Solution to a problem of equilibrium

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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%




economics




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  • 1




    "get most of the time a positive return?" What makes you think you can do this? If you mean win more than half the time, you almost assuredly cannot.
    – saulspatz
    Jul 25 at 17:46










  • well, that is why I am trying to solve this =). Maybe is not possible, maybe it is, let the numbers speak.
    – Nikko
    Jul 25 at 17:51














up vote
0
down vote

favorite












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%




economics




share|cite|improve this question

















  • 1




    "get most of the time a positive return?" What makes you think you can do this? If you mean win more than half the time, you almost assuredly cannot.
    – saulspatz
    Jul 25 at 17:46










  • well, that is why I am trying to solve this =). Maybe is not possible, maybe it is, let the numbers speak.
    – Nikko
    Jul 25 at 17:51












up vote
0
down vote

favorite









up vote
0
down vote

favorite











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%




economics




share|cite|improve this 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%





economics





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share|cite|improve this question




share|cite|improve this question








edited Jul 25 at 19:59
























asked Jul 25 at 17:32









Nikko

477




477







  • 1




    "get most of the time a positive return?" What makes you think you can do this? If you mean win more than half the time, you almost assuredly cannot.
    – saulspatz
    Jul 25 at 17:46










  • well, that is why I am trying to solve this =). Maybe is not possible, maybe it is, let the numbers speak.
    – Nikko
    Jul 25 at 17:51












  • 1




    "get most of the time a positive return?" What makes you think you can do this? If you mean win more than half the time, you almost assuredly cannot.
    – saulspatz
    Jul 25 at 17:46










  • well, that is why I am trying to solve this =). Maybe is not possible, maybe it is, let the numbers speak.
    – Nikko
    Jul 25 at 17:51







1




1




"get most of the time a positive return?" What makes you think you can do this? If you mean win more than half the time, you almost assuredly cannot.
– saulspatz
Jul 25 at 17:46




"get most of the time a positive return?" What makes you think you can do this? If you mean win more than half the time, you almost assuredly cannot.
– saulspatz
Jul 25 at 17:46












well, that is why I am trying to solve this =). Maybe is not possible, maybe it is, let the numbers speak.
– Nikko
Jul 25 at 17:51




well, that is why I am trying to solve this =). Maybe is not possible, maybe it is, let the numbers speak.
– Nikko
Jul 25 at 17:51










2 Answers
2






active

oldest

votes

















up vote
1
down vote













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.






share|cite|improve this answer





















  • Thx!. I wonder what constraints should I have
    – Nikko
    Jul 25 at 19:45






  • 1




    If $xp_1+y2p_2+z5p_3>x+y+z$, the expected return will be positive.
    – Tom M.
    Jul 25 at 19:56











  • ps: p1=42,5% p2=27%, p3=12%
    – Nikko
    Jul 25 at 19:59






  • 1




    $x,y,z$ are the money amount to each Jackpot. If $0.425x+0.54y+0.6z>x+y+z$, the expected return will be positive.
    – Tom M.
    Jul 25 at 20:05

















up vote
0
down vote













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!






share|cite|improve this answer





















  • But it must be good combinations, right?
    – Nikko
    Jul 25 at 19:45










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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote













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.






share|cite|improve this answer





















  • Thx!. I wonder what constraints should I have
    – Nikko
    Jul 25 at 19:45






  • 1




    If $xp_1+y2p_2+z5p_3>x+y+z$, the expected return will be positive.
    – Tom M.
    Jul 25 at 19:56











  • ps: p1=42,5% p2=27%, p3=12%
    – Nikko
    Jul 25 at 19:59






  • 1




    $x,y,z$ are the money amount to each Jackpot. If $0.425x+0.54y+0.6z>x+y+z$, the expected return will be positive.
    – Tom M.
    Jul 25 at 20:05














up vote
1
down vote













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.






share|cite|improve this answer





















  • Thx!. I wonder what constraints should I have
    – Nikko
    Jul 25 at 19:45






  • 1




    If $xp_1+y2p_2+z5p_3>x+y+z$, the expected return will be positive.
    – Tom M.
    Jul 25 at 19:56











  • ps: p1=42,5% p2=27%, p3=12%
    – Nikko
    Jul 25 at 19:59






  • 1




    $x,y,z$ are the money amount to each Jackpot. If $0.425x+0.54y+0.6z>x+y+z$, the expected return will be positive.
    – Tom M.
    Jul 25 at 20:05












up vote
1
down vote










up vote
1
down vote









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.






share|cite|improve this answer













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.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 25 at 17:53









Tom M.

163




163











  • Thx!. I wonder what constraints should I have
    – Nikko
    Jul 25 at 19:45






  • 1




    If $xp_1+y2p_2+z5p_3>x+y+z$, the expected return will be positive.
    – Tom M.
    Jul 25 at 19:56











  • ps: p1=42,5% p2=27%, p3=12%
    – Nikko
    Jul 25 at 19:59






  • 1




    $x,y,z$ are the money amount to each Jackpot. If $0.425x+0.54y+0.6z>x+y+z$, the expected return will be positive.
    – Tom M.
    Jul 25 at 20:05
















  • Thx!. I wonder what constraints should I have
    – Nikko
    Jul 25 at 19:45






  • 1




    If $xp_1+y2p_2+z5p_3>x+y+z$, the expected return will be positive.
    – Tom M.
    Jul 25 at 19:56











  • ps: p1=42,5% p2=27%, p3=12%
    – Nikko
    Jul 25 at 19:59






  • 1




    $x,y,z$ are the money amount to each Jackpot. If $0.425x+0.54y+0.6z>x+y+z$, the expected return will be positive.
    – Tom M.
    Jul 25 at 20:05















Thx!. I wonder what constraints should I have
– Nikko
Jul 25 at 19:45




Thx!. I wonder what constraints should I have
– Nikko
Jul 25 at 19:45




1




1




If $xp_1+y2p_2+z5p_3>x+y+z$, the expected return will be positive.
– Tom M.
Jul 25 at 19:56





If $xp_1+y2p_2+z5p_3>x+y+z$, the expected return will be positive.
– Tom M.
Jul 25 at 19:56













ps: p1=42,5% p2=27%, p3=12%
– Nikko
Jul 25 at 19:59




ps: p1=42,5% p2=27%, p3=12%
– Nikko
Jul 25 at 19:59




1




1




$x,y,z$ are the money amount to each Jackpot. If $0.425x+0.54y+0.6z>x+y+z$, the expected return will be positive.
– Tom M.
Jul 25 at 20:05




$x,y,z$ are the money amount to each Jackpot. If $0.425x+0.54y+0.6z>x+y+z$, the expected return will be positive.
– Tom M.
Jul 25 at 20:05










up vote
0
down vote













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!






share|cite|improve this answer





















  • But it must be good combinations, right?
    – Nikko
    Jul 25 at 19:45














up vote
0
down vote













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!






share|cite|improve this answer





















  • But it must be good combinations, right?
    – Nikko
    Jul 25 at 19:45












up vote
0
down vote










up vote
0
down vote









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!






share|cite|improve this answer













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!







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 25 at 18:05









Tommy-Xavier Robillard

12




12











  • But it must be good combinations, right?
    – Nikko
    Jul 25 at 19:45
















  • But it must be good combinations, right?
    – Nikko
    Jul 25 at 19:45















But it must be good combinations, right?
– Nikko
Jul 25 at 19:45




But it must be good combinations, right?
– Nikko
Jul 25 at 19:45












 

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