GTAP Sessions 11-15 Mixed Strategy

8/10/2019 GTAP Sessions 11-15 Mixed Strategy

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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

GAME THEORY & APPLICATIONS

J. Ajith Kumar, TAPMI, Manipal

Mixed Strategy


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You are asked to make an investment in ONEof two

ventures, given the following information:

2

qIf you invest in Venture A, there is a 5% chance of

making a profit of Rs. 80 crores, and a 95% chance

of no profit at all.qBut, if you invest in Venture B, you are fully assured

of a profit of Rs. 8 crores.

Which venture will youinvest in A or B?

(there is no correctanswer).

OPENING EXERCISE


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Lottery A: Is for Rs. 10 crore, and 250,000 people will

purchase it.

3

There is no one correctanswer

OPENING EXERCISE

You have Rs. 100 with you, with which you have to

purchase either of two lotteries.

The only winner of each lottery gets the full prize but a

loser loses even the Rs. 100. Which lottery will you

take?

Lottery B: Is for Rs. 2 crore, and 25,000 people willpurchase it.


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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 4

MATCHING PENNIES GAME (MULTIPLE ROUNDS)

Each player chooses Redor Black. If players choose the same

colour, Player 1 wins. If players choose different colours, Player 2wins.

We will do 30 rounds. In each round, each player must guess

what the other player will choose, and then make his own

choice.

At the end, each player will describe his thinking to all us.

What is the result?

After each round, players get to see what their respective

choices were and who won the round.

Players will be given a few seconds between rounds to think

over their choice of the next round.


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UNDERSTANDING MIXED STRATEGY

MANOJ

Park Beach

ALOK

Park 100, 80 0, 0

Beach

0, 0 60, 120

Alok and Manoj are two elderly

citizens who stay in two different

localities of the same town. Every

evening both of them visit either

the park or the beach (there is only

one of each in the town).

Does either player have a dominant strategy?

What are the PSNE? Are there any MSNE in this game?

If they both visit the same place then they enjoy each

otherscompany and that leads to positive payoffs for

both. If they visit different places, neither benefits.

The payoffs are given here.


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A mixed strategy of a given player is a probability

distribution that the player assigns to his set of pure

strategies.

e.g. in this game, Aalok = (Park, Beach)

The pure strategies that Alok can choose are Park, Beach.

However, Alok can mixthe two and choose to go to

the Park with probability p, and to the Beach with

probability1p.

If p = 0.1, then Alok = (0.1, 0.9) is a mixed strategy

played by Alok.


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Similarly, Manoj can decide to mix his strategies andplay Park and Beach with probabilities q and 1q

respectively.

If q = 0.2, then Manoj = (0.2, 0.8) is a mixed strategy

played by Manoj.

Since bothpandqcan take infinite values between 0

and 1, Alok and Manoj can choose between infinite

possible strategies.

A pure strategy is a special case of mixed strategy,

when p (or q) is 0 or 1.


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MIXED STRATEGY PROFILES

We can assign to each mixed strategy an additional row

or an additional column, depending upon the playerthat is playing it.

MANOJ

Park Manoj Beach

ALOK

Park 100, 80 ?, ? 0, 0

Alok

?, ? ?, ? ?, ?

Beach

0, 0 ?, ? 60, 120


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THE MEANING OF MIXED STRATEGYWhat does it mean to say Alok willmix strategies and play (0.1, 0.9)?

q From Aloks viewpoint: On 10% of the days, Alok will go to the Park, and on

the remaining 90%, he will go to the Beach. Thus, when the game is

played over a large number of rounds, the mixing of the two pure

strategies happens in the proportions indicated by the mixed strategies.

q However, on any given day, Alok will either go to the Park or the Beach.

Thus, whenever Alok plays the game, he actually plays only a pure strategy.

Manoj also knows this.

Simple interpretation 1: Randomization

This means that the mixed strategy is an abstraction and is never

actually playedby the player in any particular round of the game.

q From Manojs viewpoint: Manojs perception of how likely Alok is, to

choose either place. In this example, Manoj might think there is a 10%

chance that Alok will go to the Park and a 90% chance that he will go to the

beach. If so, Manoj will say that Alok is playing the mixed strategy - (0.1,

0.9).


