Game Theory Basics • Game theory is designed to model • How rational (payoff-maximizing) ``agents” will behave • When individual outcomes are determined by collective behavior. • Rules of a game specify agent payoffs as a function of actions taken by different agents.
Let’s play the median game • On the index card, write down • Your name • An integer between 0 and 100 (inclusive). • After we collect all the index cards, the person (or people) whose selected number is closest to 2/3 of the median of all the numbers (rounded down) wins a prize. • E.g., if the numbers are 3, 4, 5, 38, 60, 70, 70, 90, 100
Prisoner's Dilemma Prisoner prison response no matter what Confess betray is best silent dominates stay a dominant strategy is Betray Betray Not a Pareto optimal strategy pair Gamer of n players set of actions that player I 8 Si for each player represents can take A strategy profiles si es Vi setgsaeeigesdlstmkg.es s s n Ui 5 when payoff to playeri is players play strategy profiles the
esin 8 i strategyprofile i for all players but i strictly
S t2 t Is P l s ISP 2 to So it
P2P networks Another setting ahh that try eachhave free reeding desired by other is to upload desired decision B file or not 3 benefit of recovery file A cost of uploading file not uploading is a dominant strategy countries pollution Game n no to legislation to control Yes or pollution emissions 3 Downton control costs g cost to all countries each country that pollutes odds aren't k k I countries n are polluting don't pollute dominant strategy pollute to pollute Icts Its
aantain enter whether to Q Startup Game or not Market Startup for Microsoft Entering dominates f Microsoft out staying can safely startup Therefore Microsoft that assume will do so Microsoft stamp Nasheguilitium Stay out is Enter a to best responding player is i.e each the other
e Si si players plays n
is weakly dominated b by a ipuilb andFs O www.nfafs i yuiCb5 i bis strongly dominated V s a if by iaifa.si uicb.si deleted only only good predicting caveat dominated strongly otherwise predict strategies retunigus order depends on
deepens Back to the median game • On the index card, write down • Your name • An integer between 0 and 100 (inclusive). • After we collect all the index cards, the person (or people) whose selected number is closest to 2/3 of the median of all the numbers (rounded down) wins a prize. Yzmed 66 most median
Game Coordination Bob c f G Hia
Network coordination games each node is person useheappornot achonset 2 kw ky l OP.sn n fiIfY O dgsheEitaEog SITE x O's around all used all network cascade
Qu neo Ssi ueu T W As ProgProji Nys gratify maffine bsmynofrgy us news highest quality highest synergy synergist synergy holist highest quality random
Game Parking Inspector i s I 9 locip z p Ga p 9 inspect Effect 0 p 344 p Fete log aoa g 4 sp P p legal better than illegal l p 0 log 90kg dtp loci p p 90 loog p p a Nasihnf is 9 45 9mixedshatg.es Patent o mspeetargjnspezf.io al wihnprbtg alwin rob
Prob that Xi G playeri plays s strategy mixed strategy play Xj JPY9 Expected payef utility ofplayers s when he plays
Nash has 2pm equilibria This game NE a mixed also has It player parties where each b stays home with prob's of prob 42 I II sp lower than L p I 2 410 exp payoff both pneeg
Summary so far • A Nash equilibrium is a set of stable (possibly mixed) strategies. • Stable means that no player has an incentive to deviate given what the other players are doing. • Pure equilibrium: there may be none, unique or multiple. Can be identified with “best response diagrams”. • A joint mixed strategy for n players: • A probability distribution for each player (possibly different) • It is an equilibrium if • For each player, their distribution is a best response to the others. • Only consider unilateral deviations. • Everyone knows all the distributions (but not the outcomes of the coin flips). • Nash’s famous theorem: every game has a mixed strategy equilibrium.
Issues • Does not suggest how players might choose between different equilibria • Does not suggest how players might learn to play equilibrium. • Does not allow for bargains, side payments, threats, collusions, “pre- play” communication. • Computing Nash equilibria for large games is computationally difficult.
Other issues • Relies on assumptions that might be violated in the real world • Rationality is common knowledge. • Agents are computationally unbounded. • Agents have full information about other players, payoffs, etc.
Zero-sum games payop gain are Penalty kicks grow player Goalie Kicher says suppose Luh pnb p wer loss of Gabeygoes R it foiled tpfo.se P E gosh 20,9 Q4p 0.9 0.48 o.ee InpoxmmfEIyE7sr p
kicker goes first Suppose kicker ugh 0.8 goalie goes 0.2ps announce p 0.9 must ps o.scp 0.2 gained.gs Tuat Feast is kicker's best aap 0.8 0.9 0.410 for P choice al o 6p I Ete p p kick left af prob f p f Kick right 4 prob Z Choosing expected gain if she Kickers maximizes the p that first i.e is announce has to 0.5 pto 9 Kp maximizes p 0.811 p min guarantees herself kicker plays she If p Y Latos ft E Eg an expected payoff of goes first Goalie kicker I Supposes o.iq to 8 kickergoesnght gets to bestrespond 0 99 0.811 91 to Goalies mixed 0 8 1 asg kicker goes left dive strategy µ q 1 9 0 59 dive C g left a g www.qq.my yes 3 g for q choice 9g 3 0.025 and dire left with prob GE right by prob 73 choosing dive loss if he has to goalies expected minimizes announce first minimizes i.e 9 0.5Gt moxfoagto.SC g guarantee himself a he Goalie goes first can If 0.9 zt08Zz V2 at most loss of
kicker can guarantee himself the Y exp payoff has to go first he if guarantee himself kicker can V2 exp payoff the can go second he has he can guarantee if if he goalie hp loss go first E Vy V Viva
Summary – zero-sum games • Zero-sum games have a “value”. • Optimal strategies are well-defined. • Maximizer can guarantee a gain of at least V by playing p* • Minimizer can guarantee a loss of at most V by playing q*. • This is a Nash equilibrium. • In contrast to general-sum games, optimal strategies in zero-sum games can be computed efficiently (using linear programming).
1500 penalty kicks fractions observed actual 0.5577 0.423 in game optimal strategies 942 0.58 0.38 0.4 0.62 0.6
Extensive Form Games nd White HID
B benign A aggressive Mutual Assured Destination a gampy
Centipede so far games g perfect info
vs startup Lange Company startup announces a technology that threatens big company is prob that BC Big company can pull together coup product Startup
Prisoners Dilemma Repealed with discounting repeated game Infinitely IIcountdBp thefts PTI ayff P prob game continues
I p g Cooperate until around Gnmtriggeri then in which your opponent defects then from on defect Guthger finthgger NE Tiffortat DtfortatJ Grim Trigger I j PJ 6 8ptt2 vs P 2ptL4 pi YE 8ptt2 pJ Gj Pt when i p a 213 posts
in round 1 Cooperate Titfortat k I mercy round played play what your opponent round K l in
Recommend
More recommend
