The purpose of this short article is not to disapprove concepts and achievements of IOHK foundation. It is rather an opposite – an attempt to openly discuss issues related to the newest version ‘Shelley’ that will be tested in the upcoming months and may be implemented next year. I strongly believe that the community can and should participate in the development of better versions of consensus protocols and public incentives by challenging developers with questions. Therefore, I initiate discussion about properties of ‘Shelley’ by posting several thought-provoking questions related to the newly designed reward sharing scheme and its effect on decentralization of Cardano in the future.
For those unfamiliar with the details of the scheme I could recommend the following three-step path (start with easy concepts and finish with the details in the academic style paper):
A) relatively easy reading about new PoS implemented by Cardano: a) some very basic info about PoS principles (can be googled in 30 min); b) https://iohk.io/en/blog/posts/2018/10/23/stake-pools-in-cardano/ c) page on https://staking.cardano.org/en/;
B) quite involved reading, where some parts (if not understood) can be skipped: a) specification for Shelley version of Cardano (here hydra.iohk.io/build/902246/download/1/delegation_design_spec.pdf ); b) watch 
C) the hardest part -- read their paper https://arxiv.org/abs/1807.11218 and try to analyze it critically, discuss here, ask questions.
For those who are ready, welcome to jump to the questions. I sincerely hope that many more questions and comments will appear on this page.
1.. According to ‘Engineering Design Specification’ developed by Lars Bruenjes, different parameters may affect incentives and decentralization in Cardano. This parameters include desired number of pools $k$, parameter $a_0$ that is supposed to affect resistance to Sybil attack, as well as parameters $\eta$ and $\rho$ influencing monetary expansion. In that regard, there are several important queries:
- What are the exact values for these parameters (except $k$ which has been broadly discussed in the community) that will be set for the test version of the network and the ‘official’ mainnet?
- Is there a model(s) allowing to define optimal $a_0$, $\eta$, $\rho$?
- Could you, please, describe on how (procedure) these values can be decided (and maybe changed) in a decentralized manner? In case there is a central authority who decides these questions, doesn’t it mean that Cardano is not fully decentralized?
2.. Under certain assumptions (that are quite likely) setting parameter $a_0$ to be different from 0 has little sense. Presence of $a_0$ in the expression for pool reward is rather a heuristic move which is supposed to improve resistance to Sybil attacks. As a result, pools with larger portion of ‘stake deposited by the operator’, $s^\prime$, are favored during reward distribution. Unfortunately, it is unclear if there is a tool helping to distinguish ‘stake deposited by the operator’ from the ‘stake that is claimed to be deposited by the operator’. Intuitively, it is clear that operator can invite members to deposit ADA under ‘her name’ and to redistribute the extra gain which will serve as an incentive for such kind of cheating. Eventually, this behavior will be adopted by all pools and value of $a_0$ will not affect actual (in contrast to the described by the expression) reward distribution since secret agreements between operator and members will define it.
3.. The paper “Reward Sharing Schemes for Stake Pools” attempts at demonstrating that $k$ pools will be formed if that belief is maintained by the participants of the system meaning that such belief is consistent. However, there are other beliefs that may be consistent, too. For example, a belief that there will be less than $k$ pools may be of a particular importance. Lower than projected number of pools would reduce utilities of the members (‘stakers’) and may result in a lower rate of participation (in consensus). How would that affect ‘decentralization’ in Cardano?
Explanation. In order to see that the number of pools can indeed be larger than 1 and less than k, please, consider best responses of pool operator and members in a situation when for a particular pool (with rank $\le k$) we have condition $\hat{f}\le c$ ($\hat{f}$ is a pool reward, c is the cost to run the pool). Rewards for this condition are described on p. 40 of ‘Engineering Design Specification’. Next, take into account that in such pool, utility of a member under this condition is 0. On the other hand, there is a pool in the system with a lower rank (e.g. bigger pool) where utility of a member is positive. Therefore, the best response of a member is to join that bigger pool. In that situation, we can see that pool operator is the only person who can change condition $\hat{f}\le c$ by staking/depositing her own ADA in the pool. In case she is not able to deposit enough ADA to make $\hat{f}$ surpass cost $c$ in that pool, the pool is not going to grow. This means that the belief ‘less than k pools will be formed’ can be consistent, especially under high cost of operation $c$.
4.. Participation rate is an important indicator for PoS consensus. For instance, it can be defined as a percentage of all ADA that is staked. Intuitively, high participation rate means high interest and attention to the performance of block validators – those who run pools. In turn, this will guarantee that members actively stake with best performers which supports healthy operation of Cardano network.
Reward for participation in consensus is a form of investment made by the members, and the rate of participation depends on other available investment options. Return from alternative investments will define whether a member will stake or not (e.g. best response).
Cryptocurrency lending/depositing can be seen as a form of investment that is alternative to staking in PoS. Many services of that kind have emerged during recent year or two (see https://blockfi.com/ and https://www.coinlend.org/). In some cases, they guarantee return of 7%-8% for a year on major cryptocurrencies. How can Cardano achieve substantial participation rate under such competitive circumstances?