Basics of Game Theory - unipi.it · Basics of Game Theory Giacomo Bacci and Luca Sanguinetti Department of Information Engineering University of Pisa, Pisa, Italy {giacomo.bacci,luca.sanguinetti}@iet

Basics of Game Theory

Giacomo Bacci and Luca Sanguinetti

Department of Information Engineering

University of Pisa, Pisa, Italy

{giacomo.bacci,luca.sanguinetti}@iet.unipi.it

April - May, 2010

G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, 2010 1 / 53

OutlineEconomic applications

Cournot model of duopoly

Bertrand model of duopoly

Stackelberg model of duopoly

Collusion between Cournot duopolists

Cournot competition under asymmetric information


Economic applicationsCommon problems

Consider a group of firms in an industry competing for the market.


Economic applicationsCommon problems

Consider a group of firms in an industry competing for the market.

Many though questions to answer...

How does the outcome depends on the firms’ output, firms’ nature, costfunctions and the number of firms?

Will the benefits of competition be passed on to customers?

Will a reduction in the number of firms generate a less desirable outcome?

Many others come to mind...

The first economist attempting to answer some of these questions was Cournot(1838).


Cournot model of oligopolyIntroduction

Cournot anticipated the notion of Nash’s equilibrium in 1838.

For simplicity, we focus on a particular model of duopoly in which two firms producethe same product in two different quantities, i.e., q1 and q2.

The goal of each firm is to maximize its profit. For firm 1, we have that

u1(q1, q2) = q1P (Q) − C1(q1)

where P (Q) is the inverse demand function and C1(q1) is its cost while

Q = q1 + q2

is the firms’ total output.


Cournot model of duopolyProblem formulation

The question we pose ourselves is simply:

What are the optimal quantities chosen simultaneously maximizing profits?



The question we pose ourselves is simply:

What are the optimal quantities chosen simultaneously maximizing profits?

Assume each firm’s cost per unit is the same (no variable costs)

C1(q1) = c · q1 with 0 ≤ c < a.

If some fixed costs are present, then

C1(q1) =

0 if q1 = 0f + c · q1 if q1 > 0

with 0 ≤ c < a and f > 0.

For simplicity, in the sequel we concentrate on the first case.



Suppose the inverse demand function is given by

P (Q) =

(

a − Q if Q < a

0 if Q ≥ a.

with a being the price at which customers are willing to pay the product.

Q

P(Q)a

Under the above circumstances, a response can be easily found.


Cournot model of duopolyGame formulation

Translate the problem into a strategic form.


Cournot model of duopolyGame formulation

Translate the problem into a strategic form.

Players: firms 1 and 2

Strategies: quantities qi ∈ [0, +∞), i = {1, 2}

Utilities: each firm’s profit is given by

u1 (q1, q2) =

q1 · (a − c − Q) if Q ≤ a−cq1 if Q > a

u2 (q1, q2) =

q2 · (a − c − Q) if Q ≤ a−cq2 if Q > a


Cournot model of duopolyFirm’s profit

Firm’s profit as a function of q1 for a given q2 with a = 100 and c = 10.

0 20 40 60 80 100 120 140 160 180 200−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500

q

2 = 0

q2 = 20

q2 = 60

q2 = 95


Cournot model of duopolyNash equilibrium

Nash equilibria are such that

ui

`

q∗i , q∗\i

´

≥ ui

`

qi, q∗\i

´

∀qi ≥ 0.



Nash equilibria are such that

ui

`

q∗i , q∗\i

´

≥ ui

`

qi, q∗\i

´

∀qi ≥ 0.

The best response procedure is used to find Nash equilibria, i.e.,

q∗i = arg max0≤qi<∞

ui

`

qi, q∗\i

´

.



Consider firm 1 and assume q∗2 is given. Then,

q∗1 = arg max0≤q1<∞

u1 (q1, q∗2)

with

u1 (q1, q∗2) = q1 · (a − c − q1 − q∗2) .



Consider firm 1 and assume q∗2 is given. Then,


u1 (q1, q∗2)

with

u1 (q1, q∗2) = q1 · (a − c − q1 − q∗2) .

