More Advanced Topics

PSU CS410/510GAMES Lecture 6
May 9, 2002

• MTD(f)
  • Idea: fail to disprove AB window
  • If window wildly wrong, can quickly find subtree that proves it
  • Example: card games
  • MTD(f) (outline by Aske Plaat)
    • The narrower the AlphaBeta window, the more cutoffs you get, the more efficient the search is. Hence MTD(f) uses only search windows of zero size.
    • Zero window AlphaBeta calls return bounds. At the root of the tree the return bounds are stored in upperbound (after AlphaBeta ``failed low'') and lowerbound (after AlphaBeta ``failed high''). The bounds delimit the range of possible values for the minimax value. Each time MTD(f) calls AlphaBeta it gets a value back that narrows the range, and the algorithm is one step closer to hitting the minimax value.
    • Storing nodes in memory gets rid of the overhead inherent in multiple re-searches. A transposition table of sufficient size does the job.
    • It is more efficient to start the search close to its goal. Therefore MTD(f) needs (and can make use of) a good first guess.
• Graph/History Interaction
  • Problem: value of state may depend on how state is reached! (Canonical: draw by repetition)
  • Solutions?
• Some programming tricks
  • state representations (bit boards etc.)
  • do/undo
  • clever/incremental hashing, valuing
  • move generation and ordering
  • valuing (normalizing, true v. approx.)
• Amazons
• Rules
  • 10x10 board
  • white pieces at ends of lower "four diagonals", black symmetric. white moves first.
  • Move: move a queen, fire an arrow.
  • Loser: first person without a (full) move.
• Play
  • Try to expand territory own queens can move in
  • Try to expand mobility of queens
  • Opening book and endgame are important
• Probability
  • Problem 1: well-defined? Use expected value, require zero-sum EV, hope for best.
  • Problem 2: huge branching factor. But opponent does not control probabilities: Evaluate by Monte Carlo or related methods. Alt., do principled estimation (Yacht)
  • Monte Carlo: to get EV of a node, play the node many times and compute EV of outcomes.
    • Problem: converges very slowly (from class)
    • Problem: hard to gauge quality of estimate
    • Problem: hard SE problem
    • Examples: scrabble, bridge