Geodesic
On $k$-accessibility of the core of $TU$-cooperative game
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 8 (2016) no. 2, pp. 3-27

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In the paper, a strengthening of the core-accessibility theorem by the author is proposed. The results obtained demonstrate that for any $k \geq 1,$ and for any imputation $x$ outside of the nonempty core, a $k$-monotonic sequential improvement trajectory $\{x_r\}_{r=0}^{\infty}$ with $x_0 = x$ exists, which converges to some element of the core. Here, $k$-monotonicity means that for any $r > 0,$ an imputation $x_r$ dominates any preceding imputation $x_{r-m}$ with $r \geq m$ and $m \in [1, k].$ Note that the core-accessibility theorem, mentioned above, was established for the case $k = 1$. To show that $TU$-property is essential to provide $k$-accessibility of the core, we propose an example of $NTU$-cooperative game $G$ with a "black hole" being a closed subset $B \subseteq G(N)$ that doesn't intersect the core $C(\alpha_G)$ and contains all the sequential improvement trajectories originating at any point $x \in B$.
Mots-clés : domination, core
Keywords: dynamical system, generalized Lyapunov function, $k$-accessibility. 
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     author = {Valery A. Vasil'ev},
     title = {On $k$-accessibility of the core of $TU$-cooperative game},
     journal = {Matemati\v{c}eska\^a teori\^a igr i e\"e prilo\v{z}eni\^a},
     pages = {3--27},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MGTA_2016_8_2_a0/}
}
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Valery A. Vasil'ev. On $k$-accessibility of the core of $TU$-cooperative game. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 8 (2016) no. 2, pp. 3-27. http://geodesic.mathdoc.fr/item/MGTA_2016_8_2_a0/