Games with Chance Other Search Algorithms CPSC 315 – Programming Studio Spring 2008 Project 2, Lecture 3 Adapted from slides of Yoonsuck Choe

Game Playing with Chance Minimax trees work well when the game is deterministic, but many games have an element of chance. Include Chance nodes in tree Try to maximize/minimize the expected value Or, play pessimistic/optimistic approach

Tree with Chance Nodes For each die roll (red lines), evaluate each possible move (blue lines) … Max Min Chance

Expected Value For variable x, the Expected Value is: where Pr(x) is the probability of x occurring Example: rolling a pair of die:

Evaluating Tree Choosing a Maximum (same idea for Minimum): Evaluate same move from ALL chance nodes Find Expected Value for that move Choose largest expected value … Max Min Chance

More on Chance Rather than expected value, could use another approach Maximize worst case value Avoid catastrophe Give high weight if a very good position is possible “Knockout” move Form hybrid approach, weighting all of these options Note: time complexity increased to b m n m where n is the number of possible choices (m is depth)

More on Game Playing Rigorous approaches to imperfect information games still being studied. Assume random moves by opponent Assume some sort of model based on perfect information model Indications that often the behavior of the opponent is of more value than evaluating the board position

AI in Larger-Scale and Modern Computer Games The idealized situations described often don’t extend to extremely complex, and more continuous games. Even just listing possible moves can be difficult Larger situation can be broken down into subproblems Hierarchical approach Use of state diagrams Some subproblems are more easily solved e.g. path planning

AI in Larger-Scale and Modern Computer Games Use of simulation as opposed to deterministic solution Helps to explore large range of states Can create complex behavior wrapped up in autonomous agents Fun vs. Competent Goal of game is not necessarily for the computer to win Often a collection of ad-hoc rules Cheating allowed

General State Diagrams List of possible states one can reach in the game (nodes) Can be abstracted, general conditions Describe ways of moving from one state to another (edges) Not necessarily a set move, could be a general approach Forms a directed (and often cyclic) graph Our minimax tree is a state diagram, but we hide any cycles Sometimes want to avoid repeated states

State Diagram State A State D State E State B State F State G State C State K State J State I State H

Exploring the State Diagram Explore for solutions using BFS, DFS Depth limited search: DFS but to limited depth in tree Iterative Deepening search: DFS one level deep, then two levels (repeats first level), then three levels, etc. If a specific goal state, can use bidirectional search Search forward from start and backward from goal – try to meet in the middle. Think of maze puzzles

More informed search Traversing links, goal states not always equal Can have a heuristic function: h(x) = how close the state is to the “goal” state. Kind of like board evaluation function/utility function in game play Can use this to order other searches Can use this to create greedy approach

A* Algorithm Avoid expanding paths that are already expensive. f(n) = g(n) + h(n) g(n) = current path cost from start to node n h(n) = estimate of remaining distance to goal h(n) should never overestimate the actual cost of the best solution through that node. Then apply a best-first search Value of f will only increase as paths evaluate