1. Games with Chance Other Search Algorithms
CSCE 315 – Programming Studio
Fall 2017
Project 2, Lecture 3
Adapted from slides of
Yoonsuck Choe, 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
Chance …
Min
Chance
For each die roll (red lines), evaluate each possible move (blue lines)

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 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

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
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. General State Diagrams

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

11. 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

12. Path Planning
What would your design be?

13. 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