Is there results about the existence of a Nash equilibrium in continuous games with non-compact strategy...
$begingroup$
Consider a $n$-player continuous game $G=(P,S,U)$ where:
$P={1,2,dots,n}$ is the set of $n$ players.
$S={S_1,S_2,dots,S_n}$ where $S_i=mathbb{R}^n_+$ is the $i$- th player's set of pure strategies, and $mathbb{R}^n_+$ is the set of non-negative $n$-tuples.
$U={u_1,u_2,dots,u_n}$ where $u_i:Srightarrowmathbb{R}$ is the utility function of player $i$.
Let $sigma_iin S_i=mathbb{R}^n_+$ denote a single strategy for player $i$, $sigma={sigma_1,sigma_2,dots,sigma _n}in S$ denote a strategy profile, and $sigma_{-i}$ denote a strategy profile of all players except for player $i$.
A strategy profile $sigma^*={sigma_1^*,sigma_2^*,dots,sigma_n^*}in S$ is said to be a Nash equilibrium if the strategy $sigma_i^*$ is a local maximum for the utility function $u_i(sigma_i; sigma_{-i}^*)$ for all the players $i$.
Question:
Is there any property for the utility functions $u_i$ that could guarantee the existence of at least one Nash equilibrium?
for example, concavity or quasi-concavity.
I know that if all the utility functions are continuous and the sets $S_i$ are compact, then, the existence of a Nash equilibrium is guaranteed, but $mathbb{R}^n_+$ is not bounded.
nash-equilibrium
asked Dec 6 '18 at 18:46
GuadalupeAnimationGuadalupeAnimation
31919
$endgroup$
add a comment |
$begingroup$
Consider a $n$-player continuous game $G=(P,S,U)$ where:
$P={1,2,dots,n}$ is the set of $n$ players.
$S={S_1,S_2,dots,S_n}$ where $S_i=mathbb{R}^n_+$ is the $i$- th player's set of pure strategies, and $mathbb{R}^n_+$ is the set of non-negative $n$-tuples.
$U={u_1,u_2,dots,u_n}$ where $u_i:Srightarrowmathbb{R}$ is the utility function of player $i$.
Let $sigma_iin S_i=mathbb{R}^n_+$ denote a single strategy for player $i$, $sigma={sigma_1,sigma_2,dots,sigma _n}in S$ denote a strategy profile, and $sigma_{-i}$ denote a strategy profile of all players except for player $i$.
A strategy profile $sigma^*={sigma_1^*,sigma_2^*,dots,sigma_n^*}in S$ is said to be a Nash equilibrium if the strategy $sigma_i^*$ is a local maximum for the utility function $u_i(sigma_i; sigma_{-i}^*)$ for all the players $i$.
Question:
Is there any property for the utility functions $u_i$ that could guarantee the existence of at least one Nash equilibrium?
for example, concavity or quasi-concavity.
I know that if all the utility functions are continuous and the sets $S_i$ are compact, then, the existence of a Nash equilibrium is guaranteed, but $mathbb{R}^n_+$ is not bounded.
nash-equilibrium
asked Dec 6 '18 at 18:46
GuadalupeAnimationGuadalupeAnimation
31919
$endgroup$
add a comment |
$begingroup$
Consider a $n$-player continuous game $G=(P,S,U)$ where:
$P={1,2,dots,n}$ is the set of $n$ players.
$S={S_1,S_2,dots,S_n}$ where $S_i=mathbb{R}^n_+$ is the $i$- th player's set of pure strategies, and $mathbb{R}^n_+$ is the set of non-negative $n$-tuples.
$U={u_1,u_2,dots,u_n}$ where $u_i:Srightarrowmathbb{R}$ is the utility function of player $i$.
Let $sigma_iin S_i=mathbb{R}^n_+$ denote a single strategy for player $i$, $sigma={sigma_1,sigma_2,dots,sigma _n}in S$ denote a strategy profile, and $sigma_{-i}$ denote a strategy profile of all players except for player $i$.
A strategy profile $sigma^*={sigma_1^*,sigma_2^*,dots,sigma_n^*}in S$ is said to be a Nash equilibrium if the strategy $sigma_i^*$ is a local maximum for the utility function $u_i(sigma_i; sigma_{-i}^*)$ for all the players $i$.
Question:
Is there any property for the utility functions $u_i$ that could guarantee the existence of at least one Nash equilibrium?
for example, concavity or quasi-concavity.
I know that if all the utility functions are continuous and the sets $S_i$ are compact, then, the existence of a Nash equilibrium is guaranteed, but $mathbb{R}^n_+$ is not bounded.
nash-equilibrium
asked Dec 6 '18 at 18:46
GuadalupeAnimationGuadalupeAnimation
31919
$endgroup$
Consider a $n$-player continuous game $G=(P,S,U)$ where:
$P={1,2,dots,n}$ is the set of $n$ players.
$S={S_1,S_2,dots,S_n}$ where $S_i=mathbb{R}^n_+$ is the $i$- th player's set of pure strategies, and $mathbb{R}^n_+$ is the set of non-negative $n$-tuples.
