© 2009 Pearson Education Canada 15/1 Chapter 15 Introduction to Game Theory.

© 2009 Pearson Education Canada15/1

Chapter 15Chapter 15

Introduction to Game TheoryIntroduction to Game Theory


Game Theory: Basic DefinitionsGame Theory: Basic Definitions

Entities like firms, individuals, or Entities like firms, individuals, or governments that make choices in the governments that make choices in the game are referred to as game are referred to as playersplayers..

The actions players choose – for a firm The actions players choose – for a firm things such as output and/or price; for a things such as output and/or price; for a soldier such things as run, hide or shoot, soldier such things as run, hide or shoot, are called are called strategies.strategies.



The list of strategies a player can choose from is The list of strategies a player can choose from is the player’s the player’s strategy set strategy set (run, hide, shoot).(run, hide, shoot).

A A strategy combinationstrategy combination is just a single is just a single combination of the various player’s strategies. combination of the various player’s strategies. For two players, an example would be: (shoot, For two players, an example would be: (shoot, shoot).shoot).

The set of strategy combinations is a list of all The set of strategy combinations is a list of all possible strategies that can result in a game.possible strategies that can result in a game.



A A payoffpayoff is the outcome of a given is the outcome of a given strategy.strategy.

If each player’s payoff is determined If each player’s payoff is determined not only by their own strategy, but not only by their own strategy, but also by other players’ strategies, the also by other players’ strategies, the game is said to have game is said to have strategic strategic interdependence. interdependence.



A player’s best response to the given A player’s best response to the given strategies of other players is known strategies of other players is known as a as a best response functionbest response function..

If every player’s strategy is a best If every player’s strategy is a best response to the strategies of all other response to the strategies of all other players, the combination is known as players, the combination is known as an an equilibrium strategy equilibrium strategy combinationcombination. .


Game Theory: Basic DefinitionsGame Theory: Basic Definitions Cournot-Nash equilibrium Cournot-Nash equilibrium - An - 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 Normal forms simply represent the simply represent the

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

Extensive forms Extensive forms are aare a game tree. Each game tree. Each decision point (node) has a number of decision point (node) has a number of branches stemming from it; each one 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.



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 players, regardless of the for some players, regardless of the actions of other players, it is a actions of other players, it is a dominated strategydominated strategy..


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.


Sharing: Game of Put and KeepSharing: Game of Put and Keep

Janet decides to play a game with Janet decides to play a game with her two children Jay and Jill.her two children Jay and Jill.

Each child is to go to his/her room Each child is to go to his/her room where they have no way of where they have no way of communicating with each other.communicating with each other.

Janet gives each child $9 and the Janet gives each child $9 and the option to PUT (P) the money back or option to PUT (P) the money back or KEEP (K) it.KEEP (K) it.



Then Janet collects the two envelopes Then Janet collects the two envelopes and adds $6 whenever she finds $9.and adds $6 whenever she finds $9.

Once this is done, Janet dumps the Once this is done, Janet dumps the contents of the envelopes onto the contents of the envelopes onto the kitchen table and splits the money kitchen table and splits the money evenly between Jay and Jill.evenly between Jay and Jill.

The payoff matrix is shown in Figure The payoff matrix is shown in Figure 15.115.1


Figure 15.1 The payoff matrix for Jill and JayFigure 15.1 The payoff matrix for Jill and Jay


Figure 15.2 The payoff matrix for Jill and Jay Figure 15.2 The payoff matrix for Jill and Jay with preference arrows.with preference arrows.



Each child has a dominant strategy Each child has a dominant strategy to keep the money.to keep the money.

Thus the dominant equilibrium Thus the dominant equilibrium strategy combination is (strategy combination is (K,KK,K) with ) with payoff (9,9).payoff (9,9).

Janet’s plan to get her children to Janet’s plan to get her children to cooperate does not work.cooperate does not work.


Janet’s Second AttemptJanet’s Second Attempt

Like most mothers, Janet is not deterred Like most mothers, Janet is not deterred by one failure.by one failure.

She creates a different game (the same as She creates a different game (the same as the first), but now she tells the children if the first), but now she tells the children if they keep the money, they have to bring it they keep the money, they have to bring it to the table and throw it in with whatever to the table and throw it in with whatever comes out of the envelope.comes out of the envelope.

The payoff matrix is shown in Figure 15.3.The payoff matrix is shown in Figure 15.3.


