2

Existing implementation:
In my implementation of Tic-Tac-Toe with minimax, I look for all boxes where I can get best result and chose 1 of them randomly, so that the same solution isn't displayed each time.

For ex. if the returned list is [1, 0 , 1, -1], at some point, I will randomly chose between the two highest values.

Question about Alpha-Beta Pruning:
Based on what I understood, when the algorithm finds that it is winning from one path, it would no longer need to look for other paths that might/ might not lead to a winning case.

So will this, like I feel, cause the earliest possible box that leads to the best solution to be displayed as the result and seem the same each time? For example at the time of first move, all moves lead to a draw. So will the 1st box be selected each time?

How can I bring randomness to the solution like with the minimax solution? One way that I thought about now could be to randomly pass the indices to the alpha-beta algorithm. So the result will be the first best solution in that randomly sorted list of positions.

Thanks in advance. If there is some literature on this, I'd be glad to read it. If someone could post some good reference for aplha-beta pruning, That'll be excellent as I had a hard time understanding how to apply it.

The_Coder
  • 321
  • 5
  • 18

1 Answers1

2

To randomly pick among multiple best solutions (all equal) in alpha-beta pruning, you can modify your evaluation function to add a very small random number whenever you evaluate a game state. You should just make sure that the magnitude of that random number is never greater than the true difference between the evaluations of two states.

For example, if the true evaluation function for your game state can only return values -1, 0, and 1, you could add a randomly generated number in the range [0.0, 0.01] to the evaluation of every game state.

Without this, alpha-beta pruning doesn't necessarily find only one solution. Consider this example from wikipedia. In the middle, you see that two solutions with an evaluation of 6 were found, so it can find more than one. I do actually think it will still find all moves leading to optimal solutions at the root node, but not actually find all solutions deep down in the tree. Suppose, in the example image, that the pruned node with score of 9 in the middle actually had a score of 6. It would still get pruned there, so that particular solution wouldn't be found, but the move from root node leading to it (the middle move at root) would still be found. So, eventually, you would be able to reach it.

Some interesting notes:

  • This implementation would also work in minimax, and avoid the need to store a list of multiple (equally good) solutions
  • In more complex games than Tic Tac Toe, where you cannot search the complete state space, adding a small random number for the max player and deducting a small random number for the min player like this may actually slightly improve your heuristic evaluation function. The reason for this is as follows. Suppose in state A you have 5 moves available, and in state B you have 10 moves available, which all result in the same heuristic evaluation score. Intuitively, the successors of state B may be slightly better, because you had more moves available; in many games, having more moves available means that you are in a better position. Because you generated 10 random numbers for the 10 successors of state B, it is also a bit more likely that the highest generated random number is among those 10 (instead of the 5 numbers generated for successors of A)
Dennis Soemers
  • 8,090
  • 2
  • 32
  • 55
  • Will alpha-beta algorithm ever return multiple best solutions(all equal) for a particular state? What I thought was that it will stop at the first best solution and cut the remaining ones. – The_Coder Feb 05 '17 at 15:48
  • @The_coder edited answer with some extra information on that – Dennis Soemers Feb 05 '17 at 16:15