Mixing
Does this game have a core?
$begingroup$
I'm trying to find the core of this cooperative game:
$N = {1,2,3}$ and $v({1})=24$, $v({2})=24$, $v({3})=26$, $v({1,2})=42$, $v({1,3})=44$, $v({N})=52$.
My solution:
$x_1 ge 24$
$x_2 ge 24$
$x_3 ge 26$
$x_1+x_2 ge 42$
$x_1+x_3 ge 34$
$x_2+x_3 ge 44$
$x_1+x_2+x_3=52$
Manipulating the last four expressions I obtained the following inequalities:
$x_1 le 10$
$x_2 le 18$
$x_3 le 8$
Which (I think) are in contrast with the first three, so the core should be empty (?).
My book instead gives as a solution: $co {(24, 18, 10), (18, 24, 10), (8, 24, 20), (8, 18, 26) } $.
Solution diagram
What am I missing?
game-theory
asked Feb 1 at 1:39
ccmptccmpt
82
$endgroup$
add a comment |
$begingroup$
I'm trying to find the core of this cooperative game:
$N = {1,2,3}$ and $v({1})=24$, $v({2})=24$, $v({3})=26$, $v({1,2})=42$, $v({1,3})=44$, $v({N})=52$.
My solution:
$x_1 ge 24$
$x_2 ge 24$
$x_3 ge 26$
$x_1+x_2 ge 42$
$x_1+x_3 ge 34$
$x_2+x_3 ge 44$
$x_1+x_2+x_3=52$
Manipulating the last four expressions I obtained the following inequalities:
$x_1 le 10$
$x_2 le 18$
$x_3 le 8$
Which (I think) are in contrast with the first three, so the core should be empty (?).
My book instead gives as a solution: $co {(24, 18, 10), (18, 24, 10), (8, 24, 20), (8, 18, 26) } $.
Solution diagram
What am I missing?
game-theory
asked Feb 1 at 1:39
ccmptccmpt
82
$endgroup$
add a comment |
$begingroup$
I'm trying to find the core of this cooperative game:
$N = {1,2,3}$ and $v({1})=24$, $v({2})=24$, $v({3})=26$, $v({1,2})=42$, $v({1,3})=44$, $v({N})=52$.
My solution:
$x_1 ge 24$
$x_2 ge 24$
$x_3 ge 26$
$x_1+x_2 ge 42$
$x_1+x_3 ge 34$
$x_2+x_3 ge 44$
$x_1+x_2+x_3=52$
Manipulating the last four expressions I obtained the following inequalities:
$x_1 le 10$
$x_2 le 18$
$x_3 le 8$
Which (I think) are in contrast with the first three, so the core should be empty (?).
My book instead gives as a solution: $co {(24, 18, 10), (18, 24, 10), (8, 24, 20), (8, 18, 26) } $.
Solution diagram
What am I missing?
game-theory
asked Feb 1 at 1:39
ccmptccmpt
82
$endgroup$
I'm trying to find the core of this cooperative game:
$N = {1,2,3}$ and $v({1})=24$, $v({2})=24$, $v({3})=26$, $v({1,2})=42$, $v({1,3})=44$, $v({N})=52$.
My solution:
$x_1 ge 24$
$x_2 ge 24$
$x_3 ge 26$
$x_1+x_2 ge 42$
$x_1+x_3 ge 34$
$x_2+x_3 ge 44$
$x_1+x_2+x_3=52$
Manipulating the last four expressions I obtained the following inequalities:
$x_1 le 10$
$x_2 le 18$
$x_3 le 8$
Which (I think) are in contrast with the first three, so the core should be empty (?).
My book instead gives as a solution: $co {(24, 18, 10), (18, 24, 10), (8, 24, 20), (8, 18, 26) } $.
Solution diagram
What am I missing?
game-theory
game-theory
asked Feb 1 at 1:39
ccmptccmpt
82
asked Feb 1 at 1:39
ccmptccmpt
82
asked Feb 1 at 1:39
ccmptccmpt
82
asked Feb 1 at 1:39
ccmptccmpt
82
asked Feb 1 at 1:39
ccmptccmpt
82
82
add a comment |
add a comment |
1 Answer
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$begingroup$
You are completely right, the core must be empty. The quickest way to see that is to use the inequalites of the singleton coalitions, adding them up gives $sum_{i in N} x_{i} ge 24 + 24 +26 = 74 > 52 = v(N)$. Hence, the Pareto-efficiency constraint is violated, the core must be empty.
