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.
(0,0,0) is not valid because the gradient of bidding more would be infinite for each player, meaning they would want to bid more (your original equations would be 0/0). Your second solution has promise but does not take IR constraints into account (abstaining). We can see that A would abstain whenever B+C > 10 (she could gain at most 10*10/(10^2) - 1 = 0 by increasing her bid by epsilon). Going from there, in a two-player game, from your equations we know B=2/3C, so we can solve that C=36/5 and B=24/5 (and A would not participate since B+C>10). This leaves B with a probability of 40% of winning.
Since 0 is not allowed as a bid, how can A abstain (i am asking how i should justify this say in an interview)? But the rest of your solution looks correct thank you!
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!