Figure 15.3 The share everything gameFigure 15.3 The share everything game



It is obvious that both players again have It is obvious that both players again have a dominant strategy, which is to put the a dominant strategy, which is to put the money back in the envelope (PP).money back in the envelope (PP).

This combination is Pareto-optimal This combination is Pareto-optimal because there is no other strategy because there is no other strategy combination that makes at least one combination that makes at least one player better and does not make the player better and does not make the other worse off. other worse off.



Notice that in the first game, the Notice that in the first game, the equilibrium was not Pareto-optimal.equilibrium was not Pareto-optimal.

Had Jay and Jill placed the money back in Had Jay and Jill placed the money back in the envelope, they both would have been the envelope, they both would have been better off by $6.better off by $6.

An equilibrium can exist that is not An equilibrium can exist that is not Pareto-optimal.Pareto-optimal.


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


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?


From Figure 15.4From Figure 15.4

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.

– Note that this is a Note that this is a one shotone shot game with game with no consequences apart from the given no consequences apart from the given payoffs.payoffs.

– The equilibrium results form a The equilibrium results form a dominant strategy for both players.dominant strategy for both players.

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


Nash EquilibriumNash Equilibrium

In the next section on In the next section on coordinated coordinated gamesgames, there is no dominant strategy, , there is no dominant strategy, hence the interaction among players is hence the interaction among players is more meaningful and the more meaningful and the Nash Nash equilibriumequilibrium more useful. more useful.


Nash EquilibriumNash Equilibrium

Strategy combinations (SStrategy combinations (S11**, S, S22

**, … S, … SNN**) is ) is

a Nash equilibrium strategy combination a Nash equilibrium strategy combination if every player’s strategy is a best if every player’s strategy is a best response to the collection of strategies response to the collection of strategies of the remaining players.of the remaining players.

To find the Nash equilibrium, first To find the Nash equilibrium, first compute the best responses of all compute the best responses of all players and intersect them.players and intersect them.


Coordination GamesCoordination Games

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

This is said to be a This is said to be a coordination coordination problem.problem.

A game of coordination has multiple A game of coordination has multiple Nash equilibria and requires some Nash equilibria and requires some mechanism for determining the final mechanism for determining the final outcome.outcome.


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.


The Game of Chicken and the The Game of Chicken and the Word Processor GameWord Processor Game

Perhaps our two academic writers try Perhaps our two academic writers try to solve their dispute over which to solve their dispute over which software to use with a game of software to use with a game of chicken: The person who swerves chicken: The person who swerves first must use the other’s preferred first must use the other’s preferred software. (Figure 15.9).software. (Figure 15.9).

Unfortunately, it is also a game with Unfortunately, it is also a game with two equilibria and requires some two equilibria and requires some outside coordination as well. outside coordination as well.


Nash equilibrium in word Nash equilibrium in word processor gameprocessor game


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 known player’s strategy increases are known as as games of plain substitutesgames 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 known player’s strategy increases are known as as games of plain complementsgames of plain complements..

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


Figure 15.10 Nash equilibrium for a game of Figure 15.10 Nash equilibrium for a game of plain substitutesplain substitutes


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) subject to y) subject to y22 = y = y22**

Indifference curve Indifference curve пп11(y(y11, y, y22) is tangent to the constraint at ) is tangent to the constraint at

the Nash equilibrium (ythe Nash equilibrium (y11**, y, y22

**) in Figure 15.10.) in Figure 15.10.

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

* * elsewhere. elsewhere.



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

Figure 15.10.Figure 15.10.



For Two’s indifference curve through the For Two’s indifference curve through the Nash equilibrium. It must be tangent to Nash equilibrium. It must be tangent to the line the line yy1 1 = y= y1 1 at (y at (y11

**, y, y22**). Elsewhere it ). Elsewhere it

must lie to the left of the line ymust lie to the left of the line y11 = y = y11* * and and

the set of strategy combinations. the set of strategy combinations. Two’s preferences to the Nash Two’s preferences to the Nash

equilibrium lie to the left of this equilibrium lie to the left of this indifference curve.indifference curve.



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

Notice that in the Nash equilibrium, the Notice that in the Nash equilibrium, the players choose strategy values that are in players choose strategy values that are in some sense “too large” and a strategy some sense “too large” and a strategy combination that Pareto-dominates the combination that Pareto-dominates the Nash equilibrium, has smaller values for Nash equilibrium, has smaller values for both strategies. both strategies.


