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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     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},
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     volume = {8},
     number = {2},
     year = {2016},
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     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/

[1] Neiman Dzh., Morgenshtern O., Teoriya igr i ekonomicheskoe povedenie, Nauka, M., 1970 | MR

[2] Ouen G., Teoriya igr, Mir, M., 1971 | MR

[3] Rozenmyuller I., Kooperativnye igry i rynki, Mir, M., 1974

[4] Mashler M., Peleg B., “Stable sets and stable points of set-valued dynamic systems with applications to game theory”, SIAM J. of Control and Optimization, 14 (1976), 985–995 | DOI | MR

[5] Vasil'ev V. A., “Generalized von Neumann-Morgenstern solutions and accessibility of cores”, Soviet Math. Dokl., 36 (1987), 374–378 | MR

[6] Vasil'ev V. A., “Cores and generalized NM-solutions for some classes of cooperative games”, Russian Contributions to Game Theory and Equilibrium Theory, eds. T. Driessen, G. van der Laan, V. Vasil'ev, E. Yanovskaya, Springer-Verlag, Berlin–New York, 2006, 91–150 | DOI

[7] Wu L. S.-Y., “A dynamic theory for the class of games with nonempty cores”, SIAM J. Appl. Math., 32:2 (1977), 328–338 | DOI | MR | Zbl