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Rational Ponzi Games in Algorithmic Stablecoin

Algorithmic stablecoins (AS) are one special type of stablecoins that are not backed by any asset (equiv. without collateral). They stand to revolutionize the way a sovereign fiat operates. As implemented, these coins are poorly stabilized in most cases, easily deviating from the price target or even falling into a catastrophic collapse (a.k.a. Death spiral), and are as a result dismissed as a Ponzi scheme. However, is this the whole picture? In this paper, we try to reveal the truth and clarify such a deceptive concept. We find that Ponzi is basically a financial protocol that pays existing investors with funds collected from new ones. Running a Ponzi, however, does not necessarily imply that any participant is in any sense losing out, as long as the game can be perpetually rolled over. Economists call such realization as a \textit{rational Ponzi game}. We thereby propose a rational model in the context of AS and draw its holding conditions. We apply the model to examine: \textit{whether or not the algorithmic stablecoin is a rational Ponzi game.} Accordingly, we discuss two types of algorithmic stablecoins (\text{Rebase} \&\text{Seigniorage shares}) and dig into the historical market performance of two impactful projects (\text{Ampleforth} \&\text{TerraUSD}, respectively) to demonstrate the effectiveness of our model.


Introduction
Ponzi is making a comeback that is used to describe the hyped algorithmic stablecoins today. The most recent critique was directed at the one called Ter-raUSD (UST), which crashed by almost 98% within 24 hours on May 12 th , 2022 (cf. [1]), impacting a direct of $85B market value evaporation and the collapse of entire crypto markets. This worst-hit reveals a new type of bank run and has triggered repetitive rounds of criticism and discussion about the stablecoin and its algorithmic stabilized approaches. Stablecoins are created to share all great features that cryptocurrencies hold (e.g., decentralization, automation, cross-border), without, however, suffering from the same volatility, making them much more applicable as a store of value, medium of exchange, and unit of account [2,3,4]. In an ideal form, stablecoins are simply cryptocurrencies with stable value. Current stablecoin implementations (cf. Tab.1) aim to bridge external assets with the cryptocurrency space by anchoring a peg (e.g., USD) in price. Stablecoins in cryptocurrencies are generally tagged as either collateralized or algorithmic stablecoins in terms of their specific collaterals. The collateralized stablecoin, as the name implies, requires collateral and thus ensures the circulating token having a redemption value. Possible collateral options include fiat money, cryptocurrency, and other valuable assets. Currently, stablecoins are majorly backed by fiat money (Tab.1). USDC [5], for example, is a fully collateralized stablecoin pegged to USD and stores its US dollars in a number of banks. Such a type of stablecoins, backed by fiat money, are generally of a high degree of centralization. Instead, using cryptocurrencies as collateral can circumvent this problem. However, considering the high volatility, an over -collateralization is thereby required if relying on cryptocurrencies as collateral. One representative in this category is the stablecoin DAI [6], where a user needs to deposit Ether (the native token of Ethereum) in excess of the amount of created DAIs. Even if the collateral crypto depreciates, each unit of the stablecoin still has enough space of being redeemed for at least an equivalent amount of underlying assets.
Algorithmic stablecoin (AS), however, does not require any collateral. The money supply of algorithmic stablecoin is elastic and managed by algorithms (i.e., coded in smart contracts). Algorithmic stablecoins are thus not limited in scale by the circulating amount of the underlying collateral. For stabilization, the par value of an algorithmic stablecoin is basically preserved by expanding supply when the price is too high and contracting it when the price is too low. Two distinct AS designs are prevailing in markets [7], namely, rebase and seigniorage shares. Rebase [8] is an algorithm that automatically adjusts the money supply in response to the market demand on a routine basis. A rebase protocol can add (mint) or remove (burn) coins from circulation according to the stablecoin's price deviation from the peg, and this is normally achieved by adjusting token balance pro rata across all wallets. Instead of rebasing one coin, seigniorage shares [9] normally has a dual (or multiple) coin design: the price-stable coins and flexible investment shares. The shares algorithmically expand and contract the supply of price-stable coins much like a central bank does with fiat currencies. Seigniorage shares model has, to varying degrees, served as a foundation for TerraUSD [10], Empty Set [11], Basis Cash [12], and Frax [13].
Contributions. Unfortunately, no satisfactory algorithmic stablecoins have been healthily operated for a long-term run. Non-stable performance and non-collateral property destroy the natural trust of participants and communities. AS schemes are thus often questioned as Ponzi schemes. This motivates our observations towards this fancy but fuzzy concept. First, we comprehensively review existing in-the-wild AS projects and explore the design primitives they share in common (Sec.1). Then, we accordingly have established a rational Ponzi model in the context of algorithmic stablecoins (Sec.2) and further evaluate two mainstream projects, namely, Ampleforth and TerraUSD (the representatives of Rebase and Seigniorage share, respectively), based on six-month historical data from Mar 1 st to Oct 17 th 2022 (Sec.3). Also, we find several interesting insights (Sec.4) that are driven by our model. Last, we deliver a taking-home message saying: the rebase approach is much more likely to achieve a rational Ponzi game as long as it always exists in an active trading market whereas the seigniorage share method is, if without external interruptions, of low probability to realize it due to the uncontrollable death spirals from the overlapped effect of dual coins. Recent Studies. A series of papers and reports [4,14,15,16] establish and extend the taxonomies in stablecoins based on the peg type, the collateral type, and the collateral amount. They give an initial sight to demystify its assemblies and components. Besides, with in-depth analysis, a bunch of follow-up works is proposed. Moin et al. [2] provide a systematical classification of existing stablecoins in terms of their design elements. Ariah et al. [3] explore the economic structure of stablecoins and establish a risk-based functional characterization. Pernice et al. [17] research stablecoins from a monetary perspective. Salehi et al. [18] explore the liquidation process in stablecoins, and put forward a research agenda for alternatives to liquidation. Clements R. [19] studies the inherent fragility in algorithmic stablecoins and claims that they are built to fail. Zhao et al. [7] conduct an empirical analysis to explain the volatility of algorithmic stablecoins.