Figure 15.13: Nash equilibrium for Figure 15.13: Nash equilibrium for a game of plain complementsa game of plain complements


From Figure 15.13From Figure 15.13 A game of plain complements is a game of A game of plain complements is a game of

mutual positive externalities.mutual positive externalities. The more one firm produces, the higher is The more one firm produces, the higher is

the price and hence the profit for the other the price and hence the profit for the other firm.firm.

Notice the lens of missed opportunities lies Notice the lens of missed opportunities lies above and to the right of the Nash above and to the right of the Nash equilibrium.equilibrium.

This indicates the players chose strategy This indicates the players chose strategy values that are too small in the Nash values that are too small in the Nash equilibrium. equilibrium.


Mixed Strategies and Games of Mixed Strategies and Games of DiscoordinationDiscoordination

Games with no Nash equilibrium are Games with no Nash equilibrium are known as known as games of discoordinationgames of discoordination..

Mixed strategies are solutions to the Mixed strategies are solutions to the “no-equilibrium-in pure strategies “no-equilibrium-in pure strategies problem.”problem.”

In a In a mixed strategymixed strategy, a player’s , a player’s equilibrium consists of a probability equilibrium consists of a probability distribution over a set of actions.distribution over a set of actions.


Figure 15.15: Payoff matrix for a Figure 15.15: Payoff matrix for a discoordination gamediscoordination game


Figure 15.16 Mixed strategy space Figure 15.16 Mixed strategy space for a discoordination gamefor a discoordination game


Table 15.1: Classification of Table 15.1: Classification of outcomes and payoffsoutcomes and payoffs

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, strategy, p: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, strategy, q: 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 Her payoff increases as PP increases if 2q- increases if 2q-

1>0, or if 1>0, or if qq>1/2 and >1/2 and pp=1 is her best =1 is her best response.response.

2.2. Her payoff decreases as p increases if Her payoff decreases as p increases if 22qq - 1<0, or if - 1<0, or if qq<1/2 and <1/2 and pp=o is her best =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 2if 2q q - 1=0, or of - 1=0, or of qq=1/2, and any value of =1/2, and any value of p p is her best response.is her best response.


Mixed Strategies and Games of DiscoordinationMixed Strategies and Games of Discoordination

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

if if pp > 1/2. > 1/2.

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

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


Mixed Strategies and Games of DiscoordinationMixed Strategies and Games of Discoordination To find the Nash equilibrium, plot the best To find the Nash equilibrium, plot the best

response functions and find where they response functions and find where they intersect.intersect.

Nash equilibrium is pNash equilibrium is p00 =1/2 and =1/2 and qq00 = 1/2 (see Figure 15.17) = 1/2 (see Figure 15.17)

Note that the equilibrium is Note that the equilibrium is weak weak in that it in that it represents a strategy combination where no represents a strategy combination where no player has an incentive to deviate from his player has an incentive to deviate from his strategy given that other players do not strategy given that other players do not deviate.deviate.


Figure 15.17 Mixed strategy Nash Figure 15.17 Mixed strategy Nash equilibriumequilibrium


Sequential GamesSequential Games

The extensive form is a The extensive form is a game treegame tree that shows that shows which player moves when, what choices exist, which player moves when, what choices exist, and what payoffs result when the game is and what payoffs result when the game is finished. (see Figure 15.18).finished. (see Figure 15.18).

NodesNodes are points where either decisions are are points where either decisions are made or payoffs are received.made or payoffs are received.

Terminal nodes Terminal nodes are the final nodes on an are the final nodes on an extensive form game, representing the payoffs extensive form game, representing the payoffs for reaching that node.for reaching that node.


Figure 15.18 The sequential word Figure 15.18 The sequential word processor gameprocessor game


Figure 15.19 Pruning branches on Figure 15.19 Pruning branches on the word processor gamethe word processor game


Sequential GamesSequential Games

The The equilibrium pathequilibrium path is the sequential is the sequential combination of nodes in play that combination of nodes in play that demonstrates the game’s equilibrium. demonstrates the game’s equilibrium.

The equilibrium is found using the The equilibrium is found using the backwards induction algorithmbackwards induction algorithm. .

This equilibrium is known as a This equilibrium is known as a subgame subgame perfect Nash equilibrium.perfect Nash equilibrium.

© 2009 Pearson Education Canada 15/1 Chapter 15 Introduction to Game Theory.

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Transcript of © 2009 Pearson Education Canada 15/1 Chapter 15 Introduction to Game Theory.