Number of strategies in a sequential game Sorry for bad English, am French. My teacher and I have disagreed on this question. It was part of a homework. Specifically the a). 
The question in question
He and me have different viewpoints on number of strategies. I believe A has 20 different strategies and recieved 0 points on this question. He says A has 20 times 2 times 2 = 80 different strategies, but I think this is wrong. I cant see how he is right. This is not a homework cheating, because the homework is over and I just want to study this for the exam.
 A: I have no knowledge of Nash equilibrium theory, so I'll only address the number of strategies questions.
I assume the 20 different strategies you mentioned are the 20 options party A has when it is initially given the option to form a government (offer 1..10 perks to party B and offer 1..10 perks to party C).
However, if these offers are not accepted, then B and possibly C will also be given the option to form governments, which may include perk offers to party A, and party A needs to decide how to respond to them. That would be part of the possible strategies of party A: If party A initially offers for example 3 perks to party B (one of the 20 options party A has at this point) and this offer is rejected, party A needs to decide how it reacts if party B then offers (for example) 2 perks to party A in the next step. And if that offer is rejected, what it does when party C offers 1 perks in the last possible step.
For each of the 20 possible options initially available to party A, there are 10 possible offers each that B and C can make to party A, and A can answer each offer with 2 options: accept or reject. That means $2^{10}=1024$ possible sub-strategies for what to answer B when they make an offer. The same is true for when C offers perks to A. Of course, there is also the option that B and C don't offer perks to A but to the other when it is their turn, but as this means that there is no decision to make for A, this does not constitute new options for the strategy.
So overall, in my opinion, there are 20*1024*1024 possible strategies for A.
Of course, many of the possible 1024 sub-strategies for A when B (or C) makes them an offer are illogical. It makes no sense to accept for example an offer of exactly 3 perks but to reject exactly 5 perks, a.s.o. In practice it seems that there are 10 sub-strategies that "makes sense": Decide on a number from 1 to 10 as the minimal offer you will accept.
