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Internet Economics כלכלת האינטרנט

Internet Economics כלכלת האינטרנט. Class 2 – Introduction to Game Theory. Today. Game theory essentials . A game-theory course is 90 minutes. Mathematical modeling of games.

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Internet Economics כלכלת האינטרנט

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  1. Internet Economicsכלכלת האינטרנט Class 2 – Introduction to Game Theory

  2. Today • Game theory essentials. A game-theory course is 90 minutes. • Mathematical modeling of games. • Basic equilibrium concept that are needed for the rest of the course. (No direct connection to the Internet today…)

  3. Some more administration • Of course, participation is mandatory. • In both semesters. • Course’s blog: http://InternetEconomicsCourse.wordpress.comMaterial will be post in blog (Slides, topic suggestions, readings ).You can add comments (practical and relevant). • Reminder: send an email before coming to office hours.

  4. What is Game Theory? • Analysis of strategic situations: • Players have actions • Actions determine an outcome • Individual utilities from outcomes. • That is:my outcome depends not only on my action, but also on the actions of the others. "game theory is a sort of umbrella or 'unified field' theory for the rational side of social science, where 'social' is interpreted broadly, to include human as well as non-human players (computers, animals, plants)“ (Aumann)

  5. Applications • Economics • Essentially everywhere • Business • Pricing strategies, advertising, financial markets… • Computer science • Analysis and design of large systems. • Biology • Evolution, signaling,… • Political Science • Voting, social choice, fair division… • Law • Resolutions of disputes, regulation, bargaining… • …

  6. Today • Define “games”. • What is an “equilibrium”? • Does it always exist? • Is it unique?

  7. Example 1: coordination games Column Player שחקן העמודות Row player שחקן השורות Right number: utility for Row Player Left number: utility for Column Player Without laws, when this game is repeated, what will happen?

  8. Example 2: “chicken” Chicken!!!

  9. Example 3: Prisoner’s Dilemma • Two suspects for a crime can: • Cooperate (stay silent, deny crime). • If both cooperate, 1 year in jail. • Defect (blame the other). • If both defect, 3 years (reduced since they confessed). • If A defects (blames the other), and B cooperate (silent) then A is free, and B serves a long sentence.

  10. We saw some examples, let’s see how we formally model a game.

  11. Normal-form game • A game is actions and payoffs: • A set of actions per playerplayer-A = {a1,a2,…}, player-B={b1,b2,…},… • Payoff function for each set of chosen actionsuA(ai,bj), uB(ai,bj) • Example: • Actions:row player = {“C”,”D”}column player = {“C”,”D} • Payoffs:uA(C,C) = -1, uA(C,D) = -5, uA(D,C) = 0, uA(D,D) = -3

  12. Dominant Strategies(אסטרטגיות שולטות/דומיננטיות) • Definition: action a* is a dominant strategy for player A if:a* gains Aa higher payoff for any set of actions of the other players. • For 2 players: • Definition:(a,b) is a dominant-strategy equilibrium if a is dominant for A and b is dominant for B. • (similar definition for more players) • for every action b of player B • and for all actions a of player AuA(a*,b)≥ uA(a,b)

  13. Dominant strategies • Who has a dominant strategy in this game? We allowed ≥ in the definition. “Weakly dominant”

  14. Dominant-strategy equilibrium • Strong solution. • Why should I play anything else if I have a dominant strategy? • Main problem:Does not exist in many games….

  15. A best response • Intuitively: if I observe your strategy, what is my best action against it? • Usually, in retrospect. • A dominant strategy is a best response to any strategy!

  16. A best response • Definition:Assume that player B plays b.a* is a best response to b if no other action gains Aa higher payoff. • That is, for all strategies a of A, uA(a*,b)≥ uA(a,b). Example:If the column player plays “Left”, the best response of the row player is also “Left”.

  17. (Pure) Nash Equilibrium • Intuitively:A set of actions (strategies), where no one wants to deviate.

  18. (Pure) Nash Equilibrium • Definition: (a,b) is a (pure) Nash equilibrium if the action of each player is a best response (to the other action).

  19. Matching Pennies (זוג או פרט) Example 4: • No (pure) Nash equilibrium. • But how do we play this game? A “zero sum” games.

  20. Nash equilibrium (“pure”) • Good: • Describes “stable” outcomes. • May exist when dominant-strategy equilibria does not exist. • Simple and intuitive. • Bad: • Not unique. • What happens when multiple equilibria exist? • Does not always exist!

  21. Mixed strategies • Definition:a “mixed strategy” is a probability over actions. • If {a1,a2,…,an} are the actions (“pure strategies”) of A, then {p1,…,pn} is a mixed strategy for A if 9/10 1/4 1/3 0 1/2 1/10 1/2 2/3 1 1/2

  22. Expected payoff • When the two players play mixed strategies, the payoff is the expected payoff. (הממוצע) 1/4 3/4 2/3 1/3 • What is the payoff of the row player? • when the players play (2/3, 1/3) and (1/4,3/4)2/3 * ¼ * 3 + 1/3 * ¼ * 7 + 2/3 * ¾ * 4 + 1/3 * ¾ * 1 = 1.5

  23. Best response (w. mixed strategies) • Definition:Consider a mixed strategy sBof player B.A strategy s* for player A is a best response to sBif no other mixed strategy gains A higher expected payoff.That is, for every strategy sA, uA(s*,sB)≥ uA(sA,sB). 1/4 3/4 What is a best response to (1/4,3/4)? What would you do if you knew that your opponent plays one strategy more frequently? Best response is a pure strategy. 1 0

  24. Mixed strategies are realistic? • Do people randomize? • Computers? Evolution? Stock markets? • Model long term behavior… • Model uncertainty about the other players. • זוג או פרט • Basketball • Soccer

  25. Nash eq. with mixed strategies • Main idea: given a fixed behavior of the others, I will not change my strategy. • Definition: (SA,SB) are in Nash Equilibrium, if each strategy is a best response to the other. 1/2 1/2 1/2 1/2

  26. Example 5: Battle of the Sexes Equilibria in “battle of the sexes: • Two pure equilibria. • One mixed (2/3,1/3),(1/3,2/3)

  27. Nash’s Theorem • Theorem (Nash, 1950): every game has at least one Nash equilibrium! • (number of actions and players must be finite) • Nash was awarded the Nobel prize for this work in 1994. Nash equilibrium (with mixed strategies): • Good: always exists. Models long term stability. • Bad:Less simple and intuitive. Multiple equilibria exist.

  28. Equilibria • All we said extends to more players. Take home message: • Dominant-strategy equilibrium:my strategy is the best no matter what the others do. • Nash equilibrium:my strategy is the best given what the others are currently doing. • Dominant strategies do not always exist. But … • In the internet: we design the mechanism!

  29. Next week • Auctions for a single item: • English, Dutch • Second-price, first-price • Equilibria and revenue batman

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