Suppose a fighter only fights opponents he is considerably better than. Let's say that he only fights opponents he would defeat 8 out of 10 times, or that his "dominance" is .80. Clearly this is sandbagging, he's a can crusher, etc. If he has 5 fights, what is the probability he will go undefeated? You'd think it would be pretty high, but to get the answer, we multiply .80 * .80 * .80 * .80 * .80 = (.80)^5, which is about .33. So the fighter only has about a 1 in 3 chance of going undefeated.
In real life, obviously, the fighter will have different win probabilities against different opponents. But if we hold the win probability constant, as we have done above, the math is much simpler. So, if we hold the win probability constant, what is the probability of a fighter going 49-0 like Mayweather?
Say Floyd beats all of his opponents 9 out of 10 times. Then the probability he goes 49-0 is .90^49, which is about .006. What about the probability he loses 1 fight?
The probability he loses his first fight is .10. Then the probability he wins his next 48 is .90^48, so the probability he loses his first fights, then wins his next 48, is .10*.90^48, which is about .0006.
This also applies to a 48-1 record where he loses only his second fight, where he loses only his third fight, etc. So the total probability of a 48-1 record, given dominance .90, is about .0006*49=.0294, or about 3 percent.
The formula for the probability of a given record where the loss occurs in a given fight (the first fight, the second fight, etc.) is
And the total probability of having a given record is given by
Given this formula, we can graph what the record of a fighter of a given dominance will probably be. For a fighter with dominance .75 and 30 fights, we have
We can also graph with respect to the dominance required to have a given record. Probable dominance for a fighter with a 25-5 record:
It seems that a long undefeated record requires fighting opponents who aren't competitive, or a lot of luck.
This analysis is interesting, but there's a weakness introduced by considering all things random. Standing back and looking at a large pool of fighters, the OP argument bears a lot more weight, but when you zoom in and look at someone like Bones, GSP and Anderson, there's relatively less randomness involved, in fact you can point to very specific reasons for their success.
I think I agree with the punchline though: long winning streaks should be considered very, very special and rare... especially since these guys become huge targets for micro-optimized game planning as time goes by.
jgiveshead - This analysis is interesting, but there's a weakness introduced by considering all things random. Standing back and looking at a large pool of fighters, the OP argument bears a lot more weight, but when you zoom in and look at someone like Bones, GSP and Anderson, there's relatively less randomness involved, in fact you can point to very specific reasons for their success.
I think I agree with the punchline though: long winning streaks should be considered very, very special and rare... especially since these guys become huge targets for micro-optimized game planning as time goes by.
Well, it's not a logical certainty that any fighter will beat his opponent, no matter the difference in ability, so probability is always involved. But I agree with you: a long winning streak requires a "special" fighter, in that a long winning streak is probably due to a very high level of dominance versus each opponent. When the streak is against the best opponents, this is amazing.
in boxing I'd say most guys who are elite are probably winning 99/100x for their first like 10-20 fights. A guy like Floyd has probably had 10 or so fights than are 9/10 to win and under.
Judging is also a factor, Floyd had had at least 1 fight that was arguable
Id say it's way harder in mma due to the rules, diverse skill sets, and the amount of damage that can be inflicted with one strike.
As a fellow mathematician that's a solid model OP. Level of competition obviously plays a huge factor in potential defeat though.
Makes the incredible runs of Fedor, Khabib and JBJ even more impressive.
As a sole point of contention, we should also factor in the finish rates as you move up through the weightclasses (since, by probability, any fighter at HW faces a far higher chance of being stopped per fight than in any any other weightclass).
in boxing I'd say most guys who are elite are probably winning 99/100x for their first like 10-20 fights. A guy like Floyd has probably had 10 or so fights than are 9/10 to win and under.
Judging is also a factor, Floyd had had at least 1 fight that was arguable
Id say it's way harder in mma due to the rules, diverse skill sets, and the amount of damage that can be inflicted with one strike.
Good point. World class boxers are often built up to have undefeated records for their initial fights. And the probability must be as high as you estimate, or they'd more than likely lose. For example, if the dominance is .99, then the fighter has about an 82% chance of going undefeated over 20 fights.
But if it's .95, he has only a 36% chance of going undefeated.
If Floyd has had 10 fights that are 9/10 to win, then his probability of winning all 10 was .90^10, or about 35%. If he was 99/100 for all of his other fights, then his probability of going undefeated was .90^10*.99^39, or about 24%.
As a fellow mathematician that's a solid model OP. Level of competition obviously plays a huge factor in potential defeat though.
Makes the incredible runs of Fedor, Khabib and JBJ even more impressive.
As a sole point of contention, we should also factor in the finish rates as you move up through the weightclasses (since, by probability, any fighter at HW faces a far higher chance of being stopped per fight than in any any other weightclass).
Last part of this is a big deal.
People made a big deal about Barao but he found like 20 something absolute nobodies before getting to the big show, and in a weight class that isnt exactly known for finishes.
As a fellow mathematician that's a solid model OP. Level of competition obviously plays a huge factor in potential defeat though.
Makes the incredible runs of Fedor, Khabib and JBJ even more impressive.
As a sole point of contention, we should also factor in the finish rates as you move up through the weightclasses (since, by probability, any fighter at HW faces a far higher chance of being stopped per fight than in any any other weightclass).
Thanks. But don't forget Igor! I think most of these were at heavyweight / openweight.
