New Year's Countdown Day 17: Age is Just a Number

Three months ago on this day, I celebrated my 17th birthday. I invited 16 of my friends to attend my party, who were all younger than me and had distinct ages. After the party was over, there were a whopping 2017 pieces of cake left even after everyone had a few pieces each. In order to divide this remaining cake among ourselves, we agreed to the following procedure:

1. Everyone at the party gathers to form an after-party group.
2. The oldest person present in the group proposes a plan to divide the cake among the people in the group.
3. Each person votes on whether they are in favor of the plan or not, including the original proposer. Everyone has one vote each and can only vote in favor of or against the proposed plan. Also, only people in the group can vote.
4. If half or more of the people in the group vote in favor of the plan, then the cake is distributed according to that plan.
5. However, if less than half of the people in the group vote in favor of the plan, the original proposer is labeled a "party pooper" and is cast out of the after-party group. Then, the process above is repeated, with the next oldest person proposing the plan.
6. This procedure continues until a plan is approved or one person is remaining in the group.

Since I was the oldest of my friends, I got to propose a plan first. What is the maximum number of pieces of cake I could have gotten for myself?

\(\) Details and Assumptions:

• The pieces of cake cannot be divided any further than they are already.
• My friends and I are all perfectly logical, meaning that we can deduce as much as we can from the information we are given.
• The primary goal of everyone is to get as much cake as possible while staying in the after-party group. After that, the secondary goal is to prevent others from getting too much cake.

This problem is part of the set New Year's Countdown 2017.