Kakutani Fixed Point Theorem Puns

For example, in the game of "choose a number, whoever chooses a higher number wins", there is no nash equilibrium, even though it is zero-sum. Kakutani's fixed-point theorem requires that the domain is compact.

I'm reading the Wikipedia article and it seems that they're only applying Kakutani's fixed point theorem to the case where the number of choices is finite, so the set of strategies is "play choice i with probability p_i". Can we get the existence of nash equilibriums in more general situations?

The theorem says when correspondence function from C to C where C is convex and compact, there exists a fixed point if the correspondence function is upper hemi-continuous, non-empty valued, compact valued and convex valued.

How can we find or show a correspondence on convex and compcat set where it has only three of the following features: upper hemi-continuity, non-emptiness, compact value, convex value?

https://en.wikipedia.org/wiki/Fixed-point_theorem

"A Fixed-Point Theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms"

I am struggling to understand this, as well as why is this important?

I am almost positive that I am misunderstand this - does the Fixed Point Theorem mean that if you have a function F(x) = 2(x) , does this mean that there has to be some point where F(1) = 1? This surely can not be right and I must be misunderstanding it? Why is this theorem so important?

The second question: is the Fixed Point Theorem particularly useful in the context of "optimization and root finding"? Does the Fixed Point Theorem somehow serve to guarantee the existence of an optimum point (e.g. some extremum point)?

Thanks

Context + Terminology: In category 𝒞 with (canonical) terminal object 1,

• Say that morphism p : A × A → B is weakly globally Curry-surjective iff for all f : A → B in 𝒞, there exists a sim(f) : 1 → A (call it the simulator of f) such that Hom_𝒞(τ₁, f) = Hom_𝒞(τ₁, ⟨κₐ • sim(f), τₐ⟩ₐ • p), where
  • τ₁ : 1 → 1 is the (canonical) identity morphism of 1.
  • τₐ : A → A is the (canonical) identity morphism of A.
  • κₐ : A → 1 is the (canonical) terminal morphism from A to 1.
  • A × A is the (canonical) binary product of X with itself.
    • ⟨-,-⟩₁ : Hom_𝒞(1, A) × Hom_𝒞(1, A) → Hom_𝒞(1, A × A) is the (canonical) component of the (natural) hom isomorphism manifesting the relevant universal property at 1.
    • ⟨-,-⟩ₐ : Hom_𝒞(A, A) × Hom_𝒞(A, A) → Hom_𝒞(A, A × A) is the (canonical) component of the (natural) hom isomorphism manifesting the relevant universal property at A.
• Given morphisms s : 1 → B and g : B → B, say that s is a fixed global point of g iff s is a fixed point of Hom_𝒞(τ₁, g). (I.e., iff s • g = s.)

Theorem: In 𝒞 as above, if object B is the target of some globally Curry-surjective morphism p : A × A → B, then every morphism g : B → B has a fixed global point, namely sim(⟨τₐ, τₐ⟩ₐ • p • g) • ⟨τₐ, τₐ⟩ₐ • p.

Proof: Observe that

sim(⟨τₐ, τₐ⟩ₐ • p • g) • ⟨τₐ, τₐ⟩ₐ • p
= ⟨sim(⟨τₐ, τₐ⟩ₐ • p • g), sim(⟨τₐ, τₐ⟩ₐ • p • g)⟩₁ • p
= sim(⟨τₐ, τₐ⟩ₐ • p • g) • ⟨κₐ • sim(⟨τₐ, τₐ⟩ₐ • p • g), τₐ⟩ₐ • p
= sim(⟨τₐ, τₐ⟩ₐ • p • g) • ⟨τₐ, τₐ⟩ₐ • p • g    

. ■

I want to understand the proof for brouwers fixed point theorem. I need to understand it so i can understand the proof for existence of nash equlibrium.

My math background is nothing more than high school stuff

I dont know where to start, what resources to use and so on...

Can someone help me?

Thanks in advance

(Yes i also posted in math subreddit but im having higher hopes here)

Title says it all, i cant even understand what continous functuon mapping compact convex set to itself means

In addition, im trying to understand this so i can understand proof of nash equlibrium

If its too complicated, pointing out resources for self study would be helpful!

