To prove that not being selected for a block reward reduces wealth more than being selected increases it, observe the following example. If player A possesses 8 out of 10 total coins, having an 80% chance of being selected for a block reward (defined as $1 in this example), being selected increases their coin share to 82%, while not being selected decreases it to 73%.

Impacts of selection on wealth

To prove that the expected wealth share of a validator remains constant, observe the following example. If player A possesses 8 out of 10 total coins, then their expected wealth share after one block reward has been issued is 8/10 x 9/11 + 2/10 x 8/11 = 8/10, which was their initial wealth share.

Constant expected value of wealth

- Being selected for a block reward would not increase the wealth of a validator as much as not being selected would reduce it
- The expected wealth share for a validator does not rise or fall on average

University of Minnesota - Twin Cities

Wealth concentration is the concern that Proof-of-Stake will lead to the rich getting richer, causing a concentration of wealth.

Wealth concentration

Saleh, Fahad. (2021). Blockchain Without Waste: Proof-of-Stake. Wake Forest University. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3183935


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