intermediate macro- lecture week 2

Intermediate Microeconomics, Games and Behaviour

Week 2

Strategic interaction

Strategic interaction in single encounters

Application: Oligopoly theory

Cournot competition

Bertrand competition with homogeneous products

Bertrand with competition heterogeneous products

Application: Strategic Interaction between individuals

Provision of a public good

Question

Suppose you are in a team of 4 people. You each have 10 Euros You can either keep these 10 Euros, or invest all

or some of the 10 Euros into a joint project of your team.

The joint project will generate a profit which is twice the sum of all contributions.

The projects profit will be given in equal shares to all members of your group.

How much do you contribute??

www.shakespeak.com

Prepare to vote

TXT 1

2

Internet 1

2

Voting is anonymous

Twitter 1

2

This presentation has been loaded without the Shakespeak plug-in.

Want to download the plug-in for free? Go to

http://shakespeak.com/en/free-download/.

How much do you contribute?A. 0

B. 1

C. 2

D. 3

E. 4

F. 5

G. 6

H. 7

I. 8

J. 9

K. 10

# votes: 100 Closed

The question will open when you

start your session and slideshow.

Internet This text box will be used to describe the different message sending methods.

TXT The applicable explanations will be inserted after you have started a session.

Twitter It is possible to move, resize and modify the appearance of this text box.


Want to download the plug-in for free? Go to http://shakespeak.com/en/free-

download/.

Until now mainly: many sellers (competitive market) price takers, setting q by

equating P=MC only one seller (monopoly) price setters, Q is such that

MR=MC

In both cases: dont bother about what competition is doing.

Now (and the next two weeks):- only a few sellers (oligopoly)- to keep things as uncomplicated as possible:

N = 2 (duopoly)

Markets with only a few sellers

In oligopolistic markets, the products may or may not be differentiated

What matters is that only a few firms account for most or all of total production

In some oligopolistic markets, some or all firms earn substantial profits over the long run because barriers to entry make it difficult or impossible for new firms to enter

Oligopoly

Scale economies may make it unprofitable for more than a few firms to coexist in the market;

Patents or access to a technology may exclude potential competitors. The need to spend money for name recognition and market reputation may discourage entry by new firms.

These are natural entry barriersthey are basic to the structure of the particular market.

In addition, incumbent firms may take strategic actions to deter entry.

Why so few firms?

Crucial in case of oligopoly: - behaviour of a firm depends on how one

firm thinks that other firms will react

Suppose duopoly: which PE and which QE?

How to solve this problem?The solutions depend on the assumptions made

Oligopoly

In an oligopolistic market, a firm sets price or output based partly on strategic considerations regarding the behaviour of its competitors.

Strategic Interaction among Oligopolists

Static Games with Complete Information

Game Theory Toolbox 1

Decision Theory vs Game Theory

Decision theory:

1 decision-maker

environment exogenous

Game theory:

More than 1 decision-maker

environment endogenous (what others do, depends on what I do)

1. The players: - how many players are there?- does nature/chance play a role?

2. The rules of the game- A complete description of what the players can do, the set of all possible actions and the timing.- The information that players have available when chosing their actions.

3. A description of the payoff consequences of each player for every possible combination of actions chosen by all players playing the game.

4. A description of all players preferences over payoffs.

Elements of a game

A game of complete information requires that the following four components be common knowledgeamong all players of the game:

1. all the possible actions of all the players,

2. all the possible outcomes

3. how each combination of actions of all players affects which outcome will materialize, and

4. the preferences of each and every player over outcomes.

Games of complete information

H L

Ha

A

b

B

Lc

C

d

D

1

2players

player 1s actions

player 2s actions

player 2s payoff for HL

player 1s payoff for LL

Simultaneous decisions are represented in the strategic or normal form of the game.

Normal form representation

Firms competing in quantities: set-up a game

Example: Two firms compete for consumers

Albert Heijn and Jumbo compete for consumers by offering their products in different quantities to the market. Given the law of demand, demand and thus price will be high when the total quantity on the market is low prices, and vice versa. The managers of the two chains have to decide about the quantities, without really knowing which quantity the competitor chooses.

