Eaton Micro 6e Ch15

7/27/2019 Eaton Micro 6e Ch15

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2005 Pearson Education Canada Inc.15.1

Chapter 15

Introduction to Game Theory


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Game theory is based on the

following modelling assumptions:

There are a few producers (players) in theindustry (game).

Each player chooses an output or pricingstrategy.

Each strategy produces a result (payoff)for that player.

The payoff for each player is dependentupon the strategy he/she selects and thatselected by other players.


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Game Theory: Basic Definitions

Players-entities like individuals/firmsthat make choices.

Strategies-the choices made by theplayers (output/pricing, etc.).

Strategy combinations-a list ofstrategies for each player.

Payoff-the outcome (utility, profit,etc.) from selecting a strategy.


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Best response function-the playersbest response given the strategies ofother players.

Equilibrium strategy combination-astrategy combination where everyplayers strategy is the best response

to the strategy of all other players.


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Cournot-Nash equilibrium- Anequilibrium strategy combinationwhere there is nothing any individual

player can independently do thatincreases that players payoff. Eachplayers own strategy maximizes thatplayers own payoff.


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Normal forms-simply represents theoutcomes in payoff matrix (connects theoutcomes in an obvious way).

Extensive form description-a game tree.Each decision point (node) has a numberof branches stemming from it; each one

indicating a specific decision. At the endof the branch there is another node or apayoff.


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Game Theory: An example

A strategy better than all others,regardless of the actions of others, isa dominant strategy.

If one strategy is worse than anotherfor some player, regardless of theactions of other players, it is a

dominated strategy.


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Figure 15.1 A movement game


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From Figure 15.1

For player 2, the strategy Middle isdominated by the strategy Right.

When you find a dominated strategy,it can be eliminated from the game.

Therefore, Figure 15.1 becomesFigure 15.2.


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Figure 15.2 Game with dominated strategy award


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From Figure 15.2

For player 1, the Up strategydominates both Middle and Down.

For player 1, Up is therefore adominant strategy.

The Middle and Down rows can beeliminated from player 1s strategy.

This leaves the game shown inFigure 15.3.


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Figure 15.3 Game with last dominated strategy


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From Figure 15.3

Player 1 has no choice but to moveUp.

For Player 2, the dominant strategy

is to move Left. (Up, Left) or 4,3* is therefore the

equilibrium payoff.

It is a Nash equilibrium where bothplayers will settle on a strategy andnot want to move.


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The Prisoners Dilemma

Figure 15.4 shows payoffs for thetwo individuals suspected of cartheft.

The figures represent the jail time inmonths for Petra and Ryan.

What is the equilibrium outcome ofthis game?


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Figure 15.4 The prisoners dilemma


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From Figure 15.4

An easy way to find equilibrium is todraw arrows showing the direction ofstrategy preferences for each player.

Horizontal arrows show preferencesof player 2, vertical arrows showpreferences for player 1.

Where the two arrows meet, there isa Nash equilibrium (see Figure 15.5).


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Figure 15.5 Nash equilibrium in the PD game


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From Figure 15.5

The arrows meet where both Petraand Ryan fink (Fink, Fink) and thisis the equilibrium for the game.

Interesting aspects of the prisoners

dilemma:1. There are many real life applications.2. The equilibrium results form a dominant

strategy for both players.

3. The equilibrium outcome is not Pareto-Optimal (both would be better off if theyboth remained silent).


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Coordination Games

Often situations may have noequilibrium or they may havemultiple equilibria.

In these situations, other forms ofbehaviour must arise for a solutionto be found.


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Coordination Games: An Example

Figure 15.8 shows the payoffs forvarious strategies using MicrosoftWord (Deans preference) and

Corels WordPerfect (Richardsfavourite).

The figures represent how muchbetter/worse each author is underthe various strategies measured inmore/less papers written.


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Figure 15.8 Choosing a word processor


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From Figure 15.8

As indicated by the arrows, there aretwo equilibria in this game.