How to solve the problem?



When q∗2 < a − c, take the derivate with respect to q1 and set it to zero. This yields

a − c − q∗2 − 2q1 = 0.

The best response function is then given by

q1 =a − c − q∗2

2.



When q∗2 < a − c, take the derivate with respect to q1 and set it to zero. This yields

a − c − q∗2 − 2q1 = 0.

The best response function is then given by

q1 =a − c − q∗2

2.

On the other hand, when q∗2 ≥ a − c the maximum is achieved for

q1 = 0

since in these circumstances the profit becomes a negative-decreasing function.



Consider firm 2. Then, we have that


u1 (q∗1 , q2)

with

u2 (q∗1 , q2) = q2 · (a − c − q∗1 − q2)

from which using the same arguments it follows that

q2 =

8

<

:

12

(a − c − q∗1) if q∗1 < a − c

0 if q∗1 ≥ a − c



The Nash equilibrium is found looking for the solution of the following system

8

>

<

>

:

q∗1 =a − c − q∗2

2

q∗2 =a − c − q∗1

2



The Nash equilibrium is found looking for the solution of the following system

8

>

<

>

:

q∗1 =a − c − q∗2

2

q∗2 =a − c − q∗1

2

Solving the above system yields

q∗1 = q∗2 , q∗c =a − c

3

from which it follows that

u1 (q∗1 , q∗2) = u2 (q∗1 , q∗2) =(a − c)2

9.



Nash equilibria can also be found graphically.

Assume firm 1’s strategy satisfies q1 < a − c, firm 2’s best response is

R2(q1) =a − c − q1

2.

If q1 ≥ a − c, it takes the form

R2(q1) = 0.



q1

R2(q1)

0 a! c

a! c2



If q2 < a − c, then firm 1’s best response is

R1(q2) =a − c − q2

2.



If q2 < a − c, then firm 1’s best response is

R1(q2) =a − c − q2

2.

When q2 ≥ a − c, it follows that

R1(q2) = 0.



R1(q2)

q2

0

a! c

a! c2



The Nash equilibria are given by the intersection points.

q1

q2

0 a! c

a! c

a! c2

a! c3

a! c3

a! c2

!"

A unique Nash equilibrium exists.


Cournot model of duopolyAsymmetric costs

What does it happen if ci(qi) = ci?



What does it happen if ci(qi) = ci?

Following the same arguments yields

R1(q2) =

8

<

:

12

(a − c1 − q2) if q2 ≤ a − c1

0 otherwise

and

R2(q1) =

8

<

:

12

(a − c2 − q1) if q1 ≤ a − c2

0 otherwise



If c1 < 12(a + c2) then

q1

q2

0 a! c2

a! c1

a! c12

a! c22 !"

A unique Nash equilibrium exists given by

q∗1 =1

3(a − 2c1 + c2) and q∗2 =

1

3(a − 2c2 + c1)



If c1 ≥ 12(a + c2) then

q1

q2

0 a! c2

a! c1

a! c12

a! c22

!"

A unique Nash equilibrium exists given by

q∗1 = 0 and q∗2 =1

2(a − c2)


Cournot model of monopolyProblem formulation and solution

What if is there only one firm in the market (monopoly)?




The market price is now given by

P (Q) = a − q




The market price is now given by

P (Q) = a − q

A single-user optimization problem arises. The firm is better off producing

q∗ =a − c

2, q∗m

yielding the following monopoly profit

u(q∗m) =(a − c)2

4.


Cournot modelDuopoly vs. monopoly

Assume q1 = q2 = q∗m/2 = (a − c)/4. Each firm’s profit is then given by

u1 (q1, q2) = u2 (q1, q2) =1

8(a − c)2.

The above profit exceeds the Nash equilibrium profit, i.e.,

1

8(a − c)2 >

1

9(a − c)2.

Both firms would be better off cooperating.


Cournot modelDuopoly vs. monopoly

Assume q1 = q2 = q∗m/2 = (a − c)/4. Each firm’s profit is then given by

u1 (q1, q2) = u2 (q1, q2) =1

8(a − c)2.

