3a Normal Form Continued

8/13/2019 3a Normal Form Continued

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Nau: Game Theory 1

Introduction to Game Theory

3a. More on Normal-Form Games

Dana Nau

University of Maryland


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Nau: Game Theory 2

More Solution Concepts

! Last time, we talked about several solution concepts! Pareto optimality! Nash equilibrium! Maximin and Minimax! Dominance! Rationalizability

! Well continue with several more! Trembling-hand perfect equilibrium! !-Nash equilibrium! Rationalizability! Evolutionarily stable strategies


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Nau: Game Theory 3

Trembling-Hand Perfect Equilibrium

! A solution concept thats stricter than Nash equilibrium! Trembling hand: Requires that the equilibrium be robust against slight

errors or trembles by the agents

! I.e., small perturbations of their strategies! Recall: A fully mixed strategy assigns every action a non-0 probability! Let S= (s1, ,sn) be a mixed strategy profile for a game G! Sis a (trembling hand) perfect equilibriumif there is a sequence of fully

mixed-strategy profiles S0, S1, , that has the following properties:

! limk!"

Sk= S

! for each Sk= (s1k, ,sik, ,snk), every strategysikis a best response tothe strategies S#i

k

! The details are complicated, and I wont discuss them


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Nau: Game Theory 4

-Nash Equilibrium! Another solution concept

! Reflects the idea that agents might not change strategies if the gain wouldbe very small

! Let !> 0.A strategy profile S= (s1, . . . ,sn ) is an -Nash equilibriumif, forevery agent iand for all strategiessi"$si,

ui(si, S!i) %ui(si", S#i) #!! !-Nash equilibria always exist

! Every Nash equilibrium is surrounded by a region of !-Nash equilibriafor any !> 0

! This concept can be computationally useful! Algorithms to identify !-Nash equilibria need consider only a finite set of

mixed-strategy profiles (not the whole continuous space)

! Because of finite precision, computers generally find only !-Nashequilibria, where !is roughly the machine precision


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Nau: Game Theory 5

Problems with -Nash Equilibrium

! For every Nash equilibrium, there are !-Nash equilibria that approximate it, but theconverse isnt true

! There are !-Nash equilibria that arent close to any Nash equilibrium! Example: the game at right has just one Nash equilibrium: (D, R)

! We can use strategy elimination to get it: D dominates U for agent 1 On removing U,RdominatesL for agent 2

! (D, R) is also an !-Nash equilibrium! But theres another !-Nash equilibrium: (U, L)

! In this equilibrium, neither agents payoffis within !of the agents payoff in a Nash equilibrium

! Problem:! In the !-Nash equilibrium (U, L), agent 1 cant gain more than !by deviating! But if agent 1 deviates, agent 2 can gain more than !by best-responding to

agent 1s deviation


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Nau: Game Theory 6

Problems with -Nash Equilibrium

! Some !-Nash equilibria are very unlikely to arise! Agent 1 might not care about a gain of !/2, but might reason as follows:

Agent 2 may expect agent 1 to to playDsince it dominates U So agent 2 is likely to playR If agent 2 playsR, agent 1 doesmuchbetter by playingDrather than U

! In general, !-approximation is much messierin games than in optimization problems


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Nau: Game Theory 7

Rationalizability

! A strategy is rationalizable if aperfectly rational agentcould justifiablyplay it againstperfectly rational opponents

! The formal definition is complicated! Informally, a strategy for agent i is rationalizable if its a best response to

some beliefs that agent i could have about the strategies that the other

agents will take

! But agent is beliefs must take into account isknowledge of therationality of the others. This incorporates

the other agents knowledge of is rationality, their knowledge of is knowledge of their rationality, and so on ad infinitum

! A rationalizable strategy profile is a strategy profile that consists only ofrationalizable strategies


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Nau: Game Theory 8

Example

Matching Pennies

! Agent 1s pure strategyHeads is rationalizable! Lets look at the chain of beliefs

! For agent 1,Headsis a best response to agent 2s pure strategyHeads,! and believing that 2 would also playHeads is consistent with 2s

rationality, for the following reasons

! 2 could believe that 1 would play Tails, to which 2s best response isHeads;

! and it would be rational for 2 to believe that 1 would play Tails, forthe following reasons:

2 could believe that 1 believed that 2 would play Tails, to whichTails is a best response;

