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Understanding Normal Form Games: Equilibria and Strategies in Game Theory

This lecture by Professors Jason Hartline and Nicole Immorlica explores the fundamentals of normal form games, including dominant strategies and Nash equilibria. The session features an engaging experiment called the Median Game, where participants guess integers to win points based on their proximity to 2/3 of the median. Additionally, the lecture discusses various game scenarios such as the Prisoner's Dilemma and provides game theory definitions, strategies, and methodologies to determine equilibria. It serves as a practical introduction to reasoning in games and understanding interdependent decision-making.

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Understanding Normal Form Games: Equilibria and Strategies in Game Theory

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Presentation Transcript

  1. Social Networks 101 Prof. Jason Hartline and Prof. Nicole Immorlica

  2. Lecture Thirteen: Normal form games and equilibria notions.

  3. Let’s play a game Experiment: The median game. 1. Guess an integer between 1 and 100, inclusive. 2. Write your number and name on your card. P R I Z E : The people whose numbers are closest to 2/3 of the median win 5 points.

  4. The Median Game 25 45 0 50 69 Alok Brent Casey Dirk Ela Calculating the winner: 1. Sort the numbers: 0, 25, 45, 50, 69 2. Pick the middle one (the median): 45 3. Compute 2/3 of the median: 30

  5. The Median Game Median is 45, and Alok wins because his guess is closest to 2/3 of the median, or 30. 25 45 0 50 69 Alok Brent Casey Dirk Ela

  6. How did you play?

  7. Reasoning in games Imagine what everyone else will do, decide how to act based on that assumption.

  8. Bi-matrix games Example: prisoners’ dilemma Mrs. Column Confess Deny ( -4 , -4 ) ( 0 , -10 ) Confess ( -10 , 0 ) ( -1 , -1 ) Deny Mr. Row

  9. Prisoners’ dilemma Q. If Row confesses, what should Column do? Mrs. Column Confess Deny ( -4 , -4 ) ( 0 , -10 ) Confess ( -10 , 0 ) ( -1 , -1 ) Deny Mr. Row

  10. Prisoners’ dilemma Q. If Row denies, what should Column do? Mrs. Column Confess Deny ( -4 , -4 ) ( 0 , -10 ) Confess ( -10 , 0 ) ( -1 , -1 ) Deny Mr. Row

  11. Dominant strategies Row’s best-response was Confess no matter what Column did. Confess is a dominant strategy for row.

  12. Normal form games Definition. A normal form game for a set N of n players is described by 1. A set of strategies Si for each player i. 2. A payoff function ¼i for each player i and each profile of strategies (s1, …, sn) indicating player i’s reward for every strategy profile.

  13. Best responses Definition. A strategy si* is a best-response to strategies sj of players i ≠ j if ¼(s1, …, si*, …, sn) ¸¼(s1, …, si, …, sn) for all strategies si in Si.

  14. Dominant strategies Definition. A strategy si is a dominant strategy for player i if it is a best-response to all strategy profiles of the other players.

  15. Finding dominant strategies To find a dominant strategy for a row player, compare vectors of payoffs in each row. If (and only if) some row vector dominates coordinate-wise, it is a dominant strategy for the row player.

  16. Prisoners’ dilemma Q. Is there a dominant strategy? Mrs. Column Confess Deny ( -4 , -4 ) ( 0 , -10 ) Confess ( -10 , 0 ) ( -1 , -1 ) Deny Mr. Row

  17. Dominant strategy equilibria Definition. A strategy profile (s1, …, sn) is a dominant strategy equilibrium if, for each player i, si is a dominant strategy.

  18. Another game Q. Is there a dominant strategy? Mrs. Column High Low ( 2 , 2 ) ( 0 , 3 ) High ( 3 , 2 ) ( 5 , 1 ) Low Mr. Row

  19. Nash equilibrium Definition: A strategy profile (s1, …, sn) is a Nash equilibrium (NE)if for each player i, si is a best-response to strategies sj of players j ≠ i.

  20. Chicken

  21. Chicken Q. Is there a Nash equilibrium? Mrs. Column Swerve Stay ( 1 , 1 ) ( 0 , 2 ) Swerve ( 2 , 0 ) ( -1 , -1 ) Stay Mr. Row

  22. Finding Nash equilibria Method: Best-response (directed) graph 1. For each strategy profile s create a node su. 2. Connect node su to node sv if for some player i, his strategy sviin v is a best response to the other players’ strategies in u and for all other players j, suj= svj. 3. Search for a node with no out-going links.

  23. Chicken Swerve Stay ( 1 , 1 ) ( 0 , 2 ) Swerve (swerve, swerve) ( 2 , 0 ) ( -1 , -1 ) Stay (swerve, stay) (stay, swerve) (stay, stay)

  24. Chicken Q. Is there a Nash equilibrium? Mrs. Column Swerve Stay ( 1 , 1 ) ( 0 , 2 ) Swerve ( 2 , 0 ) ( -1 , -1 ) Stay Mr. Row

  25. Matching pennies Q. Is there a Nash equilibrium? Mrs. Column Heads Tails ( -1 , 1 ) ( 1 , -1 ) Heads ( 1 , -1 ) ( -1 , 1 ) Tails Mr. Row

  26. Matching pennies Heads Tails ( -1 , 1 ) ( 1 , -1 ) Heads (heads, heads) ( 1 , -1 ) ( -1 , 1 ) Tails (heads, tails) (tails, heads) (tails, tails)

  27. Next time Mixed Nash equilibria and fixed points.

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