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**Unformatted text preview: **Represen)ng Games •
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• Extensive form representa)on Strategies Normal form representa)on Mixed strategies Beliefs Expected payoﬀs In
Class Game: Hide and Seek • Student 1 places $3 in envelope A or envelope “B”. • Student 2 does not observe student 1’s ac)ons. • Student 2 chooses to open either A or B. • If Student 2 ﬁnds the cash, he keeps the cash. Otherwise, Student 1 gets the cash. 1 Hide and Seek A2 0, 3 B2 3, 0 A1 2 1 B1 A2 B2 3, 0 0, 3 Represen'ng Games To describe a game, formally specify: 1. the list of players 2. the possible ac)ons for each player 3. The players’ knowledge (what each player knows when he/she takes an ac)on) 4. how ac)ons lead to outcomes, and 5. the players’ preferences over outcomes (payoﬀs) 2 Example 1a: Descrip)on • A kid decides whether to be naughty or nice. • Santa observes whether a kid has been naughty or nice and then decides whether to bring the kid toys or coal. • This is a strategic situa)on. – Each person considers what the other is likely to do (or has already done) when making his own decision. Example 1a: Descrip)on • Players: – A kid and Santa. • Ac)ons: – Kid – {Be naughty, Be nice} – Santa – {Toy to naughty kid, Coal to naughty kid, Toy to nice kid, Coal to nice kid} • Knowledge: – We typically assume the players know the game and that each player is ra)onal. – Kid – Doesn’t know anything else – Santa – Knows whether the kid has been naughty or nice • Outcomes & Preferences: – Not yet speciﬁed 3 Extensive Form Representa'on A tree, featuring • Nodes – where ac)ons are taken or the game ends • Branches – ac)ons • Labels – player on the move (for decision nodes), ac)ons (for branches) • Payoﬀ numbers – represent preferences over outcomes • Informa)on sets – represent the players’ informa)on, what a player knows when he makes a decision Example 1a: Extensive Form Kid Santa Toy 6, 0 Naughty Coal Nice Toy’ Coal’ 1, 1 4, 3 0, 0 4 Example 1a: Informa)on Sets • The kid has one informa)on set. – The ini)al node. • Santa has two informa)on sets. – When Santa makes his decision he knows if the kid has been naughty or nice. • This is why his ac)ons are labeled diﬀerently. • For this example each informa)on set is a single node. • Each decision is associated with a single informa)on set. Example 1b: Descrip)on • A kid decides whether to be naughty or nice. • Santa decides whether to bring the kid toys or coal. – He doesn’t observe whether the kid has been naughty or nice. • Now Santa only has one informa)on set. – His decision cannot be con)ngent on anything. 5 Example 1b: Extensive Form Kid Santa Toy 6, 0 Naughty Coal Nice Toy Coal 1, 1 4, 3 0, 0 Example 1b: Descrip)on • Players: – A kid and Santa. • Ac)ons: – Kid – {Be naughty, Be nice} – Santa – {Toy, Coal} • Knowledge: – We typically assume the players know the game and that each player is ra)onal. – Kid – Doesn’t know anything else. – Santa – Doesn’t know anything else. • Outcomes & Preferences: – Not yet speciﬁed 6 Example 1b: Informa)on Sets • The kid has one informa)on set. – The ini)al node. • Santa also has one informa)on set. – When Santa makes his decision he doesn’t know if the kid has been naughty or nice. • He doesn’t know which node he’s at when he makes his decision. • This is why his ac)ons are labeled the same at both nodes. • Each decision is s)ll associated with a single informa)on set. Informa)on Sets 7 Strategies • Strategy: A complete con)ngent plan for a player in a game. – Prescribes an ac)on for each of this player’s informa)on sets. Strategies • Example 1a – example strategy for each player: – Kid: Be naughty. – Santa: Bring toy if the kid is naughty (T), and bring coal if the kid is nice (C’). • Ac)on for each of the informa)on sets • Example 1a – example strategy for each player: – Kid: Be naughty. – Santa: Bring toy. 8 Nota'on • Si : player i’s strategy set • si : one strategy for player i • s
