Game Theory and Strategic Behaviour

8/3/2019 Game Theory and Strategic Behaviour

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Game Theory and Strategic

BehaviourVani Borooah

University of Ulster


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Confess Deny

Confess -5,-5 0,-10

Deny -10,0 -1,-1PrisonerX

Prisoner Y


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Elements of a Game:

Players: agents participating in the game (X and Y))

Strategy Set: Actions that each player may take under anypossible circumstance (Confess, Deny)

Strategy: An action that a player takes

Outcomes: The various possible results of the game (four,each represented by one cell of matrix)

Payoffs: The cost/benefit that each player gets from eachpossible outcome of the game (the prison sentences enteredin each cell of the matrix)


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Points to note

1. The Payoff to a player depends not juston his chosen action but also on the actionchosen by the other player

So, there is inter-dependence2. Each player chooses his action, without

knowing the others choice

It is a non-co-operative game

3. It is played once: it is a one-shot game


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A Solution to a Game

Each player chooses a strategy that gives himthe highest payoff, given the choices of others

If no player wishes to change his strategy, given

the choice of the others, the game has a solutionand the players arrive at an equilibrium

The solution is the set of strategies chosen bythe players

Question is: will a game have a solution?


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Dominant Strategy

A dominating strategy is one which gives theplayers his highest payoff, regardless of theother players choice of action

In the PD game, confess is the dominating

strategy for X:1. If Y denies, X is better off confessing (0 years)2. If Y confesses, X is better off confessing (5

years) Confess is also the dominating strategy for Y

Solution to the game is for both to confessand spend 5 years in jail


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Collective versus self-interest

For both to confess (5 years in jail foreach) is not the optimal strategy

The optimal strategy (1 year in jail for

each) is for both to deny But the optimal strategy is inaccessible,

because each cannot be sure what theother will choose: there is an absence oftrust!


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Build Dont Build

Build 16, 16 20, 15

Dont Build 15, 20 18, 18GeneralMotors

Ford


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Solution to the Ford-GM Game

The strategy build dominates the strategydont build

But the optimal strategy is dont build

Think of India and Pakistan building a nuclearbomb

Same conclusion: build a bomb is thedominant strategy, but dont build is the beststrategy

But absence of trust means that best strategyis not chosen


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Dominated Strategy

The opposite of a dominant strategy is adominated strategy

A dominated strategy will always give a

player a lower payoff than the alternativestrategies, regardless of the strategychoice of the other players

A rational player will never play adominated strategy


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In the game alongside, Hondadoes not have a dominantstrategy: if Toyota builds, it isbetter for Honda to not build; if

Toyta does not build, it is betterfor Honda to build

But Toyata has a dominantstrategy: whether Honda buildsor not, it is better for Toyota tobuild

So, Toyota will build and Hondawill not build: SOLUTION!

1. Both players do not need adominant strategy. If one playerhas a dominant strategy, itsother (dominated) strategy canbe eliminated

2. Game theory teaches us to putourselves in the mind of ourrival Honda argues thatToyota will build and so Hondashould not build

18,515,6

20,312,4

Toyota

Build

Builld Do not Build

Donotbuild

Honda

Elimination of Dominated strategies


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18,1815,209,18

20,1516,168,12

18,912,80,0

Build large Build small Do not build

Build large

Build small

Do not build

Toyota

Honda

Neither Toyota nor Honda has a dominant strategy

But, for Toyota, build large is dominated by other strategies:Toyota will never choose build large

For Honda, build large is dominated by other strategies:Honda will never choose build large

So build large can be eliminated for both

Elimination of Dominated Strategies


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In the game alongside, Hondadoes have a dominant strategy:if Toyota builds small, it isbetter for Honda to build small;

if Toyota does not build, it isbetter for Honda to build small

Toyota has a dominantstrategy: whether Honda buildssmall or does not build, it isbetter for Toyota to build small

So, Toyota and Honda will bothbuild small: SOLUTION!

