I'm trying to organize a consultation game, something along the lines of "the Yanks" vs "the Cunucks", 3 players on each team. Canadian team is set-up and ready, now I'm working on the American counterpart. I will also take care of all the other organisational matters. The game should start by the end of this week.

<< mp3 of the day >>

The Grateful Dead performing an acoustic version of "Friend of the Devil" live from 1970

Actually I'm not a huge fan. I'm cheering as a matter of principle.

This is Neil Young and Crazy Horse doing "Cortez the Killer."

http://jl.reydon.free.fr/musique/sl...

Time for some bluegrass!

This is a group called "The Hillbilly Gypsies" doing "Crow Black Chicken."

A MATHEMATICAL ANALYSIS OF THE
CHALLENGER'S HANDICAP UNDER
ROBERT JAMES FISCHER'S PROPOSED
UNLIMITED DURATION MATCH FORMAT,
AND THE TRADITIONAL 24 GAME FORMAT

by "Sneaky"

<< (I) HISTORICAL BACKGROUND >>

"The whole idea is to make sure the players draw blood by winning games, and the spectators get their money's worth." R.J. Fischer

The purpose of this report is to perform a mathematical comparison of Fischer's proposal (also known as the "Pure Wins format" or "Unlimited Duration format") to the traditional 24 game match format, especially with regards to the handicap that it placed on the challenger.

It is not the purpose of this report to vindicate Fischer's proposal, to disparage Fischer's proposal, nor to make predictions about who would have won the Fischer-Karpov match had it been played. My only goal is to try to measure mathematically the degree of handicap placed upon the challenger in both systems, then to compare the two.

One of the largest criticisms of Fischer's proposed format is that it placed too great of a handicap on the challenger; this is to say, that the advantage that the champion enjoys in the Fisher format is greater than the advantage enjoyed in the traditional 24 game match. At first blush, this may seem to be almost axiomatic, as traditionally the challenger could become champion by winning the match by a single point, whereas in Fischer's proposal a two game margin was necessary. However, this comparison is invalid, because the two formats are so vastly different.

In both formats, the champion retains the title in the case of a tie match. In the traditional format, this would happen if the match resulted in a 12-12 tie. In Fischer's format, this was the result if the match should ever become tied at 9-9.

In both Fischer's format and the traditional format, the tie clause is the only tangible advantage granted to the champion. Nevertheless, it's a big one.

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?

If it can be demonstrated that a 9-9 tie under Fischer's proposal is in fact more likely than a 12-12 under the traditional system, then it is indeed true that Fischer's proposal was in some ways unfair. However, this assertion is by no means obvious. Clearly a 9-9 result would be more likely in a fixed match of 18 games compared to a fixed match of 24 games, but Fischer's proposal entails a match of unlimited duration which is very likely to last much longer than 18 games.

The challenger, of course, would prefer the format where ties are least likely, and the champion would presumably want a format where ties are commonplace. I believe that I can use mathematical models to accurately estimate the relative probability of a tied match under both systems. Once that's done, the comparison can be made, and we'll not only know which system conveyed more of an advantage to the champion, but also the extent of this advantage.

To help this article continue smoothly let us take time out to define a few terms:

- I will refer to chess game as 'drawn' if the result is a draw, and a match as 'tied' if it is declared a tie, to prevent confusion among these terms.

- A "Pure Wins Match" is a match played until either (a) one player has won g games, and is declared the winner, or (b) the score is g-1 to g-1 in which case the match is declared tied.

- A "Fischer Format Match" is a Pure Wins match where g=10.

- A "Fixed Length Match" is a match of n games, where the winner is the player with the post points at the end. If the points are equal at the end, the match is tied.

- A "Traditional Match" is a Fixed Length Match where n=24.

- The Tie% is the percent chance that a given match (either Pure Wins, or Fixed Length) will result in a tie.

