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Contributions to Game Theory and Management, 2015, Volume 8, Pages 111–136
(Mi cgtm261)
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Strategic stability of coalitions technological alliance parameters: a two-level cooperation
Nikolay V. Kolabutin St. Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Bibliotechnaya pl. 2, St. Petersburg, 198504, Russia
Abstract:
The paper is devoted to two-level cooperation in differential
games. Cooperative differential games are one of the fastest
growing parts of the game theory. They are widely used for modeling
the conflict-controlled processes in various fields, especially in
management and economics. The solution of differential game is a
cooperative agreement, and the selected principle of optimality, according to which the received payoff is distributed.
The main problem of many cooperative solutions is the instability
over time. Studies showed that initially selected cooperative
solution often loses its optimality over time. Therefore, the
question arose about the stability of the co-operative solutions.
The stability can be understood as dynamic stability (time
consistency), strategic stability or protection from irrational
behavior. The concept of dynamic stability was formalized by L.A.
Petrosyan. Cooperative solution is dynamically stable, if the
principle of optimality, selected early in the game keeps its
consistency throughout the gameplay. For dynamic stability is
necessary at each moment of time to carry out the regularization of
the chosen principle of optimality. For this regularization L.A.
Petrosyan proposed to use the redistribution of received payoff in
accordance with the "imputation distribution procedure". Petrosjan
(1993) and Petrosjan and Zenkevich (1996) presented a detailed
analysis of dynamic stability in cooperative differential games, in
which the method of regularization was introduced to construct
time-consistent solutions. Yeung and Petrosjan (2001) designed
time-consistent solutions in differential games and characterized
the conditions that the allocation-distribution procedure must
satisfy. Petrosjan (2003) employed the regularization method to
construct time-consistent bargaining procedures.
The strategic stability of cooperative solution means, that no
individual deviation from the cooperation of each member brings
benefits the decline member. This means that the outcome of this
cooperative agreement is reached at some Nash equilibrium, which
will guarantee the strategic support for such cooperation.
Recently in differential games are studied coalitional solutions in
which the
coalitions act as individual players. Coalitions can play with each other in a non-cooperative game, then payoff of each coalition is distributed
among its members in accordance with some principle of optimality. But coalitions-players can cooperate to increase the joint payoff.
In this case the joint payoff is distributed between coalitions
according to some principle of optimality then coalition's share of
joint payoff is distributed between its participants according to
maybe other principle of optimality. This cooperation is called
two-level cooperation. Optimality principles of payoff distribution
between coalitions and within coalition may be different.
To solve such cooperative models which requires at both levels of
the cooperation it is necessary to build the characteristic function
and imputation distribution procedure. This paper describes a model
of a two-level cooperation in the technological alliance
differential game. Participants of the game are the firms with the
some technology that brings profit. On the first (lower) level firms
form coalitions to increase joint profit. On the second (upper)
level coalitions act as individual players and also form the one
grand coalition to maximize the joint payoff. The top-level payoff
is distributed between coalitions-participants according to some
principle of optimality. Thus, each coalition-participants may get
more than would receive by playing individually. Then each coalition
distributes the its share of joint payoff among its firms-members.
This article also presented a stable cooperative solution in this
model. For its implementation at every level of cooperation we build
the characteristic function and prove its superadditivity. As a
principle of optimality the dynamic Shapley value is selected.
Proved the dynamic stability and the strategic stability of
cooperative soluton. The results are illustrated by a quantitative
example.
Keywords:
differential game, cooperation, imputation distribution procedure, dynamic stability, strategic stability.
Citation:
Nikolay V. Kolabutin, “Strategic stability of coalitions technological alliance parameters: a two-level cooperation”, Contributions to Game Theory and Management, 8 (2015), 111–136
Linking options:
https://www.mathnet.ru/eng/cgtm261 https://www.mathnet.ru/eng/cgtm/v8/p111
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