Hi, I want to understand the basic lattice theory that’s the basis of understanding the proof of Knaster-Tarski theorem. Can somebody point me to a good beginner friendly resource to understand the Knaster Tarski theorem?

So I'm not very good at math, US Highschool Education and 1 semester of College Business Math, and I did not have a very firm grasp on it at any point. I was watching a Vsauce video about Fixed points and he showed a checkerboard with different colors, and he states that due to the law you can manipulate the board and it's impossible to have it sit without at least one point line up with it's original position.

So I screenshotted the checkerboard and played around and ran into a scenario where this did not apply. To my understanding I didn't break the rules and it SEEMS as if this contradicts the Theorem. All I did was rotate the board 180 degrees and shrink it down so that it only fit into 2 squares.

Now I'm 100% sure that I did not make any breakthrough discovery or anything, I feel like I am misunderstanding the rule.

NOTE: The picture is not perfect quality due to me just using snapchat to adjust the photo but I think it works fine.

Picture Here

The “pop math” way of presenting this theorem is to use two sheets of paper with identical dimensions. You leave one flat on the table, crumple the other one in the air above, and then assert that there is at least one point on the crumpled sheet that is still exactly over its corresponding point on the sheet on the table.

I was thinking about this and was curious if the following manipulation of the second sheet is valid. You rotate it 180 degrees so that the only fixed point would still be the center. Then slightly fold one edge so the rest of the sheet is “pulled” toward the fold. This would move the center point off its fixed point and would leave no fixed points anywhere? Is this correct? Am I missing something? Does a 180 degree rotation not count because the resulting sheet is still isomorphic?

Thats the theorem:
Let K ⊂ R n be a compact and convex set. If f : K → K is a continuous function, then f has a fixed point x∗ , that is f(x∗ ) = x∗

I have to provide an example of a funcion where all conditions hold except the compactness of the set K. Showing in this manner how essential is the compactess in order to make the theorem always true.

I'm struggling to find one. Can someone help me?

thanks in advance guys :)

In the paper Graph Neural Networks: A Review of Methods and Applications

There is a statement: " ... it is unsuitable to use the fixed points if we focus on the representation of nodes instead of graphs because the distribution of representation in the fixed point will be much smooth in value and less informative for distinguishing each node. "

Basically, I need help with understanding that. How do we know that value will be smooth and less informative due to fixed points?

When we train any network, we expect it to converge somewhere. The fixed point theorem just tells that, a point to converge exists. What is the problem?

Thanks for the answers. I feel like I am missing very basics. Any textbook recommendations would be also great

I apologize if this is more of a r/learnmath question, but I am afraid the question won't get answered there. I can't find anything like this either so maybe I just wasn't looking at the right place or it's false.

Anyway, the obvious generalization is as follow.
Let S be a closed subset of D^n x D^n (product of 2 n-dimension unit ball) such that under the first projection, S surject onto D^n and all fiber are contractible. Then S intersect the diagonal image of D^n .

So is this true? More generally, can I replace any function f:X->X in any fixed point theorems by a compact set S in XxX that is surjective on the first component with homotopically trivial fiber?

I've used the fixed point method x_{n+1} = 1/(5x_{n} + 3). To find values for x and show that g(x) = (5x + 3)^{-1} converges. Now I have to use the inequality

n ≥ 1/ln(L) · ln [((1 − L) \epsilon ) / |x1 − x0| ]

My question is how do I find/ what are the values of L and \epsilon?

I'm trying to get to grips with the Lefschetz Fixed Point Theorem and its proof, but I'm not entirely sure I understand it. The theorem starts by talking about the trace of an arbitrary function on homology groups. What exactly is the "trace" here? Are we using the fact that the homology groups are abelian and relating them to groups over the integers using the classification of finitely generated abelian groups? If so, what is the isomorphism here? How can I envision the function in a way that the trace makes sense to me?

Apologies for the barrage of related questions; I have so much passing through my head that it's hard to really boil it down succinctly.

Brouwer's theorem states that for any continuous mapping f from a compact convex set to itself, there exists some x=fx. Can this be translated into FOL? Am I allowed to quantify over functions, or do I need second-order logic for this?