Modeling Rational Ponzi Games
Ponzi [20] has always been such a term that is instinctively resented by the public. For malicious organizers, Ponzi schemes are used to lure new investors, and leverage incoming money to pay earlier ones while keeping the rest as profit. In general, organizers will promise an abnormally high return to investors within the short term. With little or no legitimate earnings, Ponzi games require a constant flow of new funds to survive. Once becoming difficult to recruit new participants, or whenever a large number of existing investors cash out, these games tend to collapse. However, stemming from its origin, the substance of Ponzi is neutral yet subtle. Economists [21,22] discuss a perfect foresight version of the Ponzi model, where the schemes based on imperfect information or irrationality of lenders are ruled out. They call such a policy, in which all repayments are forever rolled over, i.e., financed by issuing new debt, a rational Ponzi game [23]. Following such narrative connotations, we develop the model in the context of algorithmic stablecoins.

Rational Ponzi
Game. An entity that plans to issue debt and roll it over the possibly infinite future is the borrowing side of a rational Ponzi game. Any plan will impose a set of net cash inflows I s from the lending side. The present value of borrower's net indebtedness D T of at any time T equals the present value of the stream of prospective incomes generated by lenders between 0 and T : where Γ (T ) is the discount factor of the borrowing side, it is calculated by the interest rate r t between periods t − 1 and t 3 , and Γ (s) is the discount factor applicable in period 0 to cash inflows received in the period s, where Γ (s) ≡ s j=1 (1 + r j ) −1 . We now provide the definition of a rational Ponzi game.
Definition 1 (Rational Ponzi Game). A rational Ponzi game is a sequence of debt transactions satisfying the following sense: (i) to the borrower, it is a positive net present indebtedness value such that lim T →∞ Γ (T )D T > 0, if the limit of Γ (T )D T exists; (ii) to all the participants, i.e., both the borrowing and lending sides, the game enables a Pareto improvement 4 .
The first holding condition in Definition 1 implies that debts are unnecessarily fully paid off by future cash inflows if in an infinite horizon setting. The second condition imposes restrictions that no one is being worse (assets can always be redeemed at least equal to the market value of assets deposited). Therefore, the two conditions guarantee a rational realization. We provide a general operation process of rational Ponzi schemes as in Fig 1. The red plus sign and the green minus sign represent cash inflow and cash outflow, respectively. The values in parentheses following each participant represent their corresponding utility. Scope of application. One may consider that only governments are entitled to run such rational Ponzi games. However, in a world where rational Ponzi games can exist, any infinitely lived entity can issue debt and perpetually roll it over. In principle, even a finitely lived entity can issue bonds with zero coupon, i.e., fiat money. All required is to make participants believe that there will always exist followers being willing to purchase the debt. Meanwhile, economists have proven that the introduction of perpetually rolled over debt will never make the lending economy worse off relative to a world in which no Ponzi game runs [23,24,25].

