# Minimax algorithm pseudocode

Minimax is a kind of backtracking algorithm that is used in decision making and game theory to find the optimal move for a player, assuming that your opponent also plays optimally. In Minimax the two players are called maximizer and minimizer. The maximizer tries to get the highest score possible while the minimizer tries to do the opposite and get the lowest score possible.

Every board state has a value associated with it. In a given state if the maximizer has upper hand then, the score of the board will tend to be some positive value. If the minimizer has the upper hand in that board state then it will tend to be some negative value.

The values of the board are calculated by some heuristics which are unique for every type of game. Example: Consider a game which has 4 final states and paths to reach final state are from root to 4 leaves of a perfect binary tree as shown below. Assume you are the maximizing player and you get the first chance to move, i. Which move you would make as a maximizing player considering that your opponent also plays optimally?

Since this is a backtracking based algorithm, it tries all possible moves, then backtracks and makes a decision. Being the maximizer you would choose the larger value that is 3. Hence the optimal move for the maximizer is to go LEFT and the optimal value is 3. Now the game tree looks like below : The above tree shows two possible scores when maximizer makes left and right moves. Note: Even though there is a value of 9 on the right subtree, the minimizer will never pick that.

Attention reader!This AI will consider all possible scenarios and makes the most optimal move. We shall be introducing a new function called findBestMove. This function evaluates all the available moves using minimax and then returns the best move the maximizer can make.

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The pseudocode is as follows :. Here is the pseudocode :. To check whether the game is over and to make sure there are no moves left we use isMovesLeft function. It is a simple straightforward function which checks whether a move is available or not and returns true or false respectively. Pseudocode is as follows :. One final step is to make our AI a little bit smarter. Even though the following AI plays perfectly, it might choose to make a move which will result in a slower victory or a faster loss.

Lets take an example and explain it. Assume that there are 2 possible ways for X to win the game from a give board state. Even though the move A is better because it ensures a faster victory, our AI may choose B sometimes. To overcome this problem we subtract the depth value from the evaluated score. This means that in case of a victory it will choose a the victory which takes least number of moves and in case of a loss it will try to prolong the game and play as many moves as possible. So the new evaluated value will be.

The same thing must be applied to the minimizer. Instead of subtracting the depth we add the depth value as the minimizer always tries to get, as negative a value as possible. We can subtract the depth either inside the evaluation function or outside it. Anywhere is fine. I have chosen to do it outside the function. Pseudocode implementation is as follows.

This image depicts all the possible paths that the game can take from the root board state. It is often called the Game Tree. The 3 possible scenarios in the above example are :. Remember, even though X has a possibility of winning if he plays the middle move, O will never let that happen and will choose to draw instead. Therefore the best choice for X, is to play [2,2], which will guarantee a victory for him.

We do encourage our readers to try giving various inputs and understanding why the AI chose to play that move.

Minimax may confuse programmers as it it thinks several moves in advance and is very hard to debug at times. Remember this implementation of minimax algorithm can be applied any 2 player board game with some minor changes to the board structure and how we iterate through the moves. Also sometimes it is impossible for minimax to compute every possible game state for complex games like Chess.But the great minds behind the chess computer problem had started publishing in the subject nearly 6 decades earlier.

The expansive timeframe over which the chess computer pro b lem has been pondered lends credence to the complexity of the solution. Textbooks have been written on the computer chess problem alone, and many strategies varying in complexity have been put to the test. This article, however, will focus on one general decision-making strategy used by computers across a variety of strategic games, including chess, checkers, mancala, tic-tac-toe, you name it.

This general strategy is encased in a widely-used algorithm in gaming theory, called the Minimax Algorithm.

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This article will take a brief look at how a computer decides its next move using the Minimax Algorithm, but first we need to define a few things:. Intuitively, we can see that this data structure has the potential to have an infinite number of levels. In board games, we can think of the root node Level 0 as the current state of the board, and the nodes at Level 1 as each possible state of the board depending on what the next move will be.

In other words, Level 1 nodes account for each and every possible next move. If we continue this on long enough, we can quite literally map out the future of the game. The above schematic is oversimplified in the sense that an opponent only has 3 possible moves any given turn. Often times, in chess for instance, the number of possible moves can be much, much greater, causing our game tree to become complicated in a hurry.

How utility is calculated is entirely up to the programmer. It can incorporate a large variety of factors and weigh them as the programmer sees fit. The figure below displays a tic-tac-toe board midway through the game with a very simple probably not optimal utility rule.