1, 1 1,1

1,1 1, 1

Heads Tails

Heads

Tails


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Nau: Game Theory 9

Strategies that arent rationalizable

Prisoners Dilemma

! Strategy C isnt rationalizable for agent 1! It isnt a best response to any

of agent 2s strategies

The 3x3 game we used earlier! M is not a rationalizable strategy for agent 1

! It is a best response to one of agent 2sstrategies, namelyR

! But theres no belief that agent 2 could haveabout agent 1s strategy for whichR would

be a best response

5, 0 1, 1

3, 3 0, 5


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Nau: Game Theory 10

Comments

! The formal definition of rationalizability is complicated because of theinfinite regress

! But we can say some intuitive things about rationalizable strategies! Nash equilibrium strategies are always rationalizable

! So the set of rationalizable strategies (and strategy profiles) is alwaysnonempty

! In two-player games, rationalizable strategies are simply those that survivethe iterated elimination of strictly dominated strategies

! In n-agent games, this isnt so! Rather, rationalizable strategies are those that survive iterative removal

of strategies that are never a best response to any strategy profile by the

other agents

! Example: thep-beauty contest


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Nau: Game Theory 11

The p-Beauty Contest

! At the start of my first class, I asked you to do the following:! Choose a number in the range from 0 to 100! Write it on a piece of paper, along with your name! In a few minutes, Ill ask you to pass your papers to the front of the

room

! After class, Ill compute the average of all of the numbers! The winner(s) will be whoever chose a number thats closest to 2/3 of

the average

! Ill announce the results in a subsequent class

! This game is famous among economists and game theorists! Its called thep-beauty contest! I usedp= 2/3


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Nau: Game Theory 12

The p-Beauty Contest

! Recall that in n-player games,! Rationalizable strategies are those that survive iterative removal of

strategies that are never a best response to any strategy profile by the

other agents

! In thep-beauty contest, consider the strategy profile in which everyone elsechooses 100

! Every number in the interval [0,100) is a best response! Thus every number in the interval [0,100) is rationalizable


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Nau: Game Theory 13

Nash Equilibrium for the p-Beauty Contest

! Iteratively eliminate dominated strategies! All numbers &100 => 2/3(average) < 67

=> any strategy that includes numbers %67 isnt a best response to any

strategy profile, so eliminate it

! The remaining strategies only include numbers < 67=> for every rationalizable strategy profile, 2/3(average) < 45=> any strategy that includes numbers %45 isnt a best response to any

strategy profile, so eliminate it

! Rationalizable strategies only include numbers < 45=> for every rationalizable strategy profile, 2/3(average) < 30

. . .

! The only strategy profile that survives elimination of dominated strategies:! Everybody chooses 0! Therefore this is the unique Nash equilibrium


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Nau: Game Theory 14

p-Beauty Contest Results

! (2/3)(average) = 21! winner = Giovanni


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Nau: Game Theory 15

Another Example of

p-Beauty Contest Results

! Average = 32.93! 2/3 of the average = 21.95! Winner: anonymous xx


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Nau: Game Theory 16

We arent rational

! Most of you didnt play Nash equilibrium strategies

! We arent game-theoretically rational agents! Huge literature on behavioral economicsgoing back to about 1979

!Many cases where humans (or aggregations of humans) tend to makedifferent decisions than the game-theoretically optimal ones

! Daniel Kahneman received the 2002 Nobel Prize in Economics for hiswork on that topic


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Nau: Game Theory 17

Choosing Irrational Strategies

! Why choose a non-equilibrium strategy?! Limitations in reasoning ability

Didnt calculate the Nash equilibrium correctly

Dont know how to calculate it Dont even know the concept

! Hidden payoffs Other things may be more important than winning

Want to be helpful Want to see what happens Want to create mischief

! Agent modeling (next slide)


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Nau: Game Theory 18

Agent Modeling

! A Nash equilibrium strategy is best for youif the other agents also use their Nash equilibrium strategies

! In many cases, the other agents wont use Nash equilibriumstrategies

! If you can forecast their actions accurately, you may beable to do much better than the Nash equilibrium strategy

! Ill say more about this in Session 9! Incomplete-information games


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Nau: Game Theory 19

Evolutionarily Stable Strategies

! An evolutionarily stable strategy(ESS) is a mixed strategy thats resistant toinvasion by new strategies

! This concept comes from evolutionary biology! Consider how various species relative fitness causes their proportions of the

population to grow or shrink

! For us, an organisms fitness = its expected payoff from interacting with arandom member of the population! An organisms strategy = anything that might affect its fitness

size, aggressiveness, sensory abilities, intelligence, ! Suppose a small population of invaders playing a different strategy is added to a

population

! The original strategy is an ESS if it gets a higher payoff against the mixture ofthe new and old strategies than the invaders do