Applied to Algorithmic Stablecoins
Typically, algorithmic stablecoins, based on their operating mechanisms, can be considered a Ponzi game: an on-chain protocol launches a coin (equiv. debt) claimed to be stable (investors can always obtain the equivalent fund back when selling the coin), and one joins the game by purchasing the coin and can quit the game if others are willing to buy his coins on the market. However, is such an algorithmic stablecoin game rational under our model? In this section, we instigate two typical instances, Ampleforth (rebase) and TerraUSD (seigniorage shares), to answer this question. Supposing that all the lending protocols are deployed on smart contracts on a public ledger so that all steps are perfectly enforceable (equiv. no default risk). We define the notation U as the investor's utility; it is basically the product of the number of held assets Q and the market price of the token P , in symbols as Then, we apply our model to two concrete instances by investigating their six-month historical data (Mar -Oct, 2022). We analyze the utility of investors based on extracted data covering price, quantity, and market cap.
Ampleforth. Ampleforth (AMPL) [26] is an algorithmic stablecoin protocol that applies the design of rebase. AMPL is pegged to the CPI-adjusted USD, and its rebase procedure occurs at 02:00 UTC on a daily basis. The quantity of AMPL tokens in user accounts automatically expands or contracts based on everyday price. According to Ampleforth white paper [27], the absolute supplyDelta of AMPL is equal to (price-target)*totalSupply/target. For example, at the time of rebase, if the price of AMPL is $1.1, 10% greater than the standard 1 USD peg, a wallet with a balance of 100 AMPL will be rebased as 110 accordingly. In contrast, when the AMPL price is less than 1 USD, the protocol will proportionally decrease the quantity of tokens in wallets, aiming to bring the price up to its anchor. Investor's utility analysis. Considering an investor who participates in the Ampleforth game at any time by purchasing one unit of AMPL, we analyze the changes in his utility over time. At each rebase time (cf. Fig.2), the investor's utility (red line) stays the same 5 , meaning that the investor can always redeem (trade) the identical value of his investment back. However, the price of AMPL (green line) cannot keep stable, and the AMPL balance (blue line) 6 is in the investor's wallet. In terms of a rational Ponzi model, AMPL has the potential to satisfy all listed conditions as long as there exists an lively traded market.
Since any investor can retain at least the same utility in the AMPL network, regardless of the price of the token, one may wonder why the rebase is not being adopted on a larger scale. The most obvious reason is due to its highly volatile price, resulting in holding AMPL being no different than holding a regular cryptocurrency. At the same time, one's balance varies proportionally to the price every day, making the holder fall into passive and anxious emotions. Besides, AMPL, as an ERC-20 token that works natively on Ethereum blockchain, cannot perform its rebase implementation smoothly in a centralized exchange or other blockchain platforms, significantly restricting its use scenarios. Ampleforth mitigates this constraint with the release of Wrapped-AMPL, a token that wraps AMPL similar to wrapped ETH. Wrapped-AMPL facilitates ecosystem integrations on both centralized and decentralized platforms.
TerraUSD. TerraUSD (UST) [10] is a seiniorage shares style stablecoin pegging to the USD on top of Terra's blockchain. LUNA, as the native staking token of the Terra blockchain, absorbs the volatility from UST. Aibitrage can help making UST price stable: when the UST's price is < 1 USD, arbitragers can send 1 UST to the system and receive 1 USD's worth of LUNA; whereas the UST's price becomes > 1 USD, arbitragers will send 1 USD's worth of LUNA to the system and receive 1 UST. Fig. 3: UST price, number of UST, and investor's utility over time.
Investor's utility analysis. We also consider an investor who participates in the UST game at any time by purchasing one unit of UST. For simplicity, we assume that the investor will always hold UST without converting him to LUNA in the mid-run before exiting the game. A chart of UST price (green line), the number of UST in the wallet (red line), and the investor's utility (blue line) over time are presented in Fig.3. The practice proves the ineffectiveness of UST's stabilization as the UST eventually collapsed. Investors suffered a significant loss from their initial investments, making UST unlikely to become a rational Ponzi game.
In fact, current designs of seigniorage shares are practically difficult for realizing a rational model. These stablecoins are not substantially secured by anything, other than the expectation that the stablecoin will eventually appreciate in value. Hence, a necessary condition for the seigniorage shares to work healthy is that the secondary coin has to have values. LUNA, as the secondary coin, should have, at least, trading values, as people do use Terra blockchain every day. However, in extreme cases, investors may still lose confidence in it, and the massive sell-off has accelerated both two types of coins into a death spiral faster.