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q Alok and Manoj are not particular individuals but represent two different

populations. In this example chosen, a member of each type gets value

only in interaction with a member of the other type and no value

otherwise. Further, each population has its preferred distribution.

In any given game, these are two useful ways of interpreting

to create meanings as suitable to the context of that game.

Simple interpretation 2: Population preference

q Thus, Alok plays (0.1, 0.9)means that when asked what they prefer, 10%

members of the Alok population will choose Park and 90% will choose

Beach. Thus the mixed strategy represents the preference distribution of

the population.

THE MEANING OF MIXED STRATEGYWhat does it mean to say Alok willmix strategies and play (0.1, 0.9)?


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MIXED STRATEGY & EXPECTED PAYOFFS

IfAlok = (0.1, 0.9) and Manoj= (0.2, 0.8) then what are their respective payoffs?

We answer this by computing expected values that

are probability-weighted averages of the payoffs

they get by playing their respective pure strategies

MANOJ

Park Beach

ALOK

Park 100, 80 0, 0

Beach

0, 0 60, 120

If Alok plays the pure strategy Park,given that Manoj is playing (0.2, 0.8),

Aloks payoffs will be:

Payoffs for Alok

UAlok(Park ,Manoj) = 100*0.2 + 0*0.8 = 20

Likewise, if Alok plays the pure strategy

Beach

:

UAlok(Beach,Manoj) = 0*0.2 + 60*0.8 = 48

Now, Alok himself is playing a mixed

strategy (0.1, 0.9). Hence,

UAlok(Alok,Manoj)=20*0.1 + 48*0.9 = 45.2

Payoffs for Manoj(do this yourself)

UManoj(Alok ,Manoj) = 88

Hence, the payoffs are: (45.3, 88)

Note: we work with

absolute values here, not

ordinals.


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We can now generalize the calculation of expected payoffs

MANOJ

Park (q) Beach (1-q)

ALOK

Park(p)

p.q p.(1-q)

Beach(1-p)

(1-p).q (1-p).(1-q)

Pure strategy payoff matrix Likelihood that a given strategy

profile will be playedU1(1,2) = U1(Park, Park)*p.q + U1(Park, Beach)*p.(1q) +

U1(Beach, Park)*(1p).q + U1(Beach, Beach)*(1p).(1q)

U2(1,2) = U2(Park, Park)*p.q + U2(Park, Beach)*p.(1q) +

U2(Beach, Park)*(1p).q + U2(Beach, Beach)*(1p).(1q)

MANOJ


ALOK

Park(p)

100, 80 0, 0

Beach(1-p)

0, 0 60, 120

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MIXED STRATEGY & EXPECTED PAYOFFS


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MIXED STRATEGY NASH EQUILIBRIUM

Being a rational actor, each player is keen to maximize his own

payoffs.First, we look for the PSNEs. Are there any?

Yes: (Park, Park) and (Beach, Beach) are both PSNEs. If one player

knows with certainty that the other player is choosing Park, then

he will also choose Park. Likewise, for Beach.

However, here the players cannot know with certainty what the

other player might do. They can at best make intelligent

guesses about the likelihoods that other players will attach to

their choices. In response, they can also play their choices with

probabilities, i.e. they can each randomize.

Since players can randomize, are there MSNEs also? Are there

mixed strategies for each player, that are best responses to each

other? If yes, how do we find them?

13


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LetAlokbe (p, 1p) and Manojbe (q, 1q), then

The payoffs to Alok if he chooses Park / Beach:

MANOJ


ALOK

Park(p)

100, 80 0, 0

Beach(1-p)

0, 0 60, 120

UAlok(Park ,Manoj) = 100*q + 0*(1q) = 100q

UAlok(Beach,Manoj) = 0*0.2 + 60*(1q) = 6060q

What do the utility functions tell us?

100

60

0

1

UAlok

If Manoj is playing q < 3/8, then Alok will

receive a higher payoff by playing Beach. If

q > 3/8, then playing Park gives a higher

payoff. But if Manoj plays q = 3/8, then Alok

gets the same payoffs forBeachand Park.

3/8

37.5

At q = 3/8, Alok becomes

indifferent between his choices.

