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### Adversarial Search(Games)

Chapter 6

Outline

- Summary of last lectures
- Characterizing a Game
- Optimal decisions
- Why is full exploration of the search space not feasible?
- The minimax algorithm
- α-β pruning
- Imperfect, real-time decisions
- Extensions: multi-player, chance

Summary of the past lectures

- System engineering process
- Analysis, design, implementation
- Agent’s Performance measures (non-functional requirements)
- Agents types
- Simple-reflex, model-based, goal-based, utility agents, learning agents
- Environment types
- Static/dynamic, deterministic/stochastic, fully/partially observable

Summary of the past lectures

- Search (Goal-based agents)
- Basic search algorithms and their variants
- Uninformed search strategies
- Limited information about the environment model
- Iterative Deepening search, bidirectional search, avoiding repeated states
- Informed search
- Improve time and space complexity by having additional information about the environment for search
- Heuristic function
- Greedy best-first search
- A* search, triangular inequality

Games vs. search problems

- "Unpredictable" opponent specifying a move for every possible opponent reply
- Time limits unlikely to find goal, must approximate

2-player zero-sum discrete finite deterministic games of perfect information

What does it means?

- Two player: :-)
- Zero-sum: In any outcome of any game, Player A’s gains equal player B’s losses.
- Discrete: All game states and decisions are discrete values.
- Finite: Only a finite number of states and decisions.
- Deterministic: No chance (no die rolls).
- Perfect information: Both players can see the state, and each decision is made sequentially (no simultaneous moves).
- Games: See next slide

2-player zero-sum discrete finite deterministic games of perfect information

2-player zero-sum discrete finite deterministic games of perfect information

Hidden Information

Stochastic

Not Finite

One Player

Involves Animal Behave

Mutiplayer

2-player zero-sum discrete finite deterministic games of perfect information

A Two-player zero-sum discrete finite deterministic game of perfect information is a quintuplet:

( S , I , N , T , V ) where:

Minimax Algorithm

- Optimal play for deterministic games
- Idea: choose move to position with highest minimax value = best achievable payoff against best play
- E.g., a simple 2-ply game:

Utility of a situation in a game:

- In most two-player games the termination situations have a certain value, mostly

+1 for MAX (=win)

-1 for MIN (=loose)

0 for a draw.

- Also different values possible: e.g., Backgammon (-192 to +192), etc.
- We can compute in any situation the minimax-value as follows:

Properties of minimax

- Complete? Yes (if tree is finite)
- Optimal? Yes (against an optimal opponent)
- Time complexity? O(bm)
- Space complexity? O(bm) (depth-first exploration)
- Problem: explores the whole search-space

For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible

- So, how to proceed?

b … branching factor m … maximum number of moves

Motivation for α-β pruning

- The problem with minimax algorithm search is that the number of game states it has to examine is exponential in the number of moves:
- α-β proposes to compute the correct minimax algorithm decision without looking at every node in the game tree.

PRUNING!

· 2

5

2

Pruning possible!

α-β pruning exampleNo pruning

We see: possibility to prune depends on the order of processing the successors!

Properties of α-β

- Pruning does not affect final result
- Good move ordering improves effectiveness of pruning
- With "perfect ordering," time complexity = O(bm/2)

doubles possible depth of search doable in the same time

- A simple example of the value of reasoning about which computations are relevant (a form of meta-reasoning)

Why is it called α-β?

- α is the value of the best (i.e., highest-value) choice found so far at any choice point along the path for max
- If v is worse than α, max will avoid it

prune that branch

- Define β similarly for min

Resource limits

Suppose we have 100 secs, explore 104 nodes/sec106nodes per move

even with pruning not possible to explore the whole search space e.g. for chess!

Standard approach:

- cutoff test:

e.g., depth limit (perhaps add quiescence search)

- evaluation function

= estimated desirability of position

Evaluation functions

- For chess, typically linear weighted sum of features

Eval(s) = w1 f1(s) + w2 f2(s) + … + wn fn(s)

- e.g., weight of figures on the board:

w1 = 9 with

f1(s) = (number of white queens) – (number of black queens), etc.

…

Other features which could be taken into account: number of threats, good structure of pawns, measure of safety of the king.

Cutting off search

MinimaxCutoff is identical to MinimaxValue except

- Terminal? is replaced by Cutoff?
- Utility is replaced by Eval

Does it work in practice?

bm = 106, b=35 m=4

4-ply lookahead is a hopeless chess player!

- 4-ply ≈ human novice
- 8-ply ≈ typical PC, human master
- 12-ply ≈ Deep Blue, Kasparov

Deterministic games in practice

- Checkers: Chinook ended 40-year-reign of human world champion Marion Tinsley in 1994. Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions.
- Chess: Deep Blue defeated human world champion Garry Kasparov in a six-game match in 1997. Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply.
- Othello: human champions refuse to compete against computers, who are too good.
- Go: human champions refuse to compete against computers, who are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.

Some extensions

- What if more than two players are in the game?

2-player algorithms (minimax, -, cutoff-eval) can be extended to multi-player in a straightforward way:

- Instead of 1 value use a vector of values, where each player tries to maximize its own index-value in the vector
- 2-player-zero-sum games are a special case of this, where the vector can be combined into one value since the values for both players are exactly opposite
- What if an element of chance (i.e. non-determinism) is added? E.g. rolling dice in Backgammon?

Expectiminimax next slide

Minimax with Chance Nodes:

Chance nodes have certain probabibilities.

EXPECTIMINIMAX…

- Slight variation of MINIMAX:

where P(s) is the probability of reaching s (e.g.

probability of rolling a certain number with the dice)

Summary:

- Games are fun to work on!
- They illustrate several important points about AI
- perfection is unattainable must approximate
- good idea to think about what to think about: ideas and expertise of masters deployed in evaluation functions (i.e. heuristics)

What makes Game theory interesting in practice?

- Exogenous events, i.e. non-determinism in planning can be modelled as opponent.
- Multi-agent planning: cooperative vs. competitive Can be modeled as multi-player games

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