$U={u_1,u_2,dots,u_n}$ where $u_i:Srightarrowmathbb{R}$ is the utility function of player $i$.
Let $sigma_iin S_i=mathbb{R}^n_+$ denote a single strategy for player $i$, $sigma={sigma_1,sigma_2,dots,sigma _n}in S$ denote a strategy profile, and $sigma_{-i}$ denote a strategy profile of all players except for player $i$.
A strategy profile $sigma^*={sigma_1^*,sigma_2^*,dots,sigma_n^*}in S$ is said to be a Nash equilibrium if the strategy $sigma_i^*$ is a local maximum for the utility function $u_i(sigma_i; sigma_{-i}^*)$ for all the players $i$.
Question:
Is there any property for the utility functions $u_i$ that could guarantee the existence of at least one Nash equilibrium?
for example, concavity or quasi-concavity.
I know that if all the utility functions are continuous and the sets $S_i$ are compact, then, the existence of a Nash equilibrium is guaranteed, but $mathbb{R}^n_+$ is not bounded.
nash-equilibrium
nash-equilibrium
asked Dec 6 '18 at 18:46
GuadalupeAnimationGuadalupeAnimation
31919
asked Dec 6 '18 at 18:46
GuadalupeAnimationGuadalupeAnimation
31919
edited Dec 6 '18 at 18:54
GuadalupeAnimation
asked Dec 6 '18 at 18:46
GuadalupeAnimationGuadalupeAnimation
31919
asked Dec 6 '18 at 18:46
GuadalupeAnimationGuadalupeAnimation
31919
asked Dec 6 '18 at 18:46
GuadalupeAnimationGuadalupeAnimation
31919
31919
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Concavity or quasi-concavity is not enough, since it can be linear (for example take $u_i(sigma)=sigma_i forall i$).
If ${-u_i}$ are continuous and coercive (meaning $lim_{|sigma|rightarrowinfty} u_i(sigma)=-infty forall i$) then you can restrict the problem to a compact rectangle subset of $S$ on which the functions $u_i$ are not to small and then use the result with compact strategy sets.
In particular, if ${u_i}$ are continuous and strongly concave, one can show ${-u_i}$ are also coercive.
answered Feb 17 at 11:10
MOMOMOMO
717312
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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%2f3028887%2fis-there-results-about-the-existence-of-a-nash-equilibrium-in-continuous-games-w%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Concavity or quasi-concavity is not enough, since it can be linear (for example take $u_i(sigma)=sigma_i forall i$).
If ${-u_i}$ are continuous and coercive (meaning $lim_{|sigma|rightarrowinfty} u_i(sigma)=-infty forall i$) then you can restrict the problem to a compact rectangle subset of $S$ on which the functions $u_i$ are not to small and then use the result with compact strategy sets.
In particular, if ${u_i}$ are continuous and strongly concave, one can show ${-u_i}$ are also coercive.
answered Feb 17 at 11:10
MOMOMOMO
717312
$endgroup$
add a comment |
$begingroup$
Concavity or quasi-concavity is not enough, since it can be linear (for example take $u_i(sigma)=sigma_i forall i$).
If ${-u_i}$ are continuous and coercive (meaning $lim_{|sigma|rightarrowinfty} u_i(sigma)=-infty forall i$) then you can restrict the problem to a compact rectangle subset of $S$ on which the functions $u_i$ are not to small and then use the result with compact strategy sets.
In particular, if ${u_i}$ are continuous and strongly concave, one can show ${-u_i}$ are also coercive.
answered Feb 17 at 11:10
MOMOMOMO
717312
$endgroup$
add a comment |
$begingroup$
Concavity or quasi-concavity is not enough, since it can be linear (for example take $u_i(sigma)=sigma_i forall i$).
If ${-u_i}$ are continuous and coercive (meaning $lim_{|sigma|rightarrowinfty} u_i(sigma)=-infty forall i$) then you can restrict the problem to a compact rectangle subset of $S$ on which the functions $u_i$ are not to small and then use the result with compact strategy sets.
In particular, if ${u_i}$ are continuous and strongly concave, one can show ${-u_i}$ are also coercive.
answered Feb 17 at 11:10
MOMOMOMO
717312
$endgroup$
Concavity or quasi-concavity is not enough, since it can be linear (for example take $u_i(sigma)=sigma_i forall i$).
If ${-u_i}$ are continuous and coercive (meaning $lim_{|sigma|rightarrowinfty} u_i(sigma)=-infty forall i$) then you can restrict the problem to a compact rectangle subset of $S$ on which the functions $u_i$ are not to small and then use the result with compact strategy sets.
In particular, if ${u_i}$ are continuous and strongly concave, one can show ${-u_i}$ are also coercive.
answered Feb 17 at 11:10
MOMOMOMO
717312
answered Feb 17 at 11:10
MOMOMOMO
717312
answered Feb 17 at 11:10
MOMOMOMO
717312
answered Feb 17 at 11:10
MOMOMOMO
717312
717312
add a comment |
add a comment |
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%2f3028887%2fis-there-results-about-the-existence-of-a-nash-equilibrium-in-continuous-games-w%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
