Simultaneous games with continuous strategies Suppose two players have to choose a number between 0...

Simultaneous games with continuous strategies

• Suppose two players have to choose a number between 0 and 100. They can choose any real number (i.e. any decimal). • They have continuous strategies

• If they choose the same number than player 1 wins $1000. Otherwise player 1 gets nothing.

• If the numbers sum to 100 then player 2 wins $1000. Otherwise player 2 gets nothing.

How do we model this game?

Best response function for player 1

• If player 2 plays 33, then player 1’s ‘best response’ is to play 33

• If player 2 plays 74.5 then player 1’s ‘best response’ is to play 74.5 … and so on.

• Player 1’s best response function tells us, for every possible action by player 2, what is the best response of player 1


Player 1

Player 20

100

100

The ‘45 degree line’ gives the best responses for player 1. If player 2 chooses a certain number, player 1’s best response is to choose that same number


• If player 1 plays 33, then player 2’s ‘best response’ is to play 67 so that the numbers add to 100

• If player 1 plays 74.5 then player 2’s ‘best response’ is to play 25.5 so that the numbers add to 100 … and so on.

• Player 2’s best response function tells us, for every possible action by player 1, what is the best response of player 2


Player 1

Player 20

100

100

The line with slope = -1 gives the best responses for player 2. If player 1 chooses a certain number, player 1’s best response is to choose 100 minus that number

Nash equilibrium

• To find the Nash equilibrium, put the best response functions together• A Nash equilibrium is a ‘mutual best

response’. So a Nash equilibrium occurs where the best response functions cross

Nash equilibriumPlayer 1

Player 20

100

100

The Nash equilibrium is where the best response functions cross. Here it is where both players choose 50. So each player simultaneously and independently choosing ‘50’ is a Nash equilibrium.

50

50

Solving games with continuous strategies

• For each player find their best response function

• Where the best response functions cross, we have a Nash equilibrium

Oligopoly games

• We now want to apply simultaneous games to the behaviour of a small number of firms (an oligopoly)

• Suppose there are just two firms (duopoly)• They produce identical goods. • But each firm must decide how much it will

produce before it goes to sell the product• For example, firms must simultaneously choose

plant size but then given plant size, they will operate at capacity

‘Cournot’ competition in quantities

• Firms simultaneously choose quantities• The market price then adjusts so that the total

quantity produced is sold• Each firm sets its quantity to maximise profits• Note that each firm has a continuous set of

strategies• So we need to calculate each firm’s best response

function (or reaction function)

‘Firm one’s best response function

Output of firm 2

Profit maximising quantity for

firm 1

This is the diagram we need to fill in for firm 1. GIVEN a level of output for firm 2, what is firm 1’s best response?

Firm one’s best response function

Quantity

$

Market demand

Marginal revenue

Marginal cost

QM

PM

If firm 2 produces nothing, then the profit maximising best response for firm 1 is to produce the monopoly quantity


Quantity

$

Market demand

Marginal cost

QM

PM

If firm 2 produces and sells a positive quantity then this reduces the ‘residual’ demand for firm 1.

Firm 2’s output

Firm 1’s demand

curve given firm 2’s sales


Quantity

$

Market demand

Marginal cost

QM

PM

Firm 2’s output

Marginal revenue GIVEN firm 2’s output


Quantity

$

Marginal cost

QM

PM

So if firm 2 produces more, the best response for firm 1 is to lower output. Note: firm 1 lowers output by less than firm 2’s increase so overall market price falls.

Firm 2’s output

New optimal quantity

New lower

market price


Quantity

$

Market demand

Marginal cost

PM

And if firm 2 produces ‘enough’, firm 1’s best response is to produce nothing

Firm 2’s output

Firm 1’s demand

curve given firm 2’s sales


Output of firm 2

Profit maximising quantity for

firm 1

So we can plot firm 1’s best response function. It starts at the monopoly quantity then falls with a slope less than 1.

Qm

Firm 1’s best response function


Q2

Q1We can do the same for firm 2.

Qm Firm 1’s best response function

Qm



Q2

Q1And get the Cournot equilibrium.

Qm Firm 1’s best response function

Qm


Qc2

Qc1

Symmetric firms and ‘Cournot’ competition

Q2

Q1 If the two firms are identical

then Qc1 = Qc

2. Also total

output exceeds the monopoly

quantity.

Qm


Qm


Qc2

Qc1

45 degree line

(Qm )/2

(Qm )/2

If firm 1’s costs fall then its best response function moves out

• Suppose firm 1’s marginal costs fall but not firm 2’s

• Then for any given output of firm 2, firm 1’s profit maximising output will rise

• So firm 1’s best response function shifts ‘out’ if firm 1’s costs fall

If firm 1’s costs fall then its best response function moves out

Q2

Q1

Qm (original costs)

Firm 1’s new best response function

Qm


Qm (new costs)

Firm 1’s costs fall

Q2

Q1

Original Qc2

Original Qc1

New Qc1

New Qc2

So if firm 1’s costs fall, total output rises. Firm 1 produces more in equilibrium and the other firm produces less. Firm 1 makes more profit as (a) its costs are lower and (b) its competitor produces less. Firm 2 makes less profit as (a) total production rises and (b) it produces less. Consumers gain because price falls.

