copyright© 2006 southwestern/thomson learning all rights reserved. some midterm stats ●average...

Copyright© 2006 Southwestern/Thomson Learning All rights reserved.

Some Midterm StatsSome Midterm Stats

● Average score: 82 out of 100

● 6 (six) maximum scores

● Problem 4 was the most challenging on average

● See the Answer Key published

● Go to recitation to get your MT2 back

● Average score: 82 out of 100

● 6 (six) maximum scores

● Problem 4 was the most challenging on average

● See the Answer Key published

● Go to recitation to get your MT2 back

Introduction toGame Theory


ContentsContents

●Introduction

●Simultaneous Games

●Solution Concept: Nash Equilibrium

●Sequential Games

Application to Oligopoly Models

●Introduction

●Simultaneous Games

●Solution Concept: Nash Equilibrium

●Sequential Games

Application to Oligopoly Models


IntroductionIntroduction

● Game Theory studies strategic interactions between agents

● Provides formal modeling approach to situations in which decision makers interact

● Agents choose strategies which maximize their return, given strategies others choose

● Game Theory studies strategic interactions between agents

● Provides formal modeling approach to situations in which decision makers interact

● Agents choose strategies which maximize their return, given strategies others choose


IntroductionIntroduction

● A Game consists of three elements:♦ Players (or agents)♦ Strategies for each player♦ Utilities (or payoffs) for each player

● Utilities depend on strategies chosen by ALL players (not only yourself)

● Players choose strategies to maximize payoffs given others’ strategies

● A Game consists of three elements:♦ Players (or agents)♦ Strategies for each player♦ Utilities (or payoffs) for each player

● Utilities depend on strategies chosen by ALL players (not only yourself)

● Players choose strategies to maximize payoffs given others’ strategies


Application to OligopolyApplication to Oligopoly

● Game Theory Approach to Oligopoly ♦ Each oligopolist is seen as a competing player

in a game of strategy

♦ Managers act as if they believe that their opponents will adopt the most profitable countermove to any move they make…

♦ … and try to obtain maximum profits keeping that in mind

● Game Theory Approach to Oligopoly ♦ Each oligopolist is seen as a competing player

in a game of strategy

♦ Managers act as if they believe that their opponents will adopt the most profitable countermove to any move they make…

♦ … and try to obtain maximum profits keeping that in mind


Here is where I made mistake in the lecture, so I had to change several slides because of it

All the slides between the two black ones were not shown in class.

Here is where I made mistake in the lecture, so I had to change several slides because of it

All the slides between the two black ones were not shown in class.


Example: Prisoner’s DilemmaExample: Prisoner’s Dilemma

● Simplest possible game:♦ Two players, Firm A and Firm B

♦ Two strategies: build a low-tech plant or a high-tech plant

♦ Four possible outcomes of the game:■(Low, Low), (Low, High), (High, Low), (High, High)

● Simplest possible game:♦ Two players, Firm A and Firm B

♦ Two strategies: build a low-tech plant or a high-tech plant

♦ Four possible outcomes of the game:■(Low, Low), (Low, High), (High, Low), (High, High)


Example: Prisoner’s DilemmaExample: Prisoner’s Dilemma

● We assume the following payoffs:♦ If both firms build low-tech plants, both get $10 mln

of profits♦ If both firms build high-tech plants, both get

$3 mln of profits♦ If one firm builds low-tech plant and the other builds

a high-tech plant, the former is driven out of business, having -$2 mln of loss. The other gets a $12 mln monopoly profits (high-tech rocks!)

● Can represent this using a payoff matrix

● We assume the following payoffs:♦ If both firms build low-tech plants, both get $10 mln

of profits♦ If both firms build high-tech plants, both get

$3 mln of profits♦ If one firm builds low-tech plant and the other builds

a high-tech plant, the former is driven out of business, having -$2 mln of loss. The other gets a $12 mln monopoly profits (high-tech rocks!)

● Can represent this using a payoff matrix


TABLE 1: A Payoff Matrix for Prisoner’s Dilemma

TABLE 1: A Payoff Matrix for Prisoner’s Dilemma

Copyright© 2006 South-Western/Thomson Learning. All rights reserved.

