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I have a question regarding Tarski's fixed point theorem. Consider the set $$\mathcal{X}:={0,1}^2:=\{(1,1), (0,1), (1,0), (0,0)\}$$

Consider the vector $X:=(X_1,X_2)$ taking value in $\mathcal{X}$.

Consider the function $h:\mathcal{X}\rightarrow \mathcal{X}$ $$ h:= \begin{cases} X_1:=1\{\epsilon_1+\delta X_2\geq 0\}\\ X_2:=1\{\epsilon_2+\delta X_1\geq 0\} \end{cases} $$ where $1\{\cdot \}$ is the indicator function taking value $1$ if the condition inside is satisfied and $0$ otherwise, $\delta$ is a positive parameter, and $\epsilon_1, \epsilon_2$ are any couple of real numbers.

Tarski's fixed point theorem tells me that, given some $\delta\geq 0, \epsilon_1, \epsilon_2$, the function $h$ has a greatest and a lowest fixed point. Let's call them $X^g, X^l$.

I am confused on the following: we have two elements in $\mathcal{X}$ which cannot be compared, i.e. $(1,0)$ and $(0,1)$. This means that

(A) It cannot be that $X^l=(0,1)$ and $X^u=(1,0)$ or viceversa. Correct?

(B) Other potential fixed points are only the points $X$ such that $X^l\leq X\leq X^u$, i.e., which are comparable with $X^l, X^u$. Correct?

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(A) is not correct. The Knaster-Tarski theorem states that the greatest/lowest fixed point are greatest/lowest among fixed points. Take this example

$$f:\mathcal{X}\to\mathcal{X}$$ $$f(0,0)=(0,0)$$ $$f(x,y)=(1,0)\mbox{ otherwise}$$

This function is order preserving and it has two fixed points: $(0,0)$ and $(1,0)$ which are the least and greates respectively among themselves (the set of all fixed points).

(B) is correct.

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  • $\begingroup$ Thanks. But in (A) I asked about $(0,1)$ and $(1,0)$. $\endgroup$ – STF Mar 17 '17 at 20:07
  • $\begingroup$ Can these be both fixed points? $\endgroup$ – STF Mar 17 '17 at 20:07
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    $\begingroup$ @STF Oh, at the same time? They can be fixed (e.g. the identity) but they can't be least and greatest at the same time by (B). $\endgroup$ – freakish Mar 17 '17 at 20:10

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