Strategic Substitutes

• Output (or capacity) here is a strategic variable

• Note that the output that is best for one firm falls as the other firm’s capacity increases

• For this reason, we call these type of strategic variables ‘strategic substitutes’.

Summary

• For games with continuous strategies, we model the game by looking at best response functions

• The Cournot competition game has firms simultaneously setting output• The firms produce more than monopoly in

total (but less than perfect competition)• We can capture the strategic effects of a

change in costs for one firm

‘Bertrand’ competition in prices

• Two firms simultaneously choose prices• Consumers then decide which firm to buy from• Each firm sets its own price to maximise profits• As with Cournot competition each firm has a

continuous set of strategies• Here we will consider the case where the two

firms produce imperfect substitutes


• Here we will consider the case where the two firms produce imperfect substitutes• The demand for firm 1 will increase at any price for

firm 1 if the price of firm 2 rises.• The demand for firm 1 will decrease at any price for

firm 1 if the price of firm 2 falls.• And vice-versa for firm 2

We need to find the best response functions (or reaction curves)

‘Firm one’s best response function

Price of firm 2

Profit maximising

price for firm 1

This is the diagram we need to fill in for firm 1. GIVEN a particular price set by firm 2, what is firm 1’s best response?


Quantity sold by firm 1

$

Firm 1’s demand

Marginal revenue

Marginal cost

Q1

P1*

Firm 1’s demand depends on the price set by firm 2.

Here is firm 1’s demand and profit maximising price

for a particular value of P2, the price set by firm 2.



$

Marginal cost

Q1

P1*

If firm 2’s price falls, demand for firm 1 falls.



$

Marginal cost

Original P1*

And as a result, the profit maximising price

for firm 1 also falls.

New marginal revenue

New P1*



$

Marginal cost

Q1

And if firm 2 sets a ridiculously low price (e.g.

gives its product away) then firm 1’s demand

will be very low.

Original P1*



$

Marginal cost

Q1

In this case, firm 1’s profit maximising price

is also very low – but is still positive.

New P1*

Original P1*




$

Marginal cost Q1

Original P1*

Of course, if firm 2’s price rises, then

demand for firm 1 rises.



$

Marginal cost Q1

Original P1*


New P1*

And in this case firm 1’s profit maximising

price rises.


• So if firm 2’s price goes up, the profit maximising price for firm 1 goes up

• And if firm 2’s price goes down, the profit maximising price for firm 1 goes down

• So prices are ‘strategic complements’ – they move in the same direction


Price of firm 2

Profit maximising

price for firm 1

We can plot firm 1’s best response function. It starts at a positive price and slopes up. In general, it also has a slope less than 1 for imperfect substitutes.



P2

P1We can do the same for firm 2.




P2

P1And get the Bertrand price equilibrium



PB1

PB2

If firm 1’s marginal cost falls …

• Suppose firm 1’s marginal cost falls but not firm 2’s

• Then for any given price of firm 2, firm 1’s profit maximising price will fall

• So firm 1’s best response function shifts ‘down’ if firm 1’s marginal costs fall

If firm 1’s marginal cost falls …

P2

P1

Firm 1’s new best response function with lower marginal cost


PB1

PB2

Firm 1’s costs fall

P2

P1

PB1 (original

costs)

PB2 (original costs)

PB1 (new costs)

PB2 (new costs)

If firm 1’s costs fall, both firms lower their prices. Firm 2 makes less profit as its demand has fallen. Firm 1’s profit rises as its marginal costs fall as it is cheaper to make its product. But this benefit is at least partially offset by firm 2 dropping its price.

Strategic analysis of a drop in firm 1’s marginal costs

• Suppose firm 1 can invest in new equipment (a fixed cost) that will reduce its marginal cost. • Under both Cournot and Bertrand consumers win

because prices drop• Under both Cournot and Bertrand, firm 2 loses as firm

1 becomes a stronger competitor. Firm 2 ends up with lower profits and a lower price.

• In both cases firm 1 gains because its marginal costs drop

• BUT – the strategic effect differs

Strategic analysis of a drop in firm 1’s marginal costs

• Suppose firm 1 can invest in new equipment (a fixed cost) that will reduce its marginal cost. • Under Cournot, as firm 1 expands its output, firm 2

‘backs off’ and gives up market share to firm 1.• Under Bertrand, firm 2 responds to firm 1’s increased

competitiveness by dropping its price to try and retain its customers

• So the strategic effect helps firm 1 in Cournot competition and hurts firm 1 in Bertrand competition

Which is correct?

