Another Logical Brain Teaser asked in Quant interview at DE Shaw
I guess a more elaborate explanation would be:
Consider the i_th player's action to be x_i. In order to achieve Nash Equilibrium, I apply 'assuming other players have played their strategies what is my best response?'
-> | (3/200) * ( x_1 + x_2 + ... x_50 ) - x_i | = 0
-> Summation of all other player's action = (x_i)*
where (x_i)* is i_th player's best response strategy.
So we have 50 equations like this.
(x_2)* + (x_3)* + (x_4)* ..... (x_50)* = (x_1)* ----------- Equation 1
(x_1)* + (x_3)* + (x_4)* ..... (x_50)* = (x_2)* ----------- Equation 2
(x_1)* + (x_2)* + (x_4)* ..... (x_50)* = (x_3)* ----------- Equation 3
....
...
(x_1)* + (x_2)* + (x_3)* ..... (x_49)* = (x_50)* ----------- Equation 50
Add all the equations up. You get:
49 * ((x_1)* + (x_2)* + (x_3)* ..... (x_50)*) = ((x_1)* + (x_2)* + (x_3)* ..... (x_50)*) ----------- Final equation
because on the LHS each variable is missing only once, i.e., in its own equation. This gives us:
((x_1)* + (x_2)* + (x_3)* ..... (x_50)*) = 0
which implies all the best responses have to be 0.
So each player's best response have to be 0.
Note these are simultaneous equations. Each player is playing his/her best response at the same time giving us the 50 equations.
Always coming up with great solutions ! Loved it
I guess a more elaborate explanation would be:
Consider the i_th player's action to be x_i. In order to achieve Nash Equilibrium, I apply 'assuming other players have played their strategies what is my best response?'
-> | (3/200) * ( x_1 + x_2 + ... x_50 ) - x_i | = 0
-> Summation of all other player's action = (x_i)*
where (x_i)* is i_th player's best response strategy.
So we have 50 equations like this.
(x_2)* + (x_3)* + (x_4)* ..... (x_50)* = (x_1)* ----------- Equation 1
(x_1)* + (x_3)* + (x_4)* ..... (x_50)* = (x_2)* ----------- Equation 2
(x_1)* + (x_2)* + (x_4)* ..... (x_50)* = (x_3)* ----------- Equation 3
....
...
...
(x_1)* + (x_2)* + (x_3)* ..... (x_49)* = (x_50)* ----------- Equation 50
Add all the equations up. You get:
49 * ((x_1)* + (x_2)* + (x_3)* ..... (x_50)*) = ((x_1)* + (x_2)* + (x_3)* ..... (x_50)*) ----------- Final equation
because on the LHS each variable is missing only once, i.e., in its own equation. This gives us:
((x_1)* + (x_2)* + (x_3)* ..... (x_50)*) = 0
which implies all the best responses have to be 0.
So each player's best response have to be 0.
Note these are simultaneous equations. Each player is playing his/her best response at the same time giving us the 50 equations.
Always coming up with great solutions ! Loved it