< I am assuming that I can ignore the issue that players may become fatigued after many games, in such a way that the probability of drawn games change.

I did in fact investigate both of these factors (what I call the 'fatigue factor' and also the 'make a draw' strategy) to test certain ideas which we all know are correct, first that people get tired, and second that GMs can attempt to force a draw when it is good match strategy to do so. I left this out of my report, for the following reasons:

The fatigue factor just doesn't influence the results very much. If you assume fatigue creates more errors, and therefore less draws, then the impact on the "fairness" of the traditional match is very small for any realistic values. It also could be argued that fatigue makes players less likely to take risks, therefore the draws increase. Either way, it shouldn't make much of a difference.

The strategic-draw concept works heavily against the traditional format, therefore the more you endorse that this is a real effect, the more fair Fischer's proposition seems. However if posit that this effect is only very modest, then it will not have a large impact on the results. It would be akin to picking a 'd' value slightly larger than you otherwise would.

You can not make a false assumption and then start deducing results. The fatigue factor plays a very important role as you see in the Karpov - Kasparov 1984 match and it obviously impacts the fairness of the match. A Fischer match would favor a physically fit player rather the one with the poor health who can not play a match for 4 months straight. In a sense the Fischer format rewards the physique of the player.

First of all please understand that comparing the systems by comparing the Tie% only is valid with equally matched contestants. It *might* be true with unequal contestants but it need not be.

With equally matched opponents, those two issues you bring up are the same thing. It's as if I said "This coin flips heads 60% of the time" I just told you how often it flips tails. Let's look at one of the formats as an example, the Fischer format.

There are 3 possible outcomes: (a) The challenger wins by scoring 10 wins (b) The champion wins by scoring 10 wins (c) The match is tied 9-9. Assuming equally matched contestants, if you know any one of these values you know the other two, because you know that a=b, and that a+b+c=100%. You can do the exact same thing with the 24 game format.

Another way of thinking about it is even simpler, as my report says <Because the champion's advantage is derived only from the tie clause, to fairly compare Fischer's format to the traditional format reduces to answering one question: IN WHICH FORMAT IS THE MATCH MOST LIKELY TO RESULT IN A TIE?>

In the traditional format, if either player wins by scoring more points then everybody says "Very well, he won the match fair and square." But if the match is tied 12-12 then the crowd murmers "The champion won only by the draw clause." The same can be said for the Fischer format with its 9-9 clause. The fact is, the tie-clause is the only reason why the champion is afforded any advantage.

You might say "oh, but fatigue would greatly change your results." There are only two ways that fatigue could have an effect on the calculations: if it makes for fewer draws nearing the end of a 24 game match, then it would make the traditional format seem a little more fair. if it makes for more draws nearing the end of a 24 game match, tthen the traditional format would seem a little less fair. I don't believe that a 24 game match is long enough to really witness this effect statistically therefore I chose to ignore it.

<The fatigue factor plays a very important role as you see in the Karpov - Kasparov 1984 match and it obviously impacts the fairness of the match.> But that was a Fischer format match! The fatigue effect has no bearing on our calculations of who wins a Fischer format match. <A Fischer match would favor a physically fit player rather the one with the poor health who can not play a match for 4 months straight. In a sense the Fischer format rewards the physique of the player.> Well, yes, that might be very well true, but what you just said doesn't amount of a "challenger handicap" therefore its net effect on the calculations is zero.

<You have not defended the second assumption I quoted. Just to make it more clear; a loss in a 24 game match is a bigger blow than a loss in a Fischer match. In the latter a player can lose 3 games and still have a reasonable chance of winning the match but obviously the same is not true for a 24 game match. In a 24 game match the games are highly correlated while the correlation decreases in a Fischer match.> When I say that "I am assuming that all game results are independent of each other" I mean that they are like coin flips, or die rolls; their outcome is not influenced by the previous results. I understand what you are saying, and of course the computer simulations found lots of cases where one contestant pulled ahead in the match (of either format) by two or three points and cruised into the finish line. Those cases are factored into the final statistics.

What I believe you are suggesting is that the simulation should be smart enough to know that a player, when faced with a large point deficit in a 24 game match, to "do something different" and make the result of the very next game in the match something other than the normal distribution of wins, losses, and draws. You know--he plays the Benoni, or something equally risky.

