# Liquid Proof-of-Stake in Tezos: An Economic Analysis

## Abstract

**:**

## 1. Introduction

## 2. Baking Priorities for a Single Block with No Delegation

- (a)
- $R$ is given and independent of $r$

- (b)
- $R$ is a function of $r\text{}$and $\frac{\partial R}{\partial r}=1$

## 3. Endorsement Priorities for a Single Block without Delegation

#### Baking and Endorsing Joint Probability for a Single Block

## 4. Expected Revenue for a Single Block

## 5. Expected Revenue for a Cycle of Blocks

## 6. Optimal Stake and Number of Rolls in a Cycle

#### The Staking Game

**Proposition 1.**

**Proof.**

**Proposition 2.**

**Proof.**

## 7. Delegated Proof-of-Stake for a Cycle of Blocks

- (i)
- The costs $f$ of running a full node are paid by the delegated node only, if the delegating node is not a full node.
- (ii)
- The probability of baking/endorsing exhibits a super-additivity property. That is, with delegation, the joint selection probability is at least as large as the sum of the selection probabilities without delegation.
- (iii)
- The rewards obtained by the delegated node are shared with the delegating user proportionally to the number of their rolls.

- (1)
- Suppose ${m}_{i}=7999Tz$ and ${m}_{j}=500Tz$. Therefore, neither user $i$ nor user $j$ can bake a single roll since none of them individually has at least $8000Tz$. However, if $j$ delegates $i$ of at least $1Tz$, then, jointly, the two nodes can reach at least $8000Tz$ and potentially bake/endorse blocks, which separately they could not. Given this initial symmetric situation, where neither of them can bake individually, the observed block/endorsing joint reward $\rho \left(r=1\right)$, which is a random variable, is likely to be shared exclusively according to the monetary sum at stake. Indeed, this may prevent one user from free riding on the other user, trying to convince him/her to stake as much money as possible. If ${s}_{i}$is the stake of user $i$ and ${s}_{j}$ is the stake of user $j$, then the share$$\frac{{s}_{i}}{{s}_{i}+{s}_{j}}$$

- (2)
- Suppose now that ${m}_{i}=15999Tz$ and ${m}_{j}=500Tz$. In this case, the situation is slightly different, as compared to the previous point, since user $i$ can stake one roll for baking/endorsing blocks, while user $j$ alone still cannot. If $j$ delegates at least $1Tz$ to $i$, then $j$ can participate in the baking activity, which otherwise would be impossible. User $i$, even without the support of user $j$, in this case, can be selected with a success probability given by$${P}_{i}\left(B\right)=\frac{1}{R}$$

- (c)
- Suppose now that ${m}_{i}=23999Tz$ and ${m}_{j}=1500Tz$; that is, both users can be selected separately for baking with two rolls and one roll, respectively. Moreover, in this case, if user $j$delegates user $i$ with $1Tz$, then two users, jointly, will obtain four rolls. Again, also in this case, delegation induces super-additivity in the success probabilities$${P}_{ij}\left(B\right)=4>2+1=3={P}_{i}\left(B\right)+{P}_{j}\left(B\right)$$$$E{\rho}_{i}\left(r=2\right)+E{\rho}_{j}\left(r=1\right)<{\rho}_{ij}\left(r=4\right)$$$$\frac{({s}_{i}-16000)}{({s}_{i}-16000)+({s}_{j}-8000)}$$However, (21) may not necessarily be satisfied, and the following can take place:$$E{\rho}_{i}\left(r=1\right)<{\rho}_{ij}\left(r=4\right)<E{\rho}_{i}\left(r=2\right)$$

#### Nash and Kalai–Smorodinsky Non-Cooperative Bargaining Solutions

## 8. Delegation Service Market

## 9. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Dimitri, N.
Liquid Proof-of-Stake in Tezos: An Economic Analysis. *Information* **2022**, *13*, 556.
https://doi.org/10.3390/info13120556

**AMA Style**

Dimitri N.
Liquid Proof-of-Stake in Tezos: An Economic Analysis. *Information*. 2022; 13(12):556.
https://doi.org/10.3390/info13120556

**Chicago/Turabian Style**

Dimitri, Nicola.
2022. "Liquid Proof-of-Stake in Tezos: An Economic Analysis" *Information* 13, no. 12: 556.
https://doi.org/10.3390/info13120556