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.
Game Theory Given a game, can we predict which strategies the players will play?
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 )
Conclusion In Investment Game: best strategy is to save,... no matter what other player does. This is a dominant strategy equilibrium.
Conclusion In Investment Game: best strategy is to save,... no matter what other player does,... even though it is highly sub-optimal!
( 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!
( 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
What did we do? Results of our investment game.
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.
( 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
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
( 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
( 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
Nash Equilibria Nash Equilibrium: Each players strategy is a best-response to the strategies of his opponents. (mixed if playing probabilistically, else pure)
Nash Equilibria What do you think of this prediction?
Objection to Nash equilibria There may be many Nash equilibria.
IMPORTANT ANNOUNCEMENT Lectures are MOVING to Pancoe Auditorium. (so we can accommodate more students, tell your friends to join the class!)
Your consent to our cookies if you continue to use this website.