As Ks Gametheory

8/13/2019 As Ks Gametheory

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A. S. Shaja & K. SudhakarAerospace Engg. Dept,I.I.T. Bombay,Powai, Mumbai [email protected]

Game Theory

1

t=Dt t=0

t=0

t=Dt


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Prisoners Dilemma

Two suspects are arrested for armed robbery.

If both silent, 1 year in prison

If both confess, 5 years in prison

If one confess and the other does not, the one whoconfess gets 0 years and the other gets 20 years


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Prisoners Dilemma


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Movie & Cricket

A couple deciding how to spend the eveningWife would like to go for a movie

Husband would like to go for a cricket match

Both however want to spend the time together


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What is a Game?

Agameis a formal representation of a situation in which a numberof individuals interact in setting of strategic interdependence

Welfare of each individual depends not only on own actions, but alsoon the actions of other individuals

Actions that are optimal for an individual may depend on expectedactions of other players best response

Description of a strategic interaction requires four components:

The players who are involved in playing the game

The rules available actions when moving The outcomes combinations of actions

The payoffs preferences (or gain) of players over the possibleoutcomes


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Example of a Game Matching Pennies

The Players

Two players that we will denote by 1 and 2

The Rules

Each player simultaneously puts a penny down, eitherheads up (H) or tails up (T)

The Outcomes

(H,H), (H,T), (T,H), (T,T)

The Payoffs Payoff is simply amount of money player profits

(-1,1), (1,-1)


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Normal Form Representation

Set of n players

Sets of strategies for each player

Payoffs displayed as a matrix

Player

strategy

H T

H (-1,1) (1,-1)

T (1,-1) (-1,1)

Zero Sum Game


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Prisoners Dilemma


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Prisoners Dilemma

Player

strategy

Confess Remain

Silent

Confess 5,5 0,20

Remain Silent 20,0 1,1

Prisoner1

Prisoner2


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Movie & Cricket

A couple deciding how to spend the eveningWife would like to go for a movie

Husband would like to go for a cricket match

Both however want to spend the time together


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Movie & Cricket

Player

strategy

Movie Cricket

Movie 2,1 0,0

Cricket 0,0 1,2

Wife

Husband


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Equilibirium

Definition of a equilibrium:

A situation where no player can improve his payoff byunilaterally changing his strategy

Pure strategies : Players only play a single strategy inequilibrium


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Game Theory

Advertise Don't Advertise

Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (3, 2)

Firm B

Firm A

Advertising Example


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Firm B

Firm A

What is the optimal strategy for Firm A if Firm Bchooses to advertise?

If Firm A chooses to advertise, the payoff is 4.

Otherwise, the payoff is 2. The optimal strategy is toadvertise.

Game Theory


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Firm B

Firm A

What is the optimal strategy for Firm A if Firm Bchooses not to advertise?


Otherwise, the payoff is 3. Again, the optimal strategyis to advertise.

Game Theory


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Firm B

Firm A

Regardless of what Firm B decides to do, the optimalstrategy for Firm A is to advertise. The dominantstrategy for Firm A is to advertise.

Game Theory


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Firm B

Firm A

What is the optimal strategy for Firm B if Firm Achooses to advertise?

If Firm B chooses to advertise, the payoff is 3.


Game Theory


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Game Theory



Firm B

Firm A

What is the optimal strategy for Firm B if Firm Achooses not to advertise?




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Game Theory



Firm B

Firm A

Regardless of what Firm A decides to do, the optimalstrategy for Firm B is to advertise. The dominantstrategy for Firm B is to advertise.


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Game Theory



Firm B

Firm A

The dominant strategy for Firm A is to advertise andthe dominant strategy for Firm B is to advertise. Theequilibrium is for both firms to advertise.


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MinMax Theorem

For every two-person, zero-sumgame with finitely many strategies,there exists a value V and a mixed strategy for each player, suchthat(a) Given player 2's strategy, the best payoff possible for player 1 is

V, and

(b) Given player 1's strategy, the best payoff possible for player 2 isV. Equivalently, Player 1's strategy guarantees him a payoff of Vregardless of Player 2's strategy, and similarly Player 2 canguarantee himself a payoff of V.

(c) The name minimax arises because each player minimizes themaximum payoff possible for the othersince the game is zero-

sum, he also minimizes his own maximum loss (i.e. maximize hisminimum payoff).


