Geodesic
Chain equilibria in secure strategies
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 8 (2016) no. 1, pp. 80-105.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we introduce a modification of the concept of Equilibrium in Secure Strategies (EinSS), which takes into account the non-uniform attitudes of players to security in non-cooperative games. In particular, we examine an asymmetric attitude of players to mutual threats in the simplest case, when all players are strictly ordered by their relation to security. Namely, we assume that the players can be reindexed so that each player i in his behavior takes into account the threats posed by players j > i but ignores the threats of players j i provided that these threats are effectively contained by some counterthreats. A corresponding equilibrium will be called a Chain EinSS. The conceptual meaning of this equilibrium is illustrated by two continuous games that have no pure Nash equilibrium or (conventional) EinSS. The Colonel Blotto two-player game (Borel 1953; Owen 1968) for two battlefields with different price always admits a Chain EinSS with intuitive interpretation. The product competition of many players on a segment (Eaton, Lipsey 1975; Shaked 1975) with the linear distribution of consumer preferences always admits a unique Chain EinSS solution (up to a permutation of players). Finally, we compare Chain EinSS with Stackelberg equilibrium.
Keywords: noncooperative games, equilibrium in secure strategies, asymmetric behavior, Blotto games, product competition, Stackelberg equilibrium. 
@article{MGTA_2016_8_1_a4,
     author = {Alexey B. Iskakov and Mikhail B. Iskakov},
     title = {Chain equilibria in secure strategies},
     journal = {Matemati\v{c}eska\^a teori\^a igr i e\"e prilo\v{z}eni\^a},
     pages = {80--105},
     publisher = {mathdoc},
     volume = {8},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MGTA_2016_8_1_a4/}
}
TY  - JOUR
AU  - Alexey B. Iskakov
AU  - Mikhail B. Iskakov
TI  - Chain equilibria in secure strategies
JO  - Matematičeskaâ teoriâ igr i eë priloženiâ
PY  - 2016
SP  - 80
EP  - 105
VL  - 8
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MGTA_2016_8_1_a4/
LA  - ru
ID  - MGTA_2016_8_1_a4
ER  - 
%0 Journal Article
%A Alexey B. Iskakov
%A Mikhail B. Iskakov
%T Chain equilibria in secure strategies
%J Matematičeskaâ teoriâ igr i eë priloženiâ
%D 2016
%P 80-105
%V 8
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MGTA_2016_8_1_a4/
%G ru
%F MGTA_2016_8_1_a4
Alexey B. Iskakov; Mikhail B. Iskakov. Chain equilibria in secure strategies. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 8 (2016) no. 1, pp. 80-105. http://geodesic.mathdoc.fr/item/MGTA_2016_8_1_a4/

[1] Aleskerov F. T., “Teoretiko-igrovoe modelirovanie: popytka kratkogo obsuzhdeniya i prognoza razvitiya”, Zhurnal Novoi ekonomicheskoi assotsiatsii, 17:1 (2013), 181–184

[2] Iskakov M. B., “Ravnovesie v bezopasnykh strategiyakh”, Avtomatika i telemekhanika, 2005, no. 3, 139–153

[3] Iskakov M. B., Iskakov A. B., “Ravnovesie, sderzhivaemoe kontrugrozami, i slozhnoe ravnovesie v bezopasnykh strategiyakh”, Upravlenie bolshimi sistemami, 51, 2014, 130–157

[4] Iskakov A. B., Iskakov M. B., “Ravnovesiya v bezopasnykh strategiyakh v tsenovoi duopolii Bertrana–Edzhvorta”, Matematicheskaya teoriya igr i ee prilozheniya, 6:2 (2014), 42–59

[5] d'Aspremont C., Gerard-Varet L.-A., “Stackelberg-Solvable Games and Pre-Play Communication”, Journal of Economic Theory, 23:2 (1980), 201–217 | DOI | MR | Zbl

[6] Baye M. R., Kovenock D., de Vries C. G., The solution of the Tullock rent-seeking game when $R > 2$: mixed-strategy equilibria and mean dissipation rates, Discussion Paper 1993-68, Tilburg University, Center for Economic Research, 1993

[7] Borel E., “The Theory of Play and Integral Equations with Skew Symmetric Kernels”, Econometrica, 21:1 (1953), 97–100 | DOI | MR | Zbl

[8] Dasgupta P., Maskin E., “The existence of equilibrium in discontinuous economic games. II: Applications”, Review Economic Studies, LIII (1986), 27–41 | DOI | MR | Zbl

[9] Eaton C., Lipsey R., “The Principle of Minimum Differentiation Reconsidered: Some New Developments in the Theory of Spatial Competition”, Review of Economic Studies, 42 (1975), 27–50 | DOI

[10] Iskakov M., Iskakov A., Zakharov A., Equilibria in secure strategies in the Tullock contest, CORE Discussion Paper 2014/10, Universite Catholique de Louvain, Center for Operations Research and Econometrics, Louvain, 2014

[11] Iskakov M., Iskakov A., Equilibrium in secure strategies, CORE Discussion Paper 2012/61, Universite Catholique de Louvain, Center for Operations Research and Econometrics, Louvain, 2012

[12] Iskakov M., Iskakov A., “Solution of the Hotelling's game in secure strategies”, Economics Letters, 117 (2012), 115–118 | DOI | MR | Zbl

[13] Iskakov M., Iskakov A., Asymmetric equilibria in secure strategies, Working Paper WP7/2015/03, National Research University Higher School of Economics, Higher School of Economics Publ. House, M., 2015

[14] Kats A., Thisse J.-F., “Unilaterally competitive games”, International Journal of Game Theory, 21 (1992), 291–299 | DOI | MR | Zbl

[15] Osborne M. J., Pitchik C., “Equilibrium in Hotelling's model of spatial competition”, Econometrica, 55:4 (1987), 911–922 | DOI | MR | Zbl

[16] Owen G., Game Theory, Academic Press, New York, 1968 | MR

[17] Rothschild M., Stiglitz J. E., “Equilibrium in competitive insurance markets: an essay on the economics of imperfect information”, Quarterly Journal of Economics, 90 (1976), 629–649 | DOI

[18] Shaked A., “Non-Existence of Equilibrium for the 2-Dimensional 3-Firms Location Problem”, Review of Economic Studies, 42 (1975), 51–56 | DOI | Zbl

[19] Stackelberg H., Marktform und Gleichgewicht, Springer, Wien–Berlin, 1934

[20] Wilson C., “A model of insurance markets with incomplete information”, Journal of Economic Theory, 16 (1977), 167–207 | DOI | MR | Zbl