Discrete fixed-point theorem

In discrete mathematics, a discrete fixed-point is a fixed-point for functions defined on finite sets, typically subsets of the integer grid ${\mathbb{Z}}^n$.

Discrete fixed-point theorems were developed by Iimura, Murota and Tamura, Chen and Deng and others. Yang provides a survey.

Basic concepts

Continuous fixed-point theorems often require a continuous function. Since continuity is not meaningful for functions on discrete sets, it is replaced by conditions such as a direction-preserving function. Such conditions imply that the function does not change too drastically when moving between neighboring points of the integer grid. There are various direction-preservation conditions, depending on whether neighboring points are considered points of a hypercube (HGDP), of a simplex (SGDP) etc. See the page on direction-preserving function for definitions.

Continuous fixed-point theorems often require a convex set. The analogue of this property for discrete sets is an integrally-convex set.

A fixed point of a discrete function f is defined exactly as for continuous functions: it is a point x for which f(x)=x.

For functions on discrete sets

We focus on functions $f:X\to {ℝ}^n$, where the domain X is a nonempty subset of the Euclidean space ${ℝ}^n$. ch(X) denotes the convex hull of X.

Iimura-Murota-Tamura theorem: If X is a finite integrally-convex subset of ${\mathbb{Z}}^n$, and $f:X\to \text{ch}(X)$ is a hypercubic direction-preserving (HDP) function, then f has a fixed-point.

Chen-Deng theorem: If X is a finite subset of ${ℝ}^n$, and $f:X\to \text{ch}(X)$ is simplicially direction-preserving (SDP), then f has a fixed-point.

Yang's theorems:

• [3.6] If X is a finite integrally-convex subset of ${\mathbb{Z}}^n$, $f:X\to {ℝ}^n$ is simplicially gross direction preserving (SGDP), and for all x in X there exists some g(x)>0 such that $x+g(x)f(x)\in \text{ch}(X)$, then f has a zero point.
• [3.7] If X is a finite hypercubic subset of ${\mathbb{Z}}^n$, with minimum point a and maximum point b, $f:X\to {ℝ}^n$ is SGDP, and for any x in X: $f_i(x_1,\dots ,x_{i-1},a_i,x_{i+1},\dots ,x_n)\le 0$ and $f_i(x_1,\dots ,x_{i-1},b_i,x_{i+1},\dots ,x_n)\ge 0$, then f has a zero point. This is a discrete analogue of the Poincaré–Miranda theorem. It is a consequence of the previous theorem.
• [3.8] If X is a finite integrally-convex subset of ${\mathbb{Z}}^n$, and $f:X\to \text{ch}(X)$ is such that $f(x)-x$ is SGDP, then f has a fixed-point. This is a discrete analogue of the Brouwer fixed-point theorem.
• [3.9] If X = ${\mathbb{Z}}^n$, $f:X\to {ℝ}^n$ is bounded and $f(x)-x$ is SGDP, then f has a fixed-point (this follows easily from the previous theorem by taking X to be a subset of ${\mathbb{Z}}^n$ that bounds f).
• [3.10] If X is a finite integrally-convex subset of ${\mathbb{Z}}^n$, $F:X\to 2^X$ a point-to-set mapping, and for all x in X: $F(x)=\text{ch}(F(x))\cap {\mathbb{Z}}^n$ , and there is a function f such that $f(x)\in \text{ch}(F(x))$ and $f(x)-x$ is SGDP, then there is a point y in X such that $y\in F(y)$. This is a discrete analogue of the Kakutani fixed-point theorem, and the function f is an analogue of a continuous selection function.
• [3.12] Suppose X is a finite integrally-convex subset of ${\mathbb{Z}}^n$, and it is also symmetric in the sense that x is in X iff -x is in X. If $f:X\to {ℝ}^n$ is SGDP w.r.t. a weakly-symmetric triangulation of ch(X) (in the sense that if s is a simplex on the boundary of the triangulation iff -s is), and $f(x)\cdot f(-y)\le 0$ for every pair of simplicially-connected points x, y in the boundary of ch(X), then f has a zero point.
• See the survey for more theorems.

For discontinuous functions on continuous sets

Discrete fixed-point theorems are closely related to fixed-point theorems on discontinuous functions. These, too, use the direction-preservation condition instead of continuity.

Herings-Laan-Talman-Yang fixed-point theorem:

Let X be a non-empty convex compact subset of ${ℝ}^n$. Let f: X → X be a locally gross direction preserving (LGDP) function: at any point x that is not a fixed point of f, the direction of $f(x)-x$ is grossly preserved in some neighborhood of x, in the sense that for any two points y, z in this neighborhood, its inner product is non-negative, i.e.: $(f(y)-y)\cdot (f(z)-z)\ge 0$. Then f has a fixed point in X.

The theorem is originally stated for polytopes, but Philippe Bich extends it to convex compact sets.Template:RpNote that every continuous function is LGDP, but an LGDP function may be discontinuous. An LGDP function may even be neither upper nor lower semi-continuous. Moreover, there is a constructive algorithm for approximating this fixed point.

Applications

Discrete fixed-point theorems have been used to prove the existence of a Nash equilibrium in a discrete game, and the existence of a Walrasian equilibrium in a discrete market.