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
Best response function for
player 1
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
100
0
100
Player 2
Best response function for
player 2
• 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
Best response function for
player 2
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
100
0
100
Player 2
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 equilibrium
Player 1
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.
100
50
0
50
100
Player 2
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
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?
Output of
firm 2
Firm one’s best response
function
If firm 2 produces nothing, then the profit
maximising best response for firm 1 is to
produce the monopoly quantity
$
Market
demand
PM
Marginal cost
QM
Marginal revenue
Quantity
Firm one’s best response
function
If firm 2 produces and sells a
positive quantity then this reduces
the ‘residual’ demand for firm 1.
$
Market
demand
PM
Marginal cost
Firm 1’s
demand
curve given
firm 2’s sales
QM
Firm 2’s
output
Quantity
Firm one’s best response
function
$
Market
demand
PM
Marginal cost
QM
Firm 2’s
output
Quantity
Marginal revenue GIVEN
firm 2’s output
Firm one’s best response
functionSo 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.
$
PM
New
lower
market
price
Marginal cost
QM
Firm 2’s
output
Quantity
New optimal
quantity
Firm one’s best response
function
And if firm 2 produces ‘enough’, firm 1’s
best response is to produce nothing
$
Market
demand
PM
Marginal cost
Firm 1’s
demand
curve given
firm 2’s sales
Firm 2’s
output
Quantity
Firm one’s best response
we can plot firm 1’s best
function Soresponse
function. It starts at
Profit
maximising
quantity for
firm 1
the monopoly quantity then
falls with a slope less than 1.
Firm 1’s best response
function
Qm
Output of
firm 2
Q1
‘Cournot’ competition in
quantities We can do the same for firm 2.
Firm 2’s best response
function
Qm
Firm 1’s best response
function
Qm
Q2
Q1
‘Cournot’ competition in
quantities And get the Cournot
equilibrium.
Firm 2’s best response
function
Qm
Firm 1’s best response
function
Qc1
Qc
2
Qm
Q2
Symmetric firms and ‘Cournot’
competition
Firm 2’s best response
Q1
function
45 degree
line
If the two firms are identical
then Qc1 = Qc2. Also total
output exceeds the monopoly
quantity.
Qm
Firm 1’s best response
function
Qc1
(Qm )/2
(Qm )/2
Qc
2
Qm
Q2
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
Q1
Firm 2’s best response
function
Qm (new
costs)
Firm 1’s new best
response function
Qm (original
costs)
Qm
Q2
Firm 1’s costs fall
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.
Q1
New Qc1
Original Qc1
2
Original Qc
New Qc2
Q2
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
‘Bertrand’ competition in prices
• 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
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?
Price of
firm 2
Firm one’s best response
Firm 1’s demand depends on the price set by firm 2.
function
Here is firm 1’s demand and profit maximising price
for a particular value of P2, the price set by firm 2.
$
Firm 1’s
demand
P1*
Marginal cost
Q1
Marginal revenue
Quantity sold
by firm 1
Firm one’s best response
function
If firm 2’s price falls, demand for firm 1 falls.
$
P1*
Marginal cost
Q1
Quantity sold
by firm 1
Firm one’s best response
function
And as a result, the profit maximising price
for firm 1 also falls.
$
Original P1*
New P1*
Marginal cost
New marginal revenue
Quantity sold
by firm 1
Firm one’s best response
function
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
Quantity sold
by firm 1
Firm one’s best response
function
In this case, firm 1’s profit maximising price
is also very low – but is still positive.
$
Original P1*
Marginal cost
New P1*
Q1
New marginal revenue
Quantity sold
by firm 1
Firm one’s best response
function
$
Of course, if firm 2’s price rises, then
demand for firm 1 rises.
Original P1*
Marginal cost
Q1
Quantity sold
by firm 1
Firm one’s best response
function
$
And in this case firm 1’s profit maximising
price rises.
New P1*
Original P1*
Marginal cost
Quantity sold
by firm 1
Q1
New marginal revenue
Firm one’s best response
function
• 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
Firm one’s best response
can plot firm 1’s best
function We
response function. It starts at
Profit
maximising
price for firm
1
a positive price and slopes
up. In general, it also has a
slope less than 1 for
imperfect substitutes.
Firm 1’s best response
function
Price of
firm 2
‘Bertrand’ competition in prices
P1
We can do the same for firm 2.
Firm 2’s best response
function
Firm 1’s best response
function
P2
‘Bertrand’ competition in prices
P1
Firm 2’s best response
function
PB1
And get the Bertrand price
equilibrium
Firm 1’s best response
function
PB2
P2
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 …
P1
Firm 2’s best response
function
PB1
Firm 1’s new best
response function with
lower marginal cost
PB2
P2
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.
Firm 1’s costs fall
P1
PB1 (original
costs)
PB1 (new costs)
PB2 (original costs)
PB2 (new costs)
P2
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
Case of two firms
Q1
Firm 2’s best response
function
Qm
Firm 1’s best response
function
Q1 c
Q2 c
Qm
Q2
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
P1
Firm two’s best
response function
Firm one’s best response
function
B
P2
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
Both
firms
Your
firm
Market interaction:
e.g. Cournot or
Bertrand competition
Choose strategic
variable
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
Ned
Make
promise
Ned
Confess
Confess
Don’t
Confess
-10, -10
0, -30
-30, 0
-1, -1
Kelly
Don’t
Confess
Ned promises to ‘not confess’. Should Kelly
believe him and play ‘don’t confess’? No –
because it pays Ned to cheat.
Ned
Make
promise
Ned
Confess
Confess
Don’t
Confess
-10, -10
0, -30
-30, 0
-1, -1
Kelly
Don’t
Confess
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
Firm two’s price
best
response function
P1
P1*
Firm one’s original best
response function
P1B
P2
P2B
Suppose firm one can commit not to lower its price below P1*
Example – commitment to a high
Firm two’s price
best
response function
P1
P1*
Firm one’s new best
response function
P2
This commitment by firm 1 leads firm 2 to also raise its price – so both firms
make more profit
Example – commitment to a high
price
• 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
Q1
Firm 2’s best response
function
Qm
Firm 1’s best response
function
Q1 *
Q1 e
Q2 e
Qm
Suppose that firm 1 can commit to produce at least Q1*
Q2
Example – commitment to a high
quantity
Q1
Firm 2’s best response
function
Qm
Firm 1’s new best
response function
Q1 *
Qm
Q2
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