# Quintus Questions

Just read your sybil's paper

**Q**: There are a couple of typos in the version I have which I highlighted - would you like me to send to you?

**B**: Certainly! Thank you for taking the time to read the paper and highlight the typos. I would greatly appreciate it if you could send me the version with the corrections. Thank you!

Also, could you tell me if this summary makes sense?
1) **Q**: First contribution is formalising sybil and sybil commitment games.
**B**: We have formulated Sybil proof games with symmetric payoffs, although we may remove this assumption in the future. In a subsequent version, we have included a formalization for Anonymous Sybil-proof mechanisms, which do not rely on identity. Additionally, we introduce the concept of Sybil commitments, which as far as we know, is a new concept. We demonstrate that, in general, games may not be Sybil proof with commitments even if they are Sybil proof.
2) **Q**: Apart from 2nd price auctions, there are two kinds of mechanisms that you look at which is sybil resistant: those with superlinear payoffs (need budgets for equilibrium to exist) and those where the payoff shrinks exponentially in the number of players.
**B**: Proving that pro-rata games are Sybil-proof is straightforward, and it suffices to show this for zero cost, $c=0$. We start by taking a function $f$ and its corresponding pro-rata game $U$. Suppose that player $n$ has $k+1$ Sybils and takes actions $x_n, \ldots, x_{n+k}$. Then we have:

\begin{equation}
\sum_{i=0}^k U_{n+i}(x_{n+i}, x_{-(n+i)}) = \sum_{i=0}^k \frac{x_{n+i}}{\mathbf{1}^T x} f(\mathbf{1}^T x) = \frac{\sum_{i=0}^k x_{n+i}}{\mathbf{1}^T x} f(\mathbf{1}^T x) = U_n\left(\sum_{i=0}^k x_{n+i}, x_{-n}\right)
\end{equation}

In particular, this inequality holds for $c=0$, so pro-rata games are Sybil-proof. Moreover, they are also collusive resistant, as defined in [An Axiomatic Approach to Block Rewards]().

In fact, we can prove more than this: every aggregative, symmetric, collusion-resistant, and Sybil-resistant game is a pro-rata game. For more information, see the [Notes & Questions of Pro-Rata games](https://hackmd.io/cPBzdZi2SfO-rcWUKVooUw).

3) **Q**: When you list sybil proof payoff functions, it sounds like you say that all pro rata games are sybil proof but then later you point out that f(x)=R isn't (so I'm a little confused here)
**B**: I might need to rephrase my previous statement. Can you please tell me where in the text it appears? I think I might have left out some important words. To clarify, there may be pro-rata mechanisms that allow the use of capital and those that rely on identities. The latter, however, are generally not Sybil-proof. I apologize for any confusion :') caused by my previous statement and will strive to be more specific.
4) **Q**: sybil proof mechanism for pro rata mech => PoA exponentially bad

**B**: Here I meant mechanisms that are Sybil-proof with symmetric payoffs and strategy-proof. I may need to make a slight correction to the statement.
5) **Q** you bound alpha-proportionality (also exponentially low)
7) **Q**: For cake cutting mechanisms truthful bidding=> PoA exponentially bad.
**B**: Both Yes! Also, I think there is some relation between Block builiding and cake cutting that seems interesting to explore. (Cause I think that combinatorial auctions and cake cutting are not that different problems, at least mathematically).
9) **Q**:You show that there exists a profitable sybil proof 2nd price auction bidding ring.
**B**: Yes, the most trivial example is the one were all players observe the same value $v$. Then they can use the Reward distribution mechanisms. But there are some bounds on the welfare too! (I think I haven't write them down yet).
11) **Q**: For sybil commitment games, if you upper bound the cost function and sybil (commitment) proof => PoA exponentially bad
**B**: Yes. In other words, if revealing your identity is incentive comaptible then means that the underlying game has exponentially bad price of anarchy.
13) **Q**: SCP != SP
**B**: Important!!!!
15) **Q**: you show that the CFMM pro rata game is not SCP and your proof makes me think that the assumption of known players made in their analysis is a silly assumption
**B**: In general, I believe it is a simplistic assumption, but it is necessary to obtain elegant mathematical results. A similar assumption is also required in the Flashboys 2.0 paper to establish a cooperative Nash equilibrium (However, we were able to provide a cooperative Sybil-resistant Nash equilibrium under this assumption (this result is not in the paper, but it follows easly, may be we will write a MEV version of this paper :) )). However, in reality, this assumption is clearly unrealistic, as the cost of creating credible identities is very low (for example, by creating another smart contract with slightly different opcodes and running another bot to extract the MEV). Nevertheless, if all players are utilizing a No-regret algorithm (such as gradient descent), and the underlying game possesses desirable properties (such as strict monotonicity), then the strategies will converge to the unique Nash equilibrium, even if the players have no information about the number of competitors. In this case, it is reasonable to analyze the game with an unknown number of players because the strategies in the repeated game converge to the one-shot equilibrium. However, these strategies are not Nash policies. If every player employs this type of policy, then a player has an incentive to create "Sybil instances of gradient descent" to increase their payoff. For further details, refer to the [Notes & Questions of Pro-Rata games](https://hackmd.io/cPBzdZi2SfO-rcWUKVooUw).

tl;dr sybils cause huge welfare loss in many cases and sybils with commitments is worse?

**Q**: I see that PoA isn't the best metric for understanding the impact of sybils and that you'd want to compare welfare across equilibria. The CFMM pro rata game has PoA $\Theta(n)$ though so this case at least shows that sybils make a big difference right?

**B**: We are currently in the process of determining the optimal definition for measuring the welfare loss caused by Sybils. In pro-rata games where players are aware of the number of players involved, there is no welfare loss for Sybils. However, if the number of players is unknown, the situation is different and welfare loss may occur.

**Q**: Also, I was wondering about your definition of the SC extension game and the SC equilibrium definition. It seems that the main difference between this and the sybil extension game is that the number of (pseudo) participants is known when actions are selected and that sybils act as independent rational agents. With this definition in mind, your definition of equilibrium makes sense to me, but why are sybils expected to maximise their individual payoffs? What motivated this definition?

**B**: This is a complex and technical passage, but I will try my best to rewrite it clearly.

There were multiple reasons for defining Sybil commitments. Firstly, in anonymous and permissionless environments, identities are not easily verifiable. Therefore, I asked the question: what happens if a player can convince others that they represent multiple identities, and they have an incentive to do so? How we can easly model this? Will players misreport their identity?

The second motivation for Sybil commitments was in the context of repeated games. Depending on the learning algorithm used, players' convergence to the one-shot equilibrium is independent of their priors. In such a scenario, players might have an incentive to manipulate others' learning algorithms to "learn" that there are more players. This is often true in many games.

In other words, if players use a gradient ascent policy in a strictly monotone pro-rata game, they can discover the number of players as their actions converge to the one-shot equilibrium (which can be observed in the game $\frac{x}{x+y}R-x$). However, the theorem showed that players only want to reveal their identity if the price of anarchy decreases exponentially.

Another example of the Sybil commitment game arises when an external agent is making rational decisions on behalf of the players. In this scenario, players may be incentivized to report multiple identities to the external agent in order to increase their chances of being selected. I think, this situation has been observed in Google ad auctions?

Apologies for the many questions - completely understand if you take your time to respond. Also, I read a version from ~2 weeks ago so some of my questions might stem from that.

**B**: The paper is currently undergoing revisions, and we plan to incorporate additional content, correct any errors, and make other improvements in the future.

Thank you for your questions! I'm happy to discuss them.