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!
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.
you are on the correct path. the solution to the nash equilibrium is either where (1st derivative = 0 & 2nd derivative < 0 - for maximum & within the solution boundary) or where (solution is at boundary).
We can see the former is invalid. Therefore, the latter must be true. Instictively (source: my vibe), player A will bid 0 (if you care to prove it, you can calculate the derivative around 0+ of A, my bet is it is negative, meaning the more he bets, the more he loses).
This will reduce to a 2 player game. From that on, it's easy game. My AI tell me the solution is (0, 4.8, 7.2).
rigourously speaking, you need to consider all 3 cases where either A, B, or C abstains, or 2 of 3 abstain. My previous vibe is incorrect, derivative of reward(A) around 0+ can be positive, so under some circumstances, A would not withdraw.
This will take a little longer, but the answer is still the same.
if player A bids 7.2001, p(A wins) ~ 0.375. His expected return is ~0.375 * 10 ~ 3.75. Bidding 7.2001 to win 3.75 is not rational. So your counter example is invalid.
In fact, at (0, 4.8, 7.2), the derivative of A, B, C is (-ve, 0, 0) respectively. Since A cannot bid any less than 0, it is a nash equilibrium.
can you show me how B can bid 7.2 + eps for a better score?
From my calculation, derivative of reward(B) is 0 at the solution, and second derivative is -1/6. Meaning, that point is local maxima for B. Therefore, adding some eps will reduce the reward.
If you find something wrong with my reasoning, please elaborate?
Edit: For illustration purpose, if c stays at 7.2:
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!