# Minimize loss in -EV gamble.

Betting related terms:

Multiple: a bet you make on two or more events where you win only if you guessed right all the outcomes of the events.

Odds: will be expressed in decimal, eg. in a bet at odds 2.5 if you bet 10 and win you get 25 (net won +15).

Suppose we have to make a multiple bet on n events. And we are given the possibility to bet on single events at the same time. How could we minimize loss disregarding the possible outcomes of the events?

In other terms, if I have to make a bet on a multiple, how can I find how much to bet on each single to lose the minimum?

Example

Event 2: Djokovic vs Murray; odds (Djokovic: 1.5; Murray: 2.8) P(Djokovic)=0.65 P(Murray)=0.35

If we bet a 15 euros multiple on Federer and Djokovic winning, our EV will be: 0.54* 0.65 * (15 * 2.7 - 15) - (1- 0.54*0.65) * 15 = -0.78 euro

But obviously there will be 2 different scenarios:

We win both events: net +25,5

We lose: net -15

I want to make these 2 numbers converge on the minimum loss (-0.78). In order to do that I could make single bets on single outcomes of that event in addition to the single bet. The problem is, how can I find how much to bet on every single event in order to lose the same amount for each of the possible outcomes?

Suppose you have two events $A$ and $B$ with outcomes $A_1, A_2$ and $B_1, B_2$, payoffs $a_1, a_2$ and $b_1, b_2$ and corresponding probabilities $\alpha_1, \alpha_2$ and $\beta_1, \beta_2$. We also have that $$\alpha_1 + \alpha_2 = \beta_1 + \beta_2 = 1$$ and $$a_1 < \frac{1}{\alpha_1}, ~ a_2 < \frac{1}{\alpha_2}, ~b_1 < \frac{1}{\beta_1}, ~b_2 < \frac{1}{\beta_2}$$
As far as I understand, you may bet not just on $A_1, A_2, B_1, B_2$ but also on any conjunction of these events with payoff equal to a product of included event payoffs. Usually, a bookmaker company calculates $a_1, a_2$ and $b_1, b_2$ according to $\alpha_1, \alpha_2$ and $\beta_1, \beta_2$, so if you ain't a perfect analyst (better than bookmaker asked), you have approximately the same EV for any bet. If you have your own $\alpha_1, \alpha_2$ and $\beta_1, \beta_2$, you may find the EV based on these values. For example, average payoff for $1$-euro bet $(A_1 \wedge B_2)$ is $a_1b_2\alpha_1 \beta_2 - 1$ and for $1$-euro bet $(A_2 \wedge B_1)$ it's $a_2b_1\alpha_2 \beta_1 - 1$. Taking values from your example ($A_1 =$ Federer, $A_2 =$ Nadal, $B_1 =$ Djokovic, $B_2 =$Murray) we have $a_1b_2\alpha_1 \beta_2 = 1.8 * 2.8 * 0.54 * 0.35 = 0.95256$ euro, while $a_2b_1\alpha_2 \beta_1 = 2.1 * 1.5 * 0.46 * 0.65 = 0.94185$ euro. It means that bet (Federer, Murray) is more valuable than (Nadal, Djokovic). To find the most valuable bet you have to check all such bets like $(A_1), (A_2), (B_1), (B_2), (A_1 \wedge B_1), (A_1 \wedge B_2), (A_2 \wedge B_1), (A_2 \wedge B_2)$ and perhaps some more complex bets. To avoid an exhaustive search in case of too many possibilities, you may first find the most valuable bets (strictly speaking, the least undervalued bets for which EV is closer for $0$ than for the other bets) and try to combine them or use separately.