Parrondo's Paradox: How two ugly parents can make a beautiful baby • Hacker News Board

Re: Parrondo's Paradox: How two ugly parents can make a beautiful baby

#51

Post by [deleted] » Thu, Aug 23, 2012, 5:32 PM UTC

unreal37 wrote: post #50

GavinB wrote: post #13

Isn't it a little sketchy to have one of the "games" be a game that takes into account outside information like your balanace? Here's another "paradox": Game A: You lose a dollar every time. Game B: If the last game you played was game A, you win a million dollars. Otherwise, you lose a dollar. AAAAAAA... loses, BBBBBBB... loses, but ABABABABA... makes you rich! Suddenly it doesn't seem so paradoxical to me.

Your balance isn't really outside information. In a game like Texas Holdem Poker, your balance is a key factor in how much you bet. But if we change the game so that "if you flip more than two heads in a row, your chances of flipping a third head are only 10%; if you flip two tails in a row, your chances of flipping a third tail are 90%", the result is the same, the odds turn more negative over time than Game A and y…

[deleted]

Re: Parrondo's Paradox: How two ugly parents can make a beautiful baby

#52

Post by Dylan16807 » Thu, Aug 23, 2012, 5:43 PM UTC

unreal37 wrote: post #50

GavinB wrote: post #13

Isn't it a little sketchy to have one of the "games" be a game that takes into account outside information like your balanace? Here's another "paradox": Game A: You lose a dollar every time. Game B: If the last game you played was game A, you win a million dollars. Otherwise, you lose a dollar. AAAAAAA... loses, BBBBBBB... loses, but ABABABABA... makes you rich! Suddenly it doesn't seem so paradoxical to me.

Your balance isn't really outside information. In a game like Texas Holdem Poker, your balance is a key factor in how much you bet. But if we change the game so that "if you flip more than two heads in a row, your chances of flipping a third head are only 10%; if you flip two tails in a row, your chances of flipping a third tail are 90%", the result is the same, the odds turn more negative over time than Game A and y…

It's not your balance that matters there, it's the amount of money you choose to bring into the game at the start. If I could boost my odds by bringing in only $498 dollars of my five hundred then I would do so every time.

Your actual balance is ridiculous to include in a game's calculations.

Re: Parrondo's Paradox: How two ugly parents can make a beautiful baby

#53

Post by shasta » Thu, Aug 23, 2012, 5:45 PM UTC

GavinB wrote: post #13

Isn't it a little sketchy to have one of the "games" be a game that takes into account outside information like your balanace? Here's another "paradox": Game A: You lose a dollar every time. Game B: If the last game you played was game A, you win a million dollars. Otherwise, you lose a dollar. AAAAAAA... loses, BBBBBBB... loses, but ABABABABA... makes you rich! Suddenly it doesn't seem so paradoxical to me.

Even if you don't want the games to be able to refer to the last game you played, it looks like this is greatly convoluted. Why not just consider something simpler like the following?

Game A: If you have an even number of chips, gain one. Otherwise, lose two.

Game B: If you have an odd number or chips, gain one. Otherwise, lose two.

The point being that game A (or B) always leaves you with an odd (or even) number of chips, so that if you keep playing it you lose two every turn, but if you alternate, you win one every turn (after the first, possibly). For simplicity, I'm ignoring the behavior near zero chips, but this still seems to capture the essential properties of the "paradox" in a much simpler fashion.

Re: Parrondo's Paradox: How two ugly parents can make a beautiful baby

#54

Post by Dylan16807 » Thu, Aug 23, 2012, 5:49 PM UTC

unreal37 wrote: post #49

Imagine if instead of Game B being dependent on your cash balance, let's say Game B is blackjack and your chances of winning depend on the number of 10s/face cards that have already been played. If the collective number of 10s/face cards already played is If the collective number of 10s/face cards already played is >= 35%, you have a negative chance of winning. You switch to roulette until the balance of 10s/faces wo…

I think that could be a winning strategy when playing blackjack and roulette together.

Or you can get even more money by s/roulette/taking a nap/. You're just changing your bet size (including $0) in blackjack based on the current odds. No fancy 'combining two losing strategies'.

Re: Parrondo's Paradox: How two ugly parents can make a beautiful baby

#55

Post by [deleted] » Thu, Aug 23, 2012, 5:51 PM UTC

brilee wrote: post #25

brilee wrote: post #23

Uh... Something's fishy with the first graph of the 'drunken man's walk'. It's well known that the random walk veers away from the zero-line at a rate sqrt(N), where N is number of flips. Normalized by the number of flips, the random walk converges to the zero-line at the rate 1/sqrt(N). The graph does neither, so I'm not quite sure how it was generated...

