Artificial Intelligence: Expectiminimax Algorithm

Expectiminimax Algorithm


What is it?

The Expectiminimax algorithm is an extension of the Minimax algorithm used in game theory and AI for decision-making in games. It handles games with chance elements — where outcomes can be probabilistic, not just deterministic.


When is it used?

  • In games with both adversarial players and chance events (e.g., dice rolls, card draws).

  • Examples:

    • Backgammon (dice rolls)

    • Monopoly (dice rolls, chance cards)

    • Some variants of Poker or other games with randomness


How does it differ from Minimax?

  • Minimax is for perfect-information deterministic games — where players alternate turns and outcomes are fixed.

  • Expectiminimax incorporates chance nodes in addition to min (opponent) and max (player) nodes.

  • It computes the expected utility over chance nodes instead of simply min or max values.


Key Concepts

The game tree now has three types of nodes:

  1. Max nodes — AI's turn, it chooses the move that maximizes its expected utility.

  2. Min nodes — Opponent's turn, assumed to choose moves that minimize AI's utility.

  3. Chance nodes — Represent probabilistic outcomes (like dice rolls). The value here is the expected value (weighted average) of all possible outcomes.


Algorithm Overview

  1. Max nodes:
    Choose the child node with the maximum expectiminimax value.

  2. Min nodes:
    Choose the child node with the minimum expectiminimax value.

  3. Chance nodes:
    Calculate the expected value by summing the values of all child nodes weighted by their probabilities.


Pseudocode

def expectiminimax(node):
    if node is a terminal node:
        return utility(node)

    if node is a max node:
        return max(expectiminimax(child) for child in node.children)

    elif node is a min node:
        return min(expectiminimax(child) for child in node.children)

    elif node is a chance node:
        expected_value = 0
        for child, prob in node.children_with_probabilities:
            expected_value += prob * expectiminimax(child)
        return expected_value


Example

Consider a dice roll after your move that determines the opponent’s possible responses. The chance node models the dice roll:

  • Suppose dice roll 1 happens with probability 1/6 → leads to child node A with value 3.

  • Dice roll 2 → child node B with value 5.

  • Dice roll 3 → child node C with value 2.

  • ...

The chance node value = (1/6)*3 + (1/6)*5 + (1/6)*2 + ... and so on.



Comparison with Minimax and Alpha-Beta Pruning


Minimax Algorithm
The Minimax algorithm is used for deterministic two-player games like chess, where the opponent is assumed to play optimally to minimize the AI's utility. Expectimax, in contrast, is used for games where outcomes involve randomness.

Alpha-Beta Pruning
Alpha-Beta pruning is an optimization technique for the Minimax algorithm that reduces the number of nodes evaluated in the search tree. While Expectimax can’t directly benefit from Alpha-Beta pruning (because chance nodes introduce probabilistic outcomes), heuristic approaches and evaluation functions can still be used to optimize Expectimax search.

Limitations of the Expectimax Algorithm


While Expectimax is a powerful tool for handling probabilistic decision-making, it has its limitations:

* Performance: Expectimax can be computationally expensive, as it evaluates all possible outcomes and probabilities, leading to a large branching factor at chance nodes.

* Exponential Tree Growth: Similar to Minimax, Expectimax can suffer from large search trees in games with deep decision trees and high numbers of possible outcomes.

* Accuracy of Probabilities: The quality of the decisions made by Expectimax depends on accurate modeling of the probabilities at chance nodes. Poor estimation can lead to suboptimal decisions.


Summary

Aspect Minimax Expectiminimax
Types of nodes Max and Min only Max, Min, and Chance
Game type Deterministic, perfect info Stochastic with chance/probability
Value at chance nodes N/A Expected value over probabilistic outcomes