Single-Token vs Two-Token Blockchain Tokenomics

The model takes as given two exogenous parameters. The discount factor $\delta$, a conventional element in economic models, reflects players’ preference for immediate utility over future gains, and is tied to the risk-free rate. The value of the service $S^{(t)}$ captures the usefulness of the system for users in each round $t$ and may fluctuate unpredictably due to external shocks (Remark 3). The system can set rewards $R^{(t)}$ to be distributed in each round $t$ to the validators according to their current stake at round $t$; each validator also faces a fixed cost $v>0$ associated with maintaining the required secure node configuration. As our analysis shows, the monetary policy, $R^{(t)}$, influences the equilibrium and the token price. One of our key findings (see below) is that suitably defined monetary policies can “absorb” shocks and avoid undesirable price fluctuations.

The equilibria in our model come from players who aim to maximize their own discounted utility, thus choosing strategies with time-dependent parameters, such as how many tokens to sell on the market or retain for future rounds (users need to spend tokens to access the system, while validators need to stake tokens to receive rewards). These equilibria, in turn, govern the key endogenous variables of the system, namely the staking levels of the validators, the holdings of user tokens, the fees paid to the system, the demand and supply of tokens in the spot market and the resulting trajectory or price path of the token (the market-clearing price in equilibrium as in [11]).

We give both boundary conditions and actual policies (mechanisms) that implement equilibria with all desired features: viability, decentralization, stability, and feasibility. Specifically, our analysis leads us to put forward a novel generic mechanism for blockchain “monetary policy”:

Quantitative Rewarding (QR). The amount of rewards the system gives to the validators is suitably adjusted at each step to anticipate changes in the value of the service.

This QR concept arises from our equilibrium analysis, focusing on how market demand and supply respond to changes in service value $S^{(t)}$ and rewards $R^{(t)}$. In all equilibria implementing certain desired price paths, the following conditions hold at every round (cf. Theorem 9):

• $\mathrm{■}$

  Users spend an amount of tokens whose value is a constant fraction of the current service value $S^{(t)}$.

• $\mathrm{■}$

  Validators stake an amount of tokens that is a constant fraction of current rewards $R^{(t)}$.

As these conditions and equilibria do not rule out undesirable mechanisms such as buybacks, our goal is to use QR to induce equilibria where token demand and supply match–so no buybacks are needed. Observe that if, at any point in time, the value of the service decreases, users will buy fewer tokens, thus reducing demand and threatening price stability. Instead of relying on buybacks, our QR mechanism, perhaps somewhat counterintuitively, increases the rewards allocated to validators, who react by staking more tokens, which must be acquired from the market. This action effectively restores the demand for tokens to align with the supply. The reverse effect occurs when the value of the service increases.

We show that implementing QR in single-token systems to avoid buybacks is highly non-trivial, as it requires anticipating future shocks far in advance. In contrast, for two-token systems, QR can be implemented using only a one-step lookahead of the service value. This conclusion stems from equilibrium analysis, but can be intuitively explained as follows: In single-token systems, validators must retain some tokens to stake and sell only a portion to the market to satisfy demand. In two-token systems, one token is used for staking, while the other – distributed as rewards in our mechanism – is used by users. Validators can sell all of these reward tokens to the market, making it easier to align supply with demand. Consequently, any equilibrium where rewards match the next round’s service value (current users’ demand for new tokens) avoids buybacks.

Decoupling users from validators via distinct tokens enables the system to absorb service value fluctuations more effectively, while keeping the user token’s price stable and preserving other desiderata. At equilibrium, the price of the stake token rises gradually, and validators stake a fixed amount, trading only the other token, which they receive as a reward. In contrast, a single-token setup may trigger a self-reinforcing loop of increasing validator rewards, even after service levels stabilize, underscoring a fundamental distinction between the single-token and two-token configurations.

In Section 3.3, we investigate how to implement QR via a mechanism which adjusts the rewards at every round based on the current rewards and the next round service value using a single token for accounting. An important challenge that arises in this setting is that the rewards formula is subject to a recursive condition (see the lower bound in Theorem 13 and Corollary 14) which makes rewards subject to a “feedback loop” effect that can lead to rewards explosion.

As a result, while maintaining certain (minimal) rewards to ensure decentralization and security, it is possible that the system can trigger an uncontrolled growth in the amount of rewards distributed and in the staked amounts of tokens (see Section 3.3.1 for conditions that trigger such explosion). In Section 3.4 we describe in detail the implementation of our mechanism for the case of stable prices. This implementation includes the strategies for users and validators, showing that all information needed is service value at the next round (despite their simplicity, these strategies still constitute an equilibrium in the underlining repeated game, i.e., when more complex time-dependent strategies are in principle possible).

