1. Definition 1 page 1 - It says (at the third line of the definition) that the equality holds for each player. Isn't that redundant, (and therefore wrong)? It seems that the equality holds only for the deviating player. (And the fact that each player can be the deviating player was already mentioned in the second line).
2. Equations (1) and (2) in page 2 - shouldn't the sum on the last operand be over all edges which are in P_i\P_i^? And therefore, shouldn't the text in the paragraph between the equations be "the last element of the second operand is canceled" (instead of the current version that says "not canceled").
3. What is the calculation that leads to the fact that the example for CE on page 5 is indeed CE?
4. I want to make sure I understand why the CCE on page 5 is CCE: In half of the profiles (in the support) player 1 shares his edge with another player, and on the other half he doesn't. Meaning, his cost expectation is 1/2*2+1/2*1=1.5 .
If player 1 deviates to the fixed strategy 0 then his cost expectation would be 1/72[6*2+6*3+3*1+9*2+3*1+9*2+36*1]=1.5 .
Is this calculation correct?
5. Inequality (4) in page 6 - Shouldn't it be equality? I mean, isn't the cost of a profile defined as the sum of the costs of all players?
6. Cost sharing games - what is the meaning of v in profile p? (the last line of the first paragraph in page 8)
7. In the potential definition on page 8 - shouldn't the last sum in this equation go from i=1 to i=p_e?
8. Lemma 16 page 10 - the lemma bounds (lower and upper bound) the potential function. Is this bound holds for all potential functions of fair cost-sharing games? Or just the one defined in page 8? And in general, is there only one potential function for each game ? (in case it exists)
9. the last equation on page 10 - Isn't the demand that outer sum goes over all edges in p^i redundant? If we want to run over all edges that were chosen (at least by one player) the demand p_e>=1 is enough. Also, it's confusing in the middle part of the equation when i has a slightly different meaning.
Thanks a lot