Finding Nash Equilibrium using Linear Program with strategy constraints
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Finding the Nash Equilibrium p in mixed strategies of a 2-player, symmetric zero-sum game with 3 pure strategies can be done by solving LP:
max $(0, 0, 0, 1)^{T}(p_1, p_2, p_3, epsilon)$
s.t. $A geq b$
Where A is:
begin{bmatrix}
a_{11} & a_{12} & a_{13} & -1 \
a_{21} & a_{22} & a_{23} & -1 \
a_{31} & a_{32} & a_{33} & -1 \
end{bmatrix}
and $b = (0,0,0)$
i.e. We maximize our worse payoff.
Now suppose player 2, who plays $q = (q_1, q_2, q_3)$ is under the constraint $q_1 + q_2 = c$, for some $c$ $0 leq c leq 1$.
Can this be solved? Thanks.
optimization linear-programming game-theory nash-equilibrium
asked Dec 22 '18 at 0:48
Agrim PathakAgrim Pathak
1113
$endgroup$
add a comment |
$begingroup$
Finding the Nash Equilibrium p in mixed strategies of a 2-player, symmetric zero-sum game with 3 pure strategies can be done by solving LP:
max $(0, 0, 0, 1)^{T}(p_1, p_2, p_3, epsilon)$
s.t. $A geq b$
Where A is:
begin{bmatrix}
a_{11} & a_{12} & a_{13} & -1 \
a_{21} & a_{22} & a_{23} & -1 \
a_{31} & a_{32} & a_{33} & -1 \
end{bmatrix}
and $b = (0,0,0)$
i.e. We maximize our worse payoff.
Now suppose player 2, who plays $q = (q_1, q_2, q_3)$ is under the constraint $q_1 + q_2 = c$, for some $c$ $0 leq c leq 1$.
Can this be solved? Thanks.
optimization linear-programming game-theory nash-equilibrium
asked Dec 22 '18 at 0:48
Agrim PathakAgrim Pathak
1113
$endgroup$
$begingroup$
you should provide some background info about how the linear optimization problem solves "We maximize our worse payoff." or how $q$ can appear
$endgroup$
– LinAlg
Dec 22 '18 at 13:16
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@LinAlgqis the opponent's strategy. It is not needed in the original "unconstrained" problem. For more information on the formulation, page 38 of this document might help: math.ucla.edu/~tom/Game_Theory/mat.pdf
$endgroup$
– Agrim Pathak
Dec 23 '18 at 9:04
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Hi! Did you eventually figure out what are conditions for such a constrained optimal mixed strategy to exist?
$endgroup$
– Ben Usman
Mar 5 at 14:35
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@BenUsman Hi, unfortunately I made no progress on this. There are however, iterative algorithms which can converge to NE.
$endgroup$
– Agrim Pathak
Mar 7 at 4:47
add a comment |
$begingroup$
Finding the Nash Equilibrium p in mixed strategies of a 2-player, symmetric zero-sum game with 3 pure strategies can be done by solving LP:
max $(0, 0, 0, 1)^{T}(p_1, p_2, p_3, epsilon)$
s.t. $A geq b$
Where A is:
begin{bmatrix}
a_{11} & a_{12} & a_{13} & -1 \
a_{21} & a_{22} & a_{23} & -1 \
a_{31} & a_{32} & a_{33} & -1 \
end{bmatrix}
and $b = (0,0,0)$
i.e. We maximize our worse payoff.
Now suppose player 2, who plays $q = (q_1, q_2, q_3)$ is under the constraint $q_1 + q_2 = c$, for some $c$ $0 leq c leq 1$.
Can this be solved? Thanks.
optimization linear-programming game-theory nash-equilibrium
asked Dec 22 '18 at 0:48
Agrim PathakAgrim Pathak
1113
$endgroup$
Finding the Nash Equilibrium p in mixed strategies of a 2-player, symmetric zero-sum game with 3 pure strategies can be done by solving LP:
max $(0, 0, 0, 1)^{T}(p_1, p_2, p_3, epsilon)$
s.t. $A geq b$
Where A is:
begin{bmatrix}
a_{11} & a_{12} & a_{13} & -1 \
a_{21} & a_{22} & a_{23} & -1 \
a_{31} & a_{32} & a_{33} & -1 \
end{bmatrix}
and $b = (0,0,0)$
i.e. We maximize our worse payoff.
Now suppose player 2, who plays $q = (q_1, q_2, q_3)$ is under the constraint $q_1 + q_2 = c$, for some $c$ $0 leq c leq 1$.
