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Given the following probabilities :

  • Game A : coin flip (p1 = 0.49)
  • Game B1 : coin flip (p2 = 0.09)
  • Game B2 : coin flip (p3 = 0.74)

Given the following game :

Initial balance = 0
Runs = 100000
Heads adds 1, tails adds -1 to the balance.

Do a coin flip (p1 = 0.50) (doesn't add anything to the balance yet).

If Heads :
    Then play game A
Else :
    If Balance is a multiple of 3, Then play game B1
    Else, play game B2

The result of this game tends to more or less 6000. That's Parrondo's Paradox.

But if I change the game by replacing this part :

Do a coin flip (p1 = 0.50) (doesn't add anything to the balance yet).
If Heads :
    ...
Else :
    ...

By :

If current_run%2:
    ...
Else :
    ...

Then the result tends to more or less -13000.

My question : why is that ?

I don't understand why since :

  • coin flip == heads probability = 0.50
  • current_run%2 == 1 frequency = 0.50

So the frequency of A, B1 and B2 games run should be the same.

A Jupyter Notebook of the Python implementation of this game can be found here : https://www.kaggle.com/tomsihap/notebook478df90ec5, and here is a shortened version :

def coinFlipEven():
    return 1 if random.random() <= 0.5 else -1

def coinModulo(i):
    return 1 if i%2 == 1 else -1

def coinFlipA():
    return 1 if random.random() <= 0.49 else -1

def coinFlipB1():
    return 1 if random.random() <= 0.09 else -1

def coinFlipB2():
    return 1 if random.random() <= 0.74 else -1

n = 1000000
balance1 = 0
balance2 = 0

# Try with even coin flip
for i in range(n):
    whichGame = coinFlipEven()
    if whichGame == 1:
        balance1 += coinFlipA()
    else:
        if balance1 % 3 == 0:
            balance1 += coinFlipB1()
        else:
            balance1 += coinFlipB2()

            
# Try with modulo 
for i in range(n):
    whichGame = coinModulo(i)
    if whichGame == 1:
        balance2 += coinFlipA()
    else:
        if balance2 % 3 == 0:
            balance2 += coinFlipB1()
        else:
            balance2 += coinFlipB2()
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1 Answer 1

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This is a gambler's fallacy, balance counts will not balanced out over trials but they will grow, here decrease because of -1 rewards. See other questions Regression to the mean vs gambler's fallacy.

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