1. Both players do not need adominant strategy to start with.

2. However, after eliminating theirdominated strategies, they do

have dominant strategies

18,1815,20

20,1516,16

Toyota

Build

Small Build Small Do not Build

Donotbuild

Honda

Elimination of Dominated strategies


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Obtaining a Solution to a Game

If both players have a dominantstrategy, theywill choose it: Prisoners Dilemma

If one player has a dominantstrategy, he will

choose it, the other players strategy will be hisbest response to this (Toyota-Honda)

If neither player has a dominant strategy,successively eliminate each players dominatedstrategies to simplify the game and look fordominantstrategies


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Nash Equilibrium

A dominant strategy for a player has to be hisbest strategy for every choice of strategy by therival

This is a very demanding requirement and very

often it will not be met, even after eliminatingdominated strategies A weaker requirement is that a players choice is

best for the best choice of his rival A pair of strategies is a Nash equilibrium if

player 1s choice is optimal, given 2s optimalchoice and player 2s choice is optimal, given 1soptimal choice


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Finding a Nash Equilibrium

If a dominant cant be found, identify eachplayers best strategy, given the strategyof the other player

The combination of best strategies is theNash equilibrium


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5,44,1112,14

15,20,03,10

1,313,64,2

Player 2

Strategy D Strategy E Strategy F

Strategy A

Strategy B

Strategy C

Player1

Neither player has a dominating nor a dominated strategy

For each player we identify his best strategy, given the otherplayers strategy: red for player 1, green for player 2

Two equilibrium points: (A,E) or (C,D)


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1,20,0

0,02,1

Player 2

Left Right

Top

BottomPlayer1



Two equilibrium points: (Top, Left) or (Bottom, Right)


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There are two firms: y1 and y2 are their respectiveoutputsY=y1+y2 is industry output: price (p) depends on YThe payoffs are firm profits:1=p(Y)y1-C(y1)=p(y1+y2)y1 - C(y1) = 1(y1, y2)2=p(Y)y2-C(y2)=p(y1+y2)y2 - C(y2) = 2(y1, y2)

The payoffs depend upon own output and output of rival

Firm 1 chooses y1 to maximise 1, given y2: y1*= R1(y2)

Firm 2 chooses y2 to maximise 2, given y1 :y2*= R2(y1)

Cournot Duopoly


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Cournot-Nash Equilibrium

Two Firms The dashed arrowsrepresent the pathtowards the Cournot-Nash equilibrium

At the Cournot-Nashequilibrium, no firmhas an incentive tochange

Firm

2sOutpu

t:y2

Firm 1s Output: y1

Firm 1s reaction function:

y1*

=R1(y2)

Firm 2s reaction function:

y2*

=R2(y1)

Nash Equilibrium


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Betrand Duopoly

In Cournot-Nash firms compete on output

In a Betrand model, firms compete onprice

There are two firms, producing ahomogenous product with marginal cost(MC)

They price this at p1 and p2, respectivelyD(p) is the aggregate demand curve


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P2=40

If p2=40, firm 1 will sellnothing if p1>p2 and willcapture the market if p1X, it willundercut

Then, firm 2 will undercut 1,by reducing price below$39

Process continues untilp1=p2=MC

Unprofitable to lower pricebelow MC

A

B C

E

30 60

P1=39X

Y MC

P=MC

Marginal Cost Pricing Under Betrand Equilibrium


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Reaction Functions Under Betrand Duopoly with Homogenous Product

Price charged

by firm 1

Price charged by firm 245 line

p2*

Reaction function of firm 1

Reaction function of firm 2

p1*0 40

40

MarginalCost


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Betrand Duopoly and CompetitiveEquilibrium

1. Firms price at marginal cost

2. Firms make zero profits

3. The number of firms is irrelevant to the price levelas long as more than one firm is present: two firmsare enough to replicate the perfectly competitiveoutcome!

essentially, the assumption of no capacity constraintscombined with a constant average and marginal costtakes the place of free entry


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Betrand Duopoly with Differentiated Products

There are two products: Hondas

Jazz and Renualts Clio. For eachproduct, demand depends on ownprice and rivals price:

DClio=f(pC, pJ) and DJazz=g(pC, pJ)

For each price of Jazz, Clio willhave a profit maximising price, pC

We can compute, for every Jazzprice pJ, the corresponding profitmaximising Clio price, pC: this is theClio reaction function

We can compute, for every Clioprice p

C

, the corresponding profitmaximising Jazz price, pJ: this is theJazz reaction function

Betrand equilibrium is atintersection of reaction functions

Demand Curve for Clioswhen pJ=10,000

MR MC

pC

Quantity Clios

Clio reaction function

Jazz reactionfunction

Betrandequilibrium

Price Clio

PriceJazz


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-1,31,0

0,-10,0

Player 2

Left Right

Top

BottomPlayer1



There is no Nash equilibrium


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Mixed Strategies

Suppose A plays top with probability and bottom with probability

Suppose B plays left and right with

probabilities Then p(top, left)=p(top,right)=3/8

And p(bottom, left)=p(bottom,right)=1/8

This yields a Nash equilibrium

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Game Theory and Strategic Behaviour