Stability in game theory
A principle reflecting directly or indirectly the idea of stability of a situation (or of a set of situations). One singles out the following basic concepts of stability.
1) -stability, cf. Coalitional game.
2) -stability. An optimality principle in a cooperative game, connected with the concept of stability of pairs, consisting of a partition of the set
of players into coalitions and allocations relative to the formation of new coalitions. A partition
of the set
of players is called a coalition structure. Let
be a cooperative game and
a function associating with every coalition structure
a set of coalitions
. A pair
, where
is an allocation, is called
-stable if
for all
and if
when
.
3) -stability. A special case of
-stability, when for
a set of coalitions is chosen, each of which differs from any element of
by not more than
players.
4) -stability. An optimality principle in the theory of cooperative games which formalizes the intuitive notion of stability of formation of coalitions and allocation of values
of a characteristic function
defined on the set of coalitions
relative to the possible threat of one coalition against the others. A pair
, where
is a vector satisfying the condition
,
, while
is a coalition structure, is called a configuration. A configuration is said to be individually rational if
,
. A configuration
is called coalitionally rational if the vector
satisfies
for any coalition
,
. In case
, in particular when
, for every individually rational configuration
the vector
is an allocation.
The set is called the set of partners of a coalition
in a coalition structure
. Let
be a coalitionally rational configuration and let
be disjoint coalitions. A coalitionally rational configuration
satisfying the conditions
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is called a threat of a coalition against
. By a counter-threat of
against
one understands a coalitionally rational configuration
satisfying the conditions
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A coalitionally rational configuration is called
-stable if for any pair of disjoint coalitions
and for every threat of
against
there is a counter-threat of
against
. The set of all
-stable configurations for a coalition structure
is called the
-stable set and is denoted by
or
. In the case
, the set
contains the core (cf. Core in the theory of games) of the cooperative game
. The set
often turns out to be empty, and therefore one considers further the set
which is defined analogously to
, with the following changes: one considers not only coalitionally rational configurations, but all individually rational configurations admitting only threats and counter-threats among one-element coalitions, i.e. between individual players. It can be proved that the set
is non-empty for any coalition structure. The set
for
contains the
-kernel and coincides with it and the core for a convex game
.
The concepts of -stability and
-stability have a natural generalization to cooperative games without side payments. It is known that in this case the set
may be empty; there are certain conditions for
to be non-empty.
References
[1] | R.J. Aumann, M. Maschler, "The bargaining set for cooperative games" , Advances in game theory , Princeton Univ. Press pp. 443–476 |
[2] | N.N. Vorob'ev, "The present state of the theory of games" Russian Math. Surveys , 25 : 2 (1970) pp. 77–150 Uspekhi Mat. Nauk , 25 : 2 (1970) pp. 81–140 |
[3] | R.D. Luce, , Mathematical Models of Human Behaviour , Stanford (1955) pp. 32–44 |
[4] | R.D. Luce, H. Raiffa, "Games and decisions. Introduction and critical survey" , Wiley (1957) |
[5] | B. Peleg, "Existence theorem for the bargaining of ![]() |
[6] | B. Peleg, "Quota games with a continuum of players" Israel J. Math. , 1 (1963) pp. 48–53 |
[7] | G. Owen, "The theory of games" , Acad. Press (1982) |
Stability in game theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stability_in_game_theory&oldid=14883