Games That Include An Element Of Chance

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Games That Include An Element Of Chance
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Expectiminmax

• If node is terminal node or depth 0
• return the Utility(node)
• If adversarys turn
• A -infinity
• For each child of node
• A min(A,Expectminimax(child,depth-1))
• Else if our turn
• A infinity
• For each child of node
• A max(A,max(child,depth-1))
• Else if chance node
• A 0
• For each child of node
• A A (Probability(child))
  ExpectminiMax(child, depth-1)
• Return A

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Example
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Position Evaluation Complexity

• Evaluation function must be a positive linear
  transformation
• O( bmnm ) where n is distinct dice rolls
• No Likely sequence of moves

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• Day 1 Road A leads to heap of gold Road B leads
  to a
• fork turn left and find a mound of jewels, turn
  right and
• get run over by a bus
• Day 2 Road A leads to heap of gold Road B leads
  to a
• fork turn left and get run over by a bus, turn
  right and
• find a mound of jewels
• Day 3 Road A leads to heap of gold Road B leads
  to a
• fork guess correctly and find a mound of jewels,
  guess
• incorrectly and get run over by a bus

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• Its not unreasonable to take road B on the first
  two days.
• It seems unreasonable to take road B on the third
  day.
• Averaging Over Clairvoyancy suggest to Take road
  B. Since it was optimal on the first two days one
  of those two conditions must hold on day three.
• Consider what information will be known at a
  point in time.
• It is a good idea to move to a state that will
  give you more information about the environment.

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Deep Blue

• In 1997, Deep Blue defeated Garry Kasparov in six
  game exhibition match.
• Deep Blue was a parallel computer which evaluated
  200 million possible moves per second.
• In June 1997 it was the 259th ,most powerful
  supercomputer.
• Evaluation function
• Over 8,000 parts, many designed for special
  positions.
• used a an "opening book" of about 4,000 positions
  and a database of 700,000 grandmaster games

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Arthur Samuel

• Developed a checkers program that learned its own
  evaluation function by playing itself thousands
  of times.
• Used rote learning (avoids understanding the game
  and instead focuses on memory)
• Computing equipment
• Had only 10,000 word memory
• magnetic tape for long-term storage
• a .000001-Ghz processor.