0
$\begingroup$

I'm reading about gambling in Elements of Information theory. As stated in the book, given that the gambler bets $o_i$-for-$1$ on horse $i$,

Fair odds (w.r.t some distribution): the odds is fair if $\sum_i \frac{1}{o_i} = 1$

Superfair odds: the odds is superfair if $\sum_i \frac{1}{o_i} < 1$

Subfair odds: the odds is subfair if $\sum_i \frac{1}{o_i} > 1$

Can anyone give me some intuition about these concepts (i.e. fair odds, superfaid odds and subfair odds) ?

$\endgroup$
1
$\begingroup$

From the point of the view of the house (who takes the bets) : let's assume $b_i$ is the amount bet on horse $i$, and let $B=\sum_i b_i$ be the total bet. How should we pay, so that we (the house) don't lose or win? If $i$ is the winner, we should give the total $B$ to the gamblers who bet to it, so they will get $B$-for-$b_i$, or $o_i=B/b_i$ for $1$. So, in this (fair) case, we'd have $\sum 1/o_i = 1$.

Now, typically the house will want to profit a small share of the total bet, then the winners will get a total of $\alpha B$ with $\alpha \lessapprox 1$ (conversely, if $\alpha > 1$ then the house would lose something in each game... not a very usual scenario).

Hence, in general $o_i = \alpha B/b_i$ and $\sum 1/o_i = \frac{1}{\alpha}$ or

$$\alpha = \frac{1}{\sum 1/o_i}$$

This says that ${\sum 1/o_i} > 1 \implies \alpha < 1$ , which is the usual, subfair scenario (for the gamblers).

For example, in some bet site today, the odds for the ATP match Federer-Anderson are $o_1 = 1+4/11$ , $o_2 = 1+11/5$. So $\alpha = \frac{1}{\sum 1/o_i}=0.9562$, so the house profits nearly $4.4\%$ of the bet (subfair).

(BTW: Though the textbook is about Information Theory, this question actually has little to do with it)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.