Re: Two envelopes and game theory


briggs@xxxxxxxxxxxxxxxxx wrote:
> In article <ntnmq1paao3rfkorc6rnntr5einsutqvdb@xxxxxxx>, quasi <quasi@xxxxxxxx> writes:
> > On 22 Dec 2005 17:26:56 -0800, "bill" <b92057@xxxxxxxxx> wrote:
> >>I ran a sample of 25,000 rounds. In 3532 rounds n =1 and in 2843
> >>rounds n = 2. At lerast 6375 envelops have the number '4' So the
> >>first envelop would have '4' about 3,188 times. If you always stand,
> >>the net is 12,752. If you switch every time you should get 1766 * 2 =
> >> 3532 + 1421 * 8 = 11,376, = 14,908 > 12,752
> >
> > There are some clear problems with the above calculations. For one
> > thing, if you always switch, you sometimes end up with 1 but your
> > calculation doesn't show that. Secondly, the frequencies of the rounds
> > (not the envelopes) should add up to 25,000 but the above numbers
> > don't show anything even close to that. Thirdly, there is no mention
> > of envelopes with 8 for the never-switch strategy, and no mention of
> > envelopes of 16 or 32 for either strategy, and these are not that
> > rare.
> >
> > So your calculations seem incomplete and almost certainly wrong.
>
> Rather than running a 25,000 round Monte Carlo simulation, I computed
> the distribution explicitly. I've normalized the results to the
> 25,000 scale by the simple expedient of multiplying by 25,000.
>
> Quasi is right. Bill's simulation is flawed.
>
In 25,000 rounds, n should equal "0" about 4,167 times.
This leaves 25,000-4,167 = 20,833 rounds; one-sixth of which should be
n = 1.
20,833*(1/6) = 3,472. [I got 3,532 rounds for n =1.]
This leaves 20,833-3,472 = 17,361 rounds; one-sixth of which should
be n = 2.
17,361*(1/6) = 2,894. [I got 2,843 rounds for n = 2.]
There is less than a 2% error in both cases! So what's wrong with my
simulation?.

I presume from the table that you got approximately 5,305 rounds for
n=1 and 3,070 rounds for n = 2. If these numbers are correct, how did
you calculate them?

Obviously, you should always switch when the first envelope has a "1".
You should never switch a "2" since that would negate the advantage of
switching with a "1".

Bill J.

