VBA code: Parrondo Paradox in Game Theory?
Peter Jamieson
Anyone know if VBA code has been posted to simulate this interesting paradox
on a spreadsheet?
The paradox has received attention in Nature recently courtesy of a paper of
Harmer and Abbott and is as follows.
Game A: You toss a coin that is slightly biased against you.....you win $1
with probability say 0.495 and lose $1 with probability 0.505 so that in
time your nett bankroll will decline.
Game B: You play a little more complicated affair using two coins whose
biases are different and the coin you use is dependant on some factor such
as your bankroll.
If your bankroll is a multiple of three you toss a heavily biased coin with
which you win $1 with probability 0.095 and lose $1 with probability 0.905.
If your bankroll is not a multiple of three the coin you toss is biased in
your favour so that you win $1 with probability 0.745 and lose $1 with
probability 0.255.
Game A played alone is a losing game for you.
Game B played alone is also a losing game for you.
But......!!...if game A and B are played in random order or played
alternately then on average you win!
This is the very counter-intuitive paradox discovered by Spanish physicist
Juan Parrondo.
Any takers to codify a simulation?
Cheers, Peter J.
Alan E
You could simulate this game without any VBA code as follows:
Column A: current bankroll = some start value for first row, start value
from one row up plus winnings (column E) from one row up for all other rows.
Column B: formula to randomly select game =if(RAND()<.5,"a","b")
Column C: formula to determine if bankroll is divisible by 3: =mod(A1,3)
will return 0 of value in A1 is divisible by 3
Column D: random number =RAND()
Column E: calculate winnings with this formula:
=IF(B1="A",1*(D1<0.495)-1*(D1>0.495),IF(C1=0,1*(D1<0.095)-1*(D1>0.095),1*(D1
<0.745)-1*(D1>0.745)))
Copy these formulas down for a whole bunch of rows and see what the running
bankroll is.
Peter Jamieson wrote in message ...
Alan E
0.33333 - .905 x .33333 + .745 x 0.66667 - .0.255 x .66667 = 0.05667.
Peter Jamieson <ldo...@bigpond.net.au> wrote in message
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David J. Braden
As I recall, it was also a way they devised for a shsking plstform to move an
object uphill, *but* the ratios were such that they were from a small set of
integer-valued multiples. I only saw a second-hand report; is there an on-line
version of this that bests what I read in the Wall Street Journal, because it
seems really, really cool. Also, does the bank have infinitely deep pockets? If
not, we might get into a St.Petersburg paradox were we to adjust the
probabilities to *exactly* 1/2.
Thanks, Peter, for the post.
Dave Braden
listless MVP-Excel trying to avoid some chores in the garage.
Alan E
lose on average by themselves:
On first inspection, one would expect game B by itself to win on average,
assuming that the bankroll is divisible by 3 33.333% of the time. But
what tends to happen is when the bankroll is divisible by 3, the heavily
negatively biased coin is used resulting more often than not in a $1 loss.
The next game, the bankroll is not divisible by 3, so the heavily positively
biased coin is used resulting more often than not in a $1 gain, bring the
bankroll back to a value divisible by 3. This pattern repeats itself so
often that when game B is played by itself, the bankroll is divisible by 3
approximately 38% of the time, making game B a loser on average.
However, when game A is thrown into the mix, the pattern described above
gets interrupted, so that the likelihood of the bankroll being divisible by
3 returns to approximately 33%, at which game B wins often enough to offset
the slightly negative bias of Game A.
Peter Jamieson
Nice isn't it!
Thanks for your nice formula and input!
The state of the bankroll affects the probabilities and can act as a
temporary lock or "ratchett".
There is a great deal of interest in the application of this strange
counter-intuitive result in many areas.
Several refs you may like:
http://www.eleceng.adelaide.edu.au/Personal/gpharmer/games/ps.htm
http://helix.nature.com/nsu/991223/991223-13.html
http://www.maa.org/mathland/mathtrek_3_6_00.html
http://www.ams.org/new-in-math/01-2000-media.html
An analogy for the "ratchett" effect is: shake the muesli
jar before pouring to get all the biggest nuts and fruit to rise!
Sometimes it gets the fancy title "Differential Grain Dispersion"!
I once saw an explanation along these line to account for how the massive
Easter Island heads were raised but did not believe it.
Cheers, Peter J.
"Alan E" <alane...@email.com> wrote in message
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