Firm B’s strategies

Low-tech High-tech

Firm A’s strategies

Low-tech

High-tech

A gets $10 mln

B gets $10 mln

A gets − $2 mln

B gets $12 mln

A gets $12 mln

B gets − $2 mln

A gets $3 mln

B gets $3 mln


Technical AssumptionsTechnical Assumptions

● We assume the following for simplicity:♦ Each firm knows the whole game structure

■They know their strategies, their opponent’s strategies and all the payoffs

♦ Firms take decisions simultaneously♦ Firms cannot communicate before taking a

decision

● There are games that relax these assumptions, but they are more involved

● We assume the following for simplicity:♦ Each firm knows the whole game structure

■They know their strategies, their opponent’s strategies and all the payoffs

♦ Firms take decisions simultaneously♦ Firms cannot communicate before taking a

decision

● There are games that relax these assumptions, but they are more involved


Solution ConceptsSolution Concepts

● So far, we considered a game, i.e. a model of interaction between agents

● We also want a solution concept, i.e. some prediction generated by the model

● A “good” solution is:♦ Easy to compute

♦ Unique in most cases

● So far, we considered a game, i.e. a model of interaction between agents

● We also want a solution concept, i.e. some prediction generated by the model

● A “good” solution is:♦ Easy to compute

♦ Unique in most cases


Solution Concept: Dominant StrategySolution Concept: Dominant Strategy

● Games with dominant strategies♦ Dominant strategy (DS) = one that gives the

bigger payoff to the firm that selects it, no matter which of the two strategies the competitor selects

♦ “Prisoners’ Dilemma” is a classic example of a game with a dominant strategy

♦ Let us see why

● Games with dominant strategies♦ Dominant strategy (DS) = one that gives the

bigger payoff to the firm that selects it, no matter which of the two strategies the competitor selects

♦ “Prisoners’ Dilemma” is a classic example of a game with a dominant strategy

♦ Let us see why


TABLE 2: Payoff Matrix Revisited

TABLE 2: Payoff Matrix Revisited



Low-tech High-tech


Low-tech

High-tech

A gets $10 mln

B gets $10 mln

A gets − $2 mln

B gets $12 mln

A gets $12 mln

B gets − $2 mln

A gets $3 mln

B gets $3 mln


Why Prisoner’s DilemmaWhy Prisoner’s Dilemma

● Initially this game was used to model an interaction between two criminals

● Strategies were “to confess” and “to remain silent”

● Payoffs were denoting years spent in jail♦ If both confess – get in trouble♦ If friend rats you out – you bare all the blame♦ If both silent, both get to walk away

● Initially this game was used to model an interaction between two criminals

● Strategies were “to confess” and “to remain silent”

● Payoffs were denoting years spent in jail♦ If both confess – get in trouble♦ If friend rats you out – you bare all the blame♦ If both silent, both get to walk away


Why Prisoner’s DilemmaWhy Prisoner’s Dilemma

● Dominant Strategy for everyone is to confess, no matter what your partner does

● By remaining silent, both get higher payoffs

● But none wants to remain silent, since both expect their partner to confess

● Hence both suffer from the inability to coordinate

● Dominant Strategy for everyone is to confess, no matter what your partner does

● By remaining silent, both get higher payoffs

● But none wants to remain silent, since both expect their partner to confess

● Hence both suffer from the inability to coordinate


TABLE 2 repeated: Payoff Matrix Revisited

TABLE 2 repeated: Payoff Matrix Revisited



Low-cost High-cost


Low-cost

High-cost

A gets $10 mln

B gets $10 mln

A gets $12 mln

B gets −$2 mln

A gets −$2 mln

B gets $12 mln

A gets $3 mln

B gets $3 mln


Low-tech High-tech


Low-tech

High-tech

A gets $10 mln

B gets $10 mln

A gets − $2 mln

B gets $12 mln

A gets $12 mln

B gets − $2 mln

A gets $3 mln

B gets $3 mln


Back to the Oligopoly GameBack to the Oligopoly Game

● Notice that the structure of the game for two firms is precisely the same as in PD♦ Both firms can gain from coordination on

investing into low-tech production

♦ However, each firm prefers to invest into high-tech production, lowering its profits

● That is why we refer to it as a PD-type game

● Notice that the structure of the game for two firms is precisely the same as in PD♦ Both firms can gain from coordination on

investing into low-tech production

♦ However, each firm prefers to invest into high-tech production, lowering its profits

● That is why we refer to it as a PD-type game


Here ends the edited part of the presentation

The rest of the slides were shown in class.

