© 2005 Pearson Education Canada Inc. 15.1 Chapter 15 Introduction to Game Theory

© 2005 Pearson Education Canada Inc.15.1

Chapter 15Chapter 15

Introduction to Game TheoryIntroduction to Game Theory


Game theory is based on the Game theory is based on the following modelling assumptions:following modelling assumptions:

There are a few producers (players) in the There are a few producers (players) in the industry (game).industry (game).

Each player chooses an output or pricing Each player chooses an output or pricing strategy.strategy.

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

The payoff for each player is dependent The payoff for each player is dependent upon the strategy he/she selects and that upon the strategy he/she selects and that selected by other players.selected by other players.


Game Theory: Basic DefinitionsGame Theory: Basic Definitions

Players-entities like individuals/firms Players-entities like individuals/firms that make choices.that make choices.

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

Strategy combinations-a list of Strategy combinations-a list of strategies for each player.strategies for each player.

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


Game Theory: Basic DefinitionsGame Theory: Basic Definitions Best response function-the player’s Best response function-the player’s

best response given the strategies of best response given the strategies of other players.other players.

Equilibrium strategy combination-a Equilibrium strategy combination-a strategy combination where every strategy combination where every player’s strategy is the best response player’s strategy is the best response to the strategy of all other players.to the strategy of all other players.


Game Theory: Basic DefinitionsGame Theory: Basic Definitions Cournot-Nash equilibrium- An Cournot-Nash equilibrium- An

equilibrium strategy combination equilibrium strategy combination where there is nothing any individual where there is nothing any individual player can independently do that player can independently do that increases that player’s payoff. Each increases that player’s payoff. Each player’s own strategy maximizes that player’s own strategy maximizes that player’s own payoff.player’s own payoff.


Game Theory: Basic DefinitionsGame Theory: Basic Definitions Normal forms-simply represents the Normal forms-simply represents the

outcomes in payoff matrix (connects the outcomes in payoff matrix (connects the outcomes in an obvious way).outcomes in an obvious way).

Extensive form description-a game tree. Extensive form description-a game tree. Each decision point (node) has a number Each decision point (node) has a number of branches stemming from it; each one of branches stemming from it; each one indicating a specific decision. At the end indicating a specific decision. At the end of the branch there is another node or a of the branch there is another node or a payoff. payoff.


Game Theory: An exampleGame Theory: An example

A strategy better than all others, A strategy better than all others, regardless of the actions of others, is regardless of the actions of others, is a a dominant strategydominant strategy..

If one strategy is worse than another If one strategy is worse than another for some player, regardless of the for some player, regardless of the actions of other players, it is a actions of other players, it is a dominated strategydominated strategy..


Figure 15.1 A movement gameFigure 15.1 A movement game


From Figure 15.1From Figure 15.1

For player 2, the strategy For player 2, the strategy MiddleMiddle is is dominated by the strategy dominated by the strategy Right.Right.

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

Therefore, Figure 15.1 becomes Therefore, Figure 15.1 becomes Figure 15.2.Figure 15.2.


Figure 15.2 Game with dominated strategy awardFigure 15.2 Game with dominated strategy award



For player 1, the For player 1, the Up Up strategystrategy dominates both Middle and Down.dominates both Middle and Down.

For player 1, Up For player 1, Up is therefore a is therefore a dominant strategy.dominant strategy.

The The Middle and Down Middle and Down rows can be rows can be eliminated from player 1’s strategy.eliminated from player 1’s strategy.

This leaves the game shown in Figure This leaves the game shown in Figure 15.3.15.3.


Figure 15.3 Game with last dominated strategyFigure 15.3 Game with last dominated strategy



Player 1 has no choice but to move Player 1 has no choice but to move Up.Up.

For Player 2, the dominant strategy For Player 2, the dominant strategy is to move Left.is to move Left.

(Up, Left) or (Up, Left) or 4,34,3** is therefore the is therefore the equilibrium payoff.equilibrium payoff.

It is a Nash equilibrium where both It is a Nash equilibrium where both players will settle on a strategy and players will settle on a strategy and not want to move. not want to move.


The Prisoner’s DilemmaThe Prisoner’s Dilemma

Figure 15.4 shows payoffs for the two Figure 15.4 shows payoffs for the two individuals suspected of car theft.individuals suspected of car theft.

The figures represent the jail time in The figures represent the jail time in months for Petra and Ryan.months for Petra and Ryan.

What is the equilibrium outcome of What is the equilibrium outcome of this game?this game?


Figure 15.4 The prisoner’s dilemmaFigure 15.4 The prisoner’s dilemma



An easy way to find equilibrium is to An easy way to find equilibrium is to draw arrows showing the direction of draw arrows showing the direction of strategy preferences for each player.strategy preferences for each player.

