I'm learning weighted voting methods and MathLab does a terrible job explaining this question so I'm trying to get clarification.
I don't understand how you can find a critical player without a quota. Normally these types of questions are asked like so [4:3,2,1] where 4 is the quota and 3,2,1 are P1, P2, P3 but no vlaues are given so how can you determine which is the critical player? (noted with underlines)
You read each sequential coalition from left to right, and you stop when it becomes a winning coalition. The odd thing about this problem is that it's using the same notation $\langle \rangle$ to refer to unordered winning coalitions, and ordered sequential coalitions - I'd prefer to just refer to an unordered coalition using set notation, e.g., $\{P_1,P_2,P_3\}$.
To step through an example of $\langle P_3,P_2,P_1 \rangle$:
First you have a coalition $\langle P_3\rangle$ by itself. This isn't a winning coalition yet, so we add the next member. Now we have $\langle P_3, P_2 \rangle$. This still isn't a winning coalition. Finally, we go to $\langle P_3, P_2, P_1 \rangle$. This matches one of the winning coalitions (up to ordering), so the last member added - $P_1$ - is the pivotal member.
As an aside, pivotal and critical are two different things - pivotal is looking at a coalition with an order (e.g., Shapley-Shubik), critical is looking at an unordered coalition (e.g., Banzhaf).