econs 424 - repeated games ii - washington state university

EconS 424 - Repeated Games II

Félix Muñoz-García

Washington State University

[email protected]

March 25, 2014

Félix Muñoz-García (WSU) EconS 424 - Recitation 7 March 25, 2014 1 / 28

Harrington, Ch. 14 "Check your Understanding" 1

Recall the Christie�s and Sotheby�s in�nitely repeated game fromclass:

0,0 2,1

1,2 4,4

4%

4%

Christies

Sothebys

1,4 1,7

4,1

7,1

5,5

6% 8%

6%

8%



Suppose that Christie�s and Sotheby�s want to collude to charge acommission rate of 8%, giving payo¤s of 5 for both auction houses.Suppose that if deviation were detected, the punishment is a twoperiod price war in which both auction houses set a commission rateof 4%.

Derive the conditions for this strategy pair to be a subgame perfectNash equilibrium.



First, note that the Nash equilibrium of the unrepeated game isf6%, 6%g:

0,0 2,1

1,2 4,4

4%

4%

Christies

Sothebys

1,4 1,7

4,1

7,1

5,5

6% 8%

6%

8%



Histories can be partitioned into three types.

First, suppose the history is such that both auction houses are to seta rate of 8%. Then the equilibrium condition is

51� δ

� 7+ δ� 0+ δ2 � 0+ δ3�

51� δ

�where the 7 represents the instantaneous gain in utility from deviatingtowards 6%, when the other auction house still chooses thecooperative rate of 8%. In the following two periods the grim-triggerstrategy prescribes that both players punish by setting a rate of 4%(price war), yielding a payo of zero in both of these periods.Afterwards, cooperation is reestablished with a corresponding payo¤of 5 thereafter.



Now consider a history whereby there is to be a punishment startingin the current period. Then the equilibrium condition is

0+ δ� 0+ δ2�

51� δ

�� 1+ δ� 0+ δ2 � 0+ δ3

�5

1� δ

�In this case note that in the right hand side we consider the payo¤that I obtain if I deviate towards 6% when my opponent still selects4% (implementing the price war), which constitutes my mostpro�table deviation. If I were to do that, however, I would bepunished with a price war for two periods, returning to thecooperation rate thereafter.



Finally, if the auction houses are in the second period of thepunishment, then the equilibrium condition is

0+ δ

�5

1� δ

�� 1+ δ� 0+ δ2 � 0+ δ3

�5

1� δ

�A similar situation as in the previous slide applies.



Recall the ABM Treaty in�nitely repeated game with imperfectmonitoring game from class:

10,10 6,12

12,6 8,8

NoABMs

NoABMs

USA

USSR

18,0 14,2

0,18

2,14

3,3

LowABMs HighABMs

LowABMs

HighABMs



Suppose a technological advance improves the monitoring technology,so that the probability of detecting ABMs are as follows:

Number of ABMs Probability of Detecting ABMsNone 0Low .3High .75

Using the strategy pro�le just described, derive the equilibriumconditions.

If you answer correctly, then you will �nd that the restriction factor isless stringent, indicating that better monitoring makes cooperationeasier.



No ABMs is preferred to Low ABMs when

101� δ

� 12+ δ

�.3�

�3

1� δ

�+ .7�

�101� δ

��=) δ � 2

4.1� .49

where 0.3� 31�δ = 0.3�

�3+ 3δ+ 3δ2 + ...

�indicates the expected

and discounted stream of pro�ts that the deviating country obtains ifits deviation to Low is detected and therefore punished thereafter byreverting to the psNE of the unrepeated game, (High, High), with acorresponding payo¤ of 3.



Similarly, 0.7� 101�δ = 0.7�

�10+ 10δ+ 10δ2 + ...

�represents the

expected and discounted stream of pro�ts that the deviating countryobtains if its deviation to Low is undetected, and then returns to thecooperative outcome (NoABMs, NoABMs) thereafter.



No ABMs is preferred to High ABMs when

101� δ

� 18+ δ

�.75�

�3

1� δ

�+ .25�

�101� δ

��=) δ � 8

13.25� .60

This strategy pair is a subgame perfect Nash equilibrium when thediscount factor is at least 0.6, since δ � 0.6 is more restrictive thanδ � 0.49. With the weaker monitoring technology discussed in class,the discount factor had to be at least 0.74.