Horizontal arrows show preferences Horizontal arrows show preferences of player 2, vertical arrows show of player 2, vertical arrows show preferences for player 1.preferences for player 1.

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


Figure 15.5 Nash equilibrium in the PD gameFigure 15.5 Nash equilibrium in the PD game


From Figure 15.5From Figure 15.5 The arrows meet where both Petra The arrows meet where both Petra

and Ryan fink (and Ryan fink (Fink, Fink) and this is Fink, Fink) and this is the equilibrium for the game.the equilibrium for the game.

Interesting aspects of the prisoner’s Interesting aspects of the prisoner’s dilemma:dilemma:

1.1. There are many real life applications.There are many real life applications.2.2. The equilibrium results form a dominant The equilibrium results form a dominant

strategy for both players.strategy for both players.3.3. The equilibrium outcome is not Pareto-The equilibrium outcome is not Pareto-

Optimal (both would be better off if they Optimal (both would be better off if they both remained silent).both remained silent).


Coordination GamesCoordination Games

Often situations may have no Often situations may have no equilibrium or they may have equilibrium or they may have multiple equilibria.multiple equilibria.

In these situations, other forms of In these situations, other forms of behaviour must arise for a solution to behaviour must arise for a solution to be found.be found.


Coordination Games: An ExampleCoordination Games: An Example

Figure 15.8 shows the payoffs for Figure 15.8 shows the payoffs for various strategies using Microsoft various strategies using Microsoft Word (Dean’s preference) and Word (Dean’s preference) and Corel’s WordPerfect (Richard’s Corel’s WordPerfect (Richard’s favourite).favourite).

The figures represent how much The figures represent how much better/worse each author is under better/worse each author is under the various strategies measured in the various strategies measured in more/less papers written.more/less papers written.


Figure 15.8 Choosing a word processorFigure 15.8 Choosing a word processor



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

Therefore the Nash equilibrium is Therefore the Nash equilibrium is insufficient to identify the actual insufficient to identify the actual outcome.outcome.

There exists There exists a coordination problem a coordination problem when the players must decide on when the players must decide on what equilibrium to settle on.what equilibrium to settle on.


How Do the Players Decide a How Do the Players Decide a Strategy in Coordination Games?Strategy in Coordination Games?

There is no definitive method of There is no definitive method of solving coordination games, actual solving coordination games, actual outcomes often depend upon: laws, outcomes often depend upon: laws, social customs or pre-emptive moves social customs or pre-emptive moves by players before the game.by players before the game.

In some cases there simply is no In some cases there simply is no equilibrium.equilibrium.


Games of Plain Substitutes and Games of Plain Substitutes and Plain ComplementsPlain Complements

Games in which each player’s payoff Games in which each player’s payoff diminishes as the values of the other diminishes as the values of the other player’s strategy increases are player’s strategy increases are known as known as games of plain substitutes.games of plain substitutes.

In games ofIn games of plain substitutes, plain substitutes, the the players impose negative externalities players impose negative externalities on each other.on each other.


Games of Plain Substitutes and Games of Plain Substitutes and Plain ComplementsPlain Complements

Games in which each player’s payoff Games in which each player’s payoff increases as the values of the other increases as the values of the other player’s strategy increases are player’s strategy increases are known as known as games of plain games of plain complements.complements.

In games ofIn games of plain compliments, plain compliments, the the players impose positive externalities players impose positive externalities on each other.on each other.


Games of Plain Substitutes with Games of Plain Substitutes with Simultaneous MovesSimultaneous Moves

The cross-effects in the payoff The cross-effects in the payoff functions are negative.functions are negative.

There exists mutual negative There exists mutual negative externalities.externalities.

yy110 0 and y and y22

0 0 are the Nash equilibrium are the Nash equilibrium values of the strategies.values of the strategies.

From the Nash equilibrium, yFrom the Nash equilibrium, y110 0 is a is a

best response to ybest response to y2200


Games of Plain Substitutes with Games of Plain Substitutes with Simultaneous Moves (continued)Simultaneous Moves (continued)

YY110 0 solves the constrained maximization problem:solves the constrained maximization problem:

Maximize by choice of yMaximize by choice of y11 and yand y22

пп11 (y (y11, y, y22) < y) < y22 = y = y2200

Indifference curve пIndifference curve п11 (y (y11, y, y22) is tangent to the constraint at ) is tangent to the constraint at the Nash equilibrium (ythe Nash equilibrium (y11

00, y, y2200) in Figure 15.14.) in Figure 15.14.

