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Let's imagine designing an odds pattern for a game, in which players bet for win or lose. Suppose the probablity of winning is $p$, thus the probablity of losing is $1-p$.

Now imagine $n_1$ people bet for win, $n_2$ people bet for lose, both ante is 1, and the odds for both are $1:M$ and $1:N$, respectively.

The banker doesn't want even a penny out of his wallet, so it's reasonable we have, according to Mean Value Formula: $$n_1 (M-1) p + n_2 (N-1) (1-p) \le n_2 p + n_1 (1-p)$$

specially, when $n_1 == 0$:
we have : $n_2 (N-1) (1-p) \le n_2 p$,
we get : $N \le p / (1-p) + 1$

when $n_2 == 0$, we can likewise get
$M \le (1-p) / p + 1$

To make it general, it makes sense to rewrite them like :
$N \le (p / (1-p) + 1) (n_2 / (n_1+n_2))$,
$M \le ((1-p) / p + 1) (n_1 / (n_1+n_2))$

Here comes my question: the $n_1$ and $n_2$ are influenced by the $M$ and $N$ and $P$. However, the $M$ and $N$ rely on the $n_1$ and $n_2$. How to figure out what $M$ and $N$ should be chosen?

It seems we should have a transcendental value for $n_1$ and $n_2$. FYI, there is a restriction : $n_1 + n_2 \le C - 2$, $C$ is a constant.

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I'm struggling to see a mathematical question here. How are $n_1$ and $n_2$ influenced by $M$, $N$ and $P$? –  Colin Reid Sep 18 '12 at 12:44
    
The profit(M or N) and the risk(p) will influence how people bet –  larmbr Sep 18 '12 at 13:13
    
I don't know what you mean by a transcendental value for the integers $n_1$ and $n_2$. However, your formula says you only need the banker to have a nonnegative expected return, and it is easy to ensure that. Set $M$ so that betting on Pass has a nonpositive return for the player, and set $N$ so that betting on Don't Pass has a nonpositive return. That said, I'm voting to close since I don't think the question is on the level of this site. –  Douglas Zare Sep 18 '12 at 15:48
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2 Answers

The general field you're looking for is called Game Theory. You have (slightly under-)specified a game, and you are interested in some notion of solution or optimal play.

There are many, but I'd suggest looking up Nash Equilibrium for a start. It is one of the simplest and most commonly applied, and also seems most like the concept you're after.

In general a game may have any number of Nash equilibria from zero to infinity. The non-uniqueness (a topic about which whole books have been written) can be viewed as the chicken-and-egg problem you are encountering.

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As pointed out, your model isn't completely specified. Yes $n_1$ and $n_2$ are influenced by $N$ and $M$, but how exactly depends on the participants.

For instance, you could decide to model each participant $i$ as having an estimate $p_i$ of the true probability $p$, drawn from a Beta distribution $B(p R, (1-p) R)$ for some factor $R$ and posit no risk aversion. That is however just one possible model among many.

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Thanks for your tips~ –  larmbr Sep 19 '12 at 6:54
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