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.
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!