Nonnegative Linear Combination

Nonnegative Linear Combination

Q: What is a nonnegative linear combination?

A: It is the core mathematical expression behind the arbitrage model.

Each betting leg can be treated as a payout vector:

\[ p_i=(p_{1i},p_{2i},\ldots,p_{mi}) \]

Here \(p_{ji}\) is the return multiplier of leg \(i\) under score state \(j\).

Assign stake ratios:

\[ x=(x_1,x_2,\ldots,x_n) \]

with:

\[ x_i\ge0,\quad \sum_i x_i=1 \]

The combined return under score state \(j\) is:

\[ R_j=\sum_{i=1}^{n}p_{ji}x_i \]

This is a nonnegative linear combination.

The word nonnegative matters because real betting cannot use negative stakes. You can stake less, or skip a leg, but you cannot use a negative bet to cancel risk.

The arbitrage condition is:

\[ \min_j R_j>1 \]

In plain words:

Even the worst score state returns more than the total bankroll.