- A match format is "fair" if the advantage given to the champion is small, and "unfair" if the advantage given to the champion is great. A perfectly fair format would give absolutely no advantage to the champion. When we speak of one format being "more fair" than another, we mean to say that it afford less of an advantage to the champion. This is a specialized usage of the word 'fair' which may not be in accord with common sense, but for purposes of this report it's a concise way to express the concept.

- Likewise, "unfairness" with regards to a match format is a measurement of the challenger's handicap, which is proportional to Tie%.

- An "upset" is a situation where the inferior player wins.

- The term "coloration" refers to who has the white pieces, and who has the black pieces. This will be touched upon later, but is largely ignored.

There are many ways that one may compare a Pure Wins Match to a Traditional Match. I propose that we should begin by analyzing the situation where a challenger is exactly of equal strength as the champion. This is the situation which should maximize the champion's advantage since it maximizes the likelihood of a tied match.

A related question is how the two formats compare when one of the players is measurably inferior to the other. This is an important topic, and will be treated separately.

<< (V) ASSUMPTIONS AND GIVENS >>

- I am assuming that I can safely ignore coloration issues, based on the facts that the players alternate colors, and that this advantage is shared equally.

- 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 am assuming that all game results are independent of each other.

- I am assuming the my computer's pseudo-random number generator is sufficiently rich in its output to perform an accurate simulation.

- I am assuming that for our purposes, acceptable definitions of 'superior' and 'inferior' are as follows: that for every decisive game, the superior player is 60% likely to take the point, while the inferior player is 40% likely. Some of the calculations at later stages of this article will rely upon that definition.

<< (VI) MEASURING THE TIE % >>

Based on everything stated so far, it should be obvious that that our goal is to measure the Tie% for both the Pure Wins Format, and the Fixed Length Format, and compare them. Whichever format has a least chance of a tie is the one less biased in favor of the champion, ergo, more fair.

A computer program was written to help answer many of these questions.

It's manner of operation is very simple: by using a random number generator, it simulates the many rounds of a world championship match, keeping score for both players. A function called sim_pw(g) can simulate a Pure Wins match played up to g wins (or g-1 vs g-1 tie). A similar function sim_fl(n) simulates a Fixed Length match with n games.

Both functions sim_pw(g) and sim_fl(n) return similar output: they return a -1 if the challenger wins the simulation, +1 if the champion wins, and a 0 if the match is tied.

The main loop of the program runs a loop of a large number of trials keeping track of the number of times the match was tied. It outputs the number of ties divided by the number of trials, expressed as a percentage. This is what I call Tie%.

The program also demands another variable: a value called "d" which is the probability that any single game is drawn. In the upcoming sections I will adjust the value of d to various plausible settings.

How does the computer simulate a game of chess? In "pseudo-code" the logic found within sim_fl(n) looks like this:

; simulate a game of chess
IF RAND() < d
THEN
; The game is a draw
score1 = score1 + 0.5
score2 = score2 + 0.5
ELSE
; The game is decisive
; Flip a coin to see who wins
IF RAND() < 0.5
THEN score1 = score1 + 1
ELSE score2 = score2 + 1

where d is the probability that any single game is a draw, where (score1,score2) are the scores of the two players respectively, and RAND() is a function that returns a pseudo-random number from 0 to 1. The above code is repeated n times (n=24 for our study) to simulate a single match. Millions of matches are thereby simulated and statistics can be drawn.

Using the computer model described above, I performed the simulation at various levels of d and measured the Tie% for 24 game matches. The results follow:

(TABLE 1: Odds of a tied result in a 24 game
match given various different probabilities
of drawn games.
Assumes equally matched contestants.
1,000,000 trials per data point.)

d / Tie% ( 0.01%)