Albert Heijn versus Jumbo

The strategic game that corresponds to this strategic situation is:

Players: Albert Heijn and Jumbo

Actions: Each players set of actions is {12 units, 16 units}

Preferences over outcomes:

For each player the action profile in which this player chooses 16, while the other one chooses 12, is ranked highest,

For each player the action profile in which this player chooses 12, while the other one chooses 16, is ranked lowest.

Second best is a profile in which both choose 12.

Third best is a profile in which both choose 16.

The strategic situation of two supermarkets in normal form

Pure Strategy

specific (non-probabilistic) choice available to a player

Example: AH - Jumbo Quantity competition

1s pure strategies: {12, 16}

2s pure strategies: {12, 16}

Game is written down in terms of pure strategies.

However, strategies can be probabilistic (= mixed strategies)

12 16

12288

288

320

240

16240

320

256

256

AH

Jumbo

Given the above incentives, if Jumbo offers 12, what is

best action for AH?

Payoff for AH is highest when offering 16 units.

12 16

12288

288

320

240

16240

320

256

256

AH

Jumbo


Given the above incentives, if Jumbo offers 16, what is

best action for AH?

Payoff for AH is highest when offering 16 units.

12 16

12288

288

320

240

16240

320

256

256

AH

Jumbo


Now consider Jumbo:

For Jumbo, 16 leads to the highest payoff when AH is

offering 16 units: U2(16|16) = 256 > U2(12|16) = 240

12 16

12288

288

320

240

16240

320

256

256

AH

Jumbo

Dominant strategiesA dominant strategy is always optimal no matter what an opponent does.

C D

C3

3

5

0

D0

5

1

1

1

2

Player 1 has a dominant strategy: If player 2 chooses C, for player 1 D is optimal: 5>3 If player 2 chooses D, for player 1 D is optimal: 1>0Player 2 has a dominant strategy as well: If player 1 chooses C, for player 2 D is optimal: 5>3 If player 1 chooses D, for player 2 D is optimal: 1>0

Dominant Strategies

Formal Definition: S1 strictly dominates S2 iff

(|) > (|)

If S1 strictly dominates S2, it will always give a higher payoff than S2,

regardless of the strategy player 2 uses.

Both players have dominant Strategies Nash equilibrium

Assumption needed: The players themselves are rational.

Rationality assumptions

The existence of dominated strategies may help to find the NE

No dominant strategies?

We can eliminate strictly dominated strategies

since a rational player has no incentive to play them.

Iterated Dominance

Example:

notice that D strictly dominates U cross out U

L C R

U1

0

3

-2

-1

4

M3

0

1

5

4

3

D5

1

2

4

2

6

given that, L strictly dominates C cross out C

given that, D strictly dominates M cross out M

given that, L strictly dominates R cross out R

therefore, (D, L) is an equilibrium

If we can iteratively eliminate strictly dominated strategies until only

one exists per player, then the resulting strategies lead to an

equilibrium.

Iterated eliminations of strictly dominated strategies

We can use iterated eliminations of strictly dominated strategies (IESDS) any number of rounds to come up with a reasonable prediction.

If there is a unique solution after successive elimination this solution is a Nash equilibrium!

Dominated Strategies: IESDS Nash equilibrium

The players themselves are rational, and Each player knows that the other player is rational

(i.e., their rationality is common knowledge)

Rationality assumptions

Nash Equilibrium

A pair of strategies forms a Nash Equilibrium (NEQ) iff the

strategies are best responses to each other.

Note

No player has an incentive to deviate unilaterally from the NEQ.(NEQ strategy maximizes each players utility, given that every other person is choosing a NEQ strategy)

The NEQ is not necessarily the best or most preferred outcome for each player.

John Nash (1928*)

Examples

C D

C3

3

5

0

D0

5

1

1

L R

L1

1

0

0

R0

0

1

1

Example 1:

For player 1, D strictly dominates C since

U1(D|C)=5 > 3 and U1(D|D)=1 > 0

Player 1 will not play C.

Same for player 2

Since D strictly dominates for both players,

DD is a Nash equilibrium.

1

2

1

2

Example 2:

Neither player has a dominated strategy.

However, dont need that for a Nash

equilibrium.