• They both are in the appropriate situations

• To analyse firm behaviour (or any other strategic situations) you need to study the ‘game’ carefully. ‘Small’ changes in strategic interaction can lead to ‘big’ differences in outcomes• (Of course this is why game theory and industrial

economics is interesting)

Summary

• We can capture simple strategic interaction between small numbers of firms (oligopoly) using the Cournot or Bertrand models

• Sometimes these models lead to different predictions – so use wisely

• More generally, when considering strategy, we need to carefully analyse the real world – there are no simple ‘rules’ that always apply.

Reminder: ‘Cournot’ competition in quantities

Q2

Q1

Case of two firms

Qm


Qm


Q1c

Q2c

Strategy for quantity competition

• If the other firm decreases its output then you can raise your output and increase your profits. • You can do this if you lower your rival’s best

response function. For example if you push up your rival’s costs

• If you can commit to increase your output beyond the equilibrium output then your rival will respond by lowering its output• This raises your profit so long as you do not

increase your output ‘too much’

Reminder: ‘Bertrand’ differentiated goods price competition

P1

P2


Firm two’s best response function

P1B

P2B

The case of two firms

Strategy for price competition

• If the other firm increases its price then you can raise your price and increase your profits• You can do this by pushing out your rival’s best

response function. For example if you push up your rival’s costs

• If you can commit to increase your price beyond the equilibrium price then your rival will respond by also increasing its price• This raises your profit so long as you do not increase

your price by ‘too much’

So…

• Strategy is all about changing either your best response function or your rival’s best response function• It is about changing the game that you are

playing

• The basic strategic game involves your firm ‘doing something’ before competition between your firm and your rivals

The basic strategy game

Your firm

Choose strategic variable

Both firms

Market interaction: e.g. Cournot or Bertrand competition

But what strategic variables will work?

• For a strategic variable to ‘work’ it must be a credible commitment.• In other words, choosing an action before

competition must affect the actual competition game

• It is not enough to merely threaten and bluster – that is simply cheap talk

Cheap talk in the prisoners’ dilemma

-10, -10 0, -30

-30, 0 -1, -1

Confess

Confess

Don’t Confess

Don’t Confess

Ned

Kelly

Ned Make promise

Ned promises to ‘not confess’. Should Kelly believe him and play ‘don’t confess’? No –

because it pays Ned to cheat.

-10, -10 0, -30

-30, 0 -1, -1

Confess

Confess

Don’t Confess

Don’t Confess

Ned

Kelly

Ned Make promise

Cheap talk

• Just promising to do something or threatening to do something that is not in your own interest is cheap talk.

• To have an effect a strategy must have two requirements• It must be credible – in other words it must be

something you have done and cannot undo easily• It must change the payoffs in the game so that your

rival changes its behaviour in a way that benefits you over all.

Example – commitment to a high price

P1

P2

Firm one’s original best response function


P1B

P2B

Suppose firm one can commit not to lower its price below P1*

P1*


P1

P2


Firm one’s new best response function

P1*

This commitment by firm 1 leads firm 2 to also raise its price – so both firms make more profit


• The commitment to a high price is a ‘soft’ strategy – it raises your rival’s profit as well as your profit

• So it changes the game. But how do we make it credible?• Cannot just ‘promise to raise price’• Can offer a ‘most favoured customer’ agreement to

clients at high price today. This means that if you lower your price in the future, you have to rebate money to all your old customers. It hurts you to drop your price tomorrow – so it is a credible commitment to keep a high price in the future.

Example – commitment to a high capacity

Q2

Q1

Qm


Qm


Q1e

Q2e

Q1*

Suppose that firm 1 can commit to produce at least Q1*

Example – commitment to a high quantity

Q2

Q1

Qm

Firm 1’s new best response function

Qm


Q1*

This commitment makes firm 2 reduce its output, raising profit for firm 1 and lowering profit for firm 2.

Example – commitment to a high quantity

• This is a ‘tough’ strategy – it raises your profit but reduces your rival’s profit

• So it changes the game. But how do we make it credible?• By building your plant before your rival and• By committing to a large capacity and • By making it difficult to reverse this choice (i.e. difficult to

lower capacity and• if it is difficult to run your plant below capacity, or• if it is cheap to produce output up to plant capacity

• Then you have a credible commitment to produce a high level of output. If your rival observes this commitment then your rival will reduce its plant size, raising your expected profits under competition.

• This is an example of a ‘first mover advantage’

Other examples

• Cortes and burning ships• The ‘pub fight’ from tutorials• International treaties (e.g. the WTO and

protectionism)• Mutually Assured Destruction (MAD) and

the cold war• Reputation (one ongoing player)• ‘Rational’ irrationality

Simultaneous games with continuous strategies Suppose two players have to choose a number between 0...

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