Can this be solved? Thanks.
optimization linear-programming game-theory nash-equilibrium
optimization linear-programming game-theory nash-equilibrium
asked Dec 22 '18 at 0:48
Agrim PathakAgrim Pathak
1113
asked Dec 22 '18 at 0:48
Agrim PathakAgrim Pathak
1113
edited Dec 22 '18 at 0:54
Agrim Pathak
asked Dec 22 '18 at 0:48
Agrim PathakAgrim Pathak
1113
asked Dec 22 '18 at 0:48
Agrim PathakAgrim Pathak
1113
asked Dec 22 '18 at 0:48
Agrim PathakAgrim Pathak
1113
1113
$begingroup$
you should provide some background info about how the linear optimization problem solves "We maximize our worse payoff." or how $q$ can appear
$endgroup$
– LinAlg
Dec 22 '18 at 13:16
$begingroup$
@LinAlgqis the opponent's strategy. It is not needed in the original "unconstrained" problem. For more information on the formulation, page 38 of this document might help: math.ucla.edu/~tom/Game_Theory/mat.pdf
$endgroup$
– Agrim Pathak
Dec 23 '18 at 9:04
$begingroup$
Hi! Did you eventually figure out what are conditions for such a constrained optimal mixed strategy to exist?
$endgroup$
– Ben Usman
Mar 5 at 14:35
$begingroup$
@BenUsman Hi, unfortunately I made no progress on this. There are however, iterative algorithms which can converge to NE.
$endgroup$
– Agrim Pathak
Mar 7 at 4:47
add a comment |
$begingroup$
you should provide some background info about how the linear optimization problem solves "We maximize our worse payoff." or how $q$ can appear
$endgroup$
– LinAlg
Dec 22 '18 at 13:16
$begingroup$
@LinAlgqis the opponent's strategy. It is not needed in the original "unconstrained" problem. For more information on the formulation, page 38 of this document might help: math.ucla.edu/~tom/Game_Theory/mat.pdf
$endgroup$
– Agrim Pathak
Dec 23 '18 at 9:04
$begingroup$
Hi! Did you eventually figure out what are conditions for such a constrained optimal mixed strategy to exist?
$endgroup$
– Ben Usman
Mar 5 at 14:35
$begingroup$
@BenUsman Hi, unfortunately I made no progress on this. There are however, iterative algorithms which can converge to NE.
$endgroup$
– Agrim Pathak
Mar 7 at 4:47
$begingroup$
you should provide some background info about how the linear optimization problem solves "We maximize our worse payoff." or how $q$ can appear
$endgroup$
– LinAlg
Dec 22 '18 at 13:16
$begingroup$
you should provide some background info about how the linear optimization problem solves "We maximize our worse payoff." or how $q$ can appear
$endgroup$
– LinAlg
Dec 22 '18 at 13:16
$begingroup$
@LinAlg
q is the opponent's strategy. It is not needed in the original "unconstrained" problem. For more information on the formulation, page 38 of this document might help: math.ucla.edu/~tom/Game_Theory/mat.pdf$endgroup$
– Agrim Pathak
Dec 23 '18 at 9:04
$begingroup$
@LinAlg
q is the opponent's strategy. It is not needed in the original "unconstrained" problem. For more information on the formulation, page 38 of this document might help: math.ucla.edu/~tom/Game_Theory/mat.pdf$endgroup$
– Agrim Pathak
Dec 23 '18 at 9:04
$begingroup$
Hi! Did you eventually figure out what are conditions for such a constrained optimal mixed strategy to exist?
$endgroup$
– Ben Usman
Mar 5 at 14:35
$begingroup$
Hi! Did you eventually figure out what are conditions for such a constrained optimal mixed strategy to exist?
$endgroup$
– Ben Usman
Mar 5 at 14:35
$begingroup$
@BenUsman Hi, unfortunately I made no progress on this. There are however, iterative algorithms which can converge to NE.
$endgroup$
– Agrim Pathak
Mar 7 at 4:47
$begingroup$
@BenUsman Hi, unfortunately I made no progress on this. There are however, iterative algorithms which can converge to NE.
$endgroup$
– Agrim Pathak
Mar 7 at 4:47
add a comment |
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$begingroup$
you should provide some background info about how the linear optimization problem solves "We maximize our worse payoff." or how $q$ can appear
$endgroup$
– LinAlg
Dec 22 '18 at 13:16
$begingroup$
@LinAlg
qis the opponent's strategy. It is not needed in the original "unconstrained" problem. For more information on the formulation, page 38 of this document might help: math.ucla.edu/~tom/Game_Theory/mat.pdf$endgroup$
– Agrim Pathak
Dec 23 '18 at 9:04
$begingroup$
Hi! Did you eventually figure out what are conditions for such a constrained optimal mixed strategy to exist?
$endgroup$
– Ben Usman
Mar 5 at 14:35
$begingroup$
@BenUsman Hi, unfortunately I made no progress on this. There are however, iterative algorithms which can converge to NE.
$endgroup$
– Agrim Pathak
Mar 7 at 4:47