i : strategies of players other than i • S = S1 × … × Sn : set of strategy proﬁles • s = (s1, s2, ..., sn) or s = (si, s
i): one strategy proﬁle • ui : S → R : payoﬀ func)on for player i Example 1a • SKid = {Naughty, Nice} – One speciﬁc skid is Naughty • SSanta = {Give toy to a naughty kid and give toy to a nice kid, Give toy to a naughty kid and give coal to a nice kid, Give coal to a naughty kid and give toy to a nice kid, Give coal to a naughty kid and give coal to a nice kid} – We can simplify this nota)on to {TT’, TC’, CT’, CC’} – One speciﬁc sSanta is TC’ • Strategy proﬁle: a speciﬁc play of the game, one possible way the game could unfold – One speciﬁc strategy proﬁle s is (Naughty, TC’) 9 Example 1b • SKid = {Naughty, Nice} • An example speciﬁc skid is Naughty • SSanta = {Toy, Coal} • An example speciﬁc sSanta is Toy • One speciﬁc s is (Nice, Toy) Example 2a • A pedestrian and a car are approaching a crosswalk. – First the pedestrian decides to cross or wait. – The driver observes the decision and then decides to proceed through the crosswalk or delay. 10 Example 2a: Extensive Form Proceed Delay 10, 5 Proceed’ 5, 10 Delay’ Pedestrian –100, –75 0, 0 Driver Cross Wait Con)nuous Ac)on Space • It’s possible people can make a decision from a con)nuous ac)on space. – Ie., S1 = [0, 100] 1 a 2 Yes 100 – a, a No 0, 0 11 Con)nuous Ac)on Space • We also can have informa)on sets. – Ie., S1 = [0, 100] 1 a 2 Yes 100 – a, a No 0, 0 Normal Form Representa)on • A descrip)on of players, strategy spaces and payoﬀs Given any strategy proﬁle the normal form tells us what the outcome is. 12 Normal Form • For games with two players and a ﬁnite number of strategies, the normal form can be wrilen as a table with appropriate labels. An Example 13 Another Example Back to Example 2a • A pedestrian and a car are approaching a crosswalk. – First the pedestrian decides to cross or wait. – The driver observes the decision and then decides to proceed through the crosswalk or delay. 14 Example 2a: Strategy Sets • SPed = {C, W} • SDriver = {PP’, PD’, DP’, DD’} – Each speciﬁc strategy for the driver tells either to proceed or delay if the pedestrian crosses and either to proceed’ or delay’ if the pedestrian waits. Example 2a: Normal Form Driv Ped PP’ Cross –100 , –75 Wait 5, 10 PD’ –100 , –75 DP’ 10, 5 DD’ 10, 5 0, 0 5, 10 0, 0 15 Example 2b • From some distance away a driver observes that he’s approaching a crosswalk similar to Example 2a. – He can speed up or maintain his current speed. – If he maintains his current speed the situa)on will be iden)cal to Example 2a. • First the pedestrian decides to cross or wait. – If he speeds up he’ll make the ﬁrst decision. • The pedestrian will observe this situa)on before deciding whether to cross or wait. Example 2b: Info & Strategy Sets • Informa)on sets: The driver has 4 informa)on sets and the pedestrian has 3 informa)on sets. • Strategy sets: – SDriver = {SPP’P’’, SPP’D’’, SPD’P’’, SPD’D’’, SDP’P’’, SDP’D’’, SDD’P’’, SDD’D’’, MPP’P’’, MPP’D’’, MPD’P’’, MPD’D’’, MDP’P’’, MDP’D’’, MDD’P’’, MDD’D’’} – SPed = {CC’C’’, CC’W’’, CW’C’’, CW’W’’, WC’C’’, WC’W’’, WW’C’’, WW’W’’} – We can determine how many strategies each has by mul)plying his decisions available at each informa)on set. • The driver has 24 = 16 and the pedestrian has 23 = 8. 