The above profit exceeds the Nash equilibrium profit, i.e.,

1

8(a − c)2 >

1

9(a − c)2.

Both firms would be better off cooperating.

Why do they not cooperate at the equilibrium?


Cournot model of duopolyDuopoly vs. monopoly

The problem is that qi = q∗m/2 is not the best response to q\i = q∗m/2!

Indeed, when q∗2 = q∗m/2 we have that

u1 (q1, q∗2) = q1 ·

»

3

4(a − c) − q1

–

from which it follows that

q∗1 =3

8(a − c) >

1

4(a − c).

But if q∗1 = 3(a − c)/8 then

q∗2 =5

16(a − c) >

1

4(a − c).


Cournot model of duopolyDuopoly vs. monopoly

q1

q2

0 a! c

a! c

The procedure converges to the Cournot Nash equilibrium.


Bertrand model of duopolyProblem formulation

Assume now that firms choose prices rather than quantities (Bertrand’s - 1883).

Consider two firms selling differentiated products in the same market, using twodifferent prices, i.e., p1 and p2.

The quantity that consumers demand from each firm is a function of both its priceand its competitor’s.

As in the Cournot model, assume no fixed costs of production and simultaneousactions.


Bertrand model of duopolyGame formulation

Translate the above problem into a strategic game.


Bertrand model of duopolyGame formulation

Translate the above problem into a strategic game.

Players: firms 1 and 2

Strategies: prices pi ∈ [0, +∞), i = {1, 2}

Utilities: each firm’s profit. For firm 1, we have that

u1 (p1, p2) = q1(p1, p2) · (p1 − c)

whereq1 = a − p1 + b · p2

is the quantity that consumers demand from firm 1 while b > 0 reflects the extent towhich firm i’s product is a substitute for firm 2’s product.


Bertrand model of duopolyNash equilibrium

The price p1 is the best response to p∗2 if

p∗1 = arg max

0≤p1<∞u1(p1, p

∗2) = arg max

0≤p1<∞(a − p1 + b · p∗

2)(p1 − c)




p∗1 = arg max

0≤p1<∞u1(p1, p

∗2) = arg max

0≤p1<∞(a − p1 + b · p∗

2)(p1 − c)

The solution is found as follows

du1(p1, p∗2)

dp1

˛

˛

˛

˛

p1=p∗

1

= a − 2p∗1 + b · p∗

2 + c = 0.




p∗1 = arg max

0≤p1<∞u1(p1, p

∗2) = arg max

0≤p1<∞(a − p1 + b · p∗

2)(p1 − c)

The solution is found as follows

du1(p1, p∗2)

dp1

˛

˛

˛

˛

p1=p∗

1

= a − 2p∗1 + b · p∗

2 + c = 0.

Paralleling the steps for p∗2, it is found that Nash equilibria have to satisfy

8

>

<

>

:

p∗1 = 1

2(a + c + b · p∗

2)

p∗2 = 1

2(a + c + b · p∗

1)



The solution is found to be

p∗i =

a + c

2 − b, i = 1, 2

from which it follows that b ≤ 2 (otherwise the prices would be negative).

What happens if b = 0?



The solution is found to be

p∗i =

a + c

2 − b, i = 1, 2

from which it follows that b ≤ 2 (otherwise the prices would be negative).

What happens if b = 0?

The Bertrand equilibrium yields the monopoly quantity, i.e., no player interaction.


Stackelberg model of duopolyProblem formulation

Stackelberg (1934) proposed a dynamic model of duopoly

a leader moves first, and

a follower moves second.

Assume for simplicity the firms choose quantities (Cournot model).

The fundamental difference is that here the firms’ choices are sequential rather thansimultaneous.


Stackelberg model of duopolyProblem formulation

The timing of the Stackelberg model is as follows:

firm 1 chooses a quantity q1 ≥ 0;

firm 2 observes q1 and then chooses a quantity q2 ≥ 0;

the payoff of firm i ∈ {1, 2} is given by the profit function

ui

`

qi, q\i

´

= qi · (P (Q) − c)

where

P (Q) = a − qi − q\i

and c < a is the constant marginal cost (fixed costs are zero).


