Arbitrage sports betting > Player A vs Player B. > > * Bookie 1 offers 1.36 odds on player A winning. > * Bookie 2 offers 5.5 on player B winning. > > > We have $1000 in total to bet. How would you place your bets such that profit is maximized? I have been told that this can be solved using linear programming, but I don't know how to set the problem up properly. Any ideas?

Bet $0 \leq x \leq 1000$ at the first bookie and $1000 - x$ at the second bookie. The goal is to maximize

$$\min \\{ 1.36 x - 1000, 4500 - 5.5 x \\}$$

which is the inverted "V" depicted below

![enter image description here](

The maximum is attained when

$$1.36 x - 1000 = 4500 - 5.5 x$$

The maximum is $\approx 90$, which is attained at $x \approx 802$.

There is no need to use linear programming. However, if you really, _really_ do want to use linear programming, then solve the following linear program in $x$ and $t$

$$\begin{array}{ll} \text{maximize} & t\\\ \text{subject to} & 1.36 x - 1000 \geq t\\\ & 4500 - 5.5 x \geq t\\\ & 0 \leq x \leq 1000\end{array}$$