Re: Incoherent Odds?

Protoman wrote:
What are "incoherent odds"; I've only heard of them in an unfair
wagering system called a "Dutch book".

I confess I'd never heard of the term, but you might find some relevant
info at
http://en.wikipedia.org/wiki/Coherence_%28philosophical_gambling_strategy%29

And what's the difference
between "odds" and "probability"? And how do you calculate the
probability of odds like "3:2 against"; Is it the same as "4:1
against", just take the decimal odds (5) and take the reciprocal of
that (.20)? Thanks!!!!!

Probability is a measurement of the likelihood of an event happenning.
If you carry out n identical trials, and let s be the number of times
that the event occurs, then the probability of the event happening is
the limit of the ratio s/n, as n becomes indefinitely large. "Odds" are
just a practical system designed to make it easy(ier) to work out how
much to pay as winnings (in the old days, before calculators and
computers).

Obviously in many cases (such as horse racing, stock market etc.) the
*real* probability of an event happening cannot be determined exactly
(because it is not possible to carry out identical trials). Odds are
then arrived at based on a combination of guesswork, instinct, and,
importantly, money flows ("balancing the book"). Even in cases where an
exact probability can be determined with some accuracy (such as
roulette), the odds offered probably won't reflect the actual
probability (they will favour the bookmaker, casino owner, or
whatever).

To convert from odds to probability it is common to assume that the
odds are fair (expected gain/loss for both parties is zero). Suppose
you wager b units at odds of a:b on some event with probability p.
Consider the two cases:

1. You win a units with probability p
2. You lose b units with probability 1 - p

Your expectation is a*p - b*(1-p), which should be zero. Rearranging
a*p - b*(1-p) = 0 gives p = b/(a + b). For the reasons given, this
"implied probability" may or may not bear much relation to the actual
probability (assuming that can be determined). For example, if you
factor in a margin for either party then the probability changes.

.

Relevant Pages

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