For each possible move, utility is calculated using the below utility rule. In plain English this reads:. One possible way to decide which move to make next is to simply calculate the utility of each possible next move and select the move with the highest utility. This is often times the strategy of the average human when it comes to board games, and certainly, games can be won this way.

But what differentiates the masters from the ordinary is the ability to think several moves ahead. As it turns out, computers can do this much more efficiently than even the best of the best chess masters out there.

Before diving in, we will make 2 assumptions about our game:. The premise of the algorithm is that the computer will calculate its next best move by evaluating the utility of the board several turns down the road.

In doing so, the computer assumes that the opponent always selects the best move, minimizing the utility for the computer. Of course this is not a safe assumption, but lo and behold, it tends to work out pretty well regardless. For instance, when the human player makes the best possible move, we say that utility is minimized for that turn. The algorithm incorporates three basic functions: Maximize and Minimizeas well as a Utility Calculation function.

The pseudocode looks something like this:. No need to get too wrapped up in the details for now. The two key takeaways from this schematic are:. This would call Minimize on each child of the board, which calls Maximize on each grandchild, and so on and so forth…. This means it primarily traverses vertically down the entirely length of the tree, until it reaches the terminal nodes, and then works its way back up. Secondarily, the algorithm moves horizontally, or among other sibling nodes.

The schematic below helps to illustrate this concept:. The Minimax Algorithm moves in depth-first fashion down the tree until it reaches a terminal node i.Join Stack Overflow to learn, share knowledge, and build your career. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I'm teaching myself the minimax algorithm and I just had a few questions I was hoping someone could answer. Would it return the maximum value or the minimum value found so far?

Is there a pseudocode example of this or am I misunderstanding? Since this is pseudocode, we can assume that any language you will use to implement it provides the functions max and min. If not, you can easily implement them yourself. Learn more. Understanding the minmax pseudocode Ask Question.

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Featured on Meta. Opt-in alpha test for a new Stacks editor. Visual design changes to the review queues. Related 1. Hot Network Questions. Question feed. Stack Overflow works best with JavaScript enabled.Hello people, in this post we will try to improve the performance of our Minimax algorithm by applying Alpha-Beta Pruning. Alpha-beta pruning is based on the Branch and bound algorithm design paradigm, where we will generate uppermost and lowermost possible values to our optimal solution and using them, discard any decision which cannot possibly yield a better solution than the one we have so far.

These values will be passed down to recursion calls via arguments. Let us understand this with the help of an example. Look at the sketch below —. Let us assume we have the above game tree formed by two agents max and min playing a hypothetical game. Here, the Max agent tries to maximize the score and Min agent tries to minimize the score. The topmost Min node makes the first call.

Here val is the value which will be returned.

Now what will happen next? Min has two possibilities above and the call goes to the first possibility, which is the first Max node in the above diagram. What will Max do there?

The choices for Max are 2 and 4. Since Max always maximizes the score, it will choose 4. So we update the value to be returned to 4. Now, we return value 4 from the Max node. Hmm, now the Min node sees that the first possible decision will give it a score of 4.

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Now what does Min do? It looks at the next possibility.

## Game Theory — The Minimax Algorithm Explained

So Max node starts looking at all the possibilities one-by-one. The possibilities are 6 and 8. This is important! Now, this is the case when Max has finished reading the first possibility which is 6. Now, what if the values for the choices ahead returned a value lesser than 6? Then obviously Max would choose 6 since it is the highest. Okay, what if the values for the choices ahead returned a value greater than 6, say X?Minimax sometimes MinMaxMM  or saddle point  is a decision rule used in artificial intelligencedecision theorygame theorystatisticsand philosophy for mini mizing the possible loss for a worst case max imum loss scenario.

When dealing with gains, it is referred to as "maximin"—to maximize the minimum gain. Originally formulated for n-player zero-sum game theorycovering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision-making in the presence of uncertainty. The maximin value is the highest value that the player can be sure to get without knowing the actions of the other players; equivalently, it is the lowest value the other players can force the player to receive when they know the player's action. Its formal definition is: . Calculating the maximin value of a player is done in a worst-case approach: for each possible action of the player, we check all possible actions of the other players and determine the worst possible combination of actions—the one that gives player i the smallest value.

Then, we determine which action player i can take in order to make sure that this smallest value is the highest possible. For example, consider the following game for two players, where the first player "row player" may choose any of three moves, labelled TMor Band the second player "column" player may choose either of two moves, L or R. The result of the combination of both moves is expressed in a payoff table:.