Still two pure strategy Nash equilibria.

How do we find these Nash equilibria?

Best Response to a Best Response to a

Consider what happens if two players iterate best responses:

1 chooses a strategy2 chooses best response strategy1 chooses its best response to that strategy2 chooses its best response to that strategy

Eventually, this may stabilize: 1 will have strategy Si that is a best response to 2s tj

and 2s tj will be a best response to Si

Equilibrium: no one has an incentive to deviate

Best Response to a Best Response to a

1s best response to 16 is 16

2 starts with 12





etc etc

Equilibrium: (16, 16)

Now assume that players, seeing the game, do this all in their

heads and identify the equilibrium strategies of the game.

Example:

12 16

12288

288

320

240

16240

320

256

256

AH

Jumbo

Mutual-best-response property of the Nash Equilibrium

For simple 2x2 games, just check whether anyone has incentive to deviate from a particular outcome:

Is CC a NEQ?

No. Given that 1 plays C, 2 would prefer D; same for 1.

Is DD a NEQ?

C D

C3

3

5

0

D0

5

1

1

1

2

Prisoners Dilemma

Yes. For both players, 1 > 0.

Mutual-best-response property of the Nash Equilibrium

For simple 2x2 games, just check whether anyone has incentive to deviate from a particular outcome:

Is LR a NEQ?

No. Given that 1 plays L, 2 would prefer L; same for 1.

Is LL a NEQ?

L R

L1

1

0

0

R0

0

1

1

1

2

Coordination game

Yes. For both players, 1 > 0; note that RR is also a NEQ.

Assumptions

1. Players are rational: A rational player is one who chooses his actions to maximize his payoff consistent with his beliefs about what is going on in the game.

2. Common knowledge: the fact that players are rational and intelligent is common knowledge among the players of the game.

3. Self-enforcement: Any prediction (or equilibrium) of a solution concept must be self-enforcing.

Maximin strategy

Strategy that maximizes the minimum gain that can be earned.

Thought experiment: consider one of your strategies. What is your minimum payoff for that strategy? Do this for all your possible strategies. Find the maximum of the minimum payoffs.

L C R

U300

300

400

130

450

0

M100

400

200

200

100

120

D0

420

150

100

0

20

Bob

Alice

0

120

20

120

40

Games with continuous strategies

Suppose

there are two players 1 and 2 with action sets

1 = +

and 2 = +

players payoff functions are:

1 1, 2 = (1 +1

22 1 )1

2 1, 2 = (1 +1

21 2 )2

=> We cannot analyze the game in a matrix!

41

The best response functions (reaction functions)

To find the best response of player 1 to any action of players 2,

we need to study player 1s profit as a function of its own action

s1 for given values of s2.

1 1,2

1= 1 +

1

22 1 = 0 1 2 = 1 +

1

22

2 1,2

1= 1 +

1

21 2 = 0 2 1 = 1 +

1

21

Analogously for player 2:

42

Nash equilibrium is the intersection of best-response functions

(1, 2

)

1(2)

2(1)

1

2

2

0 1

2

1

1 = 1 +1

22 1 = 1 +

1

22(1)

1 = 1 +1

2(1 +

1

21)

1 = 2

and analogously for player 2

Back to our application:

A. Strategic Interaction between firms: Oligopoly competition la Cournot

Assumptions:- homogeneous goods- each firm treats the output of the other firm as fixed - the firms decide simultaneously (meaning: without

knowledge of the competitors decision) how much to produce

So: q1 = q1 (q2)q2 = q2 (q1)

Thus we have two equations with two unknowns and, therefore, we can solve for q1 and q2 (and, given the market demand curve) for the price (with p1 = p2 because the goods are homogeneous).

Cournots model of oligopoly competition

45

Cournots model as a game between 2 players

Suppose

there are two firms 1 and 2 with action sets 1 +

and 2 + => quantity competition!

Inverse demand is given as (1+ 2) = (1+ 2)

Marginal production costs are identical and equal to c

Firms payoff functions are:

1 1, 2 = 1 + 2 1 1 = ( 1 2)1

2 1, 2 = 1 + 2 2 2 = ( 1 2)2

46

Find the best response functions

To find the best response of firm 1 to any action (output) of

firm 2, we need to study firm 1s profit as a function of its own

action (output) q1 for given values of q2.