0.20 9.00
0.25 9.27
0.30 9.72
0.35 10.05
0.40 10.42
0.45 10.94
0.50 11.49
0.55 12.16
0.56 12.25
0.57 12.41
0.58 12.47
0.59 12.67
0.60 12.81
0.61 13.03
0.62 13.23
0.63 13.41
0.64 13.56
0.65 13.75
0.66 14.00
0.67 14.17
0.68 14.39
0.69 14.70
0.70 14.86
0.71 15.17
0.72 15.48
0.73 15.73
0.74 16.06
0.75 16.41
0.76 16.78
0.77 17.10
0.78 17.49
0.79 17.96
0.80 18.39
0.81 18.98
0.82 19.48
0.83 20.08
0.84 20.78
0.85 21.48
0.90 27.04
0.95 41.29
1.00 100.00

<< (IX) WEAKNESSES IN THE ABOVE MODEL >>

While the estimates in Table 1 may be useful, there is are some possible weaknesses that should be addressed:

First, it is well understood by chess players that players can increase their chances of winning a game by taking risks which would otherwise be unattractive. Likewise, a player may choose to draw the game by purposely playing into opening variations known to be drawish. The model ignores these factors, and treats all games as independent, with equal probability of a draw or a result in any of them.

Also, there are various psychological factors in chess. Some players have difficulty winning a game immediately after they've suffered a hard loss. Such psychological factors are ignored; each game is considered to be an independent results that has no memory of previous games.

Now the computer simulation is set on the task of simulating Pure Wins matches. It is important to note that for this simulation any value of d which is less than 1 will yield the same result as any other. This is because draws don't count, so different values of d simply have no effect on the probability that the match will ultimately be tied.

We will, however, alter the value of g so we can see not only how Fischer's 10-game format performs, but also other potential g-values. (The computations other than 10 are interesting, but have no direct impact on the question we are answering.)

(TABLE 2: Odds of a tied result in a Pure Wins match: first to g-wins, where g-1 vs g-1 is considered a tie. Assumes equally matched contestants.

2,000,000 trials per data point, except for g=10 which employed 20,000,000 trials.)

g / Tie%

6 24.65
7 22.57
8 20.94
9 19.64

10 18.54 <-- Fischer's Proposal -->

11 17.62
12 16.80
13 16.08
14 15.50
15 14.94
20 12.83
25 11.43
30 10.46
40 8.99
50 8.07
75 6.53
100 5.65

That figure in the middle where g=10 is the important statistic that we needed to know: the probability that a Fischer Format Match results in a tie. Due to the importance of this figure for later calculations we ran the simulator ten times longer than normal.

The computed answer is 18.54%, or roughly 1 time in 5.4.

To help confirm this computation, let us consider for a moment a different question which is mathematically equivalent:

"Two players play a match where they flip a fair coin a number of times, until the match is over. Both players start with zero points. Heads accrues one point for the first player, while tails accrues one point for the second player. A player is declared the winner, ending the match, if they have 10 points. However, should the score ever become 9-9 then the match is immediately declared a tie. What is the probability that the match will be tied?"

This question can be answered mathematically, without the need for computer simulations.

If you think of the coin's results as a string of binary numbers, where 0=tails and 1=heads, the 9-9 tie can only exist after 18 flips where the number of 0's and 1's are exactly equal to 9. Therefore, this problem is equivalent to asking "Of all 18 digit binary numbers, how many have exactly nine 1s?"

There are 262,144 (2^18) 18 digit binary numbers.

To know how many of these numbers have nine 1s, we use the combinatorial function usually known as "choose": there are "18 choose 9" numbers with nine 1s, the formula being 18!/(18-9)!9! which equals 48,620.

Now we must take Table 1 and compare it to the value of 18.54. For each row we will compute Delta, which we define as the difference between the Traditional Format Match Tie% and 18.54. A positive number in this column indicates that under those circumstances, a 24-game match would be more favorable to the challenger than the 10-game Fischer Match. A negative number appears in this column under the circumstances that the Fischer Match provides more favorable conditions.

