# Evolution of Shares in a Proof-of-Stake Cryptocurrency

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The literature regarding Bitcoin has raised several concerns regarding its long-run viability (see, e.g., Biais et al. 2019 and Hinzen et al. 2021). Those concerns, in turn, have led to various proposals for new blockchains that seek to generate Bitcoin’s decentralization while avoiding its various documented limitations. To date, the most prominent proposal is one that changes the governance rules by which the blockchain operates from those used by Bitcoin. Bitcoin uses a set of governance rules known as Proof-of-Work whereas this alternative proposal puts forth a set of governance rules known as Proof-of-Stake.

Early work on Proof-of-Stake (see, e.g., Saleh 2020) has been encouraging, highlighting that it provides similar security guarantees as were established for Proof-of-Work within the Bitcoin White Paper (see Nakamoto 2008). Perhaps even more encouraging, Proof-of-Stake has already been deployed in over 100 blockchains, several of which have comparable or higher transaction activity than Bitcoin (see Irresberger, John and Saleh 2021). Nonetheless, an important concern regarding Proof-of-Stake has always been whether it maintains decentralization in the long-run. Conventional wisdom, without formal verification, has presumed that Proof-of-Stake would eventually generate extreme centralization, which raises foundational objections from blockchain enthusiasts. Nonetheless, a recent paper in *Management Science* entitled “Evolution of Shares in a Proof-of-Stake Cryptocurrency” by Ioanid Rosu (HEC Paris) and Fahad Saleh (Wake Forest University) formally establishes that the referenced conventional wisdom is flawed and that Proof-of-Stake, in fact, tends to maintain its level of decentralization in the long run.

To understand the findings of Rosu and Saleh, it is first important to understand how Proof-of-Stake operates. As indicated, Proof-of-Stake (like Proof-of-Work) determines the governance of a blockchain. A blockchain can be thought of as a digital append-only ledger. As many blockchains aim to achieve decentralization, neither Proof-of-Work nor Proof-of-Stake restrict the set of individuals who may provide updates to the ledger. Rather, both sets of governance rules confer authority to update the blockchain on a criterion that does not depend upon identity. However, even if “anybody” can update either a Proof-of-Work or Proof-of-Stake blockchain, that does not mean that the set of such individuals will not eventually collapse to a powerful oligarchy, thus undermining the intended “decentralization.” Responding to this wealth concentration concern in the case of Proof-of-Stake is the focus of Rosu and Saleh’s work.

Proof-of-Stake specifies the criterion for updating the blockchain as a simple lottery. The blockchain possesses a native asset, called a cryptocurrency, which is both created and settled on the blockchain. As examples, the Bitcoin blockchain possesses a native asset which is the bitcoin cryptocurrency whereas the Ethereum blockchain possesses a native asset which is the ether cryptocurrency. At each relevant time, Proof-of-Stake randomly and uniformly selects among each unit of this cryptocurrency and then grants the updating authority to the owner of the selected unit. The implication of this structure is that the ability to update the blockchain is proportional to one’s holding of the cryptocurrency. For example, holding 1% of the cryptocurrency yields an individual a 1% chance of providing the next update. New lotteries are conducted regularly so that the winner of one particular lottery does not gain centralized control of the blockchain. Nonetheless, the key concern regarding Proof-of-Stake can be described as a concern that eventually only a few individuals will be winning all the lotteries. To understand that concern, it is important to know that most blockchains provide a “block reward” to any individual who updates to the ledger. This block reward is paid in the form of newly issued units of the cryptocurrency or coins, and it is the driving force behind concerns regarding Proof-of-Stake centralization. More precisely, since Proof-of-Stake specifies a lottery over units of cryptocurrency, Proof-of-Stake can be seen as a system that rewards a lottery winner by increasing that lottery winner’s likelihood of winning a future lottery. The conventional wisdom holds that this structure of rewarding a lottery winner with a higher subsequent probability of winning the next lottery would lead to a snow-ball effect and cause only a few individuals to be winning all lotteries eventually. Rosu and Saleh’s work shows that this conventional wisdom is flawed.

