Prerequisites : Grundy Numbers/Nimbers and Mex
We have already seen in Set 2 (https://www.geeksforgeeks.org/combinatorial-game-theory-set-2-game-nim/), that we can find who wins in a game of Nim without actually playing the game.
Suppose we change the classic Nim game a bit. This time each player can only remove 1, 2 or 3 stones only (and not any number of stones as in the classic game of Nim). Can we predict who will win?
Yes, we can predict the winner using Sprague-Grundy Theorem.
What is Sprague-Grundy Theorem?
Suppose there is a composite game (more than one sub-game) made up of N sub-games and two players, A and B. Then Sprague-Grundy Theorem says that if both A and B play optimally (i.e., they don’t make any mistakes), then the player starting first is guaranteed to win if the XOR of the grundy numbers of position in each sub-games at the beginning of the game is non-zero. Otherwise, if the XOR evaluates to zero, then player A will lose definitely, no matter what.
How to apply Sprague Grundy Theorem ?
We can apply Sprague-Grundy Theorem in any impartial game and solve it. The basic steps are listed as follows:
- Break the composite game into sub-games.
- Then for each sub-game, calculate the Grundy Number at that position.
- Then calculate the XOR of all the calculated Grundy Numbers.
- If the XOR value is non-zero, then the player who is going to make the turn (First Player) will win else he is destined to lose, no matter what.
Example Game : The game starts with 3 piles having 3, 4 and 5 stones, and the player to move may take any positive number of stones upto 3 only from any of the piles [Provided that the pile has that much amount of stones]. The last player to move wins. Which player wins the game assuming that both players play optimally?
How to tell who will win by applying Sprague-Grundy Theorem?
As, we can see that this game is itself composed of several sub-games.
First Step : The sub-games can be considered as each piles.
Second Step : We see from the below table that
Grundy(3) = 3 Grundy(4) = 0 Grundy(5) = 1
We have already seen how to calculate the Grundy Numbers of this game in the previous article.
Third Step : The XOR of 3, 0, 1 = 2
Fourth Step : Since XOR is a non-zero number, so we can say that the first player will win.
Below is the program that implements above 4 steps.
Player 1 will win
Exercise to the Readers: Consider the below game.
“A game is played by two players with N integers A1, A2, .., AN. On his/her turn, a player selects an integer, divides it by 2, 3, or 6, and then takes the floor. If the integer becomes 0, it is removed. The last player to move wins. Which player wins the game if both players play optimally?”
Hint : See the example 3 of previous article.
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- Combinatorial Game Theory | Set 3 (Grundy Numbers/Nimbers and Mex)
- Predict the winner of the game | Sprague-Grundy
- Combinatorial Game Theory | Set 1 (Introduction)
- Combinatorial Game Theory | Set 2 (Game of Nim)
- Game Theory (Normal-form game) | Set 3 (Game with Mixed Strategy)
- Game Theory (Normal-form Game) | Set 7 (Graphical Method [M X 2] Game)
- Game Theory (Normal-form Game) | Set 6 (Graphical Method [2 X N] Game)
- Game Theory (Normal form game) | Set 2 (Game with Pure Strategy)
- Game Theory (Normal-form Game) | Set 5 (Dominance Property-Mixed Strategy)
- Game Theory (Normal-form Game) | Set 4 (Dominance Property-Pure Strategy)
- Game Theory (Normal - form game) | Set 1 (Introduction)
- The prisoner's dilemma in Game theory
- Expectimax Algorithm in Game Theory
- Pareto Optimality and its application in Game Theory
- Minimax Algorithm in Game Theory | Set 1 (Introduction)
- Minimax Algorithm in Game Theory | Set 5 (Zobrist Hashing)
- Minimax Algorithm in Game Theory | Set 2 (Introduction to Evaluation Function)
- Minimax Algorithm in Game Theory | Set 4 (Alpha-Beta Pruning)
- Minimax Algorithm in Game Theory | Set 3 (Tic-Tac-Toe AI - Finding optimal move)
- Game Theory in Balanced Ternary Numeral System | (Moving 3k steps at a time)