Author: Robert Hyatt
Date: 17:24:59 08/18/98
Go up one level in this thread
On August 18, 1998 at 19:06:59, fca wrote:
>On August 18, 1998 at 08:40:19, Bruce Moreland wrote:
>
>>On August 18, 1998 at 03:34:05, fca wrote:
>
>>>Say the mathematical point expectation for Ferret per game against this
>>>opponent is 0.51.
>
>>>So, how much advantage does the punter have from the ability to "walk away" as
>>>soon as he is "ahead" (whatever "ahead" means - let us forget grading as then
>>>the question is too complex for me)? Is this an advantage at all? Please
>>>ignore all psychological considerations, this is a "straight" ( ;-) ) >>math-chess question.
>
>>If I really was scoring 51% against the pool in individual games, but the
>>members of the pool want to win matches, not games, and I will accept any
>>request for a game, and never make my own requests, then I will lose most of the
>>matches. Assuming no draws (bad assumption, but this just changes the numbers,
>>but not what they mean)
>
>No need for this assumption, as draws could be ignored in analysing the sequence
>since they *cannot* affect the outcome of a series match whose duration is
>solely determined by one, single-minded ("I must win") combatant. Of course
>then we need to state 51% is the expected game-score _ignoring draws_. I'll
>interpret*everything* that follows with this simplification (please also do)...
>
>I guess this is what you meant also.
>
>>, and that everyone will stick around for exactly three games
>>, or until they are ahead
>
>if earlier
>
>>, or until they can't win
>
>also if earlier than 3
>
>>, I get the following
>>breakdown, W being a win for them and L being a loss for them:
>
>>49% W (win)
>>26% LL (loss)
>>13% LWL (loss)
>>12% LWW (win)
>
>>So in this case I lose about 61%
>
>yup, 61.2451% exactly.
>
>>of the matches even though I win 51% of the
>>games. It can only get worse if they are willing to play more games.
>
>Absolutely correct.
>
>(a) Does the 61%-type number tend to 100% as progressively longer (finally
>indefinite length) sequences are permitted? If not, to what number
>(clearly a function of the 51%-type number) does it asymptote?
>
>>So I think that if it is your goal to win a match, it is a major advantage to be
>>the one who decides when the match ends.
>
>Also of course correct. But now here is the build-up question... it is
>"random-walk" stuff as you probably knew.
>
>(b) What is the *minimum* game-winning % ignoring draws (i.e. the number that
>earlier was deemed to be 51%) (hereafter abbreviated to GWPID) that gave Ferret
>at least a 50% expected match result against such a strategy in such a match of
>*indefinite* duration?
>
>If the answer to (a) was "yes", some might suggest the answer here will need to
>be GWPID=100%.
>
>A *very* relevant question IMO, as the GWPID, together with data on draw
>frequency for that opponent (I postpone the 'pool' concept, please, which Bruce
>introduced - let me understand the single opponent problem first), translates
>into a direct ELO difference through a well-known integral (or look-up table).
>
>To show goodwill, I compute the answer x to (b) in the 3-games-at-most case ;-)
>
>(1-x) + x*(1-x)^2 <= 0.5 ( where 0 <= x <= 1 )
>
>which on solving the cubic and disregarding the two false roots gives x=
>59.6968283237+%
>
>So, if Ferret's GWPID is >= 59.6968283237%, Bruce is "OK" to promise the
>opponent 3 games (ignoring draws, here as ever) with the opponent having the
>option to truncate earlier if he chooses. By "OK" is meant, Bruce has at least
>50% chance of winning the match. For skeptics >>
>
>40.3031716763% W
>09.6968283237% LWW
>--------------
>50.0000000000%
>
>All I ask here is the extension of this to the general case i.e.
>
>W + LWW + LWLWW + LLWWW + LWLWLWW + LWLLWWW + LLWLWWW + LLWWLWW + LLLWWWW
>
>1 3 ----5---- -------------------7-------------------
>
>etc.
>
>Note it is the number of outcomes for each match-length that is the key, as each
>will always have the same basic composition. i.e. If 2k-1 is a match length
>(they are laways odd of course) then there will be k "W"s in it and k-1 "L"s in
>it. So if the GWPID=x, each constituent of the 2k-1 long match will have a
>probability of x^(k-1) * (1-x)^k. So, how many different flavours of (2k-1)
>length match are there?
>
>Length of match Number of ways match could be that long
> 1 1
> 3 1
> 5 2
> 7 5
> ' '
> 2k-1 ???
>
>Get me "???" as f(k), (the "never ahead until" problem) and I'll sum "our"
>series... I tried to see a quickie fit, but I am tired, so it might actually be
>easy (I suspect it is not).
>
>>Which is one reason I don't want to
>>play a non-specific number of games against Morovic, I might add.
>
>Good reason. I wondered whether his name would be mentioned.. ;-)
>
>btw This sub-thread is not reserved for Bruce or myself AFAIK. Dann, Dan,
>someone else?
>
>Kind regards
>
>fca
>
>
>PS: This is unrelated mathematically, but it "sounds" analogous. Draws are
>ignored for *all* purposes (i.e. as if they never occurred). Say I score <50%
>against Ferret (number is not critical here - say 49%), but have the right to an
>match of upto N games which only I can truncate at any earlier point. We bet
>(always evens) 1 chekel on the first game, 2 on the 2nd, 4 on the 3rd, 8 on the
>4th and so on (the "doubler's strategy"). Assume I have an endless supply of
>chekels. Since I have the right to end at any time, I clearly have an advantage
>because as soon as I win I can quit and I will be ahead (as 4 > (1+2), 8 >
>(1+2+4) etc.). So, how much should I pay you for the privilege of such a match
>(veniality assumed on both our parts)?
>
>My expectation in chekels if I stop as soon as I win is less than many might
>imagine:
>
>0.49*1 + 0.51*0.49*(-1+2) + 0.51*0.51*0.49*(-1-2+4) + .... to N terms
>
>= 0.49 * (1 - 0.51^N) / (1 - 0.51) = 1 - 0.51^N
>
>So, if we set N=3, and if I paid you more than 0.86735 chekels for the privilege
>of such a match, I'd have been diddled. If we set N=infinity this rises.... to
>just 1 chekel.
>
>Of course I have better strategies... like waiting till I am J > 1 game ahead,
>for example... now the fun maths starts... And the value of GWPID starts
>playing a big role. Please feel free to take a punt here too...
>
>Till next time.
Just to protest the tediousness of this thread, I am now going to remove
*all* rendom numbers from Crafty. I've taught simulation and modeling
several times, and one unit of that is probability theory and random
variates. I don't want that to filter into my computer chess hobby too.
I'd like to remain sane a while longer... :)
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