Week 4b - Adversarial Search II
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Resource Limits
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- Suppose we have 100 seconds, and we can explore 10^4^ nodes/second
- We can search 10^6^ nodes per move
- Standard Approach:
- Cutoff test
- e.g., depth limit
- Evaluation function
- = estimated desirability of position (score given for a position)

Evaluation Functions
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- For chess, evaluation functions are typically linear weighted sum of features
- $Eval(s) =w_1f_1(s) + w_2f_2(s)+\ldots+w_nf_n(s)$
- e.g. w~1~ = 9 with f~1~(s) = (number of white queens) - (number of black queens)

### Exact values don't matter
- Behavior is preserved under any monotonic transformation of Eval: it doesn't matter what is the evaluation values are, as long as they are in the same order.
- Only the order matters:
- payoff in deterministic games acts as an ordinal utility function

Cutting off search
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- MinimaxCutoff is identical to MinimaxValue except:
- Terminal Test is replaced by Cutoff Test
- Utility is replaced by Eval score

α-β Pruning
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Eliminating certain portions of the tree to reduce the search space.
Branches are only pruned if a value in the branch is smaller than the current min value in the level (vice versa for max).
i.e. if α ≥ β

### Motivating Example

### Properties of α-β Pruning
- Pruning does not affect final result
- Good move ordering improves effectiveness of pruning
- With "perfect ordering", time complexity is $O(b^{m/2})$
- Doubles depth of search
- Can easily reach depth 8 and play good chess
- A simple example of the value of reasoning about which computations are relevant

### Algorithm of α - β Pruning

### Non-deterministic Games
- Adversarial search where the actions/transitions are non-deterministic
- e.g. in backgammon, dice rolls determine the legal moves

### ExpectiMiniMax

if state is a chance node then
    return weighted average of ExpectiMinimax-Value of Successors(state)