From one point of view, there are no head starts at all in either format: both players start with zero points, both players have to reach a certain amount of wins to be declared the winner. For either format, the match is perfectly symmetrical, with both the champion and the challenger playing under identical conditions. However, in both formats the symmetry is broken wide open with the addition of a "tie clause", the fact that ties are a victory for the champion.

Of course I recognize that in both formats the ties allow the champion to retain the title, I understand the rules of the game as well as anybody. I just don't like the terminology of "head start" or "victory margin" as I think that muddles understanding the issue clearly, so I avoided them.

<i dont think these two are the same unless im misunderstanding something. or, like you were trying to say, do the statistics explain it either way?> Unless I'm misunderstanding something, the only difference in what you're talking about and what I'm talking about is semantics.

An appropriate one for today would be Tom Lehrer's funny little jingle, "That's Mathematics"

What do you take me for? Of course I meant relevant assumptions.

<Sneaky: <The fatigue factor plays a very important role as you see in the Karpov - Kasparov 1984 match and it obviously impacts the fairness of the match.> But that was a Fischer format match! The fatigue effect has no bearing on our calculations of who wins a Fischer format match.>

I can not understand what you are saying.

<Sneaky: <A Fischer match would favor a physically fit player rather the one with the poor health who can not play a match for 4 months straight. In a sense the Fischer format rewards the physique of the player.> Well, yes, that might be very well true, but what you just said doesn't amount of a "challenger handicap" therefore its net effect on the calculations is zero.>

Why? You examined which format was more fair when one player is stronger. Why not do the same thing with their physique? Otherwise you are implicitely assuming that the two players have the same physique and stamina which is obviously not true.

<Sneaky: I do believe that this is a genuine effect, but I chose to ignore it because I'm not terribly convinced that this effect will end up having a big influence on the outcome of 24 game matches.>

Suppose a player is 2 points behind after 12 games. In a 24 game match he can not afford taking draws with White. While the same situation in the Fischer match means the players are warming up! In other words in a 24 game match the "play it safe" strategy is at best a short term one, while in the Fischer match you can use it effectively like Kasparov did in 1984. The fact that the games are not that correlated in a Fischer match was one of Fischer's arguments for his proposed format, I was very surprised that you did not even consider it.

<Sneaky: Do you suppose it would make the Tie% go up or down? Remember, in addition to the "take risks" strategy, there is also the "play it safe" strategy!>

I interviewed the director of the local observatory on community radio some years ago and she brought in that song to play on air. I hadn't heard it before and had forgotten it again until you put up the mathematics song.

When I said <The fatigue effect has no bearing on our calculations of who wins a Fischer format match.> you replied <I can not understand what you are saying. > What I'm saying is, that knowing nothing about the stamina of the two competitors, the mere fact that a certain match would require endurance doesn't lead us to believe that "the challenger has a disadvantage" nor "the champion has a disadvantage." Sure, whichever one had less stamina would be at some disadvantage, but we don't know which player that is, nor can we judge how much the disadvantage. Which brings you to your next point:

<Why? You examined which format was more fair when one player is stronger. Why not do the same thing with their physique?>

A fair point. My report concluded that a stronger chess player (whether challenger or champion) should prefer the Fischer match. But what if this stronger player lacked stamina? Then it might be a toss-up, or perhaps if the fatigue factor was big enough it would even swing it back to the stronger player favoring the traditional match. What if a weaker player had more stamina? My conclusion was that the weaker player should prefer the 24 game match, but again if the stamina factor was big enough it would actually make him want to prefer the Fischer match.

"Stamina" is simply not a factor that I chose to model. Since we are talking about chess players, some measure of chess ability seemed to be a good thing to model. How fast they can run a mile was not (even though that is admittedly also relevant in an indirect way.) The model could also consider white/black coloration issues. It could consider jet-lag factors! We could even model "temperature of the venue" and then have each player rated in terms of "ability to withstand various temperatures." (Ergo, Larsen loses in Denver's crushing summer heat, while but warmblooded Mecking would be bound to fail in Sibera.) My point is, you have to draw the line somewhere.