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Nau: Game Theory 20

Evolutionary Stability! LetGbe a symmetric 2-player game

! Recall that the matrix showsu(r,r') = payoff for ragainst r'

! A strategy r' invadesa strategy rat levelx iffractionxof the population uses r'and fraction (1x) of the population uses r

! fitness(r) = expected payoff for r against a random member of the population= (1x)a+xb

! Similarly, fitness(r') = (1x)c+xd! ris evolutionarily stable against r'if there is an !> 0

such that for everyx< !, fitness(r) > fitness(r')

! i.e., (1x)a+xb > (1x)c+xd! Asx #0, (1x)a+xb#a and (1x)c+xd#c

! For sufficiently smallx, the inequality holds if either a> c, or a= c and b> d! Thus ris evolutionarily stable against r'iff one of the following holds:

! a> c! a = cand b > d

r'

r

r'r

a

c

b

d


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Nau: Game Theory 21

Evolutionary Stability

! More generally! Well use a mixed strategysto represent a population

that is composed of several different species

! Well talk aboutss evolutionary stability against all other mixed strategies! sis evolutionarily stable iff for every mixed strategys'$s,

one of the following holds: u(s,s) > u(s',s) u(s,s) = u(s',s) and u(s,s') > u(s',s')

! sis weakly evolutionarily stable iff for every mixed strategys"$s,one of thefollowing holds:

u(s,s) > u(s',s)

u(s,s) = u(s',s) and u(s,s') > u(s',s')! Includes cases where the original strategy and invading strategy have the same

fitness, so the population with the invading strategy neither grows nor shrinks

r'

r

r'r

a

c

b

d


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Nau: Game Theory 22

Example

! The Hawk-Dove game! 2 animals contend for a piece of food! The animals are chosen at random from the entire population

Each animal may be either a hawk (H) or a dove (D)! The prize is worth 6 to each! Fighting costs each 5

! When a hawk meets a dove, the hawk gets the prize without a fight:payoffs 6, 0

! When 2 doves meet, they split the prize without a fight: payoff 3, 3! When 2 hawks meet, they fight (5 for each), each with a 50% chance

of getting the prize ((0.5)(6) = 3): payoffs 2,2

! Its easy to show that this game has a unique Nash equilibrium (s,s),wheres= (3/5, 2/5)

! i.e., 60% hawks, 40% doves


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Nau: Game Theory 23

Example

! To confirm thats is also an ESS, show that, for alls"$s,u1(s, s") = u1(s", s) and u1(s, s") > u1(s", s")

! u1(s,s") = u1(s",s) is true of any mixed strategy equilibrium with full support! To show u1(s,s") > u1(s",s"), find thes"that minimizes

f (s!) = u1(s,s") #u1(s",s")

! s= play H with probability 3/5, D with probability 2/5! s"= play H with probabilityp, D with probability 1p! u1(s,s') = (3/5)[2p+ 6(1p)] + (2/5)[0p+ 3(1p)]! u1(s',s') =p[2p+ 6(1p)] + (1p)[0p+ 3(1p)]! sof (s!) = u1(s,s") #u1(s",s") is quadratic inp! Set df(s')/dp =0, solve forp => p= 3/5

So the unique minimum occurs whens"=s


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Nau: Game Theory 24

Evolutionary Stability and Nash Equilibria

Theorem. Let Gbe a symmetric 2-player game, andsbe a mixed strategy. Ifsis an evolutionarily stable strategy, then (s,s) is a Nash equilibrium of G.

Proof. By definition, an ESSs must satisfy u(s,s) %u(s",s), i.e.,sis a best

response to itself, so it must be a Nash equilibrium.

Theorem. Let Gbe a symmetric 2-player game, andsbe a mixed strategy. If(s,s) is a strict Nash equilibrium of G, thensis an evolutionarily stable

strategy.

Proof. If (s,s)is a strict Nash equilibrium, thenu(s,s) > u(s",s).

! This satisfies the first of the two alternative criteria of an ESS


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Nau: Game Theory 25

Summary

! Weve discussed several more solution concepts! trembling-hand perfect equilibria! epsilon-Nash equilibria! rationalizability

thep-Beauty Contest

! evolutionarily stable strategies Hawk-Dove game

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