Beach

Park

q 14



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Now, computing utilities for Manoj if Alok plays (p, 1p)

MANOJ


ALOK

Park(p)

100, 80 0, 0

Beach(1-p)

0, 0 60, 120

UManoj(Alok, Park) = 80*p + 0*(1p) = 80pUManok(Alok, Beach) = 0*p + 120*(1p) = 120120p

120

80

01

If Alok is playing p < 3/5, then Manoj

will receive a higher payoff by playing

Beach. If p > 3/5, then playing Park

gives a higher payoff. But if Alok plays

p = 3/5, then Manoj gets the samepayoffs for Beachand Park.

3/5

48

At p = 3/5, Manoj becomes

indifferent between his choices.

Beach

Park

UManoj

p15



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From the utility functions, we can extract the best response

functions.

1

p = BAlok(q)

3/8

p=0

beach q

p=1

park1

q = BManoj(p)

3/5

q=0

beach

p

q=1

park

q

Best response of Alok to Manoj

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We can get the Nash equilibrium by overlapping the best response

functions.

Best response of Manoj to Alok



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Overlapping the best response functions:

p = BAlok(q)

3/8

p=0beach

q

p=1

park

q = BManoj(p)3/5

q=0

beachq=1

park

The best response functions

intersect at: p = 3/5, q = 3/8

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Hence, the Mixed Strategy Nash Equilibrium =

{ (3/5, 2/5), (3/8, 5/8) }

This means that the

equilibrium strategies of Alok

and Manoj are:

*Alok = (3/5, 2/5) and

*Manoj = (3/8, 5/8)

respectively.

Can also be expressed as = { (0.6, 0.4), (0.375, 0.625) }



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Making sense of the MSNE

The MSNE is: { (0.6, 0.4), (0.375, 0.625) }. What does ittell us?

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The likelihood is 60% that Alok will go to the Park and

40%, the Beach. The corresponding likelihoods forManoj are 37.5% and 62.5%.

If the other person sticks to his strategy, then neither

has an incentive to change his strategy. For eachperson, his strategy is now his best response to the

others strategy.


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Making sense of the MSNE

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Now suppose, Alok increases his likelihood of going to

the Park to 75%. Then how should Manoj respond?

What is his best response?

As per our model, Manoj should respond by only going to thePark: (1, 0).

For any likelihood of Aloks going to Park that is < 60%, Manojs

payoff is always higher in going to the Beach than to the Park.

Review Questions

If Alok plays (0.55, 0.45), what is Manojs best response?

If Manoj plays (0.45, 0.55), what is Aloks best response?

If Manoj plays (0.35, 0.65), what is Aloks best response?


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Opponents Indifference Property

The strategy chosen by each player in a MSNE makes the

other player indifferent between his choices (this can be

proved as a general result.)

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e.g. when Alok plays (0.6, 0.4), then Manoj gets the

same payoff regardless of what strategy he plays.

whether he plays (0.1, 0.9), (0.375, 0.625), (0.8, 0.2) or

anything else he gets a payoff of exactly 48.

Question: If that is true, then why should Manoj play

(0.375, 0.625), his equilibrium strategy?Answer: If Manoj plays anything other than (0.375, 0.625), then

Alok will respond by moving away from (0.6, 0.4) towards one of

the pure strategiesThat will cause a disequilibriumand Manoj

will then have to respond by changing his strategyand so on


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Aloks equilibrium strategy (p*, 1p*) is such that it

makes Manoj indifferent between his pure strategies.

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80.p* + 0.(1p*) = 0.p* + 120.(1p*)

200.p* = 120; p* = 0.6

Hence, Aloks equilibrium strategy: (0.6, 0.4).



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Manoj

s equilibrium strategy (q*, 1

q*) is such that itmakes Alok indifferent between his pure strategies.

100.q* + 0.(1q*) = 0.q* + 60.(1q*)

160.q* = 60; q* = 0.375

By this route, we dont need to determine the best

response functions etc.!

Hence, Manojs equilibrium strategy: (0.375, 0.625).



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23

LESSON: Each players strategy in a MNSE is suchthat it makes the other player indifferent between

her pure strategies that she plays with a positive

probability.

In a MSNE, given the other players strategies, the

focal players expected payoff is,

q equal for all pure strategies to which sheassigns a

positive probability,

q which is at least as good as those for pure

strategies to which sheassigns a zero probability.


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24

Helps us find the MSNE in a game.