This theorem is mostly unknown. I only found one book at my university that mention it but I think it's a cool one It says: In a vectorial space let's K be a convex compact. If you got two non-expansive maps that commutes they got a common fixed-point. The demonstration I know is pretty cool and use Banach point-fixed theorem and Ascoli. Nevertheless, I can't find any non trivial application of this theorem. The only non-expansive maps that commutes that i can think of got an obvious fixed-point. Can you help me or tell me how I should do my research? Thanks a lot

Some sort of fixed-point theorem phrased like this: if in a city, there is a to-scale model of the city, then some point in the city is at the exact same position as its counterpart in the model.

It sounds like some sort of fixed-point theorem. I've tried looking up the list of fixed-point theorems on Wikipedia, but none of them are phrased like this.

Also, the main TOMT afflicting me is, where have I encountered this? I vaguely remember it being used for some sort of plot point in a short-ish story, maybe a video. Someone discovered someone built a model town or city in his basement or something. Does anyone know what story? I'm very curious how this could have been used as a plot point, and why I can't remember it.

The statement of Brouwer's Fixed-Point Theorem from §2.2 of Guillemin-Pollack is is stated as follows.

"Any smooth map f of the closed unit ball B^n ⊂ R^n into itself must have a fixed point; that is, f(x) = x for some x ∈ B^n ."

As the question suggests, does the fixed point in the above statement of Brouwer's Fixed-Point Theorem have to be an interior point? Many thanks in advance.

I submitted a bug report a long time ago, the little orbs only lasted 10 seconds which wasn't what the arcane description was. Early October they corrected the duration and apparently the actual damage/debuff to match the description. So now its a bit stronger and lasts long enough to actually do something! I know its not even close to meta but its nice to pop on for fun. I looked around and I don't think anyone has the arcanes equipped to notice the fix.

As the title might suggest, I've been struggling with the notion of hemicontinuity and fixed point thereoms, so this might be a troublesome several-part question here.

First off, we've been studying Brouwer's Fixed Point Theorem, which as far as I understand simply implies that any function going from R^n to R^n has a fixed point. That seems simple enough, but we are asked to verify the assumptions made by this theorem. I'm honestly not sure what assumptions we are making in proving this theorem, but if anyone could direct me to a proof that would be helpful. As far as I understand, the only assumption we need to verify is that the function is continuous.

Secondly, Kakutani's Fixed Point Theorem. I really have no idea where to begin with this one. Most of the resources I've looked at online don't provide me with much help and seem pretty dense. I understand that it means that such and such a function with a certain domain and range has a fixed point, but I have no idea what a fixed point means when you are going from say, R^2 to R^4. Any insight on this would be incredibly helpful.

Lastly, hemicontinuity. This is the one I've been struggling with most. I can pull up definitions online and apply them to a correspondence, but I've no idea what upper and lower hemicontinuity actually means or what it looks like on a graph. General insight on this, any sort of easy way to understand it would be awesome (for example, it's easy to look at a set in R^2 and determine whether or not it is convex - is there a similar intuition for upper and lower hemicontinuity?)

As always, thanks for the help. You guys are great.

I'm having trouble with a problem: prove any A invariant subspace W contains at least one eigenvalue of A. It occurred to me that if it's an A invariant subspace, it must be a contraction mapping, as every point in W is mapped to another point in W. Therefore, by the Banach fixed point theorem, there must be a fixed point in W under A, which is a point with eigenvalue 1.
I think this works, but I'm not sure. Does anyone know exactly when that theorem holds? We discussed it in another class, but only in the context of the reals.
Thanks in advance!

https://en.wikipedia.org/wiki/Fixed-point_theorem

"A Fixed-Point Theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms"

I am struggling to understand this, as well as why is this important?

I am almost positive that I am misunderstand this - does the Fixed Point Theorem mean that if you have a function F(x) = 2(x) , does this mean that there has to be some point where F(1) = 1? This surely can not be right and I must be misunderstanding it? Why is this theorem so important?

The second question: is the Fixed Point Theorem particularly useful in the context of "optimization and root finding"? Does the Fixed Point Theorem somehow serve to guarantee the existence of an optimum point (e.g. some extremum point)?

Thanks