A concrete example helps to flesh out both the conventional wisdom and the flaw within it. Suppose there exist only two cryptocurrency units, one held by Alice and one held by Bob. Suppose also that each reward per lottery is one new coin. The conventional wisdom worries that Alice begins with a 50% probability of winning the first lottery, but winning that lottery escalates Alice’s subsequent win probability to approximately 67% and further wins only heighten that probability more until Alice is almost certain to always win. Rosu and Saleh’s first main result can be understood as follows: just as there is a chance that Alice has a streak of victories moving her close to a 100% holding of coins, there is an equally likely probability that Alice has a streak of equally long losses leading her close to a 0% holding of coins. More generally, the potential increase in an individual’s coin share is offset by the potential decrease in that individual’s coin share. Formally, Rosu and Saleh establish that each individual’s share of coins is a “martingale” or a “fair game”, as it is sometimes called in the probability literature. The second main result is that each individual coin share has a stable limiting distribution without point masses (i.e., without concentrating towards one dominating share). The first result, that each individual’s coin share is a martingale, means that while an individual’s coin share can grow or shrink in the future, it does neither *on average,* because the potential increase and the potential decrease exactly offset each other. The second result, that the proportion of coins has a stable limiting distribution without point masses, implies that the likelihood of extreme outcomes such as one individual holding all coins is zero.

Rosu and Saleh further show that mature blockchains, with large numbers of coins already in circulation, produce very stable limiting distributions for the coin shares. Intuitively, this result arises because lucky streaks of consecutive lottery wins by the same individual are both less likely and less impactful when there are many coins already in circulation. Consequently, a high initial number of coins in circulation leads to a tighter limiting distribution for an individual’s coin share. To understand this result, consider the previous example in which Alice and Bob initially own one coin each (Case A) and compare it with a case in which Alice and Bob initially own one thousand coins each (Case B). In both cases, Alice’s initial share is the same, 1/2. In Case A, the likelihood of a lucky streak of five lottery wins in a row is (1/2) x (2/3) x (3/4) x (4/5) x (5/6) = 1/6, and such a streak would bring Alice’s coin holdings to over 85% of all coins. In Case B, however, Alice’s holding would remain close to 1/2 even after five consecutive lottery wins, and the probability of such a streak (which is slightly higher than 1/32) is less than a fifth of the probability in Case A (which is 1/6). Rosu and Saleh’s third main result generalizes this intuition, providing useful guidance for the case when block reward schedules are decreasing, constant, or increasing.

Rosu and Saleh’s fourth main result is that their previous findings remain robust if trading in the cryptocurrency is permitted. One may worry that investors have an incentive to amass the cryptocurrency in order to increase their probability of earning even more coins by winning the Proof-of-Stake lottery. Rosu and Saleh show that this intuition is incorrect. While a purchase of coins does increase the probability that an investor wins a subsequent lottery (which would then increase the investor’s coin holdings), the purchased coins also face *dilution,* as all outstanding coins decline in value once the block reward from the lottery inflates the supply of existing coins. Rosu and Saleh prove that in equilibrium the dilution effect exactly offsets the probability increase, which implies that investors have no incentive to amass large cryptocurrency holdings via trading.

Overall, Rosu and Saleh provide important insights about Proof-of-Stake, one of the most promising ideas within the cryptocurrency community in recent years.

**Read the full article at: https://doi.org/10.1287/mnsc.2020.3791.**

**References**

Biais B, Bisiere C, Bouvard M, Casamatta C (2019) The blockchain folk theorem. Rev. Financial Stud. 32(5):1662–1715.

Hinzen F, John K, Saleh F (2021) Bitcoin’s fatal flaw: The limited adoption problem. Working paper, New York University Stern, New York.

Irresberger F, John K, Saleh F (2021) The public blockchain ecosystem: An empirical analysis. Working paper, New York University Stern, New York.

Nakamoto S (2008) Bitcoin: A peer-to-peer electronic cash system. White paper. https://bitcoin.org/bitcoin.pdf.

Rosu I, Saleh F (2021). Evolution of Shares in a Proof-of-Stake Cryptocurrency. Management Science 67(2): 661-672.

Saleh F (2020) Blockchain without waste: Proof-of-stake. Rev. Financial Stud., ePub ahead of print July 7, https://academic.oup.com/rfs/advance-article-abstract/doi/10.1093/rfs/hhaa075/5868423?redirectedFrom=fulltext.

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