1. Games with Chance Other Search Algorithms
CPSC 315 – Programming Studio
Spring 2013
Project 2, Lecture 3
Adapted from slides of
Yoonsuck Choe

2. 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

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

7. Accounting for Time Often, real-world clock time is a factor in a game
Limit time per move
Limit total time allowed
We might not be able to predict how far ahead we can look in a tree in a given time
Different computers, board states, etc.
A non-conservative estimate might violate the time constraint
A conservative estimate is likely to waste time

8. Dealing with Time Iterative Deepening Compute to one level
Then to two levels (repeating level 1)
Order based on best move found previously…
Then to three, etc. until time runs out
Return the best known move so far at time limit
Wastes time in repeat of earlier levels
But, can be surprising small, especially with large branching factors
Ordering from prior evaluations helps

9. 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

10. 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

11. 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

12. 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

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

14. 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

15. 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