The above results apply to the single token setting which is the most common in PoS systems (e.g., Ethereum and Cardano operate in this way, for example, as PoS cryptocurrencies) and paint a rather negative picture: one has to either accept tokenomics with widely fluctuating token prices (the most common outcome that is observed in these systems as deployed in the real world), implement drastic measures such as token buybacks, or suffer a potential explosion in rewards accounting (unless future shocks in service value can be predicted – a rather unreasonable assumption). Note that QR mechanisms are, in general, more complex than standard “fee burning” policies used in practice (see Remark 16).

Poised to obtain a more favorable outcome, we next explore the implementation of QR in two token PoS systems in which one token is used to get the service and another token is used for staking (and governance). While less common, this type of setup has been adopted in a few occasions: e.g., in the NEO cryptocurrency [19] and can be implemented in other systems as well (e.g., in the Cosmos SDK [8]). To investigate the potential of QR in this setting, we adapt our model to the two tokens setting (Section 4). In order to maintain feasibility, while maintaining desirable prices, we identify the necessary conditions that equilibria (mechanism) must guarantee in terms of rewards.

The two token mechanism for QR is presented in Section 4.3. The main advantage of the resulting rewards formula is that it is simpler than in the single token setting and the mechanism does not need to know “global information” about the service value time series – merely a one step lookahead suffices (cf. Theorem 24). As an important corollary, the mechanism achieves our objectives and is capable of avoiding the feedback loop effect of the single token case that leads to an exponential explosion for rewards.

Let us first observe that the total amount of tokens bought by the users (demand) is

where $ℛ=1+\mathrm{Rew2Stake}\cdot {\kappa }_R$. The total amount of tokens sold by the validators (supply) is

Hence, the no buyback condition $m{b}_U^{(t)}\ge n{s}_V^{(t)}$ is equivalent to

that is

By rearranging the terms, the theorem follows. $◀$

The above theorem provides a recursive (algorithmic) formula for the rewards, which leads to our mechanism described in Section 3.4 below. By unfolding the recursion (18) in same theorem, we get the following bound on the growth of the rewards necessary to guarantee the no buyback.

We stress that the minimal rewards satisfying the no buyback condition must satisfy the no buyback condition tightly, and thus must coincide with the rewards $R_{single}^{(t)}$ in Corollary 14.

As we discuss below, implementing quantitative rewarding can be difficult due to the possibility of an uncontrolled rewards growth. Indeed, Corollary 14 states that the initial rewards $R^{(0)}$ cannot exceed a certain value unless rewards grow exponentially. This value, critically, depends on future service values. In Section 3.4, we describe an implementation under the assumption that the initial rewards $R^{(0)}$ can be set properly by the system (either having full knowledge about future service values, a good estimate, or some partial control over them).

1. 1.

   The risk free rate $r$ and thus the discount factor $\delta =1/(1+r)$.

2. 2.

   The total number $m$ of users and the total number $n$ of validators (both constant over time).

3. 3.

   There is no need for the system to buyback tokens (Definition 12).

The system design depends on essentially two parameters:

1. 1.

   Total value of the service $S^{(t)}$ which is divided among the users at each step $t$.

2. 2.

   Total amount of rewards $R^{(t)}$ which is distributed to the validators at each step $t$.

At each round $t\ge 0$,

1. 1.

   Users strategies:

   1. (a)

      Start with $𝕋{𝕂}_U^{(t)}=\frac{S^{(t)}}{m}\cdot \delta \cdot {\kappa }_S$ tokens, where ${\kappa }_S=g(n)\cdot \frac{m-1}{m}\cdot s^{\prime }(1/m)$.

   2. (b)

      Spend all these tokens to get a service value equal to $S^{(t)}\cdot s(1/m)$.

   3. (c)

      Given the next round service value $S^{(t+1)}$, buy $b_U^{(t)}=𝕋{𝕂}_U^{(t+1)}=\frac{S^{(t+1)}}{m}\cdot \delta \cdot {\kappa }_S$ tokens required for the next round according to Item 1a.

2. 2.

   Validators strategies:

   1. (a)

      Start with $𝕋{𝕂}_V^{(t)}=\frac{R^{(t)}}{n}\cdot \frac{\delta }{1-\delta }\cdot {\kappa }_R$ tokens, where ${\kappa }_R=\frac{n-1}{n}\cdot r^{\prime }(1/n)$.

   2. (b)

      Stake all these tokens to get $R^{(t)}\cdot r(1/n)$ new tokens as reward.