Here ends the edited part of the presentation

The rest of the slides were shown in class.


Dominant Strategies: Pro et ContraDominant Strategies: Pro et Contra

● DS is intuitively appealing:♦ If you have a choice that is better than any

other, clearly you should stick to it

● DS is not so good sometimes:♦ Many interesting games do not have DS at all

● Consider the following example where we only changed the payoffs of the firms

● DS is intuitively appealing:♦ If you have a choice that is better than any

other, clearly you should stick to it

● DS is not so good sometimes:♦ Many interesting games do not have DS at all

● Consider the following example where we only changed the payoffs of the firms


TABLE 3: A Payoff Matrix for a game without DS

TABLE 3: A Payoff Matrix for a game without DS



Low-cost High-cost


Low-cost

High-cost

A gets $10 mln

B gets $10 mln

A gets $7 mln

B gets $3 mln

A gets $3 mln

B gets $7 mln

A gets $8 mln

B gets $8 mln


Games without DSGames without DS

● Notice that no firm has a DS in this game:♦ If Firm A knows that Firm B will build a low-

cost plant, it is better for A to build a low-cost plant as well

♦ And if Firm A knows that Firm B will build a high-cost plant, it is better for A to build a high-cost plant also

● So there are no DS in this game

● Notice that no firm has a DS in this game:♦ If Firm A knows that Firm B will build a low-

cost plant, it is better for A to build a low-cost plant as well

♦ And if Firm A knows that Firm B will build a high-cost plant, it is better for A to build a high-cost plant also

● So there are no DS in this game


Solution Concept: Nash EquilibriumSolution Concept: Nash Equilibrium

● Introduce a different solution concept called a Nash Equilibrium (NE)♦ Named after the Economics Nobel Prize

winning mathematician John Nash

● To define NE precisely, we need another concept – Best Response

● Introduce a different solution concept called a Nash Equilibrium (NE)♦ Named after the Economics Nobel Prize

winning mathematician John Nash

● To define NE precisely, we need another concept – Best Response


Best ResponseBest Response

● Suppose you know that your opponent will play some strategy 1. What would you do?♦ You will choose your strategy so that to

maximize the payoff you will get! (At least if you apply economic thinking )

● Best response: Given a strategy for the opponent, your best response (BR) is the strategy that gives you the highest payoff

● Suppose you know that your opponent will play some strategy 1. What would you do?♦ You will choose your strategy so that to

maximize the payoff you will get! (At least if you apply economic thinking )

● Best response: Given a strategy for the opponent, your best response (BR) is the strategy that gives you the highest payoff


Best Response and NEBest Response and NE

● A Nash Equilibrium is a combination of strategies such that each player plays his best response to the opponent’s best response.

● Put differently, if you know that your opponent will not change his/her behavior, playing a NE strategy means that you do not want to change your actions as well

● A Nash Equilibrium is a combination of strategies such that each player plays his best response to the opponent’s best response.

● Put differently, if you know that your opponent will not change his/her behavior, playing a NE strategy means that you do not want to change your actions as well


Back to the Oligopoly GameBack to the Oligopoly Game

● Here is how we find NE in a given game:♦ Suppose you’re Firm A and you know Firm B

will build a low-cost plant■Then you want to build a low-cost plant as well■So the BR of firm A to the strategy of firm B

“build a low-cost plant” is a strategy “build a low-cost plant”

♦ Similarly for all three other cases

● Consider the payoff matrix again:

● Here is how we find NE in a given game:♦ Suppose you’re Firm A and you know Firm B

will build a low-cost plant■Then you want to build a low-cost plant as well■So the BR of firm A to the strategy of firm B

“build a low-cost plant” is a strategy “build a low-cost plant”

♦ Similarly for all three other cases

● Consider the payoff matrix again:


TABLE 4: A Payoff Matrix for a game without DS revisited

TABLE 4: A Payoff Matrix for a game without DS revisited



Low-cost High-cost


Low-cost

High-cost

A gets $10 mln

B gets $10 mln

A gets $7 mln

B gets $3 mln

A gets $3 mln

B gets $7 mln

A gets $8 mln

B gets $8 mln


Interpretation of the GameInterpretation of the Game

● Two NE in this game:♦ Both firms build low-cost plants…♦ … or both firms build high-cost plants

