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Peter Lam Discrete Math CS

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Outline Use game theory to solve strictly determining games Non strictly games Create models for games Find optimal mixed strategies such as expected values of or pay off values

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Basic Principles Decision Makers (Players) Information States at Decision Time Collection of Possible Moves Procedure to Determine All Possible Moves Possible Outcomes Utility or Payoff

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Decision Makers (Players) Two Ways Players Make Moves Chance Choice These affect either State of Perfect Information State of Imperfect Information Rules Limit and Determine Moves and Outcome

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Information States/Possible Moves State of Perfect Information State of Imperfect Information When Moves are Known to All Players Players Use Pure Strategy All Moves are Thought Out When Moves are Made by Chance Players Use Mixed Strategy Moves Based off of Probability Distribution

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Payoff State of the Game at its Conclusion Examples: Win/Loss Material: Money Ranking

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Determining Maximum Payoff Create a Matrix List Players Outcomes vs. Others i.e. Two Players each with six sided die Players roll and loser pays the winner the difference in numbers

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Strictly Determined Game Two Player Zero-Sum Game Consisting Nash Equilibrium Both Players Using Pure Strategies Maximin payoff = Minimax Payoff Value of Game is Determined by Value of Equilibrium Outcome

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Non Strictly Determined Game Two Player Zero-Sum Game Consisting Both Players Use Mixed Strategies Maximin payoff < Minimax Payoff Not Ideal but Both Players Have Opportunity in Changing their Strategy Payoff Value Continuously Changes

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Example: Prisoner’s Dilemma

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Example: Let’s Make a Deal http://www.stat.sc.edu/~west/javahtml/Lets MakeaDeal.html http://www.stat.sc.edu/~west/javahtml/Lets MakeaDeal.html

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In General Game Theory based on Choices of Others Probability of Outcome Based on Decisions and Rules Factors That Ultimately Determine Outcome Rules Player Choices/Decision

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Sources http://econ2.econ.iastate.edu/classes/econ30 8/tesfatsion/gamedef.308.pdf http://library.thinkquest.org/26408/math/pris oner.shtml http://mathchaostheory.suite101.com/article. cfm/what_is_game_theory http://userpages.umbc.edu/~nmiller/POLI388 /MIXED.Q&A.htm

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1. problem set 6 from Osborne’s Introd. To G.T. p.210 Ex. 210.1 p.234 Ex. 234.1 p.337 Ex. 26,27 from Binmore’s Fun and Games.

1. problem set 6 from Osborne’s Introd. To G.T. p.210 Ex. 210.1 p.234 Ex. 234.1 p.337 Ex. 26,27 from Binmore’s Fun and Games.

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