The basics of Game
Theory
Understanding strategic
behaviour
The basics of Game Theory

As we saw last week, oligopolies are a
problem for classical theory
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
The best strategy for a firm depends on what the
other firm decides to do
Unless some assumption is made, the solution
can’t be found...
Game theory is the study of the strategic
behaviour of agents

Not just useful in economics, but also in
international relations, games of money, etc.
The basics of Game Theory
The prisoner’s dilemma
Nash equilibrium and welfare
Mixed strategy equilibria
Retaliation
The prisoner’s dilemma

The prisoner’s dilemma is the « historical » game
that founded game theory as a specific area of
study:
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This is because the solution to this game is suboptimal from the point of view of the players. This
means that there is a solution that makes both
players better off, but the rationality of the agents
does not lead to it.
The prisoner’s dilemma shows quite elegantly how
difficult it is to get agents to cooperate, even when
this cooperation is beneficial to all agents.
The prisoner’s dilemma

A typical prisoner’s dilemma:

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Two suspected criminals are caught by the
police, but the police lacks the hard evidence to
charge them.
They can only sentence them to 1 year for minor
misdemeanours.
The police needs to get them to confess their
crimes in order to be able to charge them both
to 20 years.
How do the police get the suspects to
cooperate ?
The prisoner’s dilemma

They offer the criminals a “deal”...

If one of them “spills the beans” on his colleague,
he gets a reduced sentence (6 months), and the
other guys gets a extended one (25 years)
1st criminal
Payoff Matrix
Confess
Deny
Confess
20
20
25
0.5
Deny
0.5
25
1
1
2nd criminal
The prisoner’s dilemma
The prisoner’s dilemma applied to a duopoly

Two firms competing on a market can:
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Compete (This leads, for example, to the
Cournot solution)
Collude and share monopoly profits (cartel).
Profit in a cartel > profit in a duopoly.
If collusion is not illegal, then it is clearly the
optimal situation from the point of view of
these two firms. But is it the equilibrium the
market ends up in ?
The prisoner’s dilemma

2 players :

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2 firms (A and B) producing the same
good (Airbus/Boeing fits well!!)
2 strategies :
Produce at the duopoly level
 Produce at the cartel level (which is lower)


Given 2 players and 2 strategies, there
are 4 possible market configurations
 These are listed in the payoff matrix
The prisoner’s dilemma

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Let’s put some numbers on the different possible profits:
For the Cartel case:
 Each firm earns a share of the monopoly profits:
Πc = 10
For the duopoly competition case :
 Each firm earns duopoly profits, which are lower:
Πd = 2
For the « cheating » case:
 The firm producing at duopoly level captures the market
share of the other firm, and makes very high profits :
Πt = 15
 The other firm is penalised and earns minimum profits :
Πm = 0
The prisoner’s dilemma
Firm B
Payoff
Matrix
Qd
Qc
Qd
2
2
0
15
Qc
15
0
10
10
Firm A
What is the best strategy
for each firm?
For firm A:
Qd if firm B chooses Qd
Qd if firm B chooses Qc
For firm B:
Qd if firm A chooses Qd
Qd if firm A chooses Qc
Note: the game is symmetric, so the dominant
strategy is to produce the duopoly quantity.
The basics of Game Theory
The prisoner’s dilemma
Nash equilibrium and welfare
Mixed strategy equilibria
Retaliation
Nash equilibrium and welfare
Definition of a Nash equilibrium:

