the basics of game theory

The basics of Game Theory

Understanding strategic behaviour


As we saw last week, oligopolies are a problem for classical theory 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 prisoner’s dilemma

Nash equilibrium and welfare

Mixed strategy equilibria

Retaliation


The prisoner’s dilemma is the “historical” game that founded game theory as a specific area of study: This is because the solution to this game is sub-

optimal 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.


A typical prisoner’s dilemma: 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 confess ?


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)

Payoff Matrix1st criminal

Confess Deny

2nd criminal

Confess2020

250.5

Deny0.525

11


The prisoner’s dilemma applied to a duopoly

Two firms competing on a market can: 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 ?


2 players :2 firms (A and B) producing the same

good (Airbus/Boeing fits well!!) 2 strategies :

Produce at the duopoly levelProduce at the cartel level (which is lower)

Given 2 players and 2 strategies, there are 4 possible market configurationsThese are listed in the payoff matrix


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


Payoff Matrix

Firm B

Qd Qc

Firm A

Qd

22

015

Qc

150

1010

For firm A:Qd if firm B chooses Qd

Qd if firm B chooses Qc

Note: the game is symmetric, so the dominant strategy is to produce the duopoly quantity.

What is the best strategy for each firm?

For firm B:Qd if firm A chooses Qd

Qd if firm A chooses Qc





Retaliation


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)


Let’s go back to the Duopoly example:

Payoff Matrix

Firm B

Qd Qc

Firm A

Qd

22

015

Qc

150

1010

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


Payoff Matrix

Firm B

Qd Qc

Firm A

Qd

22

015

Qc

150

1010

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





Retaliation


A pure-strategy Nash equilibrium does not exist for all games…

Example of a penalty shoot-out: 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?


For the striker:R if the keeper goes LL if the keeper goes R

Payoff Matrix

Goalkeeper

L R

Striker

L10

01

R01

10

For the goalkeeper:L if the striker shoots LR if the striker shoots R

No pure-strategy Nash equilibrium !

Whatever the outcome, one of the players can increase his sucess by changing strategy


Payoff Matrix

Goalkeeper

L R

Striker

L10

01

R01

10

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 0.25o The striker scores one out of two, the other is stopped by the goalkeeper


Let’s check that this is actually a Nash equilibrium:

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 !





Retaliation

Retaliation

Finally, the stability of the equilibrium also depends on whether the game is repeated or not. 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: 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 If one firm cheats, it captures all the profits for

that period What do you think happens in the next period?

Retaliation

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) 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. The threat of retaliation can bring the social optimum

The optimal retaliation strategy is also the simplest one: “tit for tat” 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.

the basics of game theory

Documents

prisoners dilemmafor

prisoners dilemmathey

prisoners dilemma2 players

duopoly profits

historical game

monopoly profits cartel

cartel case

high profits