Oops, I misinterpreted how the graph was generated... but the complaint still holds. The averaging of a million trials will cut the amplitude of the fluctuations by a factor of 1000, but the sqrt(N) effect should still be visible. (Note that if you multiply the Y-axis by sqrt(million) = 1000, then you get a scale of about 20, which is approx sqrt(500))

[deleted]

Re: Parrondo's Paradox: How two ugly parents can make a beautiful baby

#56

Post by boris007 » Thu, Aug 23, 2012, 5:54 PM UTC

guilloche wrote: post #5

I am not sure whether the conclusion is right. But the explanation is lame since the second game depends on the your cash status which could be affected in a favored way to make the second game a winning one.

I really like his blog. I started a fan jar for him, on Propster

Re: Parrondo's Paradox: How two ugly parents can make a beautiful baby

#57

Post by Dylan16807 » Thu, Aug 23, 2012, 5:58 PM UTC

legaultmarc wrote: post #26

GavinB wrote: post #13

Isn't it a little sketchy to have one of the "games" be a game that takes into account outside information like your balanace? Here's another "paradox": Game A: You lose a dollar every time. Game B: If the last game you played was game A, you win a million dollars. Otherwise, you lose a dollar. AAAAAAA... loses, BBBBBBB... loses, but ABABABABA... makes you rich! Suddenly it doesn't seem so paradoxical to me.

The author does say that the Parrondo's paradox only works if the games are not independent. It still is paradoxal that "A combination of losing strategies becomes a winning strategy".

It's only 'paradoxical' when the details of the games are sufficiently obscured.

If you word it as "a combination of losing strategies becomes a winning strategy" many people will be surprised and ask you to explain.

If you word it as "losing in A adds to the prize in B, so playing both beats the house" people aren't going to be impressed. note: used a simpler A/B mechanic than the blog post for illustration purposes

Re: Parrondo's Paradox: How two ugly parents can make a beautiful baby

#58

Post by ljscharen » Thu, Aug 23, 2012, 5:58 PM UTC

softbuilder wrote: post #38

Struggling with this one. Help me out: If I play roulette and bet black, red, red, black, red, red, etc., I'm going to win?

Another concrete example is to define BL and BW as any point at which a game B has a negative or positive expectation. A real-life example of using the A, BL, BW strategy were the MIT blackjack teams of the 80's and 90's.

They had a spotter at each table that would count cards and wait until the odds were in the player's favor before calling in the big money player. Assuming that the bets placed by the spotter are negligible, the the big money player could choose from the following games:

   A : Do nothing. E[x] = 0  (break-even)
   BL: Play blackjack when the deck favors the casino, E[x]  0

Obviously playing blackjack has negative expectation in the long run, and we could choose another casino game with very-close-to even odds for game A (like Baccarat), rather than doing nothing.

Re: Parrondo's Paradox: How two ugly parents can make a beautiful baby

#59

Post by nohat » Thu, Aug 23, 2012, 7:25 PM UTC

shasta wrote: post #53

GavinB wrote: post #13

Isn't it a little sketchy to have one of the "games" be a game that takes into account outside information like your balanace? Here's another "paradox": Game A: You lose a dollar every time. Game B: If the last game you played was game A, you win a million dollars. Otherwise, you lose a dollar. AAAAAAA... loses, BBBBBBB... loses, but ABABABABA... makes you rich! Suddenly it doesn't seem so paradoxical to me.

Even if you don't want the games to be able to refer to the last game you played, it looks like this is greatly convoluted. Why not just consider something simpler like the following? Game A: If you have an even number of chips, gain one. Otherwise, lose two. Game B: If you have an odd number or chips, gain one. Otherwise, lose two. The point being that game A (or B) always leaves you with an odd (or even) number of…

Exactly - most of the effort was making a game that hid that simple mechanic so that it took some (admittedly interesting) math to figure out where the dependence was.

Re: Parrondo's Paradox: How two ugly parents can make a beautiful baby

#60

Post by roblev » Thu, Aug 23, 2012, 7:44 PM UTC

Dylan16807 wrote: post #52

unreal37 wrote: post #50

Earlier quoted context omitted.

Your balance isn't really outside information. In a game like Texas Holdem Poker, your balance is a key factor in how much you bet. But if we change the game so that "if you flip more than two heads in a row, your chances of flipping a third head are only 10%; if you flip two tails in a row, your chances of flipping a third tail are 90%", the result is the same, the odds turn more negative over time than Game A and y…

It's not your balance that matters there, it's the amount of money you choose to bring into the game at the start. If I could boost my odds by bringing in only $498 dollars of my five hundred then I would do so every time. Your actual balance is ridiculous to include in a game's calculations.

In texas hold'em your actual balance is a very significant factor in your optimal behaviour in a particular hand. Short-stack tactics are different to big stack tactics.