2. example 1 page 8 - when I calculate p (the probability in which the rows player plays B in NE) I get the result p=3/5, in contrast to the result in the scribe : p=2/5. So I'm trying to understand if there is is a mistake in my calculation, and if there is - what is it?
The calculation was done as follows:
We assume we get NE when player 1 (rows player) plays B with probability p and plays O with probability 1-p.
We can calculate p using the fact that each pure strategy choice of player 2 (column player) leads to the same utility expectation for player 2, meaning E[u_2| player 2 plays B] = E[u_2| player 2 plays O]. (The utility expectation for player 2 when he plays B is equal to the utility expectation for player 2 when he plays O). So we have 2p+0(1-p) = 0p+3(1-p) leading to p=3/5.
Where is the mistake here?
3. The function in Brouwer fixed-point theorem is from domain C to range C, when C is a subset of R^t. (So any element in C can be viewed as a vector of size t). The domain and range used in the proof of the existence of NE are not a subset of some R^t. (Any element in that domain is a "vector of vectors"). So the implementation of Brouwer fixed-point theorem seems to be not very trivial.
Anyway, I was asking: can we interpret a profile as a vector of size t=|S_1|+..+|S_n| (and then the using of Brouwer theorem will be more trivial)?
4. in page 10, after the minmax theorem (lemma 15) it said "Immediate implication: with mixed strategies, the order of the turns does not matter - the players will take the same strategy sets."
How does it implies from that lemma? (The lemma above is an inequality (unlike the minmax theorem in lecture 2) and it seems that the only thing implies from it is that each player is better playing second).
Thank you
