Surely I was not clear ennough because you above take my point.Bill Spight wrote: Indeed, it does.
As you may recall, early on I suggested an environment where the number of plays of each size was unknown, thus its parity was unknown. I did not appeal to a random number generator, but the effect is basically the same. We may estimate the value of making the top play in the environment at ambient temperature t as t/2. Also, my original environment where each successive temperature is less than or equal to its previous temperature has the same property.
One drawback to all of these is that there may be sizable drops in temperature in them. That is why Berlekamp preferred what he called a rich environment, where each temperature has an odd number of plays, but the temperature drops are small. He even suggested an environment where each temperature drop is 0.01 point. He did not envision players actually making successive plays in the environment, but making sealed bids of the temperature at which they are willing to make a play in the game.
Let's s the two following proposals for building on environment:
1) you take as an environment at temperature t, say t=4, the environment made of the 8 gote points 4, 3½, 3, 2½, 2, 1½, 1, ½. It looks fine because the expected gain by playing first in this environment is :
gain(8 gote points) = 4 - 3½ + 3 - 2½ + 2 - 1½ + 1 - ½ = 2.
But as soon as you have played the highest got move it remains 7 gote points 3½, 3, 2½, 2, 1½, 1, ½ and this time the gain when playing first in this remaining environment at temperature 3½ is:
gain(7 gote points) = 3½ - 3 + 2½ - 2 + 1½ - 1 + ½ = 2 which if different from t/2 = 3½ / 2 = 1¾
2) you take as an environment at say temperature t=4, a set of N gote points each with a value of 4 points. Because you take N at random you cannot say if N is even or odd and as a consequence the gain expected from this environment is clearly t/2. That is fine but this environment cannot be ideal because the temperature can only drop from 4 to 0 without any intermediate temperature.
My idea is mixing the ideas of this two environments. To create my environment I need only a random generator (a piece of money?) which gives as a result the value "true" or the value "false".
As an example let's create an environment at temperature t=4.
First of all I put in this environment a 4 points gote point.
Then I initialise a variable v to v=t, this variable being the current value of the gote point I will add to my environment
and now I continue the building of the environment by the following loop:
loop
...I ask for the result of the random generator :
...if the result is "true" I add to the environment a gote point with value v
...if the result is "false" I do not add a gote point in the environment but instead I decrease the v value to v -> v-½
...You continue the loop until v reach 0
end of loop
The purpose of my previous post was to proof the gain expected from such environment at temperature t is equal to t/2.
Note the following other caracteristic of this environment:
For any value v < t the mean number of gote points of value v is equal to 1. More precisely, the number of gote points on this level v is 0 with probability ½, 1 with probability ¼, 2 with probability ⅛ ...
You can now see why this environment is a kind of mixture of the two previous environments I mentionned above but this time with good caracteristics:
gain in the environment equal to t/2, temperature droping slowly from t to 0, small number of gote points (about 2t gote points).
OC, if you may prefer, instead of decreasing v by the value ½, you can decrease this value by 0.01 but it is another point.