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Firm B

Firm A

Game Theory

A Second Advertising Example


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Game Theory

What is the optimal strategy for Firm A if Firm Bchooses to advertise?





Firm B

Firm A


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Game Theory

What is the optimal strategy for Firm A if Firm Bchooses not to advertise?


Otherwise, the payoff is 6. In this case, the optimalstrategy is not to advertise.



Firm B

Firm A


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Firm B

Firm A

Game Theory

The optimal strategy for Firm A depends on whichstrategy is chosen by Firms B. Firm A does not have adominant strategy.


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Game Theory

What is the optimal strategy for Firm B if Firm Achooses to advertise?





Firm B

Firm A


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Game Theory

What is the optimal strategy for Firm B if Firm Achooses not to advertise?





Firm B

Firm A


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Game Theory

Regardless of what Firm A decides to do, the optimalstrategy for Firm B is to advertise. The dominantstrategy for Firm B is to advertise.



Firm B

Firm A


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Game Theory



Firm B

Firm A

The dominant strategy for Firm B is to advertise. IfFirm B chooses to advertise, then the optimal strategyfor Firm A is to advertise. The Nash equilibrium is forboth firms to advertise.


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Prisoners Dilemma

Confess Don't Confess

Confess (5, 5) (0, 10)

Don't Confess (10, 0) (1, 1)

Individual B

Individual A


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Prisoners Dilemma

The prisoners dilemma is the historical game thatfounded game theory as a specific area of study: This is because the solution to this game is sub-optimal

from the point of view of the players. This means that

there is a solution that makes both players better off,but the rationality of the agents does not lead to it.

The prisoners dilemma shows quite elegantly how

difficult it is to get agents to cooperate, even whenthis cooperation is beneficial to all agents.


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PayoffMatrix

Firm B

Qd Qc

Firm A

Qd2

2

0

15

Qc150

1010

What is the Nash Equilibrium for this game?


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PayoffatrixFirm B

Qd

Qc

Firm A

Qd2

2

0

15

Qc150

1010

Is the Qd-Qdequilibrium a Nashequilibrium ?Can firm A or B improvetheir outcome byshifting aloneto thecartel quantity Qc?

Qd-Qd is indeed aNash equilibrium

What is the Nash Equilibrium?


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Mixed Strategy

A pure-strategy Nash equilibrium does not existfor all games

Example of a penalty shoot-out:

2 players:a goal-keeper and a striker 2 strategies :shoot / dive to the left or the right We assume that the players are talented: The

striker never misses and the goalkeeper alwaysintercepts if they choose the correct side.

This is not required for the game, but it simplifiesthings a bit!

What is the payoff matrix?


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Mixed Strategy

For the striker:R if the keeper goes LL if the keeper goes R

ayoffatrix GoalkeeperL R

Striker

L1

0

0

1

R0

1

1

0

For the goalkeeper:

L if the striker shoots LR if the striker shoots R

No pure-strategy Nash equilibrium !

Whatever the outcome,one of the players can

increase his sucess bychanging strategy


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Mixed Strategy

Payoffatrix GoalkeeperL R

Striker

L1

0

0

1

R0

1

1

0

There is, however, amixedstrategy equilibrium

Strategy for both players:

Go L and R 50% of the time (1out of two, randomly)

That way :o Each outcome has a

probability of 0.25o The striker scoresone out of two, theother is stopped bythe goalkeeper


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Mixed Strategy

Lets check that this is actually a Nash equilibrium:

The goalkeeper plays L and R 50% of the time. Can thestriker increase his score by changing his strategy?

The striker decides to play 60% left and 40% right. His new

success rate is:

(0.60.5) + (0.40.5) = 0.5(0.3) + (0.2) = 0.5

By choosing 60-40, the striker scores more on the left handside, but less on the right. His success rate is the same, hissituation has not improved. This corresponds to a Nashequilibrium !


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Nash Equilibirium

Definition of a equilibrium:

A situation where no player can improve his payoff byunilaterally changing his strategy

Central properties: Every game has at least one Nash equilibrium:

Either in pure strategies : Players only play a single strategyin equilibrium

Or in mixed strategies: Players play a combination of several

strategies with a fixed probability Theproofof this result is the main contribution of John Nash

(and the reason why it is called a Nashequilibrium)


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Types of games

Cooperative / Non-cooperative

Simultaneous / Sequential

Perfect / Imperfect information

Finite / Infinitely long games

Repeated games


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As Ks Gametheory

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