Stackelberg model of duopolyGame formulation

Translate the above problem into an extensive game.

Players: the leader (firm 1) and the follower (firm 2);

Terminal histories: (q1, q2), with q1, q2 ≥ 0;

Player function: P (∅) = leader, P (q1) = follower;

Payoffs: ui

`

qi, q\i

´

= qi · (p − c) = qi ·`

a − c − qi − q\i

´


Stackelberg model of duopolySubgame perfect equilibrium

Apply the backward-induction procedure.




Compute first firm 2’s reaction to an arbitrary quantity q1 by firm 1:

R2(q1) = arg max0≤q2<∞

u2(q1, q2) = arg max0≤q2<∞

q2 · (a − c − q1 − q2).






u2(q1, q2) = arg max0≤q2<∞

q2 · (a − c − q1 − q2).

This yields the same result obtained for the strategic Cournot game, i.e.,

R2(q1) =a − c − q1

2

provided that q1 < a − c.






u2(q1, q2) = arg max0≤q2<∞

q2 · (a − c − q1 − q2).

This yields the same result obtained for the strategic Cournot game, i.e.,

R2(q1) =a − c − q1

2

provided that q1 < a − c.

The difference is that R2(q1) is now firm 2’s reaction to firm 1’s observed quantity.



Firm 1 knows that q1 will be met with reaction R2(q1).

Then, firm 1’s problem in the first stage of the game amounts to solving


u1 (q1, R2(q1)) = arg max0≤q1<∞

q1 · [a − c − q1 − R2(q1)]

= arg max0≤q1<∞

q1 ·a − c − q1

2.



Firm 1 knows that q1 will be met with reaction R2(q1).

Then, firm 1’s problem in the first stage of the game amounts to solving


u1 (q1, R2(q1)) = arg max0≤q1<∞

q1 · [a − c − q1 − R2(q1)]

= arg max0≤q1<∞

q1 ·a − c − q1

2.

Solving the above problem yields

q∗1 =a − c

2.



Then, in the Stackelberg game we have that

q∗1 =a − c

2and q∗2 =

a − c

4.

Firm 1’s profit results given by

u1(q∗1 , q∗2) =

1

8(a − c)2

while firm 2’s profit is

u1(q∗1 , q∗2) =

1

16(a − c)2.


Stackelberg model of duopolyStackelberg vs. Cournot

By contrast in the strategic Cournot game each firm produces

q∗c =1

3(a − c)

and each profit is given by

u1 (q∗c , q∗c ) = u2 (q∗c , q∗c ) =(a − c)2

9.



Then, firm 1 is better off in the subgame perfect equilibrium, i.e.,

u1 (q∗1 , q∗2) =(a − c)2

8> u1 (q∗c , q∗c ) =

(a − c)2

9.



Then, firm 1 is better off in the subgame perfect equilibrium, i.e.,

u1 (q∗1 , q∗2) =(a − c)2

8> u1 (q∗c , q∗c ) =

(a − c)2

9.

On the other hand, firm 2 is always worse off

u2 (q∗1 , q∗2) =(a − c)2

16< u2 (q∗c , q∗c ) =

(a − c)2

9.

Differently from single-player problems, in multi-player problems more informationcan make a player worse off.

What is the impact of this information? Just try to remove it.



Consider firm 2 chooses q2 without observing q1.

What happens now? Firm 2 has to form a belief.

If firm 2 supposes that firm 1 will choose its Stackelberg quantity, then q∗2 = (a− c)/4.






But firm 1 knows that firm 2 will have this belief, then it prefers to choose its bestresponse to (a − c)/4. This yields

q∗1 =3

8(a − c)






But firm 1 knows that firm 2 will have this belief, then it prefers to choose its bestresponse to (a − c)/4. This yields

q∗1 =3

8(a − c)

Firm 2 knows this and then prefers to choose its best response to 3(a − c)/8, i.e.,

q∗2 =5

16(a − c)

Iterating this procedure, we end up with the Cournot equilibrium!G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, 2010 38 / 53

Collusion between Cournot duopolistsProblem formulation

Consider again a Cournot model for duopoly without a leader and a follower.

