Chapter 17: Making Complex Decisions

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Chapter 17 Making Complex Decisions

• April 1, 2004

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17.6 Decisions With Multiple Agents Game Theory

• Assume that agents make simultaneous moves
• Assume that the game is a single move game.

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Uses

• Agent Design (2 finger Morra)
• Mechanism Design

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Game Components

• Players
• Actions
• Payoff Matrix e.g. rock-paper-scissors

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Terminology

• Pure Strategy deterministic policy
• Mixed Strategy randomized policy, p a
  (1-p) b
• Outcome result of game
• Solution player adopts a strategy profile that
  is a rational strategy

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Prisoners Dilemna
B testifies B refuses
A testifies A -5 B -5 A 0 B -10
A refuses A -10 B 0 A -1 B -1
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Terminology

• (testify, testify) is a dominant strategy
• s strongly dominates s s is better than s for
  all other player strategies
• s weakly dominates s s is better than s for
  one other strategy and is at least as good as all
  the rest

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Terminology

• An outcome is Pareto optimal if there is no other
  outcome that all players would prefer
• An equilibrium is a strategy profile where no
  player benefits by switching strategies given
  that no other player may switch strategies
• Nash showed that every game has an equilibrium
• Prisoners Dilemna!

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Example Two Nash Equilibria
no dominant strategy! B dvd B cd
A dvd A 9 B 9 A -4 B -1
A cd A -1 B -4 A 5 B 5
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Von Neumanns Maximin

• zero sum game
• E maximizer (2 finger Morra)
• O minimizer (2 finger Morra)
• U(E 1, O 1) 2
• U(E 1, O 2) -3
• U(E 2, O 1) -3
• U(E 2, O 2) 4

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Maximin

• E reveals strategy, moves first
• p one 1-p two
• O chooses based on p
• one 2p -3(1-p)
• two -3p 4(1-p)
• p 7/12
• UE,O -1/12

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Maximin

• O reveals strategy, moves first
• q one 1-q two
• E chooses based on q
• one 2q -3(1-q)
• two -3q 4(1-q)
• q 7/12
• UO,E -1/12

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Maximin

• 7/12 one, 5/12 two is the Maximin equilibrium
  or Nash equilibrium
• Always exists for mixed strategies!
• The value is a maximin for both players.

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Repeated Move Games

• Application packet collision in an Ethernet
  network
• Prisoners Dilemna fixed number of rounds no
  change!
• Prisoners Dilemna variable number of rounds
  (e.g. 99 chance of meeting again)
• perpetual punishment
• tit for tat

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Repeated Move Games

• Partial Information Games games that occur in a
  partially observable environment such as blackjack

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17.7 Mechanism Design

• Given rational agents, what game should we design
• Tragedy of the Commons

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Auctions

• Single Item
• Bidderi has a utility vi for the item
• vi is only known to Bidderi
• English Auction
• Sealed Bid Auction
• Sealed Bid Second Price or Vickrey auction
  (no communication, no knowledge of others)