Because Because пп11 (y (y11, y, y22) decreases as y) decreases as y2 2 increases, this increases, this indifference curve must lie below the line yindifference curve must lie below the line y22 = y = y22

0 0 elsewhere. elsewhere.


Games of Plain Substitutes with Games of Plain Substitutes with Simultaneous MovesSimultaneous Moves

For the same reason, the set of strategy For the same reason, the set of strategy combinations that One prefers to the Nash combinations that One prefers to the Nash equilibrium lies below this indifference curve, as equilibrium lies below this indifference curve, as indicated by the downward–pointing arrows in indicated by the downward–pointing arrows in the figure.the figure.

For Two’s indifference curve through the Nash For Two’s indifference curve through the Nash equilibrium. It must be tangent to the line equilibrium. It must be tangent to the line yy1 1 = = yy1 1 at (y at (y11

00, y, y2200). Elsewhere it must lie to the left ). Elsewhere it must lie to the left

of the line yof the line y11 = y = y110 0 and the set of strategy and the set of strategy

combinations. combinations. Two’s preferences to the Nash equilibrium lie to Two’s preferences to the Nash equilibrium lie to

the left of this indifference curve.the left of this indifference curve.


Figure 15.14 Nash equilibrium for a game of Figure 15.14 Nash equilibrium for a game of plain substitutesplain substitutes



All strategy combinations in the All strategy combinations in the Lense of Missed OpportunityLense of Missed Opportunity are are preferred by both players to the Nash preferred by both players to the Nash equilibrium.equilibrium.

When players impose mutual When players impose mutual negative externalities on one negative externalities on one another, they produce too much and another, they produce too much and would be better off cutting back on would be better off cutting back on their strategy values. their strategy values.


Mixed Strategies and Games of Mixed Strategies and Games of DiscoordinationDiscoordination

Possible Possible OutcomesOutcomes

Claire’sClaire’s

PayoffPayoffProbability of Probability of

Each OutcomeEach OutcomeZak’s Payoff Zak’s Payoff

for Each for Each OutcomeOutcome

(A,A)(A,A) 11 pqpq 00

(A,B)(A,B) 00 p(1- q)p(1- q) 11

(B,A)(B,A) 00 (1- p)q(1- p)q 11

(B,B)(B,B) 11 (1- p)(1- q)(1- p)(1- q) 00


Mixed Strategies and Games of DiscoordinationMixed Strategies and Games of Discoordination

Claire’s payoff is the probability Claire’s payoff is the probability weighted average of the payoffs weighted average of the payoffs associated with each outcome: associated with each outcome: ΠΠ11(p,q)=1(p,q) (p,q)=1(p,q) +0(p(1-q))+0((1-p)q) +1((1-p)(1-q))+0(p(1-q))+0((1-p)q) +1((1-p)(1-q))

Claire’s payoff is a linear function of her Claire’s payoff is a linear function of her strategy, p: strategy, p: ΠΠ11(p,q)=(1-q)+p(2q-1)(p,q)=(1-q)+p(2q-1)

Zak’s payoff is a linear function of his Zak’s payoff is a linear function of his strategy, q: strategy, q: ΠΠ2 2 (p,q)= p+q(1-2p)(p,q)= p+q(1-2p)


Mixed Strategies and Mixed Strategies and Games of DiscoordinationGames of Discoordination

Claire’s best response function:Claire’s best response function:1.1. Her payoff increases as P increases if 2q-Her payoff increases as P increases if 2q-

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

2.2. Her payoff decreases as p increases if Her payoff decreases as p increases if 2q - 1<0, or if q<1/2 and p=o is her best 2q - 1<0, or if q<1/2 and p=o is her best response.response.

3.3. Her payoff doesn’t change as p increases Her payoff doesn’t change as p increases if 2q - 1=0, or of q=1/2, and any value if 2q - 1=0, or of q=1/2, and any value of p is her best response.of p is her best response.



Zak’s best response functions:Zak’s best response functions:1.1. q=0 is his best response if (1 - 2p)<0, or q=0 is his best response if (1 - 2p)<0, or

if p > 1/2.if p > 1/2.

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

3.3. Any q in the interval [0,1] is best Any q in the interval [0,1] is best response if p = 1/2response if p = 1/2



To find the Nash equilibrium, plot the To find the Nash equilibrium, plot the best response functions and find best response functions and find where they intersect.where they intersect.

Nash equilibrium is pNash equilibrium is p00 =1/2 and =1/2 and

qq00 = 1/2 (see Figure 15.21). = 1/2 (see Figure 15.21).


Figure 15.21 Mixed strategy Nash equilibriumFigure 15.21 Mixed strategy Nash equilibrium

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© 2005 Pearson Education Canada Inc. 15.1 Chapter 15 Introduction to Game Theory