Social Networks 101 P ROF. J ASON H ARTLINE AND P ROF. N ICOLE I MMORLICA

Lecture Thirteen: Normal form games and equilibria notions.

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.

The Median Game AlokBrentCaseyDirkEla Calculating the winner: 1. Sort the numbers: 0, 25, 45, 50, Pick the middle one (the median): Compute 2/3 of the median: 30

The Median Game Median is 45, and Alok wins because his guess is closest to 2/3 of the median, or 30. AlokBrentCaseyDirkEla

How did you play?

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

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

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

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

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

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

Best responses Definition. A strategy s i * is a best-response to strategies s j of players i ≠ j if ¼ (s 1, …, s i *, …, s n ) ¸ ¼ (s 1, …, s i, …, s n ) for all strategies s i in S i.

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

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.

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

Dominant strategy equilibria Definition. A strategy profile (s 1, …, s n ) is a dominant strategy equilibrium if, for each player i, s i is a dominant strategy.

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

Nash equilibrium Definition: A strategy profile (s 1, …, s n ) is a Nash equilibrium (NE) if for each player i, s i is a best-response to strategies s j of players j ≠ i.

Chicken

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

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

( 1, 1 ) SwerveStay Swerve Chicken ( 2, 0 )( -1, -1 ) ( 0, 2 ) (swerve, swerve) (swerve, stay) (stay, swerve) (stay, stay)

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

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

( -1, 1 ) HeadsTails Heads Matching pennies ( 1, -1 )( -1, 1 ) ( 1, -1 ) (heads, heads) (heads, tails) (tails, heads) (tails, tails)

Next time Mixed Nash equilibria and fixed points.