Therefore the Nash equilibrium isinsufficient to identify the actualoutcome.

There exists a coordination problemwhen the players must decide onwhat equilibrium to settle on.


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How Do the Players Decide a

Strategy in Coordination Games?

There is no definitive method ofsolving coordination games, actual

outcomes often depend upon: laws,social customs or pre-emptive movesby players before the game.

In some cases there simply is noequilibrium.


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Games of Plain Substitutes and

Plain Complements

Games in which each players payoffdiminishes as the values of the other

players strategy increases areknown as games of plain substitutes.

In games ofplain substitutes, the

players impose negative externalitieson each other.


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Games of Plain Substitutes and

Plain Complements

Games in which each players payoffincreases as the values of the other

players strategy increases areknown as games of plaincomplements.

In games ofplain compliments, theplayers impose positive externalitieson each other.


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Games of Plain Substitutes with

Simultaneous Moves

The cross-effects in the payofffunctions are negative.

There exists mutual negativeexternalities.

y10 and y2

0 are the Nash equilibriumvalues of the strategies.

From the Nash equilibrium, y10 is a

best response to y20


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Simultaneous Moves (continued)

Y10 solves the constrained maximization

problem:Maximize by choice of y1

and y2

1 (y1, y2) < y2 = y20

Indifference curve 1 (y1, y2) is tangent to theconstraint at the Nash equilibrium (y1

0, y20) in

Figure 15.14.

Because 1 (y1, y2) decreases as y2 increases,this indifference curve must lie below the liney2 = y2

0 elsewhere.


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Simultaneous Moves For the same reason, the set of strategy

combinations that One prefers to the Nashequilibrium lies below this indifference curve,as indicated by the downwardpointing arrows

in the figure. For Twos indifference curve through the Nash

equilibrium. It must be tangent to the line y1 =y1 at (y1

0, y20). Elsewhere it must lie to the left

of the line y1 = y10 and the set of strategy

combinations. Twos preferences to the Nash equilibrium lie to

the left of this indifference curve.


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Figure 15.14 Nash equilibrium for a game of

plain substitutes


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From Figure 15.14

All strategy combinations in theLense of Missed Opportunityarepreferred by both players to the

Nash equilibrium.When players impose mutual

negative externalities on oneanother, they produce too much andwould be better off cutting back ontheir strategy values.


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Mixed Strategies and Games of

DiscoordinationPossible

OutcomesClaires

Payoff

Probability ofEach Outcome

Zaks Payofffor EachOutcome

(A,A) 1 pq 0

(A,B) 0 p(1- q) 1

(B,A) 0 (1- p)q 1

(B,B) 1 (1- p)(1- q) 0


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Mixed Strategies and Games of Discoordination

Claires payoff is the probabilityweighted average of the payoffsassociated with each outcome:1(p,q)=1(p,q) +0(p(1-q))+0((1-p)q) +1((1-p)(1-q))

Claires payoff is a linear function of herstrategy, p: 1(p,q)=(1-q)+p(2q-1)

Zaks payoff is a linear function of hisstrategy, q: 2 (p,q)= p+q(1-2p)


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Mixed Strategies and

Games of Discoordination

Claires best response function:1. Her payoff increases as P increases if 2q-

1>0, or if q>1/2 and p=1 is her best

response.2. Her payoff decreases as p increases if

2q - 1


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Zaks best response functions:

1. q=0 is his best response if (1 - 2p) 1/2.

2. q=1 is his best response if (1 - 2p)>0,or if p < 1/2.

3. Any q in the interval [0,1] is best

response if p = 1/2


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To find the Nash equilibrium, plot thebest response functions and findwhere they intersect.

Nash equilibrium is p0 =1/2 and

q0 = 1/2 (see Figure 15.21).


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2005 Pearson Education Canada Inc15 36

Figure 15.21 Mixed strategy Nash equilibrium

Eaton Micro 6e Ch15

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