   3. (c)

      Given the rewards $R^{(t+1)}$ of next round, sell this amount of tokens (if negative, buy tokens): $s_V^{(t)}=\left(R^{(t)}-R^{(t+1)}\right)\cdot \frac{\delta }{1-\delta }\cdot {\kappa }_R+R^{(t)}\cdot r(1/n)$, which yields the required $𝕋{𝕂}_V^{(t+1)}=\frac{R^{(t+1)}}{n}\cdot \frac{\delta }{1-\delta }\cdot {\kappa }_R$ tokens for next round $t+1$ according to Item 2a.

3. 3.

   System actions:

   1. (a)

      Given the next round service value $S^{(t+1)}$, set the next total rewards $R^{(t+1)}$ so that

      $m\cdot S^{(t+1)}\cdot \delta \cdot {\kappa }_S\ge n\cdot \left(R^{(t)}-R^{(t+1)}\right)\cdot {\displaystyle \frac{\delta }{1-\delta }}\cdot {\kappa }_R+n\cdot R^{(t)}\cdot r(1/n)$

      which guarantees the no buyback condition, that is, $m\cdot b_U^{(t)}\ge n\cdot s_V^{(t)}$.

   2. (b)

      Announce the next rewards $R^{(t+1)}$ before the token buy or sell (Subround II) begins.

   3. (c)

      Sell the required amount of tokens, $m𝕋{𝕂}_U^{(t+1)}-n{s}_V^{(t)}$, to match demand during Subround II.

4. 4.

   Prices: all tokens are exchanged at a stable price $\mathbb{P}ℝ𝕀ℂ{𝔼}^{(t)}=1$.

The first three items in the definition of generic symmetric equilibrium (Definition 17) follow from Lemma 20, from the definition of symmetric equilibrium requiring that users selling constraints of token $𝐁$ are non-strict (Definition 17), and from the assumption about the prices (16).

The last condition in Definition 17 is a rewriting of the prices in Lemma 19 and of definition of symmetric equilibrium requiring that validators selling constraints of token $𝐀$ are non-strict (Definition 17). $◀$

We consider the two inequalities of the no buyback condition (Definition 23) separately.

1. 1.

   For tokens of type $𝐁$, we observe that, for $\mathbb{P}ℝ𝕀ℂ{𝔼}_{𝐁}^{(t-1)}\ne \delta \cdot \mathbb{P}ℝ𝕀ℂ{𝔼}_{𝐁}^{(t)}$, the users buying strategies satisfy $mb_i^{}{}_{𝐁}{}^{(t)}=m{𝐁}_U^{(t+1)}=\frac{S^{(t+1)}}{Ser2Fee{s}_{𝐁}^{(t+1)}\cdot \mathbb{P}ℝ𝕀ℂ{𝔼}_{𝐁}^{(t+1)}}=S^{(t+1)}\cdot \frac{\delta }{\mathbb{P}ℝ𝕀ℂ{𝔼}_{𝐁}^{(t)}}\cdot {\kappa }_S.$ Moreover, the validators’ selling strategies satisfy $s_j^{}{}_{𝐁}{}^{(t)}=R_{𝐁}^{(t)}\cdot r(1/n),$ since the validators selling constraint for tokens of type $𝐁$ must be strict, and thus ${𝐁}_i^{(t)}=0$ for all $t$. Therefore the no buyback condition for tokens of type $𝐁$ is equivalent to $n\cdot R_{𝐁}^{(t)}\cdot r(1/n)\le S^{(t+1)}\cdot \frac{\delta }{\mathbb{P}ℝ𝕀ℂ{𝔼}_{𝐁}^{(t)}}\cdot {\kappa }_S.$

2. 2.

   We also require no buyback for the tokens of type $𝐀$, which means that $s_j^{}{}_{𝐀}{}^{(t)}=0$ because, in any generic symmetric equilibrium users never hold (and thus never buy) tokens of type $𝐀$: If users hold tokens $𝐀$, the second part Lemma 20 implies $\mathbb{P}ℝ𝕀ℂ{𝔼}_{𝐀}^{(t-1)}=\delta \cdot \mathbb{P}ℝ𝕀ℂ{𝔼}_{𝐀}^{(t)}$, thus contradicting (31) in Definition 21. Hence, ${𝐀}_V^{(t+1)}={𝐀}_V^{(t)}+R_{𝐀}^{(t)}\cdot r(1/n)+b_j^{}{}_{𝐀}{}^{(t)}$ with $b_j^{}{}_{𝐀}{}^{(t)}\ge 0$.