Recoverable Identifier

arXiv:2605.13883 · detector doi_compliance · incontrovertible · 2026-05-19 09:05:34.954497+00:00

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Evidence text

That is, when 2𝑎6U+𝑎U6>0, and 2𝑎6U+𝑎U6𝑎6U+𝑎U6+𝑎9U+𝑎U9−𝑎69−𝑎96𝑎6U+𝑎U6+𝑎9U+𝑎U9−𝑎69−𝑎962𝑎U9+𝑎9U>0, then 𝑝∗ is an ESS. In other words, if B does not satisfy these conditions, then 𝒑∗ is a Nash equilibrium, but not an ESS. When B is a positive semi-definite matrix (B is a positive semi-definite matrix if and only if all principal minors of B are non-negative), that is, when 2𝑎6U+𝑎U6≥0,2𝑎U9+𝑎9U≥0，and 2𝑎6U+𝑎U6𝑎6U+𝑎U6+𝑎9U+𝑎U9−𝑎69−𝑎96𝑎6U+𝑎U6+𝑎9U+𝑎U9−𝑎69−𝑎962𝑎U9+𝑎9U≥0， the fixed point 𝑥∗ is Lyapunov stable. Particularly, if 𝐵=0, then 𝑥,pa∗≡𝑐.,@6. When B is a negative definite matrix (B is negative definite matrix if and only if all the odd-order leading principal minors of B are less than 0 and all the even-order leading principal minors of B are greater than 0), that is, when 2𝑎6U+𝑎U6<0 and 18 2𝑎6U+𝑎U6𝑎6U+𝑎U6+𝑎9U+𝑎U9−𝑎69−𝑎96𝑎6U+𝑎U6+𝑎9U+𝑎U9−𝑎69−𝑎962𝑎U9+𝑎9U>0, all orbitals except 𝑥≡𝑥∗ tends to 𝐵𝑑𝑆.. Now let us consider the Rock-Scissors-Paper game with matrix： 𝐴=0𝑎−1−10𝑎𝑎−10 where 𝑎>0, Eqn. (1) has a unique fixed point in 𝐼𝑛𝑡SU, that is, 𝑥∗=6U,6U,6U and 𝐵=2𝑎−1𝑎−1𝑎−12𝑎−1. (i) When 𝑎>1, B is a positive definite matrix and 𝑥∗ is asymptotically stable, and 𝑝∗=6U𝑅+6U𝑆+6U𝑃 is an ESS. All the inner orbits converge to 𝑥∗ . (ii) When 𝑎=1, B is a positive semi-definite matrix and 𝐵=0, 𝑥∗is Lyapunov stable, but not asymptotically stable, and 𝑥,d£≡𝑐U,@6, that is: 𝑥6𝑥9𝑥U≡𝐶. (iii) When 𝑎<1, B is a negative definite matrix and 𝑥∗ is unstable, all the orbits except 𝑥≡𝑥∗ will tend to 𝐵𝑑𝑆U. Obviously, mo

Evidence payload

{
  "printed_excerpt": "That is, when 2\ud835\udc4e6U+\ud835\udc4eU6>0, and 2\ud835\udc4e6U+\ud835\udc4eU6\ud835\udc4e6U+\ud835\udc4eU6+\ud835\udc4e9U+\ud835\udc4eU9\u2212\ud835\udc4e69\u2212\ud835\udc4e96\ud835\udc4e6U+\ud835\udc4eU6+\ud835\udc4e9U+\ud835\udc4eU9\u2212\ud835\udc4e69\u2212\ud835\udc4e962\ud835\udc4eU9+\ud835\udc4e9U>0, then \ud835\udc5d\u2217 is an ESS. In other words, if B does not satisfy these conditions, then \ud835\udc91\u2217 is a Nash equilibrium, but not an ESS. When B is a positive s",
  "reconstructed_doi": "10.1109/tcss.2025.3634842.kMaynard",
  "ref_index": 7,
  "resolved_title": null,
  "verdict_class": "incontrovertible"
}