I give short and constructive proofs of Tarski's fixed-point theorem, and of a much-used extension of Tarski's fixed-point theorem to set- valued maps.tarski, fixed-point theorem, supermodular, supermodular games, strategic complementarities, equilibrium existence

Federico Echenique

English

Research Papers in Economics

A SHORT AND CONSTRUCTIVE PROOF OF
TARSKI’S FIXED-POINT THEOREM
FEDERICO ECHENIQUE
Abstract. I give short and constructive proofs of Tarski’s ﬁxed-
point theorem, and of a much-used extension of Tarski’s ﬁxed-point
theorem to set-valued maps.
1. Introduction
I give short and constructive proofs of two related ﬁxed-point the-
orems. The ﬁrst is Tarski’s ﬁxed-point theorem: If F is a monotone
function on a non-empty complete lattice, the set of ﬁxed points of
F form a non-empty complete lattice. The second is a much-used
extension of Tarski’s ﬁxed-point theorem to set-valued functions: If
ϕ : X → 2X is monotone—when 2X is endowed with the induced set
order—the set of ﬁxed-points of ϕ form a non-empty complete lattice.
Tarski’s [4] original proof is beautiful and elegant, but non-constructive
and somewhat uninformative. Cousot and Cousot [1] give a construc-
tive proof of Tarski’s ﬁxed-point theorem. Their proof is long and quite
involved, though. The proof I present is much simpler, and ﬁts in a
napkin. On the other hand, [1] obtain certain sub-products from their
approach that I do not obtain; I shall only be concerned with Tarski’s
ﬁxed-point theorem, and its extension to set-valued functions.
The extension to set-valued functions was developed by Smith [3] and
Zhou [7]. I give a constructive proof of Zhou’s version of the result.
Zhou’s version of the result is important in game theory ([5],[6]). Smith
has a weaker monotonicity requirement than Zhou, but Smith does not
obtain a lattice structure on the set of ﬁxed-points. In addition, Smith
needs a continuity assumption.
2. Definitions
A set X endowed with a partial order ≤ is denoted hX,≤i. hX,≤i
is a complete lattice if, for all nonempty B ⊆ X,
V
B and
W
B exist in
Date: May 5, 2003.
1991 Mathematics Subject Classiﬁcation. Primary 47H10; Secondary 06B23.
12 F. ECHENIQUE
X. If A ⊆ X, say that A is a subcomplete sublattice of hX,≤i if, for all
nonempty B ⊆ A,
V
B ∈ A, and
W
B ∈ A.
Say that A ⊆ X is smaller than B ⊆ X in the induced set order
(denoted A  B) if
(x ∈ A,y ∈ B) ⇒ (x ∧ y ∈ A,x ∨ y ∈ B).
The induced set order is a partial order on the set of sublattices of X.
Denote by  the usual linear order on ordinal numbers.
Let hX,≤i be a lattice and hY,=i be a partially ordered set. A
function F : X → Y is monotone if x ≤ y implies F(x) = F(y). Say
that a set-valued map ϕ : X → 2X is monotone if it is monotone when
ϕ(X) is ordered by the induced set order.
3. Theorems
Let hX,≤i be a lattice and F : X → X be monotone. The set of
ﬁxed points of F is E(F) = {x ∈ X : x = F(x)}.
Lemma 1. hE(F),≤i has a smallest element.
Proof. Let η be an ordinal number with cardinality greater than X, let
ξ = η + 1. Deﬁne f : ξ → X by transﬁnite recursion as f(0) =
V
X,
and
f(β) =
_
α<β
F(f(α))
for β > 0.
That (β ≺ α) ⇒ (f(β) ≤ f(α)) is immediate from the deﬁnition of
f. Then, for any α ∈ η, f(α + 1) = F(f(α)), as f(β) ≤ f(α), for all
β < α, and F is monotone. Since η has cardinality greater than X,
there is γ ∈ η such that f(γ) = f(γ +1). Let e = f(γ), then e = F(e).
So e ∈ E(F).
I shall prove that e is the smallest element in hE(F),≤i. Let e ∈
E(F), and consider the proposition Pα : f(α) ≤ e. Pα implies that
f(α + 1) = F(f(α)) ≤ F(e) = e. By transﬁnite induction, then, e ≤
e. 
A version of Lemma 1 is also crucial in [1]’s proof of Tarki’s ﬁxed-
point theorem. It was apparently ﬁrst proven in [2]; my proof is more
direct than the one in [2].
Theorem 2. hE(F),≤i is a non-empty complete lattice.
Proof. By Lemma 1, E(F) is nonempty. Let E ⊆ E(F) be nonempty.
I shall ﬁnd
W
E(F) E.TARSKI’S THEOREM 3
Let x =
W
X E, and let Y = {z ∈ X : x ≤ z} be the set of upper
bounds on E. If z ∈ Y , then, for any e ∈ E, e ≤ F(z), as e = F(e) ≤
F(z). Thus F(Y ) ⊆ Y . Let G = F|Y; so G maps Y into Y , and G is
monotone.
By Lemma 1, hE(G),≤i has a smallest element. By deﬁnition of G,
this smallest element is
W
E(F) E.
