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) ?


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)


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