If qi = q∗c = (a − c)/3

ui

`

qi, q\i

´

, uc =1

9(a − c)2

while if qi = q∗m/2 = (a − c)/4

ui

`

qi, q\i

´

,um

2=

1

8(a − c)2.

Although

1

8(a − c)2 >

1

9(a − c)2

the absence of cooperation leads both firms to choose qi = q∗c .G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, 2010 39 / 53

Collusion between Cournot duopolistsProblem formulation

How to force cooperation in the Cournot model? Playing the game repeatedly!

Friedman (1971) showed that cooperation could be achieved in an infinitely repeatedgame by using trigger strategies.

The original application was for a Cournot duopoly.


Collusion between Cournot duopolistsTrigger strategy

Consider the infinitely repeated game based on the Cournot stage-game.

Assume both firms have the same discount factor γ.

Compute the values of γ for which the following trigger strategy is a subgame-perfectequilibrium.

Start producing q∗m/2. At time t, produce q∗m/2 if both firms have produced

q∗m/2 in the t − 1 previous periods; otherwise produce the Cournot quantity.



If firm \i is going to produce q∗m/2 at time t, then the quantity that maximizes firmi’s profit at time t is

q∗d =3

8(a − c)

where the subscript d stands for deviation.



If qi = q∗c = (a − c)/3 for i = 1, 2

ui

`

qi, q\i

´

, uc =1

9(a − c)2

If qi = q∗m/2 = (a − c)/4 for i = 1, 2,

ui

`

qi, q\i

´

,um

2=

1

8(a − c)2

If qi = q∗d and q\i = q∗m/2

ui

`

qi, q\i

´

, ud =9

64(a − c)2



The trigger strategy is a subgame perfect equilibrium provided that

um

2≥ (1 − γ) · ud + γ · uc



The trigger strategy is a subgame perfect equilibrium provided that

um

2≥ (1 − γ) · ud + γ · uc

Using all the above results yields γ ≥ 9/17.


Collusion between Cournot duopolistsModified trigger strategy

How can cooperation be enforced for γ < 9/17?



How can cooperation be enforced for γ < 9/17?

Simply computing for any γ the most profitable quantity q∗γ the firms can produce ifthey both play trigger strategies.

To compute this quantity, let us consider the following trigger strategy:

Start producing q∗γ . At time t, produce q∗γ if both firms have produced q∗γ in

the t − 1 previous periods; otherwise produce the Cournot quantity.



If both firms play q∗γ

ui

`

qi, q\i

´

, uγ = q∗γ · (a − c − 2q∗γ)

If one deviates, the deviation payoff is now q∗d′ = (a − c − q∗γ)/2 with

ui

`

qi, q\i

´

, u′d =

1

4(a − c − q∗γ)2



As before, the following inequality has to be satisfied

uγ ≥ (1 − γ) · u′d + γ · uc.




uγ ≥ (1 − γ) · u′d + γ · uc.

Solving with respect to γ produces

q∗γ =9 − 5γ

3(9 − γ)(a − c).




uγ ≥ (1 − γ) · u′d + γ · uc.

Solving with respect to γ produces

q∗γ =9 − 5γ

3(9 − γ)(a − c).

How does q∗γ vary with γ?



How does q∗γ vary with γ?

q∗γ =9 − 5γ

3(9 − γ)(a − c)

!

q*

0 9/17

a! c3

a! c2

!

Clearly, for γ = 0 the Cournot strategy is played.