A similar intuition as above is now applicable for the expected streamof payo¤s that the deviating country obtains if its deviation to High isdetected, 0.75� 3

1�δ = 0.75��3+ 3δ+ 3δ2 + ...

�, or undetected,

0.25� 101�δ = 0.25�

�10+ 10δ+ 10δ2 + ...

�. Therefore, cooperation

can be sustained for a larger set of discount factors as the monitoringtechnology becomes more accurate (closer to perfect monitoring).


Oligopoly and Collusion when �rms compete a la Cournot

Assume that there are two �rms competing in quantities (a laCournot) with zero marginal costs and a demand function given by:

p(q1, q2) = 1� q1 � q2

so the pro�t function for �rm 1 is:

π1(q1, q2) = p � q1 = (1� q1 � q2) � q1= q1 � q21 � q1q2



Find the equilibrium if the game is played just one time.



The FOCs with respect to q1 are

1� 2q1 � 2q2 � 0

assuming we have an interior solution (meaning that both q1 and q2are positive), this FOC holds with equality. Solving for q1, we have�rm 1�s best response function

q1 =12� 12q2

Similarly by �rm 2 (since �rms 1 and 2 are symmetric):

q2 =12� 12q1



Hence,q1

q2

BR2(q1)

BR1(q2)

1

1

½

½

⅓

⅓



The Nash equilibrium is given by:

q1 =12� 12

�12� 12q1

�=) qCournot1 = qCournot2 =

13

So the total production in the market is QCournot = 13 +

13 =

23 and

the market price is

pCournot = 1� 13� 13=13

In terms of pro�ts, each �rm i will get:

πi = p � qiπCournoti =

13� 13=19



Find the equilibrium in a cartel (that is if the two �rms collude)



In a cartel, they maximize their joint pro�ts, as a monopoly.

Π(Q) = p �Q = (1�Q) �Q= Q �Q2

The FOC with respect to Q is:

1� 2Q = 0



Solving for Q

QCartel =12

So each �rm will product half of QCartel :

qCartel1 = qCartel2 =QCartel

2=14

Thus, the price is

pCartel = 1� 12=12



So the pro�ts for each �rm i in the cartel are

πCarteli =12� 14=18

and as we can see

πCarteli =18> πCournoti =

19



Can the two �rms achieve cooperation with the cartel quantity?



Using the following Grim-Trigger strategy, in time period t:

If t = 1qi = q

Carteli =

14, thus πi =

18

If t > 1

qi =�qCarteli = 1

4 as long as qi = qj =14 in every previous period

qCournoti = 13 forever otherwise

with corresponding pro�t levels

πi =

�πCarteli = 1

8 as long as qi = qj =14 in every previous period

πCournoti = 19 forever otherwise



Thus, by cooperating:

18+ δ

18+ δ2

18+ ... =

18� 11� δ



Let us now evaluate the �rm�s pro�ts from deviating : (But towardswhat?)The optimal deviation is:

maxqi

�1� qCartelj � qi

�qi

where �rm j sticks to the collusive agreement (producing qCartelj = 14 )

and �rm i deviates. The pro�t maximization problem for �rm i is now

maxqi

�1� 1

4� qi

�qi

We are now ready to �nd out which value of qi minimizes �rm i�spro�ts given that the other �rm (�rm j) repects the collusiveagreement (j cooperates). i.e.,

maxqi

�qi � qi

14� q2i

�Félix Muñoz-García (WSU) EconS 424 - Recitation 7 March 25, 2014 26 / 28


The FOC with respect to qi is

1� 14� 2qi � 0

and like before, we assume an interior solution and solve for qi ,

qDevi =38

With pro�ts as

πDevi =

�1� 1

4� qDevi

�qDevi

=

�1� 1

4� 38

�38=964



So the cooperation will be observed if and only if:

Payo¤ fromcooperatingz }| {

πCarteli � 11� δ

�

Payo¤ fromdeviatingz}|{πDevi +

Punishmentthereafterz }| {

πCournotiδ

1� δ18� 11� δ

� 964+19� δ

1� δ

Solving for δ, this condition holds when

δ � 917

so collusion is still possible among �rms is they do not discount thefuture too much.


econs 424 - repeated games ii - washington state university

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