I guess that this is a misprint in the book. By the way, due to the above constraints set the value of coalition ${1,3}$ should be $34$ and that of coalition ${2,3}$ must be $44$.
answered Feb 1 at 9:43
Holger I. MeinhardtHolger I. Meinhardt
798147
$endgroup$
$begingroup$
Thank you for your help!
$endgroup$
– ccmpt
Feb 1 at 9:57
add a comment |
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
You are completely right, the core must be empty. The quickest way to see that is to use the inequalites of the singleton coalitions, adding them up gives $sum_{i in N} x_{i} ge 24 + 24 +26 = 74 > 52 = v(N)$. Hence, the Pareto-efficiency constraint is violated, the core must be empty.
I guess that this is a misprint in the book. By the way, due to the above constraints set the value of coalition ${1,3}$ should be $34$ and that of coalition ${2,3}$ must be $44$.
answered Feb 1 at 9:43
Holger I. MeinhardtHolger I. Meinhardt
798147
$endgroup$
$begingroup$
Thank you for your help!
$endgroup$
– ccmpt
Feb 1 at 9:57
add a comment |
$begingroup$
You are completely right, the core must be empty. The quickest way to see that is to use the inequalites of the singleton coalitions, adding them up gives $sum_{i in N} x_{i} ge 24 + 24 +26 = 74 > 52 = v(N)$. Hence, the Pareto-efficiency constraint is violated, the core must be empty.
I guess that this is a misprint in the book. By the way, due to the above constraints set the value of coalition ${1,3}$ should be $34$ and that of coalition ${2,3}$ must be $44$.
answered Feb 1 at 9:43
Holger I. MeinhardtHolger I. Meinhardt
798147
$endgroup$
$begingroup$
Thank you for your help!
$endgroup$
– ccmpt
Feb 1 at 9:57
add a comment |
$begingroup$
You are completely right, the core must be empty. The quickest way to see that is to use the inequalites of the singleton coalitions, adding them up gives $sum_{i in N} x_{i} ge 24 + 24 +26 = 74 > 52 = v(N)$. Hence, the Pareto-efficiency constraint is violated, the core must be empty.
I guess that this is a misprint in the book. By the way, due to the above constraints set the value of coalition ${1,3}$ should be $34$ and that of coalition ${2,3}$ must be $44$.
answered Feb 1 at 9:43
Holger I. MeinhardtHolger I. Meinhardt
798147
$endgroup$
You are completely right, the core must be empty. The quickest way to see that is to use the inequalites of the singleton coalitions, adding them up gives $sum_{i in N} x_{i} ge 24 + 24 +26 = 74 > 52 = v(N)$. Hence, the Pareto-efficiency constraint is violated, the core must be empty.
I guess that this is a misprint in the book. By the way, due to the above constraints set the value of coalition ${1,3}$ should be $34$ and that of coalition ${2,3}$ must be $44$.
answered Feb 1 at 9:43
Holger I. MeinhardtHolger I. Meinhardt
798147
edited Feb 1 at 10:16
answered Feb 1 at 9:43
Holger I. MeinhardtHolger I. Meinhardt
798147
answered Feb 1 at 9:43
Holger I. MeinhardtHolger I. Meinhardt
798147
answered Feb 1 at 9:43
Holger I. MeinhardtHolger I. Meinhardt
798147
798147
$begingroup$
Thank you for your help!
$endgroup$
– ccmpt
Feb 1 at 9:57
add a comment |
$begingroup$
Thank you for your help!
$endgroup$
– ccmpt
Feb 1 at 9:57
$begingroup$
Thank you for your help!
$endgroup$
– ccmpt
Feb 1 at 9:57
$begingroup$
Thank you for your help!
$endgroup$
– ccmpt
Feb 1 at 9:57
add a comment |
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