● We now see that:♦ NE is good since it exists (it is always true)♦ But NE may not be unique! (as here)

● There are many refinements of NE that deal with multiplicity, but we will not get into it

● Two NE in this game:♦ Both firms build low-cost plants…♦ … or both firms build high-cost plants

● We now see that:♦ NE is good since it exists (it is always true)♦ But NE may not be unique! (as here)

● There are many refinements of NE that deal with multiplicity, but we will not get into it


Game Theory and CartelsGame Theory and Cartels

● We can apply BR and NE machinery to illustrate cartel instability.

● This is also a good illustration of a game where agents have many strategies.

● We will consider a simple duopoly where firms compete in a Cournot fashion

● We can apply BR and NE machinery to illustrate cartel instability.

● This is also a good illustration of a game where agents have many strategies.

● We will consider a simple duopoly where firms compete in a Cournot fashion


Cournot CompetitionCournot Competition

● Cournot was the first to propose such a model (competition in quantities)

● Two firms produce homogeneous good

● Each decides how much output to produce without any coordination with the rival♦ we will relax this assumption later

● Face a market demand curve and cost curve

● Cournot was the first to propose such a model (competition in quantities)

● Two firms produce homogeneous good

● Each decides how much output to produce without any coordination with the rival♦ we will relax this assumption later

● Face a market demand curve and cost curve



● Let P = 8 − Q be the inverse demand

● The marginal revenue curves are:♦ MR1 = 8 − 2q1 − q2

♦ MR2 = 8 − 2q2 − q1

● Constant marginal cost of 2 per unit for each firm

● What are the best responses?

● Let P = 8 − Q be the inverse demand

● The marginal revenue curves are:♦ MR1 = 8 − 2q1 − q2

♦ MR2 = 8 − 2q2 − q1

● Constant marginal cost of 2 per unit for each firm

● What are the best responses?



● Firm 1 treats q2 as fixed. What’s the best q1?

♦ As usual, MR = MC works

♦ MR1 = 8 − 2q1 − q2 = 2 = MC, so q1 = 3 − ½q2

● Similarly, we determine the best q2

♦ MR2 = 8 − 2q2 − q1 = 2 = MC, so q2 = 3 − ½q1

● NE is when firm 1 plays a BR to firm 2 and vice versa, by definition

● Firm 1 treats q2 as fixed. What’s the best q1?

♦ As usual, MR = MC works

♦ MR1 = 8 − 2q1 − q2 = 2 = MC, so q1 = 3 − ½q2

● Similarly, we determine the best q2

♦ MR2 = 8 − 2q2 − q1 = 2 = MC, so q2 = 3 − ½q1

● NE is when firm 1 plays a BR to firm 2 and vice versa, by definition



● We solve these equations in q1 and q2 to get q1 = q2 = q = 2

● The price then is P = 8 − (2 + 2) = 4

● The profits are π = (P − AC)q = (4 − 2)2 = 4

● So in Cournot competition, each firm earns a profit of 4

● What if firms form a cartel?

● We solve these equations in q1 and q2 to get q1 = q2 = q = 2

● The price then is P = 8 − (2 + 2) = 4

● The profits are π = (P − AC)q = (4 − 2)2 = 4

● So in Cournot competition, each firm earns a profit of 4

● What if firms form a cartel?


CartelCartel

● If firms form a cartel, they will act as monopoly, splitting profits and production

● Suppose that the monopoly marginal revenue curve is MR = 8 − 2Q

● The marginal cost curve does not change

● Optimal output for monopoly is when MR = MC, as always

● If firms form a cartel, they will act as monopoly, splitting profits and production

● Suppose that the monopoly marginal revenue curve is MR = 8 − 2Q

● The marginal cost curve does not change

● Optimal output for monopoly is when MR = MC, as always


CartelCartel

● MR = 8 − 2Q = 2 = MC, so Q = 3

● Then P = 8 − Q = 8 − 3 = 5

● Cartel profits are π = (P − AC)Q = 9

● Each firm produces q = 1.5 and gets π = 4.5

● Note this is greater than in the Cournot case

● Eq’m price is higher, and output is lower if firms collude (i.e. form a cartel)

