Revision 210743847b809eaa1dd2a4518c0b950fa8e7f729 (click the page title to view the current version)

Adversarial Search


R&N Chapter 6

  • The fundamental concept is two-player, zero-sum games
    • The basic solution technique is minimax search
  • Minimax search grows exponentially
    • heuristic searches are important (Section 6.3)
  • Video lecture mp4/ogg provides a few anchor points for the material. It does not replace the text book.


Two-player, Zero Sum games

  • State-machine
    • to-move: state \(\to\) player
    • actions: state \(\to\) {action}
    • result: state \(\times\) action \(\to\) state
    • utility: state \(\times\) player \(\to\mathbb{R}\)
  • Minimax algorithm:
    • maximises utility for the player currently to move
    • Tree Search: optimise utility bottom-up
      • min nodes and max nodes
    • exhaustive research
  • Caveats and variations
    • Branching factor
    • multi-player games - alliances and trust
    • co-operative games
    • \(\alpha\beta\) pruning
    • heuristic searches
    • Type A and Type B:
      • move generation
      • move evaluation
  • Monte Carlo Tree Search
    • Monte Carlo \(\sim\) Stochastic Simulation
    • Balance Exploitation and Exploration
  • More Caveats
    • Stochastic Games - Chance Nodes
    • Partially Observable Games
      • the percept sequence is no longer the opponent’s move.
    • Limitations - Section 6.7
      • Intractible
    • Approximations and Assumptions
    • Individual moves - no sight of the bigger picture


Tic Tac Toe

I was not able to find suitable exercises on CodinGame, so instead, I have provided a simulator for you. You should

  1. Clone the git repo, git clone
  2. Change to the TicTacToe subdirectory
  3. Modify the template to implement your intelligent agent. You should use the minimax algorithm as described for two-player, zero sum games.
  4. Play the game, using the test scripts: python3
  5. Consult the README file for details.

This assumes that you have git and python3 installed.


From R&N Exercises

  • Exercise 1
  • Exercise 3


All the algorithms we are studied are based on (discrete) state machine models and tree search algorithms in the graph defined by the states and transitions in the state machines. It is crucial that we understand the basic structure of these models and algorithms well, so that we can adapt them to the different special cases we encounter.

  • exhaustive searches in small state spaces
    • BFS
    • DFS
  • non-deterministic environments lead to random transitions
    • modelled by and-or trees
  • partial observation leads to belief states
  • in offline search we can jump to arbitrary nodes to restart the search
    • in online searches we need to backtrack by actual actions in the environment
  • two-player games lead to adversarial transitions
    • Minimax algorithm considers both own moves (max) and adversarial moves (min)
  • Pruning may be essential to speed up the algorithm
    • Dijkstra prunes subtrees which are known to give longer paths than the best one known
    • alpha-beta pruning applies to minimax search
  • Heuristics are estimates of the value/cost of a node. This allows
    • Greedu algorithms and best first searches
    • Heuristic pruning, discarding subtrees which look bad, but with no guarantee