John Smith wrote:
>
> I was wondering if anyone could help me with what is probably a fairly
> basic question.
>
> I want to investigate the significance of head-to-head records of
> chess players, versus their ELO rating. In other words, does player A,
> who has lost 3 of 10 games to player B and 7 of 10 games to player C,
> have more of a chance of beating player B than player C, if their ELO
> ratings were all equal?
>
It sounds to me as if this relates closely to the well-observed
phenomenon of intransitivity. That is, in many sports you may have three
players such that A usually beats B, who usually beats C, who usually
beats A.
Efron showed (via an example of four oddly-numbered dice, which can be
reduced to three) that this does *not* need to involve interaction.
[This first appeared in Martin Gardner's Scientific American column in
1970; Steinhaus and Trybula examined a similar phenomenon in 1959.]
That is, the same phenomenon could appear (eg) among darts players. It
seems certain that it would appear among chess players, where
interaction is at the heart of the game and many playing styles are
possible.
However, it's easy to see that the ability of a single-number rating to
predict probable victors - and the necessity of an undisputable champion
among any group - is equivalent to transitivity. So we can't necessarily
draw any conclusion EITHER from the ELO rating or from head-to-head
records against a third party.
Head-to-head records against the players involved (as in your original
question) do of course have some predictive value - although strictly we
cannot say that A's probability (in a strict frequentist sense) of
beating B is higher, as it is also possible that A was very lucky
against B in those ten games and unlucky against C, and would do
differently in the long run.
-Robert Dawwson
.
.
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