Definition of secure computation with more that two parties In Definition of secure computation in m-party case with respect to semi-honest adversary (Definition 7.5.1 in Foundations of Cryptography: Volume 2, Basic Applications by Oded Goldriech) we say protocol $\Pi$ is secure if there exists simulator $S$, such that for every $I\subset [m] = \{1,2,...,m\}$ output of simulation and $VIEW^\Pi_I$ be indistinguishable.

I don't understand why it is not enough to construct a simulator only for every party $p\in{1,2,...,m}$ (instead of any subset of parties). Besides order of quantifiers seems confusing to me too. It is mentioned in the footnote that (page 697):


  Note that for a fixed $m$, it may make as much sense to reverse the order of quantifiers (i.e., require that “for
  every $I$ there exists an algorithm $S_I$ ”).


What does "may make as much sense" mean? The original order of quantifiers doesn't make sense to me, but the reverse does. (Reverse order just seems like definition of two-party case.)

I would appreciate:

1) explanation (maybe including simple example) that give me the intuition about why security is defined this way.

2) Suggestion of good reference (almost as good as Yehuda Lindell's tutorial) as an example of security proof in multiparty case.
It is not enough to be able to simulate the view of a single party, because corrupted parties may cooperate. As mentioned in the book, it may be easier to view all the corrupted parties as a single entity. It is the view of this single entity that we must simulate.

If $m$ is fixed, the order of quantifiers does not matter, because the number of subsets $I$ is fixed and so a single simulator $S$ can of course incorporate the code of the simulator $S_I$ for each $I$ (since $S$ is given $I$, it knows which of the $S_I$ to run as a subroutine).

(Edited after a comment.) If $m$ is not fixed, this may not be possible (for example if $m$ grows polynomially with the security parameter, the number of subsets grows exponentially), so you get two possibly different definitions depending on the order of the quantifiers. The one with a single simulator is stronger (if there is a single simulator that can generate a valid view, there trivially is a family of simulators that can, but not the other way around), so it is the one that was used. In practice, no protocol is known which is secure for the first definition and not the second.