The construction of
V
E(F) E is symmetric. 
Let ϕ : X → 2X be a set-valued map such that, for all x ∈ X,
ϕ(x) is a non-empty subcomplete sublattice of X. Suppose that ϕ is
monotone. The set of ﬁxed points of ϕ is E(ϕ) = {x ∈ X : x ∈ ϕ(x)}.
Lemma 3. hE(ϕ),≤i has a smallest element.
Proof. Let F(x) =
V
X ϕ(x). Note that, ∀x, F(x) ∈ ϕ(x), and that
F is monotone. By Lemma 1, there is a smallest element, say e of
hE(F),≤i. Note that e = F(e) ∈ ϕ(e), so e ∈ E(ϕ).
I shall prove that e is the smallest element in hE(ϕ),≤i. Let e ∈
E(ϕ). Let f be as in the proof of Lemma 1. Consider the proposition
Pα : f(α) ≤ e. Pα implies that f(α + 1) = F(f(α)) ≤ F(e) ≤ e,
as F(e) =
V
X ϕ(e), and e ∈ ϕ(e). By transﬁnite induction, then,
e ≤ e. 
Theorem 4. hE(ϕ),≤i is a non-empty complete lattice.
Proof. Lemma 3 implies that E(ϕ) is nonempty. I shall prove that it is
a complete lattice.
Let E ⊆ E(ϕ) be nonempty. I shall prove that
W
E(ϕ) E exists.
Let x =
W
X E, and let Y = {z ∈ X : x ≤ z}. Deﬁne ψ : Y → 2Y by
ψ(z) = Y ∩ ϕ(z).
First, I show ψ(z) 6= ∅. Note that x ≤ z implies that, for any e ∈ E,
there is ˆ ze ∈ ϕ(z) with e ≤ ˆ ze, as e ∈ ϕ(e) and ϕ is monotone. But
ϕ(z) is subcomplete, and thus
x ≤
_
e∈E
ˆ ze ∈ ϕ(z),
so ψ(z) 6= ∅.
Second, I show that ψ is monotone. Let z ≤ z0, and ﬁx y ∈ ψ(z)
and y0 ∈ ψ(z0). ϕ is monotone, so y ∧ y0 ∈ ϕ(z) and y ∨ y0 ∈ ϕ(z0).
Let e ∈ E, e ∈ ϕ(e) implies e ∨ (y ∧ y0) ∈ ϕ(z). But y ∧ y0 ∈ Y
so ϕ(z) 3 e ∨ (y ∧ y0) = y ∧ y0. Similarly, e ∨ (y ∨ y0) ∈ ϕ(z0) and
y∨y0 ∈ Y implies y∨y0 ∈ ψ(z0). Thus y∧y0 ∈ ψ(z) and y∨y0 ∈ ψ(z0).
Third, ψ(z) is a subcomplete sublattice because ψ(z) = Y ∩ ϕ(z)
and ϕ(z) is a subcomplete sublattice.4 F. ECHENIQUE
Thus, ψ satisﬁes the hypothesis of Lemma 3. Let e∗ ∈ E(ψ) be the
smallest ψ-ﬁxed point. If e ∈ E(ϕ) is an upper bound on E, then
e ∈ Y and thus e ∈ E(ψ). We have then e∗ ≤ e. But e∗ ∈ E(ϕ), so
e∗ =
W
E(ϕ) E.
The proof that
V
E(ϕ) E exists is symmetric. 
4. Continuous F
The proof of Tarski’s Theorem is elementary when F is order-continuous,
in addition to monotone.
First, the proof of Lemma 1 goes: Let x =
V
X, the smallest point in
X, and let {xn} be the sequence of F-iterates from x; so xn = F(xn−1)
and x0 = x. Since F is monotone, {xn} is a monotone sequence, and
thus converges to a point e. The continuity of F implies that e is a
ﬁxed point, as x2n+1 = F(x2n), and both {x2n} and {x2n+1} converge
to e. Further, if e is any other ﬁxed point of F, x ≤ e, and xn ≤ e
implies xn+1 = F(xn) ≤ F(e) = e. By induction, e is the smallest ﬁxed
point.
Second, Lemma 1 is used to prove Tarski’s Theorem as in the proof
of Theorem 2 above.
References
1. Patrick Cousot and Radhia Cousot, Constructive versions of tarski’s
ﬁxed point theorems, Paciﬁc J. of Math. 82 (1979), no. 1, 43–57.
2. V. Devid´ e, On monotonous mappings of complete lattices, Funda-
menta Mathematicae 53 (1964), no. 2, 147–154.
3. R. E. Smithson, Fixed points of order preserving multifunctions,
Proc. Amer. Math. Soc. 28 (1971), no. 1, 304–310.
4. Alfred Tarski, A lattice theoretical ﬁxpoint theorem and its applica-
tions, Paciﬁc J. of Math 5 (1955), 285–309.
5. Donald M. Topkis, Equilibrium points in nonzero-sum n-person sub-
modular games, SIAM Journal of Control and Optimization 17
(1979), no. 6, 773–787.
6. , Supermodularity and complementarity, Princeton Univer-
sity Press, Princeton, New Jersey, 1998.
7. Lin Zhou, The set of nash equilibria of a supermodular game is a
complete lattice, Games and Economic Behavior 7 (1994), no. 2,
295–300.
HSS 228-77, California Institute of Technology, Pasadena CA 91125.
E-mail address: fede@hss.caltech.edu

A lattice theoretical ﬁxpoint theorem and its applications,

Constructive versions of tarski’s ﬁxed point theorems,

Fixed points of order preserving multifunctions,

On monotonous mappings of complete lattices,

Supermodularity and complementarity,

The set of nash equilibria of a supermodular game is a complete lattice,

Topkis, Equilibrium points in nonzero-sum n-person submodular games,

A Short And Constructive Proof of Tarski's Fixed-Point Theorem

http://128.118.178.162/eps/ge/papers/0305/0305001.pdf

A Short And Constructive Proof of Tarski's Fixed-Point Theorem

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