Chapter15: Oligopoly And Strategic Behavior
Section: Chapter Questions
Problem 17P
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Only help with 3.5, 3.6, 3.7 and 3.8 please
![3. Consider the two-player simultaneous game below:
P2
Rock
Раper
Scissors
Rock
0,0
-1, 1
1, -1
P1
Раper
1, -1
0,0
-1, 1
Scissors
-1, 1
1, -1
0,0
3.1) Does the game have any NEs? Explain.
Now consider a three-player game G = (S¡,u¿)ie[131 where S; is the strategy space for player i
and uj = u;(S1, S2, S3) is the utility function for player i, with s; being the strategy played by
player i. Let' S; = {Rock, Paper, Scissors} V i E {1,2,3}. Then let,
Z = {(Rock, Scissors), (Scissors, Paper), (Paper, Rock)}
1 if (x1, x2) E Z
-1 if (x2, x1) E Z
0 otherwise
P(x1,X2)
u;(S1, S2, S3) = Eje{1,2,3}|j#i P(Sį, S;) v ie [1,3].
This fully specifies G. To check your understanding, make sure that you can see, for example,
that the above imply that u2 (Scissors, Paper, Scissors) = -2.
3.2) Give a brief intuitive explanation of how this game works.
3.3) What payoffs do the players get when they choose the joint strategy (Rock, Paper,
Scissors)?
3.4) If i = 1, enumerate (list all the items in) S-i, the set of joint strategies of all players except
player i.
3.5) Are there any strictly dominated strategies in this game?
3.6) Is the joint strategy (Rock, Paper, Scissors) a NE? Explain.
3.7) Now find all the pure strategy NE(s) of this game. Do not draw tables or graphs; use the
game's symmetry to simplify your analysis. [Hint: Even though there are in principle 27
possible joint strategies that need to be considered, these can be categorised into four
equivalence classes (categories):
1: All players play the same strategy (there are three such joint strategies),
1 Recall that the symbol V means "for all".
2: All players play different strategies (there are six such joint strategies),
3: A joint stratey such as (Paper, Paper, Rock) where the "odd" player loses against the
other two (there are nine joint strategies in this class),
4: Finally, a joint strategy such as (Rock, Rock, Paper) where the "odd" player wins
against the other two (nine more joint strategies).
All you need to do is to determine whether a single joint strategy in each of these four classes
represent a NE.]
3.8) (Optional) Would increasing the number of players make a difference? [A comprehensive
analysis is not needed.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87b53351-2067-4a4d-a81c-a5cf0ffaa5a8%2Ff43600e0-da6a-43f6-88a7-5b6e9c52e522%2Fc4lrgea_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Consider the two-player simultaneous game below:
P2
Rock
Раper
Scissors
Rock
0,0
-1, 1
1, -1
P1
Раper
1, -1
0,0
-1, 1
Scissors
-1, 1
1, -1
0,0
3.1) Does the game have any NEs? Explain.
Now consider a three-player game G = (S¡,u¿)ie[131 where S; is the strategy space for player i
and uj = u;(S1, S2, S3) is the utility function for player i, with s; being the strategy played by
player i. Let' S; = {Rock, Paper, Scissors} V i E {1,2,3}. Then let,
Z = {(Rock, Scissors), (Scissors, Paper), (Paper, Rock)}
1 if (x1, x2) E Z
-1 if (x2, x1) E Z
0 otherwise
P(x1,X2)
u;(S1, S2, S3) = Eje{1,2,3}|j#i P(Sį, S;) v ie [1,3].
This fully specifies G. To check your understanding, make sure that you can see, for example,
that the above imply that u2 (Scissors, Paper, Scissors) = -2.
3.2) Give a brief intuitive explanation of how this game works.
3.3) What payoffs do the players get when they choose the joint strategy (Rock, Paper,
Scissors)?
3.4) If i = 1, enumerate (list all the items in) S-i, the set of joint strategies of all players except
player i.
3.5) Are there any strictly dominated strategies in this game?
3.6) Is the joint strategy (Rock, Paper, Scissors) a NE? Explain.
3.7) Now find all the pure strategy NE(s) of this game. Do not draw tables or graphs; use the
game's symmetry to simplify your analysis. [Hint: Even though there are in principle 27
possible joint strategies that need to be considered, these can be categorised into four
equivalence classes (categories):
1: All players play the same strategy (there are three such joint strategies),
1 Recall that the symbol V means "for all".
2: All players play different strategies (there are six such joint strategies),
3: A joint stratey such as (Paper, Paper, Rock) where the "odd" player loses against the
other two (there are nine joint strategies in this class),
4: Finally, a joint strategy such as (Rock, Rock, Paper) where the "odd" player wins
against the other two (nine more joint strategies).
All you need to do is to determine whether a single joint strategy in each of these four classes
represent a NE.]
3.8) (Optional) Would increasing the number of players make a difference? [A comprehensive
analysis is not needed.]
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Step 1: Define Nash equilibrium
VIEWStep 2: Explain any strictly dominated strategies in this game
VIEWStep 3: Explain weather the joint strategy (Rock, Paper, Scissors) a NE
VIEWStep 4: Explain the pure strategy NE(s) of this game
VIEWStep 5: Explain Would the increase in the number of players make a difference
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