Lecture 20 Zero Sum Games p2 14482

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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