There's another reason why I chose to keep the model simple. Overly complex models allow the operator (the one who programs the parameters of the simulations) to be able to achieve any result they like, because all manner of fuzzy variables are arbitrarily chosen. At that point, the output of the model is useless, unless your only goal is to trick people.

However on closer inspection, relaxing these constraints will likely only refine your results. For assumption (i), one need only look at the case where one player becomes relatively weaker across the match. In this case, the solution is likely to be a hybrid of the cases you have treated: the simple equal strength case, and the unequal strength case.

Assumption (ii) is more difficult, since it requires nonzero correlations between games. It's not clear immediately - to me at least - what form this correlation ought to take. Do players gain strength from their wins, or are they expended (or complacent)? This is delving into psychology, which is never a lot of fun to quantify, for the simple reason that there are no hard and fast rules. People like to give reasons in retrospect, but where are the predictions? It's like trying to tease rules out of random noise, and perhaps that's exactly what it is! In that case, the correlations may tend to wash out on average, and yours would be a well justified approximation. I'm not sure what the truth is, but at least you've taken a constructive step in addressing the problem.

One thing I thought I could add, is an attempt to estimate 'd' in the case of equal strength. To start, your 60:40 assumption in the case of one player stronger than the other, corresponds to a 70pt rating difference under the Elo/chessmetric systems. A 55:45 ratio corresponds to a 35 point differece. So, I went through and found all WC matches where the two players' chessmetrics ratings were within 35pts of each other at the time of the match. The results are below, 'd' being draw fraction, 'R' the rating difference at the time of the match.

<Match, d, R>

Steinitz-Zukertort (1886), 0.25, 20
Steinitz-Gunsberg (1890), 0.47, 10
Steintiz-Chigorin (1892), 0.22, 20
Lasker-Tarrasch (1908), 0.31, 35
Capablanca-Alekhine (1927), 0.74, 32
Euwe-Alekhine (1937), 0.44, 24
Botvinnik-Smyslov (1954), 0.42, 11
Smyslov-Botvinnik (1957), 0.59, 20
Botvinnik-Smyslov (1958), 0.48, 33
Botvinnik-Petrosian (1963), 0.68, 34
Petrosian-Spassky (1966), 0.71, 31
Petrosian-Spassky (1969), 0.57, 20
Karpov-Korchnoi (1978), 0.66, 16
Karpov-Korchnoi (1981), 0.56, 35
Karpov-Kasparov (1984), 0.83, 33
Karpov-Kasparov (1985), 0.67, 8
Kasparov-Karpov (1986), 0.63, 23
Kasparov-Karpov (1987), 0.67, 33
Kramnik-Leko (2004), 0.71, 10

<My range would imply it is in general just slightly advantageous for the challenger to use the traditional format; since it corresponds to a tie percentage of <12.3-20.1%>, compared to the fixed 18.5% for the Fischer Format.> It might be appropriate to throw out the highest and lowest figures as anomalies, but I too concluded it is just slightly advantageous for the challenger to use the traditional format. I didn't write that in my report because the proper value for 'd' is up in the air, and I didn't want to let personal opinion guide me to a conclusion.

As my report states, "The purpose here is not to calculate some sort of ultimate answer but to get a broad feel for what we can expect from these different match formats."

In the end, my statement is this: I tried to find a significant mathematical advantage of one format over the other, and I couldn't.

So your defense is that you could not do it so you did not? If I were you, I would not do the simulations without taking that factor into account considering that this was one of Fischer's arguments for his format being more fair.

<Sneaky: Different people can get over jet lag better than others, for some it takes weeks for their body to change its rhythm. Imagine if you had to play Topalov and your body was telling you "It's 4:00am, get some sleep!">

This band blows me away. I see them every time they come to South Florida. Dubbed "The Codetalkers", from Atlanta Georgia, they feature Colonel Bruce Hampton, and they perform a weird mixture of bluegrass, jazz, country, and good-natured insanity.

This song is called "Arkansas."

http://www.thecodetalkers.com/arkan...

<Hesam> Look, "fatigue" is not a question which can be mathematically answered. If you want to argue that fatigue will hurt some players' performance more than others, you may be very well right, but you've left the realm of pure mathematics, and moreover, you've also departed from trying to answer the question I attempted to answer.