# The intuition of fair odds (information theory)

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) ?

## 1 Answer

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)