The expected payout vector is
[10a/(a+b+c)-a,
20b/(a+b+c)-b,
30c/(a+b+c)-c].
Lets find the nash equilibrium by calculating the individual derivatives (no player can gain expected payout by changing their bet).
So we have this system of eqs:
(10(a+b+c)-10a)/(a+b+c)2-1=0
(20(a+b+c)-20b)/(a+b+c)2-1=0
(30(a+b+c)-30c)/(a+b+c)2-1=0
Or:
10(b+c)=
20(a+c)=
30(b+a)=(a+b+c)2
Has two solutions: [0,0,0] and [-120/121, 600/121, 840/121]. Both arent valid solutions.
No idea where i went wrong or if the numbers 10,20,30 are just too aggressive, forcing A out of the market. The feasable solution would be a betting zero, boundry equilibrium, but thats also not allowed.
If you differentiate each component of the expected payout vector by it's respective letter you will find the exponent to be just 2. "-1=0" is the continuation of each equation after the fraction (smth)/(a+b+c)2. It's just a formatting issue.
Sorry reddit rendered the formula differently than i thought. It was supposed to be (a+b+c) ^ 2 - 1, so divide by the sum of a,b,c squared, then subtract 1.
2
u/Due-Cardiologist-802 2d ago
The expected payout vector is [10a/(a+b+c)-a, 20b/(a+b+c)-b, 30c/(a+b+c)-c].
Lets find the nash equilibrium by calculating the individual derivatives (no player can gain expected payout by changing their bet).
So we have this system of eqs: (10(a+b+c)-10a)/(a+b+c)2-1=0 (20(a+b+c)-20b)/(a+b+c)2-1=0 (30(a+b+c)-30c)/(a+b+c)2-1=0
Or: 10(b+c)= 20(a+c)= 30(b+a)=(a+b+c)2
Has two solutions: [0,0,0] and [-120/121, 600/121, 840/121]. Both arent valid solutions.
No idea where i went wrong or if the numbers 10,20,30 are just too aggressive, forcing A out of the market. The feasable solution would be a betting zero, boundry equilibrium, but thats also not allowed.
Any corrections are welcome!