A situation where no player can improve his outcome by
unilaterally changing his strategy
Central properties:
 The Nash equilibrium is generally stable
 Every game has at least one Nash equilibrium:
 Either in pure strategies : Players only play a single
strategy in equilibrium
 Or in mixed strategies : Players play a combination of
several strategies with a fixed probability
 The proof of this result is the main contribution of John
Nash (and the reason why it is called a Nash equilibrium)
Nash equilibrium and welfare
Let’s go back to the Duopoly example:
Firm B
Payoff
Matrix
Qd
Qc
Qd
2
2
0
15
Qc
15
0
10
10
Firm A
Is the “Qd-Qd” equilibrium
a Nash equilibrium ?
Can firm A or B improve
their outcome by shifting
alone to the cartel quantity
Qc ?
“Qd-Qd” is indeed a Nash
equilibrium
Nash equilibrium and welfare
Firm B
Payoff
Matrix
Qd
Firm A
Qc
Qd
Qc
2
2
0
15
15
0
10
10
So the dominant strategy
is to produce “Qd”
But the “Qd-Qd”
equilibrium is not socially
optimal
With a small number of
agents, individual
rationality does not
necessary lead to a social
optimum
The basics of Game Theory
The prisoner’s dilemma
Nash equilibrium and welfare
Mixed strategy equilibria
Retaliation
Mixed strategy equilibria
A pure-strategy Nash equilibrium does not exist
for all games…

Example of a penalty shoot-out:
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2 players: a goal-keeper and a striker
2 strategies : shoot / dive to the left or the right
We assume that the players are talented: The
striker never misses and the goalkeeper always
intercepts if they choose the correct side.
This is not required for the game, but it
simplifies things a bit!
What is the payoff matrix?
Mixed strategy equilibria
Payoff
Matrix
Goalkeeper
L
Striker
R
L
R
1
0
0
1
0
1
1
0
For the striker:
R if the keeper goes L
L if the keeper goes R
For the goalkeeper:
L if the striker shoots L
R if the striker shoots R
Whatever the outcome,
one of the players can
increase his sucess by
changing strategy
No pure-strategy Nash equilibrium !
Mixed strategy equilibria
Payoff
Matrix
Goalkeeper
L
R
L
1
0
0
1
R
0
1
1
0
Striker
There is, however, a mixed
strategy equilibrium
Strategy for both players:
Go L and R 50% of the time (1
out of two, randomly)
That way :
o
Each outcome has a
probability of 0.25
o
The striker scores
one out of two, the
other is stopped by the
goalkeeper
Mixed strategy equilibria
Let’s check that this is actually a Nash equilibrium:

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The goalkeeper plays L and R 50% of the time. Can the
striker increase his score by changing his strategy?
The striker decides to play 60% left and 40% right. His new
success rate is:
(0.6 ✕ 0.5) + (0.4 ✕ 0.5) = 0.5
(0.3)
+
(0.2) = 0.5

By choosing 60-40, the striker scores more on the left
hand side, but less on the right. His success rate is the
same, his situation has not improved. This corresponds to
a Nash equilibrium !
The basics of Game Theory
The prisoner’s dilemma
Nash equilibrium and welfare
Mixed strategy equilibria
Retaliation
Retaliation

Finally, the stability of the equilibrium also
depends on whether the game is repeated
or not.
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The very concept of a mixed strategy equilibrium
depends on the repetition of the game through
time.
Even for a pure strategy equilibrium, the ability
to replay the game can influence the outcome
 Players can retaliate, and thus influence the
decisions of other players
Retaliation

Back to the duopoly case:
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The 2 firms agree to form a cartel, and maximise
joint profits.
There is, however, the temptation to cheat on
this agreement
Imagine now that the game is played several
times
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If one firm cheats, it captures all the profits for
that period
What do you think happens in the next period?
Retaliation
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Actually, this depends on whether the game
is repeated a fixed number of times or
indefinitely (open-ended)...
Let’s say that our 2 firms decide to play the
game 5 times (5 years)
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What is the best strategy on year 5 ?
What about year 4, given what we know about
year 5 ?
This process shows that the equilibrium cannot
be stable
Retaliation

Lets imagine now that our 2 firms have an
open-ended agreement.
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The optimal retaliation strategy is also the
simplest one: “tit for tat”
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The threat of retaliation can bring the social optimum
Robert Axelrod: just choose what your opponent
did last period: cooperate if he cooperated, cheat
if he cheated.
But the threat needs to be credible i.e. the
opponent needs to believe that it will
effectively be carried out.