For the sake of example, we consider only pure strategies. Check each player in turn:. The minimax value of a player is the smallest value that the other players can force the player to receive, without knowing the player's actions; equivalently, it is the largest value the player can be sure to get when they know the actions of the other players.

The definition is very similar to that of the maximin value—only the order of the maximum and minimum operators is inverse. In the above example:. For every player ithe maximin is at most the minimax:.

Intuitively, in maximin the maximization comes before the minimization, so player i tries to maximize their value before knowing what the others will do; in minimax the maximization comes after the minimization, so player i is in a much better position—they maximize their value knowing what the others did. Conversely for maximin.

In two-player zero-sum gamesthe minimax solution is the same as the Nash equilibrium. In the context of zero-sum games, the minimax theorem is equivalent to:  [ failed verification ]. For every two-person, zero-sum game with finitely many strategies, there exists a value V and a mixed strategy for each player, such that.

The name minimax arises because each player minimizes the maximum payoff possible for the other—since the game is zero-sum, they also minimize their own maximum loss i. See also example of a game without a value. The following example of a zero-sum game, where A and B make simultaneous moves, illustrates maximin solutions. Suppose each player has three choices and consider the payoff matrix for A displayed on the right.

Assume the payoff matrix for B is the same matrix with the signs reversed i. Then, the maximin choice for A is A2 since the worst possible result is then having to pay 1, while the simple maximin choice for B is B2 since the worst possible result is then no payment.

However, this solution is not stable, since if B believes A will choose A2 then B will choose B1 to gain 1; then if A believes B will choose B1 then A will choose A1 to gain 3; and then B will choose B2; and eventually both players will realize the difficulty of making a choice. So a more stable strategy is needed. Some choices are dominated by others and can be eliminated: A will not choose A3 since either A1 or A2 will produce a better result, no matter what B chooses; B will not choose B3 since some mixtures of B1 and B2 will produce a better result, no matter what A chooses.

These mixed minimax strategies are now stable and cannot be improved. Frequently, in game theory, maximin is distinct from minimax. Minimax is used in zero-sum games to denote minimizing the opponent's maximum payoff. In a zero-sum gamethis is identical to minimizing one's own maximum loss, and to maximizing one's own minimum gain.

In non-zero-sum games, this is not generally the same as minimizing the opponent's maximum gain, nor the same as the Nash equilibrium strategy. The minimax values are very important in the theory of repeated games.

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One of the central theorems in this theory, the folk theoremrelies on the minimax values. In combinatorial game theorythere is a minimax algorithm for game solutions. A simple version of the minimax algorithmstated below, deals with games such as tic-tac-toewhere each player can win, lose, or draw. If player A can win in one move, their best move is that winning move.Hello people!

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In this post we will look at one of the most basic Artificial Intelligence algorithm, the MiniMax algorithm. MiniMax algorithm is used to implement basic AI or game logic in 2 player games. The most common scenario is implementing a perfect Tic-Tac-Toe player. So, in this article we will look at how to implement it. Given that two players are playing a game optimally playing to winMiniMax algorithm tells you what is the best move that a player should pick at any state of the game.

So, the input to MiniMax algorithm would be —. So now, the bigger the number score has, the better it is for X, which means X will try to maximize score as much as possible. Similarly, the lesser the score, the better it is for O, so O will try to minimize the score as much as possible. To understand this better, imagine you are player X and this is the current state of your game —. So if you are in state 1 from the above diagramyou have 3 possible moves which lead to state 2, 3 and 4.

For these moves the scores are —. So conceptually we know that player X must choose the winning move. This is done programmatically bu choosing the move which will return the maximum score. So, X will always try to maximize the score and will always choose that move which will fetch X the maximum score. Thus, in the above scenario X chooses the move which goes to state 2. Now, to make sure you understand both sides of this algorithm.

Can you draw a state diagram which will depict all the possible moves and the scores which will be returned by playing them? You can! Did you get a diagram like this? Player O has 3 possible moves and the scores are —. Player O will always try to minimize the score, so player O must select a move which will either lead to state 2 or 3 to win. Firstly, have a clarity on the smaller pieces of logic and write methods for them first. You will need these 3 helper methods for your code —.

So now you have a little clarity over the smaller pieces, code them first. Now, you are ready to write the MiniMax algorithm method. It is a recursive method which takes in 2 inputs —.

Let's Learn Python #21 - Min Max Algorithm

We can make a new class to return all the information we need.