Adversarial Search¶

Also called as Game Playing

Game¶

A game can be defined as a type of search in AI which can be formalized of the following elements

1. Initial state
2. Terminal state
3. Player\((s)\)
4. Action\((s)\)
5. Result\((s, a)\) - It is the transition model, which specifies the result of moves in the state space.
6. Terminal-Test\((s)\) - Terminal test is true if the game is over, else it is false at any case.
7. Utility\((s, p)\) - A utility function gives the final numeric value for a game that ends in terminal states \(s\) for player \(p\).

A game tree is a tree where nodes of the tree are the game states and Edges of the tree are the moves by players. Game tree involves initial state, actions function, and result Function.

Types of Games¶

Zero Sum game¶

• In Zero-sum game each agent's gain or loss of utility is exactly balanced by the losses or gains of utility of another agent.
• One player of the game tries to maximize one single value, while other player tries to minimize it.
• Examples are tic tac toe and chess.

Mini-Max Algorithm¶

Algo to determine optimal moves for utility maximizing agent in fully-observable, deterministic games

Min-max is complete and optimal

Assumption¶

Opponent behaves optimally, ie always perform the move that is worst for us

Logic¶

• Utility of each node is computed bottom up from leaves toward root.
• At each MAX node, pick move w/ max utility
• At each MIN node, pick move w/ min utility

Limitations¶

Really expensive for trees with large branding factor (complex games such as Chess, Go)

Complexity = \(O(b^m)\)

Can be overcome with \(\alpha \beta\) Pruning

\(\alpha \beta\) Pruning¶

Alpha Beta Pruning

• Optimized mini-max
• Form of meta-reasoning
• Used to reduce branching factor, hence handles complex games as well
• Maintain two parameters in depth-first search

• \(\alpha =\) highest value found yet for MAX along any path

  • \(\alpha_\text{root} = - \infty\)

• \(\beta =\) lowest value found yet for MIN along any path

  • \(\beta_\text{root} = + \infty\)

• Prune (skip) a subtree once it is known to be worse than current \(\alpha\) or \(\beta\)

• If \(\alpha > \beta\), stop evaluating children

Implications¶

• Solution does not change
• Complexity = \(O(b^{m/2})\)

Cut-off Search¶

• Cutoff search tree before terminal state reached
• Use heuristic of minimax value at leaves, instead of utility

Deep Blue¶

• Minimax
• \(\alpha\beta\) pruning
• Progressive deepening
• Parallel computing
• uneven tree development

14-16 levels deep

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