Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
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.]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
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
VIEWSolution
VIEW
Step by step
Solved in 6 steps with 19 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Recommended textbooks for you
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Managerial Economics & Business Strategy (Mcgraw-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education