(TABLE 3: Comparison of Tie% of a 10-game Pure Wins match compared to a 24-game Fixed Length match.)

d / Tie% / Delta

0.20 9.00 9.54
0.25 9.27 9.27
0.30 9.72 8.82
0.35 10.05 8.49
0.40 10.42 8.12
0.45 10.94 7.60
0.50 11.49 7.05
0.55 12.16 6.38
0.56 12.25 6.29
0.57 12.41 6.13
0.58 12.47 6.07
0.59 12.67 5.87
0.60 12.81 5.73
0.61 13.03 5.51
0.62 13.23 5.31
0.63 13.41 5.13
0.64 13.56 4.98
0.65 13.75 4.79
0.66 14.00 4.54
0.67 14.17 4.37
0.68 14.39 4.15
0.69 14.70 3.84
0.70 14.86 3.68
0.71 15.17 3.37
0.72 15.48 3.06
0.73 15.73 2.81
0.74 16.06 2.48
0.75 16.41 2.13
0.76 16.78 1.76
0.77 17.10 1.44
0.78 17.49 1.05
0.79 17.96 0.58
0.80 18.39 0.15
0.81 18.98 -0.44
0.82 19.48 -0.94
0.83 20.08 -1.54
0.84 20.78 -2.24
0.85 21.48 -2.94
0.90 27.04 -8.50
0.95 41.29 -22.75

Based on section (X), the expect results from a Fischer Format match, with equally matched contestants, should always look like this:

Challenger wins outright 40.73%
Champion wins outright 40.73%
Tie (champion retains title) 18.54%

Let us suppose for a moment that we suggest that 3/4ths of all games played at that level result in draws. 3/4ths = 0.75, so we go to the row where d=0.75 and see that the Tie% is 16.41. That means that in a Traditional Format Match, two equal contestants would experience the following results:

Challenger wins outright 41.79%
Champion wins outright 41.79%
Tie (champion retains title) 16.41%

We can see here that the Fischer format is slightly less fair for the challenger than the Traditional format, with d=0.75. From the challenger's viewpoint, he would have a 41.79% chance of winning in the traditional format, and a 40.73% chance of winning under Fischer's proposal. So assuming 3/4th of games will be drawn, the challenger has a very tiny (1.06%) advantage playing under the traditional format.

Next let's look at one that favors the Fischer format. If the number of draws is expected to be 85%, then

Challenger wins outright 39.26%
Champion wins outright 29.26%
Tie (champion retains title) 21.48%

Here, the challenger's chances are only 39.26%, but the Fischer format would provide 40.73%. In this scenario, the challenger would have a better chance of becoming champion under the Fischer format.

We see now that the question of how frequently draws occur at this level is integrally related to the our question of how likely the traditional format should produce a 12-12 tie.

One poster at chessgames.com pointed out: "If you take all matches from FIDE era (1951-1990) which were played on limited number of games, there were 133 decisive games in 320 (without the second game of Fischer-Spassky match) that is 41.6%." (Honza Cervenka, Robert James Fischer page at chessgames.com, May-27-2005.)

However, analyzing historical figures may be misleading. For Table 1 and Table 3, the value of d assumes that the contestants are perfectly equally matched. Clearly, not every FIDE world championship bout has been between two equally matched opponents. It is logical therefore that a good estimate for the number of decisive games should be lower than 41.6% if we could somehow identify and select only the matches where neither player is significantly stronger than the other.

Some values are so out of bounds they can be safely dismissed. For d=0.50 and lower, we would be suggesting a level of draws typical of the early 19th century. For d=0.9 and greater we would be suggesting a level of draws so high that even in the modern era it would be regarded as a highly unusual outcome.

The frequency of draws is also related to the temperament of the players themselves. Some players are simply more content to offer draws, to accept draws, and to play into drawish variations. If both players fit this profile, then we could expect d to be very large. Other match-ups would likely produce a low value for d. From this point of view, the exact value of d is not something that can really be computed from any amount of historical research, but is instead a function of the specific match-up which you are examining.

It is outside the scope of this research to pinpoint the very best value of d to use. Therefore, our final answer is a spectrum of results based on different assumptions of the d-value. It should be stressed, however, than 'd' is not an estimate for the odds that any game is drawn at a WC match, only the odds that a game is drawn when you presuppose that both players are of equal strength, since this is the assumption we have adopted. (The discussion of unequal opponents will follow.)