● MR = 8 − 2Q = 2 = MC, so Q = 3

● Then P = 8 − Q = 8 − 3 = 5

● Cartel profits are π = (P − AC)Q = 9

● Each firm produces q = 1.5 and gets π = 4.5

● Note this is greater than in the Cournot case

● Eq’m price is higher, and output is lower if firms collude (i.e. form a cartel)


Cartels Are UnstableCartels Are Unstable

● Recall the BR of firm 1: q1 = 3 − ½q2

● If q2 = 1.5, then optimal q1 = 2.25 > 1.5● Price will be P = 8 − 2.25 − 1.5 = 4.25● Profits then are:

♦ π1 = (4.25 − 2) 2.25 ≈ 5.05♦ π2 = (4.25 − 2) 1.5 ≈ 3.37

● If firms collude, each one wants to renege on the collusion agreement

● Recall the BR of firm 1: q1 = 3 − ½q2

● If q2 = 1.5, then optimal q1 = 2.25 > 1.5● Price will be P = 8 − 2.25 − 1.5 = 4.25● Profits then are:

♦ π1 = (4.25 − 2) 2.25 ≈ 5.05♦ π2 = (4.25 − 2) 1.5 ≈ 3.37

● If firms collude, each one wants to renege on the collusion agreement


Cartels are unstableCartels are unstable

● Here are the results we found so far:♦ If both firms compete, they get π = 4 each

♦ If both collude, they get π = 4.5 each

♦ If one firm produces cartel output, and the other firm breaks the agreement, the latter gets π = 5.05 and the former gets π = 3.37

● Let us write this as a payoff matrix as before to see this is a Prisoner’s Dilemma game

● Here are the results we found so far:♦ If both firms compete, they get π = 4 each

♦ If both collude, they get π = 4.5 each

♦ If one firm produces cartel output, and the other firm breaks the agreement, the latter gets π = 5.05 and the former gets π = 3.37

● Let us write this as a payoff matrix as before to see this is a Prisoner’s Dilemma game


TABLE 5: A Payoff Matrix for a cartel instability game

TABLE 5: A Payoff Matrix for a cartel instability game


Firm 2’s strategies

Collude Compete

Firm 1’s strategies

Collude

Compete

1st gets 4.5

2nd gets 4.5

1st gets 3.37

2nd gets 5.05

1st gets 5.05

2nd gets 3.37

1st gets 4

2nd gets 4


Cartels Are UnstableCartels Are Unstable

● The dominant strategy for each firm is to compete

● And if both collude, they get higher profits

● So this is in fact a Prisoner’s Dilemma game

● Why might we like this result?♦ Because this game does not account for

consumers and their losses caused by collusion

● The dominant strategy for each firm is to compete

● And if both collude, they get higher profits

● So this is in fact a Prisoner’s Dilemma game

● Why might we like this result?♦ Because this game does not account for

consumers and their losses caused by collusion


Interpretation of the GameInterpretation of the Game

● A market with a duopoly serves the public interest better than a monopoly because of the competition created between the duopolists.

● It is damaging to the public to allow rival firms to collude on what prices to charge for their products and what quantity of product to supply.

● A market with a duopoly serves the public interest better than a monopoly because of the competition created between the duopolists.

● It is damaging to the public to allow rival firms to collude on what prices to charge for their products and what quantity of product to supply.


Sequential GamesSequential Games

● In many situations in economics, players do not get to act simultaneously♦ Consider entry problem – a firm wants to start

operating in a market where some other firm is already operating for a while

♦ After the entry, the old firm may try to hurt the new one in some way (e.g. start a price war)

● Sequential games (SG) can be used to model such situations

● In many situations in economics, players do not get to act simultaneously♦ Consider entry problem – a firm wants to start

operating in a market where some other firm is already operating for a while

♦ After the entry, the old firm may try to hurt the new one in some way (e.g. start a price war)

