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

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IMPORTANT ANNOUNCEMENT Lectures are MOVING to Pancoe Auditorium. (so we can accommodate more students, tell your friends to join the class!)

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The investment game

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Experiment: You may invest one of your points in the community. 1. In your envelope is a piece of paper. Write your name and whether you wish to invest or save on the paper. DO NOT SHOW ANYONE. 2. Put the paper in your envelope, pass it to the TAs. We will match the contributions at 50% (hence every invested point becomes 1.5 points) and then redistribute the points evenly among everyone.

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Lecture Two: How do we play games?

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What is a game? A set of players their possible strategies, and a function relating strategy choices to payoffs.

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Im Mrs. Column. Lets play a game! Normal-form games two players Hi, my name is Mr. Row.

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Mr. Row Mrs. Column Mr. Row and Mrs. Column each have 4 quarters to invest. 2-player Investment Game

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Mr. Row Mrs. Column InvestSave Column strategies 2-player Investment Game

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InvestSave Invest Row strategies 2-player Investment Game Mr. Row Mrs. Column

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( ?, ? ) InvestSave Invest 2-player Investment Game Mr. Row Mrs. Column Investments

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( ?, ? ) InvestSave Invest 2-player Investment Game Mr. Row Mrs. Column Returns ( 3, 7 )

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( 6, 6 ) InvestSave Invest Payoff matrix. 2-player Investment Game ( 7, 3 )( 4, 4 ) ( 3, 7 ) Mr. Row Mrs. Column

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Game Theory Given a game, can we predict which strategies the players will play?

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If Column invests, I am better off not investing. If Column doesnt invest, I am still better off not investing. I SHOULD NOT INVEST! Same here! What should row do? InvestSave Invest Prediction: Players will end up not investing. Mr. Row Mrs. Column ( 6, 6 ) ( 7, 3 )( 4, 4 ) ( 3, 7 )

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Conclusion In Investment Game: best strategy is to save,... no matter what other player does. This is a dominant strategy equilibrium.

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Conclusion In Investment Game: best strategy is to save,... no matter what other player does,... even though it is highly sub-optimal!

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( 6, 6 ) InvestSave Invest Social Optimum ( 7, 3 )( 4, 4 ) ( 3, 7 ) Mr. Row Mrs. Column Social optimum: Each player gets 2 more quarters than in equilibrium!

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( 6, 6 ) InvestSave Invest Price of anarchy ( 7, 3 )( 4, 4 ) ( 3, 7 ) Mr. Row Mrs. Column How societal value much is lost due to lack of coordination? Total val. in equil.: 8q Total val. in soc. opt.:12q PoA:2/3

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What did we do? Results of our investment game.

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Dominant strategies What do you think of this prediction? Dominant Strategy Equilibrium: Each players strategy is her best choice no matter what her opponent does.

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John Nash

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Movie Time

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The dating game Mr. Row Mrs. Column ( 0, 0 ) BlondeBrunettes Blonde ( 1, 2 )( 1, 1 ) ( 2, 1 ) A blonde and two brunettes are sitting in the computer lab …

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The dating game If Column goes for the blonde, Row is better off going for the brunette. Mr. Row Mrs. Column ( 0, 0 ) BlondeBrunettes Blonde ( 1, 2 )( 1, 1 ) ( 2, 1 )

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The dating game But if Column goes for the brunette, Row definitely wants to go for the blonde. Mr. Row Mrs. Column ( 0, 0 ) BlondeBrunettes Blonde ( 1, 2 )( 1, 1 ) ( 2, 1 )

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The dating game There is no dominant strategy equilibrium! Mr. Row Mrs. Column ( 0, 0 ) BlondeBrunettes Blonde ( 1, 2 )( 1, 1 ) ( 2, 1 )

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( 0, 0 ) BlondeBrunettes Blonde The dating game ( 1, 2 )( 1, 1 ) ( 2, 1 ) Mr. Row Mrs. Column

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How to play the dating game? What did you do?

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How to play the dating game? If you think the competition is going to go for the blonde, then go for the brunettes. …but if you think the competition will go for the brunettes, hit on the blonde!

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Nash equilibria Each person is playing a mutual best-response. This is a Nash equilibrium.

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( 0, 0 ) BlondeBrunette Blonde The dating game ( 1, 2 )( 1, 1 ) ( 2, 1 ) Mr. Row Mrs. Column Equilibria of the dating game Are there any other equilibria?

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Time for

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( 0, 0 ) BlondeBrunette Blonde The dating game ( 1, 2 )( 1, 1 ) ( 2, 1 ) Mr. Row Mrs. Column Mixed Nash equilibria: Players choose strategies probabilistically. q (1-q) (1-p) p

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q (1-q) ( 0, 0 ) BlondeBrunette Blonde The dating game ( 1, 2 )( 1, 1 ) ( 2, 1 ) Mr. Row Mrs. Column Observation: For Row to play both strategies, payoff must be equal. 1/2 (1-p) p

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( 0, 0 ) BlondeBrunettes Blonde The dating game ( 1, 2 )( 1, 1 ) ( 2, 1 ) Mr. Row Mrs. Column Observation: For Column to play both strategies, payoff must be equal. 1/2

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( 0, 0 ) BlondeBrunettes Blonde The dating game ( 1, 2 )( 1, 1 ) ( 2, 1 ) Mr. Row Mrs. Column Mixed equilibrium: Each player flips a fair coin to decide whether to chat up the blonde or the brunettes. 1/2

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Nash Equilibria Nash Equilibrium: Each players strategy is a best-response to the strategies of his opponents. (mixed if playing probabilistically, else pure)

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Nash Equilibria What do you think of this prediction?

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Objection to Nash equilibria There may be many Nash equilibria.

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IMPORTANT ANNOUNCEMENT Lectures are MOVING to Pancoe Auditorium. (so we can accommodate more students, tell your friends to join the class!)

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Next time markets * * in Pancoe Auditorium.

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