Discussion
Market cap vs price stability. The investors involved in stablecoin markets often have intuitive feelings that the larger the market cap of the stablecoin, the better stability the stablecoin can achieve with regard to its peg. However, according to the price and market cap data from CoinGecko [28], this conjecture may not be valid as imagined for algorithmic stablecoins. The statement of ineffective application to the rebase design is explicit, as its market capitalization will vary as the supply of the currency is adjusted. For the seigniorage share design, this statement will be invalid as well. Due to its relative complexity, there are a lot of possibilities resulting in this situation, one of which is the loss of confidence. The recent joint collapse of LUNA/UST was a loop in which the market lost confidence and entered a death spiral, pushing them out of the market cap top 10 within merely several hours. Thus, a larger market cap in algorithmic stablecoin does not necessarily guarantee better stability. Price oracle risk. Price oracle provides the price feed for on-chain DeFi protocols [29], and it is crucial to the secure operation of an algorithmic stablecoin. Algorithmic stablecoins may have special price-feeding needs due to the existence of algorithms, which further need to establish their own oracles. An internal oracle, however, will inevitably lead to an additional layer with potential centralization and incentivize the coin issuer to update the price in his favor. In contrast, opting for an external price oracle may mitigate this problem, but it is still vulnerable to external price manipulation, being out of the control of the coin issuer. Relevant attacks include flash loan price manipulation attack, pump, and dump [30]. Selecting a trustworthy price oracle is not easy for an algorithmic stablecoin, and a comprehensive ex-ante risk assessment is required. Reconsider the necessity of a peg. Stablecoins are designed to be pegged to some underlying assets, with the USD being a popular choice. Anchoring a benchmark is an effective way to achieve stability in stablecoins. People may thus generally assume a fact that the stablecoin is perfectly fungible for its peg. However, this is not the full truth. Stablecoins can be, in fact, rather considered as their own free-floating assets that closely track the value of a peg. Besides, it is also possible that stablecoins could provide the desired stability without pegging to an asset. In one possible future, once the economy develops surrounding the stablecoin itself, and such stablecoins have also been widely used as a medium of exchange, maintaining a perfect peg becomes unnecessary.

Concluding Remarks
This study creatively applies O'Connell & Zeldes's model (1988, cf. [23]) for rational Ponzi games to the context of algorithmic stablecoins. Our analyses of investors' utilities indicate that the rebase approach has the potential to realize a rational Ponzi game as long as it always exists in an active trading market. Notably, by leveraging smart contracts, DeFi protocols will be much easier to realize a rational Ponzi model than those in centralized finance, by means of many advanced and innovative financial tools like automated market makers (AMM), flash loans, and enforceable payment. In contrast, the seigniorage share method is difficult to guarantee a rational Ponzi, as dealing with the selling pressure in a death spiral for dual coins is intrinsically complicated. Beyond that, we point out the hurdles existing in current algorithmic stablecoin implementations and further identify the potential opportunities for future endeavors.