> switch_never
> 2083.33333333333 3819.44444444444 3182.87037037037
> 2652.39197530864 2210.32664609054 1841.93887174211
> 1534.94905978509 1279.12421648758 1065.93684707298
> 888.280705894152 740.233921578460 616.861601315383
> 514.051334429486 428.376112024572 356.980093353810
> 297.483411128175 247.902842606813 206.585702172344
> 172.154751810287 143.462293175239 119.551910979366
> 99.6265924828047 83.0221604023373 69.1851336686144
> 57.6542780571787 48.0452317143156 40.0376930952630
> 33.3647442460525 27.8039535383771 23.1699612819809
> 19.3083010683174 16.0902508902645 13.4085424085538
> 11.1737853404615 9.31148778371789 7.75957315309824
> 6.46631096091520 5.38859246742933 4.49049372285778
> 3.74207810238148 3.11839841865124 2.59866534887603
> 2.16555445739669 1.80462871449724 1.50385726208104
> 1.25321438506753 1.04434532088961 0.870287767408008
> 0.725239806173340 0.604366505144450 0.503638754287041
> 0.419698961905868 0.349749134921557 0.291457612434631
> 0.242881343695525 0.202401119746271 0.168667599788559
> 0.140556333157133 0.117130277630944 9.760856469245337E-002
> 8.134047057704448E-002 6.778372548087040E-002 5.648643790072534E-002
> 4.707203158393779E-002 3.922669298661483E-002 3.268891082217903E-002
> 2.724075901848252E-002 2.270063251540210E-002 1.891719376283509E-002
> 1.576432813569591E-002 1.313694011307993E-002 1.094745009423327E-002
> 9.122875078527727E-003 7.602395898773108E-003 6.335329915644256E-003
> 5.279441596370214E-003 4.399534663641846E-003 3.666278886368204E-003
> 3.055232405306837E-003 2.546027004422365E-003 2.121689170351971E-003
> 1.768074308626642E-003 1.473395257188869E-003 1.227829380990724E-003
> 1.023191150825603E-003 8.526592923546696E-004 7.105494102955580E-004
> 5.921245085796318E-004 4.934370904830266E-004 4.111975754025222E-004
> 3.426646461687685E-004 2.855538718073071E-004 2.379615598394226E-004
> 1.983012998661855E-004 1.652510832218213E-004 1.377092360181844E-004
> 1.147576966818203E-004 9.563141390151696E-005 7.969284491793081E-005
> 6.641070409827566E-005 3.018668368103440E-005
>
> switch_always
> 2083.33333333333 3819.44444444444 3182.87037037037
> 2652.39197530864 2210.32664609054 1841.93887174211
> 1534.94905978509 1279.12421648758 1065.93684707298
> 888.280705894152 740.233921578460 616.861601315383
> 514.051334429486 428.376112024572 356.980093353810
> 297.483411128175 247.902842606813 206.585702172344
> 172.154751810287 143.462293175239 119.551910979366
> 99.6265924828047 83.0221604023373 69.1851336686144
> 57.6542780571787 48.0452317143156 40.0376930952630
> 33.3647442460525 27.8039535383771 23.1699612819809
> 19.3083010683174 16.0902508902645 13.4085424085538
> 11.1737853404615 9.31148778371789 7.75957315309824
> 6.46631096091520 5.38859246742933 4.49049372285778
> 3.74207810238148 3.11839841865124 2.59866534887603
> 2.16555445739669 1.80462871449724 1.50385726208104
> 1.25321438506753 1.04434532088961 0.870287767408008
> 0.725239806173340 0.604366505144450 0.503638754287041
> 0.419698961905868 0.349749134921557 0.291457612434631
> 0.242881343695525 0.202401119746271 0.168667599788559
> 0.140556333157133 0.117130277630944 9.760856469245337E-002
> 8.134047057704448E-002 6.778372548087040E-002 5.648643790072534E-002
> 4.707203158393779E-002 3.922669298661483E-002 3.268891082217903E-002
> 2.724075901848252E-002 2.270063251540210E-002 1.891719376283509E-002
> 1.576432813569591E-002 1.313694011307993E-002 1.094745009423327E-002
> 9.122875078527727E-003 7.602395898773108E-003 6.335329915644256E-003
> 5.279441596370214E-003 4.399534663641846E-003 3.666278886368204E-003
> 3.055232405306837E-003 2.546027004422365E-003 2.121689170351971E-003
> 1.768074308626642E-003 1.473395257188869E-003 1.227829380990724E-003
> 1.023191150825603E-003 8.526592923546696E-004 7.105494102955580E-004
> 5.921245085796318E-004 4.934370904830266E-004 4.111975754025222E-004
> 3.426646461687685E-004 2.855538718073071E-004 2.379615598394226E-004
> 1.983012998661855E-004 1.652510832218213E-004 1.377092360181844E-004
> 1.147576966818203E-004 9.563141390151696E-005 7.969284491793081E-005
> 6.641070409827566E-005 3.018668368103440E-005
>
> The relevant part of the code is makes it quite clear that the two
> distributions must be identical:
>
> /* pair() is the distribution of envelope pairs
> /* pair(i) is the probability that the lesser of the two envelopes
> /* has value 2^(i-1)
> /*
> /* Half the time you'll randomly choose the lesser of the pair
> /* and half the time you'll choose the greater.
> /*
> /* If you switch always then half the time you'll end up with the
> /* greater of the pair and half the time the lesser.
> /*
This should have been obvious to all from the very beginning, myself
included!

Bill J

> /* Note that I'm using 1-based indexing (Fortran default) and quasi is
> /* using a 0-based convention.
>
> do i = 1, 100
>
> ! Do switch_never
> switch_never ( i ) = switch_never ( i ) + 0.5d0 * pair(i)
> switch_never ( i+1 ) = switch_never ( i+1 ) + 0.5d0 * pair(i)
>
> ! Do switch_always
> switch_always ( i+1 ) = switch_always ( i+1 ) + 0.5d0 * pair(i)
> switch_always ( i ) = switch_always ( i ) + 0.5d0 * pair(i)

.

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