I will not opine upon what the best value of d is, but I can made some broad statements about a reasonable range of values. Because 0.6 and 0.9 both seem to me unreasonable, I suggest therefore a reasonable range for d falls within 0.61 and 0.89.

Assuming equally matched contestants, if d is 0.61, then the Fischer format affords the challenger 2.75% less of a chance than he would otherwise stand. If d is 0.89 then the Fischer format favors the challenger by 4% more than the traditional format. And at 80% the odds are nearly exactly equal.

At this stage I would like to take a "time out" to examine some other possible match formats, to help us understand how altering various parameter influences the results.

Let us consider a Fixed Length Format with n=14, as was employed for the Kramnik-Leko match in Brissago, 2004.

(TABLE 4: Odds of a tied result in a 14 game
match given various different probabilities
of drawn games.
Assumes equally matched contestants.
1,000,000 trials per data point.)

d / Tie%

0.60 16.75
0.65 18.01
0.70 19.59
0.75 21.59
0.80 24.39
0.85 28.90
0.90 36.97

As we can see, the 14 game format places a much larger burden on the challenger than the 24 game match, and for most reasonable values of 'd' it appears to be less fair than the Fischer format as well.

Next, we will examine a Fixed Length Match where n=48.

(TABLE 5a: Odds of a tied result in a 48 game
Fixed Length match. Tie% given for various different probabilities of drawn games. Assumes equally matched contestants.
1,000,000 trials per data point.)

d / Tie%

0.60 9.07
0.65 9.73
0.70 10.55
0.75 11.56
0.80 12.89
0.85 14.99
0.90 18.56

This is the most fair system yet analyzed. For most reasonable values of d is the challenger's handicap is smaller than either the Traditional Format or Fischer's Format.

Now, for illustrative purposes, let us consider a format where 48 games are played instead of 24, and furthermore the challenger needs to win the match by a minimum of 2 points. I.e., not only will 24-24 be considered a tie, but 24.5-23.5 will be considered a tie as well.

(TABLE 5b: Odds of a tied result in a 48 game
match with a 'win by 2 points' requirement.
Tie% given for various different probabilities
of drawn games.
Assumes equally matched contestants.
1,000,000 trials per data point.)

d / Tie%

0.60 17.90
0.65 19.13
0.70 20.68
0.75 22.60
0.80 25.23
0.85 28.98
0.90 35.12

Not surprisingly, this is the most unfair format we have yet to examine. It is substantially less fair than the Fischer format, the Traditional Format, and even the 14-Game Format for all reasonable values of d.

So far, we have been assuming for simplicity that both the challenger and the champion are exactly equal. Now it's time to examine these formats when one of the competitors is measurably superior to the other. For each of the below tables, Chall% is the percent chance that the challenger wins, Champ% is the probability that the champion wins the match outright, and Tie% is, as usual, the probability that the match is tied.

(TABLE 6: Fischer Format Match where the
challenger is superior.*
10,000,000 trials.)

d / Tie% / Chall% / Champ%

any 12.83 73.74 13.43

(TABLE 7: Traditional 24-Game Match where
the challenger is superior.*
1,000,000 trials per data point.)

d / Tie% / Chall% / Champ%

0.60 10.64 67.92 21.45
0.65 11.67 65.96 22.37
0.70 12.97 63.82 23.21
0.75 14.60 61.32 24.08
0.80 16.91 58.29 24.80
0.85 20.19 54.31 25.49
0.90 26.07 48.59 25.34

* = We define 'superior' as follows: for every decisive (i.e. non-drawn) game, the result is 60% likely to be in favor of the superior player, and only 40% likely to be a win for the inferior player.

Now let's compare the data of the previous section to an analysis of an inferior challenger.

