Rational Number
Another Logical Brain Teaser asked in Quant interview at DE Shaw
Calvin is one of the members of a group of 50 people playing Squid Game. Everyone must choose a number between 0 and 1000. The person who chooses the number that is closest to 3/4th of the group's average response wins. Which number should Calvin choose to increase his chances of winning?
Try to solve this problem yourself before moving on to the solution below
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Solution
This problem is often used in introductory Game Theory lectures.
The logical response would be for all players to play 0. This is due to the following logic: All players have access to the same information and are considered to be equally reasonable. So if a person does x, then every other player wants to do x. However, the first player believes that if I play 3/4*x, I would win, so he plays 3/4*x.
This applies to all players, and so the iterated value of x decreases until we approach zero, at which point if every other player plays zero, I, too, would want to play zero. So, Calvin should pick 0 as well.
In game theory, this is known as a Nash equilibrium.
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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.