Abstract
The Gale–Nikaido–Debreu lemma plays an important role in establishing the existence of competitive equilibrium. In this paper, we use Sperner’s lemma and basic elements of topology to prove the Gale–Nikaido–Debreu lemma.
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Notes
Looking back at history, Debreu [9] used the Eilenberg–Montgomery fixed point theorem to prove the existence of a social equilibrium. Then, by using this social equilibrium existence theorem, Arrow and Debreu [1] proved the existence of a general equilibrium for a competitive economy with productions. See Debreu [12] and Florenzano [16] for excellent treatments of the existence of equilibrium. See also Duppe and Weintraub [13], Khan [25] for discussions about the history of the general equilibrium theory.
Another important lemma in the general equilibrium theory is Gale and Mas-Colell’s lemma introduced and proved by Gale and Mas-Colell [21, 22]. Their proofs are based on the Kakutani fixed point theorem and Michael selection theorem [30]. See Florenzano [17] for the role of these two lemmas in the general equilibrium theory.
Sperner’s lemma [37] can be viewed as a combinatorial variant of the Brouwer fixed point theorem [5, 23] and actually equivalent to it. For instance, Knaster, Kuratowski and Mazurkiewicz [25] used the Sperner lemma to prove the Knaster–Kuratowski–Mazurkiewicz theorem which implies the Brouwer theorem. Meanwhile, Yoseloff [40], Park and Jeong [34] proved the Sperner lemma by using the Brouwer theorem. The reader is referred to Park [33] for a more complete survey of fixed point theorems and Ben-El-Mechaiekh, Bich, and Florenzano [3] for a survey of general equilibrium and fixed point theory.
Recall that if \(\varDelta _i = [[ x^{i_1}, x^{i_2}, \ldots , x^{i_m}]]\), then \(\hbox {ri}(\varDelta _i) \equiv \{ x| x=\sum _{k=1} ^m \alpha _k x^k (i);\sum _k \alpha _k =1\); and \(\forall k: \alpha (k)>0\}\).
These functions \(\alpha _i\) constitute a partition of unity subordinate to the covering \(\big (B \left( x^i (\epsilon ), \epsilon \right) \big )_{i=1, \ldots , I(\epsilon )}\). See, for instance, Section 2.19 in Aliprantis and Border [1].
Carathéodory’s convexity theorem states that: In an n-dimensional vector space, every vector in the convex hull of a non-empty set can be written as a convex combination using no more than \(n+1\) vectors from the set. For a simple proof, see Proposition 1.1.2 in Florenzano and Le Van [19] or Theorem 5.32 in Aliprantis and Border [1].
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Le, T., Le Van, C., Pham, NS. et al. A Direct Proof of the Gale–Nikaido–Debreu Lemma Using Sperner’s Lemma. J Optim Theory Appl 194, 1072–1080 (2022). https://doi.org/10.1007/s10957-022-02067-2
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DOI: https://doi.org/10.1007/s10957-022-02067-2