Cournot competition under asymmetric informationProblem formulation

Consider again a Cournot duopoly model with

pi(Q) = pi

`

qi, q\i

´

=

(

a − Q, Q < a

0, Q ≥ a

The payoff function for both firms is equal to

ui

`

qi, q\i

´

= qi · (pi − ci)

and the cost for firm 1 is c1(q1) = c (and this is known to both firms) while firm 1knows only that firm 2’s marginal cost c2(q2) is given by

c2(q2) =

(

cH , with probability θ

cL, with probability 1 − θ


Cournot competition under asymmetric informationNash equilibrium

The Nash equilibrium is computed as it were a three-player game.




If firm 2 cost is cH , it will choose q∗2(cH) to solve

q∗2(cH) = arg max0≤q2<∞

u2 (q∗1 , q2)

= arg max0≤q2<∞

(a − q∗1 − q2 − cH) q2

=a − cH − q∗1

2




If firm 2 cost is cH , it will choose q∗2(cH) to solve

q∗2(cH) = arg max0≤q2<∞

u2 (q∗1 , q2)

= arg max0≤q2<∞

(a − q∗1 − q2 − cH) q2

=a − cH − q∗1

2

If firm 2 cost is cL instead, it will choose q∗2(cL) to solve

q∗2(cL) = arg max0≤q2<∞

u2 (q∗1 , q2)

= arg max0≤q2<∞

(a − q∗1 − q2 − cL) q2

=a − cL − q∗1

2



Firm 1 chooses q∗1 to solve the following optimization problem

q1 = arg max0≤q1<∞

u1 (q1, q∗2)

= arg max0≤q1<∞

θ · [(a − q1 − q∗2(cH) − c) q1] + (1 − θ) · [(a − q1 − q∗2(cL) − c) q1]

=θ · (a − c − q∗2(cH)) + (1 − θ) · (a − c − q∗2(cL))

2



Firm 1 chooses q∗1 to solve the following optimization problem

q1 = arg max0≤q1<∞

u1 (q1, q∗2)

= arg max0≤q1<∞

θ · [(a − q1 − q∗2(cH) − c) q1] + (1 − θ) · [(a − q1 − q∗2(cL) − c) q1]

=θ · (a − c − q∗2(cH)) + (1 − θ) · (a − c − q∗2(cL))

2

The Nash equilibrium is found solving the system

8

>

>

>

>

<

>

>

>

>

:

q∗1 =θ · (a − c − q∗2(cH)) + (1 − θ) · (a − c − q∗2(cL))

2

q∗2(cH) =a − cH − q∗1

2

q∗2(cL) =a − cL − q∗1

2



It turns out that

q∗1 =a − 2c + θ · cH + (1 − θ) · cL

3

q∗2(cH) =a − 2cH + c

3+

1 − θ

6(cH − cL)

q∗2(cL) =a − 2cL + c

3−

θ

6(cH − cL)



It turns out that

q∗1 =a − 2c + θ · cH + (1 − θ) · cL

3

q∗2(cH) =a − 2cH + c

3+

1 − θ

6(cH − cL)

q∗2(cL) =a − 2cL + c

3−

θ

6(cH − cL)

Compare this solution to the Cournot equilibrium under complete information withunequal costs c1 and c2, i.e.,

q∗1 =a − 2c1 + c2

3

q∗2 =a − 2c2 + c1

3

For example, how about q∗2(cH) and q∗2(cL)?G. Bacci and L. Sanguinetti (IET) Basics of Game Theory April - May, 2010 52 / 53


It is easily seen than q∗2(cH) > q∗2 while q∗2(cL) < q∗2 . Why?




This occurs because firm 2 not only tailors its quantity to its cost but also respondsto the fact that firm 1 cannot do so.




This occurs because firm 2 not only tailors its quantity to its cost but also respondsto the fact that firm 1 cannot do so.

For instance, assume firm 2’s cost is high.

The best quantity q2 is higher than that in the complete-information case becausefirm 2 knows that firm 1 will produce a quantity that maximizes its expected profit.

This expected profit is smaller than that firm 1 would produce if it knew firm 2’s costto be high.


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Basics of Game Theory - unipi.it · Basics of Game Theory Giacomo Bacci and Luca Sanguinetti Department of Information Engineering University of Pisa, Pisa, Italy {giacomo.bacci,luca.sanguinetti}@iet