● Sequential games (SG) can be used to model such situations


Sequential GamesSequential Games

● We can represent SG with a payoff matrix♦ But it is hard since the number of strategies is

usually huge in SG

● A better thing to do is to draw something that is called a game tree♦ A game tree is a simplified diagram that

illustrates all the key points of the game

♦ The important thing is the order of moves

● We can represent SG with a payoff matrix♦ But it is hard since the number of strategies is

usually huge in SG

● A better thing to do is to draw something that is called a game tree♦ A game tree is a simplified diagram that

illustrates all the key points of the game

♦ The important thing is the order of moves


A Game of EntryA Game of Entry

● An old firm is operating in the industry, and is afraid of entry from competitors

● It considers two options:♦ Build a new small factory♦ Build a new big factory

● A big factory is expensive to build, but a small factory won’t produce enough to satisfy the whole market demand

● An old firm is operating in the industry, and is afraid of entry from competitors

● It considers two options:♦ Build a new small factory♦ Build a new big factory

● A big factory is expensive to build, but a small factory won’t produce enough to satisfy the whole market demand


A Game of EntryA Game of Entry

● Effectively a small factory opens the possibility of entry for competitors

● A potential entrant has two options:♦ Enter♦ Do not enter

● New firm does not want to enter if old firm builds a big factory, and vice versa.

● Things get involved, so here’s the game tree

● Effectively a small factory opens the possibility of entry for competitors

● A potential entrant has two options:♦ Enter♦ Do not enter

● New firm does not want to enter if old firm builds a big factory, and vice versa.

● Things get involved, so here’s the game tree

FIGURE 1: Entry and Entry-Blocking Strategy

FIGURE 1: Entry and Entry-Blocking Strategy

6 0

2 2

4 0

–2 –2

Profits (millions $) Old Firm New Firm

Possible Reactions of New Firm

Possible Choices of Old Firm

Don’t Enter

Don’t Enter

Enter

Big Factory


Small Factory Enter


Solving Sequential GamesSolving Sequential Games

● We can still use the NE concept to solve SG

● There is a very useful algorithm to find NE in such games, called backward induction♦ Also called “cut-down-the-tree” method

● The only trick is to start solving the game from the end, not from the beginning

● Here is how it works:

● We can still use the NE concept to solve SG

● There is a very useful algorithm to find NE in such games, called backward induction♦ Also called “cut-down-the-tree” method

● The only trick is to start solving the game from the end, not from the beginning

● Here is how it works:


FIGURE 2: Backward Induction Analysis of a SG

FIGURE 2: Backward Induction Analysis of a SG

6 0

2 2

4 0

–2 –2

Profits (millions $) Old Firm New Firm

Possible Reactions of New Firm

Possible Choices of Old Firm

Enter

Don’t Enter

Don’t Enter

Enter

Small Factory

Big Factory



Solving Sequential GamesSolving Sequential Games

● To summarize:♦ Old firm builds small factory – new firm enters♦ Old firm builds big factory – new firm doesn’t

● By our payoff structure, old firm doesn’t want a competitor, so it build a big factory

● Could have assumed that cost of building a big factory is so big that old firm would still prefer to have competition

● To summarize:♦ Old firm builds small factory – new firm enters♦ Old firm builds big factory – new firm doesn’t

● By our payoff structure, old firm doesn’t want a competitor, so it build a big factory

● Could have assumed that cost of building a big factory is so big that old firm would still prefer to have competition


Repeated GamesRepeated Games

● Most markets feature repeat buyers. ● Repeated games give players the

opportunity to learn about other’s behavior patterns and, perhaps, to arrive at mutually beneficial arrangements.

● Threats and credibility♦ Induce rivals to change their behavior♦ Threat must be credible

● Most markets feature repeat buyers. ● Repeated games give players the

opportunity to learn about other’s behavior patterns and, perhaps, to arrive at mutually beneficial arrangements.

● Threats and credibility♦ Induce rivals to change their behavior♦ Threat must be credible


Repeated GamesRepeated Games

● Repeated games are a combination of simultaneous and sequential games♦ Every day agents play a simultaneous game

♦ They get to observe the rival’s behavior yesterday and can act accordingly

● Repeated games are hard to analyze, so we stop here.

● Repeated games are a combination of simultaneous and sequential games♦ Every day agents play a simultaneous game

♦ They get to observe the rival’s behavior yesterday and can act accordingly

● Repeated games are hard to analyze, so we stop here.


The EndThe End

??????

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