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Lecture 20 Zero Sum Games p2 14482
Announcements
Hw 6 due Friday llate option Is TEffsunSolus posted Sun 5pm
Exam topics up later todayMon March 22 7pm Same
Exam 2 proTaeme
ReadingsLect2f Gtf part need
VanderbeiLeete GM 24,26
Lecter GM 2.5
Finite choice 2 player Zero som games
Payoff matrix
Nij amount Alice CRous wins if
Alice plays pure statesy iBob plays pure statesyj
Expected payoff for mixed strategies x yx'My
pad myinxthly given fixed x
acy m xxT1hy given fixed y
Alice wants to minimize padBob wants to maximize 9cgI is el for Alice if
PCE Max poxis worst case optimal for Bob if
y2cg minacy
Good use 2 P to find I g given M
Lemme Last time
If Alice's mixed strategy is fixed
Bob's best response is a pure strategy
Ie worst case payoff for Alice's xis min nine XTMez Mean
Alice wants to maximize the worst case
playoff over all possible mixedstrategies probablity distributions
MIX min name XTMez xTMem
Unfortunately this isnt a linear Program
12cg for maximizing a minimum
Introduce new variable U
maximize 2h is smaller than
each option
TRICK
MIXmin fi CD fund fmCx subj to
e pIfunctions
Max V toxx a
subject toa C optimum
If
U FmCxX C p
Our problem becomes
Tq.usubiect.to xelRUE xTµej for all i EEE Rm
Zxi I XiZ
Let I in IRN
Max U subj to
Em f i
XI 0
maxusubjtou.ITXTMEOT 0 13x J
earn
I E T
Similarly Bob wants to minimize daysminvsubj.to
Iv My20I I
EtExercise check these are dual
By strong duality PCI 2cg at
optimum I JDef The values of the game is
pcxT acy
A game with value 0 is called fair
Example Easter Bunny_RPS_BunnyR P fer
R 0 43,43
n
awe 43
Lipford what is a basicmin V subj.to
I7
Yi 192 1 is alwaysYi 20 tight
yz.IO Need 2 more
Check I
Poke rds Is bluffing usefuldetails in
ante each player potsin 1
v chop
dead 1 card to each
bidding A starts bidding bet add 1 to poteach player
passstops when
bet followed by bet best card wins
pass followed by passbet followed by pass player whobet wins
5 things can happen
A p B p1 to highest card
Ap B b Ap H to B
Ia
Ab Bb 2 to Itc
PurestrategiesA can
Lpass if B bets passpass if B bets betbet
B can pass no matter whatdo what A diddo opp of what A didbet no matter what
A cyyyz.gsyo _which line of belting to use if havecardis
A has 27 pure strategies
B has 64 pure strategies
Eliminate bad choices_tosh P
If holding 3 dont pass smartIf holding 1 don't bet
Assume other player is smart
If A holds 2 and B bets
A knows B doesnt have 1 wait bet ETC
Shrink to 8 pure Strasies for A 4 for 13
Solve LP Yes bluffing is useful
other kinds of games extra not part ofclass
Bimatrixgames Alice and Bobget own payoff matrix A B
Zero sons A B
Exagple Prisoner's Dilemma
2 convictschoice ret or staysitt
both setIf both stay silent Small punishment
both rot bothget med punone rats one silent rat goes free
silent gets big pun
MixedNasticiumwherepair I g each is best response
against each other
x tay mgxxTAy ffhafdu.ITXTAy mayxxTBy
Eg rat is best stragety agonist rat's
For a ten game the mixed EasyTNash eouilibrium is just pair ofworst case optimal solutions
d
Office hours Grot recorded
veysimplepmofofffactsgge.mepcxs mgnxTMy
Lamina For any mixed strategy for Alice
BC muinxTMejIe Bex minimum over pure strategies Ej for Bob
ProofusingLPfacts
If we fix can use LP to find pix
xTµ isJ myin Ethyl Egil Yi 30 a fixedvector
optimum is exactly padknow that some optimum will be at
corner of polyhedron
Ey c Rml Egil go 203
93 corners are bfs
ya correspond to pure strategies
Ty