p>The candidate shrugs and produces the money. The interviewer notes the sanguine candidate and asks if this is the usual ending.<p>The candidate laughs and says that he had a $10k bet with the interviewer&#x27;s competitor that he&#x27;d have his finger up the interviewer&#x27;s backside in under an hour.<p>In summary, the paradox in The Famous Article seems to boil down to <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Arbitrage" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Arbitrage</a>

I don’t really understand how this is a paradox, but it’s definitely surprising and non intuitive.<p>It seems like if you have 2 games A and B, the second you start playing them together you’ve effectively created a new game C, which is a game of A and B combined.

It sounds counterintuitive, but it&#x27;s not that hard to come up with an example.<p>Suppose you have a game where your score is A*B. The strategy to only increase A or only increase B are losing ones, but combining them gives a winning strategy.

It&#x27;s not really a paradox. Once you start playing Game A + Game B, then it becomes Game C (a new game entirely) which resembles the original two games but now has a different rule for winning.

I&#x27;ve always found this paradox interesting, as it was discovered fairly recently (1996) compared to other paradoxes in math.

Life itself is a loosing game:-)<p>We die at the end and all we&#x27;ve accumulated doesn&#x27;t worth a thing. Game over.<p>But paradoxically, by being alive, playfull and involved in playing the loosing game of life we win... moment after moment.