The limitations of this approach are immediately evident. It is
definitely not geared to cope well with more complex, multi-player,
semi-cooperative (semi-competitive), imperfect information situations.
Von Neumann proved that there is a solution for every ZSG with 2
players, though it might require the implementation of mixed strategies
(strategies with probabilities attached to every move and outcome).
Together with the economist Morgenstern, he developed an approach to
coalitions (cooperative efforts of one or more players - a coalition of
one player is possible). Every coalition has a value - a minimal amount
that the coalition can secure using solely its own efforts and
resources. The function describing this value is super-additive (the
value of a coalition which is comprised of two sub-coalitions equals,
at least, the sum of the values of the two sub-coalitions). Coalitions
can be epiphenomenal: their value can be higher than the combined
values of their constituents. The amounts paid to the players equal the
value of the coalition and each player stands to get an amount no
smaller than any amount that he would have made on his own. A set of
payments to the players, describing the division of the coalition's
value amongst them, is the "imputation", a single outcome of a
strategy.
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