Helps us test whether a given profile is a MNSE.


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TESTING FOR MSNE

[ (3/4, 0, 1/4), (0, 1/3, 2/3) ]

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PLAYER 2L C R

PLAYER1T

1, 2 3, 3 1, 1

M 2, 1 0, 1 2, 0

B 0, 4 5, 1 0, 7

U1(T ,2) = (0)*1 + (1/3)*3 + (2/3)*1 = 5/3

U1(M ,2) = (0)*2 + (1/3)*0 + (2/3)*2 = 4/3

U1(B,2) = (0)*0 + (1/3)*5 + (2/3)*0 = 5/3

U2(1, L) = (3/4)*2 + (0)*1 + (1/4)*4 = 5/2

U2(1, C) = (3/4)*3 + (0)*1 + (1/4)*1 = 5/2

U2(1, R) = (3/4)*1 + (0)*0 + (1/4)*7 = 5/2


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TESTING FOR MSNE

[ (3/4, 0, 1/4), (0, 1/3, 2/3) ]

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are the same (i.e. 5/3) for the two pure strategies, T and B, to

which she has assigned positive probability,

which are at least as good as that for M (i.e. 4/3), to which shehas assigned zero probability.

Player 1s expected payoffs, given Player 2 is playing (0, 1/3, 2/3):

are the same (i.e. 5/2) for the two pure strategies, C and R, to

which she has assigned positive probability,which are at least as good as that for L (i.e. 5/2), to which she

has assigned zero probability.

Player 2s expected payoffs, given Player 1 is playing (3/4, 0, 1/4):

Hence, the given profile is a MSNE


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TESTING FOR MSNE

[ (1/3, 1/3, 1/3), (1/4, 1/2, 1/4) ]

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PLAYER 2L C R

PLAYER1T 1, 2 3, 3 1, 1

M 2, 1 0, 1 2, 0

B 0, 4 5, 1 0, 7

U1(T ,2) = (1/4)*1 + (1/2)*3 + (1/4)*1 = 2

U1(M ,2) = (1/4)*2 + (1/2)*0 + (1/4)*2 = 1

U1(B,2) = (1/4)*0 + (1/2)*5 + (1/4)*0 = 5/2

U2(1, L) = (1/3)*2 + (1/3)*1 + (1/3)*4 = 7/3

U2(1, C) = (1/3)*3 + (1/3)*1 + (1/3)*1 = 5/3

U2(1, R) = (1/3)*1 + (1/3)*0 + (1/3)*7 = 8/3


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TESTING FOR MSNE

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are different across the three pure strategies, T, M and B, to

which she has assigned positive probabilities,

Player 1s expected payoffs, given Player 2 is playing (1/4, 1/2, 1/4):

are also different across the three pure strategies, L, C and R, to

which she has assigned positive probabilities,

Player 2s expected payoffs, given Player 1 is playing (1/3, 1/3, 1/3):

Hence, the given profile is not a MSNE

[ (1/3, 1/3, 1/3), (1/4, 1/2, 1/4) ]


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FINDING THE MSNE

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The method of using best responses is easy to use in 2-

player 2-action games, but becomes difficult to apply

when the number of actions is more than two.

In 2-player games with more than 2 actions we can use a

trail and error method, in conjunction with theOpponents Indifference Property to find the MSNE.


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FINDING THE MSNE

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PLAYER 2B S X

PLA

YER1B 4, 2 0, 0 0, 1

S 0, 0 2, 4 1, 3

Find all the NE in this game:

Enumerate all possible profile types, by varying each

players playing his pure strategies with a positive or zero

probability.

A1: {B, S} and A2: {B, S, X}


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FINDING THE MSNE

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(1, 0), (1, 0, 0)

(1, 0), (0, 1, 0)

(1, 0), (0, 0, 1)

(1, 0), (q, 0, 1q)

(1, 0), (0, q, 1q)(1, 0), (q, 1q, 0)

(1, 0), (q1, q2, 1q1q2)

(0, 1), (1, 0, 0)

(0, 1), (0, 1, 0)

(0, 1), (0, 0, 1)

Both players assign positive probability to

only one of their actions.

Player 1 assigns positive probability to only one action; Player 2 to

more than one action.