1 1, 0 = ( 1)1If q2=0 : A quadratic function that is zero if q1=0 and when q1 = a-c

q2=0

a-c q1

1(1, 2)

0

2

1 1, 0

1= 21

1(0) =1

2( )

47


1 1, 2 = ( 1 2)1If q2>0 : A quadratic function that is zero if 1 = 0 and when 1 = 2

q2=0

q2>0

a-ca-c q2 q1

1(1, 2)

0 22

2

1 1, 21

= 21 2

1(2) =1

2( 2)

48

1 1,2

1= 21 2 = 0 1(2) =

1

2( 2)

2 1,2

2= 22 1= 0 2(1) =

1

2( 1)

Find the best response functions of both firms

49

Nash equilibrium as the intersection of best-response functions

2

2

3

3

0

(1, 2

)

1(2)

2(1)

1

2 1 2 =

1

2( 2 (1))

1 =1

2(

1

2( 1))

1 =1

3( )

and analogously for firm 2

Cournot equilibrium is a Nash equilibrium:Each firm does the best it can, given what its competitors are doing.

Cournot model does not specify adjustment process when not in equilibrium.

How do firms reach the equilibrium?

In P&R, the best response function is labelled the reaction curve.

For an example of how to translate the theoretical model into an empirical one and find the reaction curves and the Cournot equilibrium, see p. 461


Example from P&R:

Inverse market demand: P = 30 QMC1 = MC2 = 0 (NB: this is an exception!)

Firm 1:

Firm 2:

Intersection leads to Cournot equilibrium: 1 = 2 = 10


1 1, 2 = 30 1 + 1 1 = (30 1 2)1

1 1,2

1= 30 21 2 = 0 1(2) =

1

2(302)

1 1,2

2= 30 22 1= 0 2(1) =

1

2(301)

2 1, 2 = 30 1 + 1 2 = (30 1 2)2

53

Nash equilibrium and other outcomes

2

2

3

3

0

(1, 2

)

1(2)

2(1)

1

2Firm 2 in a monopolist

Cournotoutcome

Cartel with 50-50 profit sharing

Outcome under perfect

competition

Our example: Payoff Matrix in terms of profit

16

12

1612

Jumbo

AH

24

24

288, 288

320, 240

288, 144

240, 320

256, 256

192, 128

144, 288

128, 192

0, 0

1 =1

3( )

1 =1

2( )

1 =1

4( )

With = 48

Another application:

B. Strategic Interaction between firms: Oligopolistic pricing la Bertrand

Albert Heijn lowers price of a thousand brand articles

Albert Heijn starts attack on LidlSaturday 7 sep 2013, Volkskrant

Jumbo follows Albert Heijn in new price warThursday 12 sep 2013, NRC

"Ook nu is het de marktleider, Albert Heijn, die met veelbombarie prijsverlagingen aankondigt. En ook nu zietde concurrentie zich genoodzaakt daarin mee te gaan."

Also now, it is the market leader, Albert Heijn, which kicks up a fuss by announcing price cuts. And once again the competitors see no other solution than to follow

NRC, September 12

Early 2000s: the leading Dutch supermarket chain Albert Heijn suffered from an unfavourable and deteriorating price image, which was especially troublesome in the light of the rise of hard discounters (Aldi and Lidl) and worsening economic conditions.

After several years of a sliding market share, on October 20, 2003, Albert Heijndecided to slash its prices for more than 1000 products.

Although Albert Heijns operation to decrease prices was undertaken in complete secrecy, within two days, all major competitors matched or even exceeded the price reductions.

The price war that followed was nationwide, entailing an 8.2% reduction in food prices and resulting in the lowest inflation level in 15 years.

The loss in added value for the Dutch retailing industry is estimated to be 900M in one year, and more than 30,000 employees in the grocery industry lost their jobs.