(TABLE 8: Fischer Format Match where the
champion is superior.*
10,000,000 trials.)

d / Tie% / Chall% / Champ%

any 12.83 13.43 73.74

Notice that the above table is exactly like table 6, except that the Chall% and Champ% are inverted, as we should expect.

(TABLE 9: Traditional 24-Game Match where
the champion is superior.*
1,000,000 trials per data point.)

d / Tie% / Chall% / Champ%

0.60 10.66 21.49 67.87
0.65 11.71 22.33 65.96
0.70 12.94 23.21 63.82
0.75 14.64 24.06 61.31
0.80 16.90 24.84 58.26
0.85 20.23 25.54 54.31
0.90 26.12 25.37 48.52

Notice that the above table is very close to table 7, except that the Chall% and Champ% are inverted. (Mathematically and logically, it should be the exact invert of table 7, but with "only" one million trials per data point there is a small random variance between program runs.)

* = We use the same definition of 'superior' from above.

Some analysis was performed to determine the expected number of games from Fischer matches at various values of d. While this does not effect our estimates of the Tie% it helps in our understanding of the nature of the proposal. The sheer length of a Fischer Format match, while perhaps not a desirable feature for practical reasons, makes it more accurate at determining which player is actually superior.

We perform this computation twice, once assuming equally matched contestants, and again using our previously stated definitions of 'superior' and 'inferior'. Because some people believe that a Fischer Format match would produce fewer draws, we will extend the calculations into a range of draws lower than in previous analyses.

(TABLE 10a: Average match length, in games, of a Fischer format match given that both contestants are equally matched.
1,000,000 trials per data point.)

d / avg. length

0.30 23.27
0.35 25.06
0.40 27.16
0.45 29.62
0.50 32.60
0.55 36.20
0.60 40.71
0.65 46.53
0.70 54.32
0.75 65.16
0.80 81.45
0.85 108.59

(TABLE 10b: Average match length, in games, of a Fischer format match given that one contestant is superior.
1,000,000 trials per data point.)

d / avg. length

0.30 22.36
0.35 24.07
0.40 26.07
0.45 28.46
0.50 31.29
0.55 34.77
0.60 39.10
0.65 44.71
0.70 52.13
0.75 62.61
0.80 78.21
0.85 104.35

In summary:

(1) IF CHALLENGER AND CHAMPION ARE OF EQUAL STRENGTH

- In terms of the handicap placed upon the challenger, both Fischer's proposal and the traditional 24-game match are similar.

- Depending on your initial assumptions of d (the percentage of games expected to be drawn), Fischer's proposal could be interpreted as being more fair, or less fair, or equal to, the traditional format.

(2) IF CHALLENGER AND CHAMPION ARE OF UNEQUAL STRENGTH

- IF CHALLENGER IS SUPERIOR TO CHAMPION, it is strongly in his favor to play a Fischer Match Format, as he is significantly more likely to win than with the Traditional Format.

- IF CHALLENGER IS INFERIOR TO CHAMPION, it is strongly in his favor to play a Traditional Format, as he is almost twice as likely to achieve an upset than with the Fischer Format.

- THEREFORE, the stronger player benefits from the Fischer Format, while the inferior player should desire the fixed length match with its proclivity for upsets.

(3) OTHER OBSERVATIONS

- In cases where the contestants are of unequal strength, the Fischer Match format is a more reliable test, with the better player more likely to prevail.

I would like to address some possible criticisms below.

1. Comment: Since you don't pinpoint an exact value for d, you never actually answered the question of which format is more fair.

Reply: This is true, but intentional. By presenting many different answers for many different plausible estimates of d, we get a better understanding of the relative fairness of the two systems across a wide spectrum of initial assumptions. Furthermore, the value of 'd' is something that changes historically. (Few chess players would argue with the observation that games between top players are more likely to be drawn today than in the 19th century!)

2. Comment: Your model is very simplistic. With effort, many more factors could be taken into consideration, such as fatigue, psychological factors, the effects of coloration, strategies where one player 'tries to draw', etc.

