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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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Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (3, 2)
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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Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (3, 2)
Firm B
Firm A
What is the optimal strategy for Firm A if Firm Bchooses not to advertise?
If Firm A chooses to advertise, the payoff is 5.
Otherwise, the payoff is 3. Again, the optimal strategyis to advertise.
Game Theory
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Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (3, 2)
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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Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (3, 2)
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.
Otherwise, the payoff is 1. The optimal strategy is toadvertise.
Game Theory
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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
What is the optimal strategy for Firm B if Firm Achooses not to advertise?
If Firm B chooses to advertise, the payoff is 5.
Otherwise, the payoff is 2. Again, the optimal strategyis to advertise.
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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
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
Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (3, 2)
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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Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (6, 2)
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?
If Firm A chooses to advertise, the payoff is 4.
Otherwise, the payoff is 2. The optimal strategy is toadvertise.
Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (6, 2)
Firm B
Firm A
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Game Theory
What is the optimal strategy for Firm A if Firm Bchooses not to advertise?
If Firm A chooses to advertise, the payoff is 5.
Otherwise, the payoff is 6. In this case, the optimalstrategy is not to advertise.
Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (6, 2)
Firm B
Firm A
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Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (6, 2)
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?
If Firm B chooses to advertise, the payoff is 3.
Otherwise, the payoff is 1. The optimal strategy is toadvertise.
Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (6, 2)
Firm B
Firm A
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Game Theory
What is the optimal strategy for Firm B if Firm Achooses not to advertise?
If Firm B chooses to advertise, the payoff is 5.
Otherwise, the payoff is 2. Again, the optimal strategyis to advertise.
Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (6, 2)
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.
Advertise Don't Advertise
Advertise (4, 3) (5, 1)Don't Advertise (2, 5) (6, 2)
Firm B
Firm A
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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
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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