16 Example 2b: Extensive Form * Payoﬀs are given (uPed, uDriver). 5, 10 Cross’ 10, 5 Wait’ Pedestrian –100, –75 Wait Driver Cross 0, 0 Proceed Speed Delay Driver Example 2b: Extensive Form Proceed’ Driver Maintain Delay’ Pedestrian –100, –75 10, 5 Proceed’’ 5, 10 Delay’’ 0, 0 Driver Cross’’ Wait’’ * Payoﬀs are given (uPed, uDriver). 17 Back to Example 2b • Consider informa)on and strategy sets • What will the normal form look like for example 2b? • For prac)ce, try ﬁlling in some other entries into the normal form matrix. Ped Driv CC’C’’ SPP’P’’ –100, –75 CC’W’’ CW’C’’ CW’W’’ –100, –75 100, –75 –100, –75 SPP’P’’ –100, –75 –100, –75 –100, –75 –100, –75 SPD’P’’ –100, –75 –100, –75 –100, –75 –100, –75 SPD’D’’ –100, –75 –100, –75 –100, –75 WC’C’’ WC’W’’ WW’C’’ WW’W’’ –100, –75 SDP’P’’ 10, 5 10, 5 10, 5 10, 5 SDP’D’’ 10, 5 10, 5 10, 5 10, 5 SDD’P’’ 10, 5 10, 5 10, 5 10, 5 SDD’D’’ 10, 5 10, 5 10, 5 10, 5 MPP’P’’ 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 5, 10 MPP’D’’ MPD’P’’ MPD’D’’ MDP’P’’ MDP’D’’ MDD’P’’ MDD’D’’ 18 How many informa)on sets does F have? How many strategy proﬁles are there in this game? Normal Form • The normal form contains all the relevant informa)on when the players move simultaneously and independently. – Neither observes any ac)ons chosen by the other player before making own decision. – There are several classic normal form games that ﬁt this descrip)on. 19 Classic Normal Form Games Matching Pennies 2 1 H T H 1, –1 –1, 1 T –1, 1 1, –1 Prisoner’s Dilemma 2 1 C D H 0, 0 1, 3 D 3, 1 2, 2 D 0, 3 1, 1 Pigs Hawk
Dove/Chicken 2 1 H D C 2, 2 3, 0 S D P W P 4, 2 6, –1 W 2, 3 0, 0 Classic Normal Form Games Coordina)on 2 1 A B A 1, 1 0, 0 B 0, 0 1, 1 Pareto Coordina)on 2 1 A B B 2, 1 0, 0 F 0, 0 1, 2 B 0, 0 1, 1 Stag Hunt Balle of the Sexes 2 1 B F A 2, 2 0, 0 2 1 S R S 2, 2 1, 0 R 0, 1 1, 1 20 Mixed Strategies • The strategies we looked at earlier are called pure strategies. – A player chooses one speciﬁc strategy with certainty. • A mixed strategy assigns probabili)es on a player’s pure strategies. – Technically a pure strategy is a mixed strategy with a probability of 1 on that pure strategy and a probability of 0 on all other pure strategies. Beliefs • In strategic situa)ons guesses about what the other players will do are important. – We use beliefs to represent these. • When one player chooses a strategy, she will consider what strategies the other player is likely to choose. • Beliefs look a lot like mixed strategies (for the other player) because both of them are chosen from the set of probability distribu)ons over that other player’s strategies. 21 Beliefs • Bart vs. Lisa in Rock, Paper, Scissors • hlp://www.youtube.com/watch?
v=NMxzU6hxrNA • Rock, Paper, Scissors World Championships • hlp://www.youtube.com/watch?
v=nGYqSqf0yCY 44 22 Beliefs • θi is a speciﬁc belief about player i • Bart vs. Lisa in Rock, Paper, Scissors • Could write θB=(1,0,0) over R,P,S or θB (R)=1 • uL(P, θB)=1 uL(S, θB)=
1 • NYT ar)cle on RPS Tournament men)ons some believe that journalists oten throw paper • Could write θJ=(¼, ½, ¼) over R,P,S Expected Payoﬀs 23 ...

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