Abstract
We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in Ω ⊂ ℝn. The method of the proof is based on a game-theoretic idea to estimate the value of a related game defined in Ω ×Ω via couplings.
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Open access funding provided by University of Jyväskylä (JYU). The authors have been supported by the Academy of Finland project #298641. Á. A. was partially supported by the grant MTM2017-85666-P.
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Appendix A: Orthogonal Transformations
Appendix A: Orthogonal Transformations
Lemma A.1
Let |νx| = |νz| = 1.There exist\(\mathbf {P}_{\nu _{x}},\mathbf {P}_{\nu _{z}}\in O(n)\)suchthat\(\mathbf {P}_{\nu _{x}}\mathbf {e}_{1}=\nu _{x}\),\(\mathbf {P}_{\nu _{z}}\mathbf {e}_{1}=\nu _{z}\)and
for every\(\zeta \in {\mathbb B}^{\mathbf {e}_{1}}\).
Proof
In order to show this result, we construct explicit orthogonal matrices satisfying the required conditions. For each fixed |νx| = |νz| = 1, we choose \(\mathbf {P}_{\nu _{x}}\) and \(\mathbf {P}_{\nu _{z}}\) in O(n) as follows. First, we denote by {νx}⊥, {νz}⊥ and {νx,νz}⊥ the vector spaces
Then dim {νx}⊥ = dim {νz}⊥ = n − 1. If νx = ±νz, then {νx}⊥ = {νz}⊥, otherwise dim {νx,νz}⊥ = n − 2. In both cases, we can find a (n − 2)-dimensional vector space contained in {νx,νz}⊥. Then, let {r3,r4,…,rn} be a collection of n − 2 unitary column vectors in ℝn that form an orthonormal basis for such subspace. Let R ∈ ℝn×(n− 2) be the matrix containing all the elements of the basis as column vectors, i.e.,
Note that, therefore, the vector space
defines a (2-dimensional) plane containing the unitary vectors νx and νz. In addition, for νx ∈ {R}⊥, there exist a unique unitary vector ϱx ∈ {R}⊥∩{νx}⊥ such that
Analogously, let ϱz ∈ {R}⊥∩{νz}⊥ the unique unitary vector such that
Then,
and, for any \(\zeta \in {\mathbb B}^{\mathbf {e}_{1}}\), ζ1 = 0 and
Finally, we show that, for this particular choice of the vectors ϱx and ϱz, it holds
By the properties of the n-dimensional orthogonal group, the matrix \(\mathbf {P}_{\nu _{x}}^{\top }\mathbf {P}_{\nu _{z}}\) is also in O(n) with determinant \(\det (\mathbf {P}_{\nu _{x}}^{\top }\mathbf {P}_{\nu _{z}})=-1\), and it takes the form
where
has determinant det Q = − 1, that is, Q is a reflection matrix in ℝ2 and, thus, there exists σ ∈ [0, 2π) such that
Then, in particular, \(\langle {\varrho _{x}}\rangle {\varrho _{z}}=-\langle {\nu _{x}}\rangle {\nu _{z}}\) and
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Arroyo, Á., Luiro, H., Parviainen, M. et al. Asymptotic Lipschitz Regularity for Tug-of-War Games with Varying Probabilities. Potential Anal 53, 565–589 (2020). https://doi.org/10.1007/s11118-019-09778-8
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DOI: https://doi.org/10.1007/s11118-019-09778-8