Solution to a problem of equilibrium

Question

I am not sure how to write this problem. But here it goes. I have a fixed amount of money. I want to distribute it in three different jackpots. Each one with a unique way of payout. Jackpot1 pays the same amount that you stake. Jackpot2 pays the double. Jacpot3 pays 5 times the amount. Every one of them with certain probability p1,p2,p3 of winning (p1>p2>p3; p1+p2+p3<1).

How can I find the right amount of money that I need to put in every spot to get most of the time a positive return?

It must be an equilibrium I suppose, but I can't find the right way to write the equations. Any help is appreciated =) Thx!

ps: p1=42,5% p2=27%, p3=12%

Answer 1

If there is a constraint that you need to put at least one dollar into each machine, the optimal distribution will be as follows:

If $p_{1}>2p_{2},5p_{3}$, put 1 dollar into Jackpot 2 and 3 and the rest of your money into Jackpot 1.

If $2p_{2}>p_{1},5p_{3}$, put 1 dollar into Jackpot 1 and 3 and the rest of your money into Jackpot 2.

If $5p_{3}>p_{1},2p_{2}$, put 1 dollar into Jackpot 1 and 2 and the rest of your money into Jackpot 3.

However, under the constraints you wrote, there is no guarantee that the expected return is positive.

Answer 2

The thing is ,to put it into simple words, that there will always be a "better" payout structure than the others, or it will be equal to the others. So either you will have to put all of your money into one Jackpot, either it will not matter, because the expected return will be the same whichever you choose!