Games with Chance Other Search Algorithms

1
Games with ChanceOther Search Algorithms

• CPSC 315 Programming Studio
• Spring 2009
• Project 2, Lecture 3

Adapted from slides of Yoonsuck Choe
2
Game Playing with Chance

• Minimax trees work well when the game is
  deterministic, but many games have an element of
  chance.
• Include Chance nodes in tree
• Try to maximize/minimize the expected value
• Or, play pessimistic/optimistic approach

3
Tree with Chance Nodes
Max
Chance

Min
Chance

• For each die roll (blue lines), evaluate each
  possible move (red lines)

4
Expected Value

• For variable x, the Expected Value is
• where Pr(x) is the probability of x occurring
• Example rolling a pair of die

5
Expectiminimax Evaluating Tree
Max
Chance

Min
Chance

• Choosing a Maximum (same idea for Minimum)
• Evaluate all chance nodes from a move
• Find Expected Value for that move
• Choose largest expected value

6
More on Chance

• Rather than expected value, could use another
  approach
• Maximize worst case value
• Avoid catastrophe
• Give high weight if a very good position is
  possible
• Knockout move
• Form hybrid approach, weighting all of these
  options
• Note time complexity increased to bmnm where n
  is the number of possible choices (m is depth)

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More on Game Playing

• Rigorous approaches to imperfect information
  games still being studied.
• Assume random moves by opponent
• Assume some sort of model based on perfect
  information model
• Indications that often the behavior of the
  opponent is of more value than evaluating the
  board position

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AI in Larger-Scale and Modern Computer Games

• The idealized situations described often dont
  extend to extremely complex, and more continuous
  games.
• Even just listing possible moves can be difficult
• Consider writing the AI controller for a
  non-player opponent in a modern strategy game
• Larger situation can be broken down into
  subproblems
• Hierarchical approach
• Use of state diagrams
• Some subproblems are more easily solved
• e.g. path planning

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AI in Larger-Scale and Modern Computer Games

• Use of simulation as opposed to deterministic
  solution
• Helps to explore large range of states
• Can create complex behavior wrapped up in
  autonomous agents
• Fun vs. Competent
• Goal of game is not necessarily for the computer
  to win
• Often a collection of ad-hoc rules
• Cheating allowed (e.g. Civilization)

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General State Diagrams

• List of possible states one can reach in the game
  (nodes)
• Can be abstracted, general conditions
• Describe ways of moving from one state to another
  (edges)
• Not necessarily a set move, could be a general
  approach
• Forms a directed (and often cyclic) graph
• Our minimax tree is a state diagram, but we hide
  any cycles
• Sometimes want to avoid repeated states

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State Diagram
State C
State I
State A
State D
State B
State J
State E
State H
State G
State K
State F
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Exploring the State Diagram

• Explore for solutions using BFS, DFS
• Depth limited search
• DFS but to limited depth in tree
• Iterative Deepening search
• DFS one level deep, then two levels (repeats
  first level), then three levels, etc.
• If a specific goal state, can use bidirectional
  search
• Search forward from start and backward from goal
  try to meet in the middle.
• Think of maze puzzles

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More informed search

• Traversing links, goal states not always equal
• Can have a heuristic function h(x) how close
  the state is to the goal state.
• Kind of like board evaluation function/utility
  function in game play
• Can use this to order other searches
• Can use this to create greedy approach

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A Algorithm

• Avoid expanding paths that are already expensive.
• f(n) g(n) h(n)
• g(n) current path cost from start to node n
• h(n) estimate of remaining distance to goal
• h(n) should never overestimate the actual cost of
  the best solution through that node.
• Then apply a best-first search
• Value of f will only increase as paths evaluate