Adversary Search

Adversary Search

  • So far, we have only fought the instance

  • Consider a two-player game with:

    • Perfect knowledge (for now)

    • No "luck" (for now)

    • Turn-based

    • Always ends

    • Well-defined final score + "zero sum"

  • Examples: Checkers, Chess, Connect-4, Tic-Tac-Toe

  • We can use state-space search to find "good" or maybe even "best" plays in that game at a given state

Game State Space

  • State:

    • "Board" position (where are the checkers)

    • Side on move (red or black)

    • Maybe history of game (ugh, not checkers)

  • State space is directed, neighbors are states after legal moves

  • State space may be cyclic (which is where history rules sometimes come from)

Zero-Sum and Negamax

  • Minimax Theorem (Von Neumann, Nash): Any state in the game has a well-defined value assuming "best play" by both sides

  • Basic Negamax idea:

    • Choose a next state with minimum value for opponent (moves are boring)

    • Determine this by valuing each opponent state from opponent's point of view (recursively)

    • Zero-sum, so your value = negation of opponent's value

    • Thus, pick state that maximizes negation of opponent's value

negamax search tree

Win Pruning

  • If on some search you find a move (successor state) that is a win for the side on move — no point in searching further

  • This is true whether the win is immediate or by forced play

  • Can remove an "exponential" number of states from the search

Game Value

  • Value of a game is value of starting state to side on move

    • Game's value is value of starting state

    • Game is "solved" if the value of every state is known (recorded or trivially calculated)

    • Game is "weakly solved" if the value of every state that could occur in perfect play is known

  • Value is known: Hex

  • Weakly solved: Checkers

  • Solved: Tic-Tac-Toe, Connect-4, Awari, many others

Last modified: Sunday, 18 October 2020, 10:50 PM