Finding Nash Equilibrium using Linear Program with strategy constraints
$begingroup$
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
$endgroup$
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
$begingroup$
@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 |
$begingroup$
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 |
0
active
oldest
votes
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3049043%2ffinding-nash-equilibrium-using-linear-program-with-strategy-constraints%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3049043%2ffinding-nash-equilibrium-using-linear-program-with-strategy-constraints%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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