(0, 1), (q, 0, 1q)

(0, 1), (0, q, 1q)(0, 1), (q, 1q, 0)

(0, 1), (q1, q2, 1q1q2)


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FINDING THE MSNE

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(p, 1p ), (1, 0, 0)

(p, 1p ), (0, 1, 0)

(p, 1p ), (0, 0, 1)

(p, 1p ), (q, 0, 1q)

(p, 1p ), (0, q, 1q)

(p, 1p ), (q, 1q, 0)

(p, 1p ), (q1, q2, 1q1q2)

Both players assign

positive probability to

more than one action.

Player 2 assigns positiveprobability to only one action;

Player 1 to more than one action.


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FINDING THE MSNE

33

PLAYER 2B S X

P

LAYER1B 4, 2 0, 0 0, 1

S 0, 0 2, 4 1, 3

This game has three MSNEs, of which two are PSNEs and

one is not:

[ (1, 0), (1, 0, 0) ],

[ (0, 1), (0, 1, 0) ],

[ (3/4, 1/4), (1/5, 0, 4/5) ]


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NASH EQUILIBRIUM PURE & OTHERWISE

NASH EQUILIBRIUM

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Those that are PSNE

(made up of only pure

strategies)

Those that are not PSNE

(at least one player is not

playing a pure strategy)

Remember: a pure strategy is a special case of a mixed strategy.


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NASH EXISTENCE THEOREM

Every finite game has at least one equilibrium.

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Finite game: has a finite number of players, each with a

finite number of pure strategies (actions).

Is it possible that in a game that is not finite, there is

no NE at all?


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EXERCISE

Find all the NEs in this game.

36

MANOJ


ALOK P

ark(p)

40, 40 10, 70

Beach(1-p)

70, 10 20, 20

q We cannot find any that is not a PSNE.

q We only have one NE that is a PSNE [(0,1), (0,1)].


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EXERCISE


37

SAILESH

Red (q) Black (1q)

KAVISH R

ed(p)

1, 1 1, 1

Black(1

p)

1, 1 1, 1

q Only NE is: [(0.5, 0.5), (0.5, 0.5)}there is no PSNE.

q Recollect the results we got in the class??

MANISH

Right (q) Left (1q)

NIKHIL R

ight(p)

1, 1 1, 1

Left(1

p)

1, 1 1, 1


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EXERCISE


38

WIFE

C (q) M (1q)

HUSBA

ND C

(p)

30, 20 0, 0

M

(1p)

0, 0 20, 30

q Three NEs:[(1, 0), (1, 0)], [(0, 1), (0, 1)], [(0.6, 0.4), (0.4, 0.6)].


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GENERAL REALIZATIONS

39

q Sometimes, a finite game can have more than one equilibrium.

q A PSNE is a degenerate case of a MSNE, where in the player

chooses one action with probability 1 and all others withprobability 0.

qWe can have three types of games:

q Games in which all equilibria are only PSNE.

q Games in which no equilibrium is a PSNE (e.g. Matching

Pennies),

q Games in which some equilibria are PSNE and some are

not (e.g. Battle of the Sexes).


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EXERCISE SET C

40

q Problem 6: Hawk Dove Game.

q Problem 9: Swimming with sharks.

q Problem 10: Testing for MSNE.

q Problem 13: Defending Territory.


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Do people actually

play mixed

strategies as

predicted by Game

Theory?

Palacious-Huerta, I. (2003),

Professionals Play

Minimax, Review of

Economic Studies, Vol. 70,

pg. 395-415.


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Do people actually play mixed strategies as

predicted by Game Theory?

qAnalyzed data from 1417 Penalty kicks from FIFA games: Spain,England, Italy.

The payoff (success) matrix.

q Given this payoff matrix, identify the kickers and goalies

probabilities of going left and right, in a Nash Equilibrium.

Is this Nash Equilibrium actually observed in practice?


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Do people actually play mixed strategies as

predicted by Game Theory?

This was observed in the data!

qPeople dont use payoff matrices to calculate

equilibrium values, before deciding how to act.

qYet, in this study, Kicker and Goalie aggregatedbehaviour over time is consistent with the Nash

Equilibrium prediction!


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Does randomization happen in the business

and administration?

qPolice checking of drunken driving and motorists

strategies.

qTax raids on companies & companiesstrategies.

qVVIP security.


