Game Playing
Adversarial Search
How do we model the opponent?
As an aggregate economy ?
As part of a nondeterministic environment?
As explicitly adversarial agents
THE ENVIRONMENT
Multi-agent
Competitive
Zero-sum
Turn-based
Perfect information
How do you approach game playing?
Analysis of features
If-Then Rules/Decision Trees
Brute Force
Look ahead, and evaluate
Consider 2-players in a zero-sum game
Opposing objectives
Player A is the Maximizer and moves first
Player B is the Minimizer and follows Player A
Consider 2-players in a zero-sum game
Construct search-trees
Starting with MAX, we can expand the entire game tree
The leaves are terminal states with 1 winner, or draw states
Minimax
MAX chooses the move that maximizes the minimum payoff
MIN chooses the move that minimizes the maximum payoff
Assumes both players play optimally
Minimax
Start at bottom, evaluate utility at all leaves
Back up the tree, alternating between MAX and MIN
Once we hit root, we get optimal play sequence
Problems with
Minimax
Expensive - computes $b^d$ leaves
Wastes computation on obviously bad moves
Recall Uniform Cost Search
Key Idea: Branch and Bound
Alpha Beta Pruning
Idea: Don't compute the entire tree
Compute only what is necessary
Don't compute obviously bad moves
Alpha Beta Pruning
How much less work do we need to do?
Optimal case: $S = 2b^{d/2}$
Average case: pretty close!
More on branching factors
Varies widely between and within games
We can get stuck at some level $d$
Guarantee solution from previous level
Heuristic Alpha Beta Pruning
Use an evaluation function instead of utilities
Especially useful when we can't expand the whole tree