The History

Assumptions:- homogeneous goods- each firm treats the price of the other firm

as fixed- the firms decide simultaneously what price

to charge (meaning without knowledge of the competitors decision)

- All consumers buy at the firm offering the lowest price (no market frictions)

Bertrands model of oligopoly competition

62

Bertrands model as a game between 2 players

Suppose

there are two firms 1 and 2 with action sets p1=+

and p2= + => price competition!

Demand is given as = for > , with pibeing the lowest price.

Marginal production costs are identical and equal to

< .

firms payoff functions are:

1, 2 =

( )( )1

2()( )

0

< = >

with j = 2 if i = 1, and j = 1 if i = 2.

63

The Nash equilibrium of the Bertrand game

1. (1 , 2) = , is a Nash equilibrium: if one firm charges a price c, then the other firm can do no better than charge the price c also, because if it raises its price it sells nothing and when it lowers its price it incurs a loss.

2. No other pair (1, 2) is a Nash equilibrium: If < : profit is negative and can be increased by raising price

to c. If = and > : firm i can increase its profit from zero to

positive by raising the price. If > and > : suppose , then firm i can increase

its payoff by lowering its price below or even to .

Assumptions:- heterogenous goods- each firm treats the price of the other firm as

fixed- the firms decide simultaneously what price to

chargeSo: 1 = 1 (2)

2 = 2 (1)

I.e. two equations with two unknowns

Price competition with heterogeneous goods

Two duopolists, fixed costs = 20, no variable costs (= MC =0)

Demand firm 1: 1 (1, 2) = 12 21 + 2

Demand firm 2: 2(1, 2) = 12 22 + 1

1 = 11(1, 2) 20 = 121 2 12 + 12 20

11

= 12 4 1 + 2 = 0

Firm 1s reaction curve: 1 = 3 +1

42

Firm 2s reaction curve: 2 = 3 +1

41

Price competition with heterogeneous goods: example

66

Bertrand Nash equilibrium with heterogeneous products

3

4

4

0

(1, 2

)

1(2)

2(1)

1

2

3

1 = 3 +1

42(1) 1 = 3 +

1

4(3 +

1

41) 1 = 4

and analogously for firm 2

Another application:

C. Strategic Interaction between individuals: Provision of a public good

Assumptions:- There are 4 citizens , , , - each citizen can contribute k 1,10 units to

the provision of a public good, with , , , =1,2,3,4.

- The citizens benefit of contributing to the public good is

1, 2, 3, 4 = 10 +2

4(1 + 2 + 3 + 4)

Provision of a public good

To maximize total wealth of all citizens:

1 1, 2, 3, 4 + 2 1, 2, 3, 4 + 3 1, 2, 3, 4 + 3 1, 2, 3, 4

= 40 1 2 3 4 +8

4(1 + 2 + 3 + 4)

Maximized when 1= 2 = 3 = 4 = 10 Contributing to the public good has a higher social benefit than

keep the contribution on the private accounts.

What if each citizen gets to decide whether to contribute?

Each citizen will contribute as long as the cost of contributing to that citizen is outweighed by the gains for that citizen.

70


To find the best response of citizen 1 to any contribution of

citizens 2 and 3 we need to find the citizens best response

function:

1, 2, 3, 4

= 1

2 = 0

In the Nash equilibrium of this game each citizen will not contribute anything to the public good.

A conflict over scarce resources

Game theory can be used to explain overuse of shared resources.

Extend the Prisoners Dilemma to more than two players.

How much do you contribute?

A.

B.

C.

D.

E.

F.

G.

H.

I.

J.

K.

0

1

2

3

4

5

6

7

8

9

10

23.0%

3.0%

2.0%

4.0%

2.0%

14.0%

3.0%

4.0%

3.0%

4.0%

38.0%

Closed

Internet This text box will be used to describe the different message sending methods.

TXT The applicable explanations will be inserted after you have started a session.

Twitter It is possible to move, resize and modify the appearance of this text box.


Want to download the plug-in for free? Go to http://shakespeak.com/en/free-

download/.

Thank you for your attention!

Next week:

Sequential strategic interaction

Applications

1. Dynamic oligopoly theory (van Stackelberg, Entry)

2. Bargaining

intermediate macro- lecture week 2

Documents

new firms

incumbent firms

firms account

shakespeak plugin

strategic considerations

strategic actions

market reputation

market patents