# Arbitrage Theorem

Let $X=(X_1, \dots, X_n)$ be the amount bet on each of the n experiment. Each experiment has possible outcome $j=1,2, \dots, m$, $r_i(j)$= return function of bet $i$ when outcome is $j$ the expected return when outcome is j: $= E[X| j]= \sum_{i=1}^{n} X_i r_{i}(j)$

There must be a set of probabilities $P=(P_1, \dots, P_n)$ for which, the expected return $= \sum_{j=1}^m P_j r_i(j)=0$, for all $i=1, \dots, n$.

Otherwise, an arbitrage would exists: $\sum_{i=1}^n X_i r_i(j) >0$ for all $j=1, \dots, m$.

The set of P are called risk-neutral probabilities, resulting in all bets being fair.