Games with Chance Other Search Algorithms

1. Games with ChanceOther Search Algorithms CSCE 315 – Programming Studio Spring 2019 Project 2, Lecture 3 Robert Lightfoot Adapted from slides of YoonsuckChoe, John Keyser

2. 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

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

4. 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?

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

6. 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)

7. 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 • Example: Hide and seek • Larger situation can be broken down into subproblems • Hierarchical approach • Use of state diagrams • Some subproblems are more easily solved • e.g. path planning

8. 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

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

10. 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

11. 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