Mixing
The smallest-sum game
$begingroup$
The game is a function of an integer $ngeq 1$ and a number $tin(0,n)$.
An adversary picks $n$ numbers in $[0,1]$ whose total sum is $t$. You divide the numbers into two subsets and the adversary picks one of them. Your score is the sum of numbers in the remaining subset.
What is the largest sum $s(n,t)$ that you can guarantee?
$s(1,t) = 0$ - one subset will always be empty.
$s(2,t) = max(0, t-1)$ - each subset will contain one number. The adversary will try to make them as unbalanced as possible, and the worst he can do is make the larger number equal 1 so you get the smaller one which is $t-1$.
$s(3,t) = min(t/3, max(t-1,0))$ - the optimal division for you is apparently to put the two smallest numbers in one subset and the largest number in the other subset, so the adversary will try to make these subsets as unbalanced as possible. One way to do it is to make all numbers equal, in which case your score is $t/3$; another way is to make the largest number 1, in which case your score is $t-1$. The adversary will pick the worst of these two options.
For $ngeq 4$, the game becomes much more complex to analyze, since there are many different possible partitions. Do you see any pattern? Any way to make it simpler?
combinatorial-game-theory
asked Jan 19 at 19:39
Erel Segal-HaleviErel Segal-Halevi
4,33011861
$endgroup$
add a comment |
$begingroup$
The game is a function of an integer $ngeq 1$ and a number $tin(0,n)$.
An adversary picks $n$ numbers in $[0,1]$ whose total sum is $t$. You divide the numbers into two subsets and the adversary picks one of them. Your score is the sum of numbers in the remaining subset.
What is the largest sum $s(n,t)$ that you can guarantee?
$s(1,t) = 0$ - one subset will always be empty.
$s(2,t) = max(0, t-1)$ - each subset will contain one number. The adversary will try to make them as unbalanced as possible, and the worst he can do is make the larger number equal 1 so you get the smaller one which is $t-1$.
$s(3,t) = min(t/3, max(t-1,0))$ - the optimal division for you is apparently to put the two smallest numbers in one subset and the largest number in the other subset, so the adversary will try to make these subsets as unbalanced as possible. One way to do it is to make all numbers equal, in which case your score is $t/3$; another way is to make the largest number 1, in which case your score is $t-1$. The adversary will pick the worst of these two options.
For $ngeq 4$, the game becomes much more complex to analyze, since there are many different possible partitions. Do you see any pattern? Any way to make it simpler?
combinatorial-game-theory
asked Jan 19 at 19:39
Erel Segal-HaleviErel Segal-Halevi
4,33011861
$endgroup$
add a comment |
$begingroup$
The game is a function of an integer $ngeq 1$ and a number $tin(0,n)$.
An adversary picks $n$ numbers in $[0,1]$ whose total sum is $t$. You divide the numbers into two subsets and the adversary picks one of them. Your score is the sum of numbers in the remaining subset.
What is the largest sum $s(n,t)$ that you can guarantee?
$s(1,t) = 0$ - one subset will always be empty.
$s(2,t) = max(0, t-1)$ - each subset will contain one number. The adversary will try to make them as unbalanced as possible, and the worst he can do is make the larger number equal 1 so you get the smaller one which is $t-1$.
$s(3,t) = min(t/3, max(t-1,0))$ - the optimal division for you is apparently to put the two smallest numbers in one subset and the largest number in the other subset, so the adversary will try to make these subsets as unbalanced as possible. One way to do it is to make all numbers equal, in which case your score is $t/3$; another way is to make the largest number 1, in which case your score is $t-1$. The adversary will pick the worst of these two options.
For $ngeq 4$, the game becomes much more complex to analyze, since there are many different possible partitions. Do you see any pattern? Any way to make it simpler?
combinatorial-game-theory
asked Jan 19 at 19:39
Erel Segal-HaleviErel Segal-Halevi
4,33011861
$endgroup$
The game is a function of an integer $ngeq 1$ and a number $tin(0,n)$.
An adversary picks $n$ numbers in $[0,1]$ whose total sum is $t$. You divide the numbers into two subsets and the adversary picks one of them. Your score is the sum of numbers in the remaining subset.
What is the largest sum $s(n,t)$ that you can guarantee?
$s(1,t) = 0$ - one subset will always be empty.
$s(2,t) = max(0, t-1)$ - each subset will contain one number. The adversary will try to make them as unbalanced as possible, and the worst he can do is make the larger number equal 1 so you get the smaller one which is $t-1$.
$s(3,t) = min(t/3, max(t-1,0))$ - the optimal division for you is apparently to put the two smallest numbers in one subset and the largest number in the other subset, so the adversary will try to make these subsets as unbalanced as possible. One way to do it is to make all numbers equal, in which case your score is $t/3$; another way is to make the largest number 1, in which case your score is $t-1$. The adversary will pick the worst of these two options.
For $ngeq 4$, the game becomes much more complex to analyze, since there are many different possible partitions. Do you see any pattern? Any way to make it simpler?
combinatorial-game-theory
combinatorial-game-theory
asked Jan 19 at 19:39
Erel Segal-HaleviErel Segal-Halevi
4,33011861
asked Jan 19 at 19:39
Erel Segal-HaleviErel Segal-Halevi
4,33011861
asked Jan 19 at 19:39
Erel Segal-HaleviErel Segal-Halevi
4,33011861
asked Jan 19 at 19:39
Erel Segal-HaleviErel Segal-Halevi
4,33011861
asked Jan 19 at 19:39
Erel Segal-HaleviErel Segal-Halevi
4,33011861
4,33011861
add a comment |
add a comment |
0
active
oldest
votes
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%2f3079724%2fthe-smallest-sum-game%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%2f3079724%2fthe-smallest-sum-game%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
