Basics of Game Theory

Some Basics of Game Theory.
Q1.Let us suppose that you have 74 coins and you can pick minimum 2 coins and maximum 6 coins and There are two players A and B
how much coins should a pick at start to win the game.
Ans 1: You have to reduce 74 to multiple of (2+6)

so A will reduce 74 to 72.Now its Finders winners means the player who plays
last turn wins the match At each turn

A will reduce the game to multiple of 8 to win the game at last suppose at last
the number will arrive at 8 now B will pick any thing

between 2 and 6 coins a will pick the left over coin and wins the match.
But if it would be a Keepers losers game than
we have to reduce the number to multiple of 8 + x (Where x is equal to 1<x<=a ).

**************************** Composite Game / Grundy Number *********

*****************************************************

Imp-> We will use sprague grundy theorem in that case in

which we are not suppose to choose any no of coins we want we have a limit

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Q2.Calculate Grundy Number.
/* A Dynamic Programming (Memoization-based) approach to
calculate Grundy Number of a Game
Game Description-
Just like a one-pile version of Nim, the game starts with
a pile of n stones, and the player to move may take any positive number of stones.
The last player to move wins. Which player wins the game? */

#include<bits/stdc++.h>

using namespace std;



// A Function to calculate Mex of all the values in that set

// This function remains same

int calculateMex(unordered_set<int> Set)

{

int Mex = 0;


while (Set.find (Mex) != Set.end())

Mex++;



return (Mex);

}

// A function to Compute Grundy Number of ‘n’

// Only this function varies according to the game

int calculateGrundy(int n, int Grundy[])

{

if (n == 0)

return (0);



if (Grundy[n] != -1)

return (Grundy[n]);



unordered_set<int> Set; // A Hash Table



for (int i=0; i<=n-1; i++)

Set.insert(calculateGrundy(i, Grundy));

// Store the result

Grundy[n] = calculateMex (Set);



return(Grundy[n]);

}

// Driver program to test above functions

int main()

{

int n = 10;

// An array to cache the sub-problems so that

// re-computation of same sub-problems is avoided

int Grundy[n+1];

memset (Grundy, -1, sizeof(Grundy));
printf (“%d”, calculateGrundy(n, Grundy));
return (0);

}

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->IMPORTANT :

Now we know that when we have to calculate grundy number now if We have N piles and we have to pick minimum x coins and maximum

y coins so here we will use Sprague grundy theorem. We will calculate XOR of all grundy theorem if it comes zero 2nd PLayer

would win either A.



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