Reply: This too is intentional. The more sophisticated a model becomes, the more easy it becomes for an operator to adjust values so that it can produce almost any desired outcome. It was one of my primary goals in this report to never be accused of such statistical trickery. Furthermore, in practice, tiny alterations in the model tend to negate each other, therefore you can achieve a very similar and perhaps more accurate answer with a simple model.

3. Comment: Your definition of 'superior' and 'inferior' with the 60/40 rule is completely arbitrary. If you picked different values you would get different results.

Reply: True, but different results would not necessarily lead to different conclusions. The research could easily be expanded to examine other settings such as 55/45 or 70/30, but I had to stop somewhere. Furthermore, the analysis of the matches where one player is superior to the other is only half of the picture: it is also very important, if not supremely important, to consider the case of equal competitors.

4. Comment: "One should not read too much into simple models. " (Gypsy, from Chessgames.com May-27-2005)

I welcome all honest criticisms to my analysis, however I wish to save time by addressing a few complaints which may result from misunderstandings.

1. Criticism: "In a Fischer format match, because draws do not count, there should be fewer draws." (Alternately, "Karpov and Kasparov adopted a format similar to Fischer's and experienced a very large number of draws.") This fact is very important and ignored by your model.

Reply: It is ignored by the model because it makes absolutely no difference. The percentage of draws in a Fischer format match does not influence the computed figure of 18.54. Only in the 24 game match does the frequency of draws have a measurable effect on the Tie%. This was empirically demonstrated during testing, when several values of 'd' were used in Fischer match simulations. As expected, changing d had no impact on the computed value of the Tie%.

2. Criticism: There are always factors that your model ignores, therefore it tells us nothing.

Reply: One does not follow the other. All models are necessarily incomplete, by definition. That does not mean that calculations based on the model are worthless; on the contrary, they are the very best estimates, to my knowledge, that have been derived to date.

3. Criticism: You began this project with a preconceived notion and then manipulated the numbers to yield the answer you wanted.

Reply: If you want to accuse me of falsifying data then the least you can do is to point out a specific instance where I might have done this. I meticulously programmed the model, tracked its output, and reported the results, without knowing beforehand what results it would yield. I listed all of my assumptions and even pointed out criticisms to my own methods. I refused to endorse any single value of 'd' because I knew that the entire conclusion could be skewed by carefully picking this value to meet some preconceived notion.

If you have a specific complaint about any of my methods, I am very much interested, but vague accusations and ad hominem attacks will be duly ignored.

4. Criticism: How do I know what numbers your computer spit out? You could be making this whole thing up.

Reply: Like any good scientific experiment, my experiment is 100% reproducible. You are welcomed to duplicate my efforts. If you do, use trial sizes as large as mine, and I expect that you should find your results to equal mine to the nearest 0.03.

5. Criticism: Even a child can see that it's harder to win by two points than to win by one point. If you reach any other conclusion then your math must be wrong.

Reply: I question whether you have read this article. Please refer to section II starting at "If you have trouble understanding..."

6. Criticism: You say that Fischer's format is "less fair" (for some values of 'd') because it produces more ties when the contestants are equally matched. But isn't a tie the proper result? It seems that makes it MORE fair.

Reply: I am using a specialized definition of 'fair'. Please see section III.

7. Criticism: Fischer's format is untenable from an economic and practical viewpoint. How will you rent a venue if you don't know how long you'll be needing it? What happens if the match turns into an endurance contest, as was witnessed between Kasparov and Karpov?

Reply: That may be true, but it's not the question that I sought to answer. Please refer to the title of this document: "A MATHEMATICAL ANALYSIS OF THE CHALLENGER'S HANDICAP..." It is not designed to either vindicate nor disparage Fischer's proposal. It is not an analysis of venue rental costs, or any other issue beyond than the handicap placed on the challenger.

8. Criticism: There are other possible match formats which would be superior to Fischer's proposal.

I thank Chessgames user Gypsy for inspiring me to do this.

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You can direct comments about this report to my home page at Chessgames.com here: Sneaky chessforum