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Exercise: Swimming with SharksProblem 9 of Exercise Set C

45

Alternatives: Yes (swim on the first day),

No (dont swim on first day)

Friend

You

Yes No

No

Yes

What doesc represent?

cis the ratio of the magnitude of displeasure experienced in a shark attack,

to the magnitude of pleasure experienced in swimming without being

attacked.

What does represent?

is players perceived likelihood of sharksexisting in the water.

TRY THIS OUT ON YOUR OWN. AFTER YOU HAVE ATTEMPTED IT

THOROUGHLY, COMPARE YOUR WORK WITH MY EFFORTS HERE.


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ap Prof. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 46

Possible situations that can be modeled by this type of game

q You are considering watching the first day first show of a new movie (there

is a certain thrill in doing so and ideally, you wouldnt want to lose that). If

you find the movie good, you will surely watch it a second time. But you

also realize that there is a probability the movie will not be good and

associate a pain cwith a bad experience. Will you watch it first and then

tell your friend, or wait for friends to watch it and then tell you?

q Two firms from the same country are considering investing in a new

country, about whose market there is some uncertainty. Both can benefit

from investing in the first year itself, however there is a possibility of things

going wrong. Thus each might benefit by waiting for the other firm to

invest, watching the result, and then deciding whether to invest in the

second year or not.

q Any situation where being the first mover can sometimes be better andsometimes worse than being a late mover.

Other situations where this can be used?


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CREATIVE EXERCISE

q groups of 4 we will have 15 posters from each

section.

q Form your groups & enter into Google Spreadsheet

(Monday, Oct 20th, 11:59 pm). No change after that.

q Two PLs in each section to plan & execute the

whole project.

qA project of GTAPers of 2013-15 batch.

q demonstrate that Game Theory is useful to

management in organizations.

qIn-class presentation & Poster exhibition.

MORE INSTRUCTIONS BY OCT 23RD


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Iterated Elimination of Strictly Dominated Strategies

Consider the Prisoners Dilemma game PRISONER BQuiet Admit

PRISONERA

Quiet

1, 1 10, 0

Admit

0, 10 5, 5

Instead of examining each action profile and testing to

see if it is a Nash Equilibrium, we can instead proceed

with reasoning as follows:

Player 1 thinks: Is there any strategy that Player 2 will surely not play?

He examines Player 2s payoffs and finds that Player 2s strategy Quiet is

strictly dominated by his strategy Admit. Hence, Player 1 concludes that

Player 2 will never play Quiet. Which means that Player 2 can play only

Admit. If that is so, Player 1 asks what should my response be?

Obviously, Admitsince, for him too,Quietis strictly dominated by Admit.

He then asks: Player 2 will think about what I will do. How will he think?. He

realizes that Player 2 will reason about him (Player 1) similar to the way he

reasoned about Player 2 and conclude that he (Player 1) would playAdmit.


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More generally speaking, each player can think over

q What is the best strategy for me / others as well as what is never a best

strategy for myself / others?

q What the other player(s) think(s) about what I think as the best strategy

for me / themselves as well as what is never a best strategy for me /

themselves?

q What the other player(s) think(s) about what I think about their thinking

about the best strategy for me / themselves as well as what is never a

best strategy for me / themselves? on so on

It becomes clear from this way of thinking that each player caneliminate all strategies of himself / others that can never be a best

strategy since that strategy(-ies) will never be played.


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Consider this game.L C R

U 2, 0 2, 1 0, 0

M 1, 1 1, 1 5, 0

D 0, 1 4, 2 0, 1

Identify all the strictly dominated strategies

and eliminate them one by one.

1. R is strictly dominated by C

Eliminate R and create a smaller game. In the

smaller game, we find,

L C

U 2, 0 2, 1

M 1, 1 1, 1

D 0, 1 4, 2

2. M is strictly dominated by UEliminate M and create a smaller

game. In the smaller game,

L C

U 2, 0 2, 1

D 0, 1 4, 2

3. L is strictly dominated by C

Eliminate L and create a smaller

game. In the smaller game,C

U 2, 1

D 4, 2

C

D 4, 2

4. U is strictly

dominated by D

Eliminate U.


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L C R

U 2, 0 2, 1 0, 0

M 1, 1 1, 1 5, 0

D 0, 1 4, 2 0, 1

By iteration, the procedure led us to a

single strategy profile. Hence, we infer

that this is the unique Nash Equilibrium of

the game.

