Games with Chance Other Search Algorithms CSCE 315 – Programming Studio Fall 2017 Project 2, Lecture 3 Adapted from slides of Yoonsuck Choe, John Keyser

Game Playing with Chance Minimax trees work well when the game is deterministic, but how about games with an element of chance Solitaire: Shuffles are random Monopoly: Next move depends on die rolling outcome How do we analyze games with chance? Include Chance nodes in tree Try to maximize/minimize the expected value Or, play pessimistic/optimistic approach

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

Expected Value For variable x, the Expected Value is: where Pr(x) is the probability of x occurring Example 1: When you roll a dice, what is average value on the face? Example 2: How about rolling a pair of dice?

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

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 O(bmnm) where n is the number of possible choices (m is depth)

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 Example: Walking 10 steps to the right or 11, or 1000 Larger situation can be broken down into subproblems Hierarchical approach Use of state diagrams Some subproblems are more easily solved e.g. path planning

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 of 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

General State Diagrams

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

Exploring the State Diagram Explore for solutions using BFS, DFS Depth limited search: DFS but to limited depth in tree Iterative Deepening search: As described before, but with DFS on graph, not just tree 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

Path Planning What would your design be?

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