If there were other Nash Equilibria, the game would not terminate this way,

in a single strategy profile.


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L C R

U 3, 1 0, 1 0, 0

M 1, 1 1, 1 5, 0

D 0, 1 4, 1 0, 0

A different example 1. R is strictly dominated by both C and L

Eliminate R.

2. M is strictly dominated by (, 0, ) and others

Eliminate M.

L C

U 3, 1 0, 1

M 1, 1 1, 1

D 0, 1 4, 1L C

U 3, 1 0, 1

D 0, 1 4, 1

This game has infinite Nash

Equilibria. Find them!

But now there is no strategy that is strictly

dominated by another pure strategy. Can we find

a mixed strategy that strictly dominates a pure

strategy?

By trial and error, we find that if Player 1 plays

(, 0, ) or any (p, 0, 1p) such that 1/3 < p < 3/4,

then this mixed strategy strictly dominates M.

Why?

Now we can find no more pure / mixed strategies that

dominate an existing pure strategy. Terminate.


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q Helps us narrow down a complex game to a simpler one. Hence can be

used as a pre-processing step, before using another technique to find Nash

Equilibria.

q The process preserves the games Nash Equilibria, since in every iteration

we eliminate only those strategies that can never be best responses.

q Dominance Solvability: Games that are fully solvable using this technique

are said to be dominance solvable. In such a game, we terminate at a

unique strategy profile after the iterative elimination procedure. Only

some games are dominance solvable.

q Order of Removal: If there are multiple strictly dominated strategies, then

does the order of their removal matter to the solution? No. Verify this

by using a different order for the first game demonstrated.

q Weakly dominated strategies: Will this procedure work if we eliminate

weakly dominated strategies also? IT MAY OR MAY NOT. A Nash

Equilibrium can consist of weakly dominated strategies!.... At least one

equilibrium will be preservedbut the order of removal can matter.


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You are asked to make an investment

in ONEof two ventures, given the

following information:

q If you invest in Venture A, there

is a 5% chance of making a

profit of Rs. 80 crores, and a

95% chance of no profit at all.

q But, if you invest in Venture B,you are fully assured of a profit

of Rs. 8 crores.

Which venture should you invest in

A or B? There is no one correct

answer. Both answers can be correct,

and we are looking at how individualsmake choices.

Think and answer Which venture

will you invest in? Why?

Lottery A: Is for Rs. 10 crore, and 250,000

people will purchase it.

There is no one correctanswer. Each

person can have different reasons for his

choice.

What is your answer?

You have Rs. 100 with you, with which you

have to purchase either of two lotteries.

The only winner of each lottery gets the fullprize but a loser loses even the Rs. 100.

Which lottery will you take?

Lottery B: Is for Rs. 2 crore, and 25,000

people will purchase it.

Experiment done in classroom.


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Experiment done in the classroom

? ?

? ?Venture

Lottery

A B

B

A

In solving for mixed strategies, we used the concept ofexpected value. But do people actually make their

preference on the basis of expected value?


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Expected Value Calculation

For the decision to purchase lottery A

EV(A) = 0.000004*100000000 + 0.999996*(-100) = 400 99.9996

EV(A) = ~300.

Investment

Lottery

For the decision to invest in Venture A

EV(A) = 0.95*0 + 0.05*80 = 4

For the decision to invest in Venture BEV(B) = 1.00*8 = 8

Are peoples preferences consistent with EV?

For the decision to purchase lottery B

EV(B) = 0.00004*20000000 + 0.99996*(-100) = 800 99.996

EV(B) = ~700.

Are peoples preferences consistent across experiments?


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p P f J Ajith K TAPMI M i l GAME THEORY NOTES

How does Expected Value Relate to Preference?

In both experiments, EV(B) > EV(A).

If people are consistent with their choices, those who choseVenture A should choose Lottery A and those who chose Venture

B should choose Lottery B.

? ?

? ?Investment

Lottery

A B

B

A 4, 300 4, 700

8, 300 8, 700Investment

Lottery

A B

B

A

No. of people who played the

different choice combinationsThe payoffs associated with

the different choice

combinations

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GTAP Sessions 11-15 Mixed Strategy

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