How do proposers take turns to submit new state roots on L1?

Arbitrum One currently has an allowlist for validators (The state of Arbitrum's progressive decentralization | Arbitrum DAO - Governance docs). My question is what determines when is the turn for which validator to submit new state roots to L1? Furthermore, once the allowlist is removed, how should such coordination among (permissionless) validators be conducted?

Right now, many of the allowlisted validators do not seem to be doing anything (e.g, 0x0fF813f6BD577c3D1cDbE435baC0621BE6aE34B4), while other validators seem to be fulfilling most of the proposer role (e.g., 0x83215480dB2C6A7E56f9E99EF93AB9B36F8A3DD5). Apparently, the allocation of load among proposers is not even, but I cannot find any information about what the allocation rule is.

Relatedly, if in the future we were able to figure out a fair and orderly way to coordinate the actions of proposers, then why do we need the sequencer anymore? That is, it seems that the proposers can combine the task of sequencing and proposing assuming an orderly proposing sequence?

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There is no fixed coordination mechanism. Any validator can post an assertion.

If validators are rational, then one validator will post all of the assertions, and the other validators will just sit back and watch. As long as the validator who is posting continues to post correct assertions, the others will be happy to just watch.

The reason this outcome is rational is that a validator needs to be staked if it has posted any assertions in the last week. That means that if Validator A is already staked, and Validator B is not already staked, the cost of A posting again is lower than the cost of B posting for the first time.

On Arbitrum One today, in practice there is a single validator that does all of the posting. If it stops posting or posts a dishonest assertion, the other validators can jump in and post. But that’s not the expected normal case.

There is no restriction or enforcement that prevents any validator from posting if it wants to. That kind of restriction is not needed–and would be a bad idea.

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Thanks for the explanation. I agree with you that “there is no restriction or enforcement that prevents any validator from posting if it wants to. That kind of restriction is not needed–and would be a bad idea.”

However, I am not sure if only-one-staked-validator-and-everyone-else-watch is necessarily the equilibrium outcome. Presumably, the validators will be rewarded for their services, then it is unclear why all potential validators will just sit there and watch the incumbent take all the reward. As another example, suppose validator A ever goes down, if there is no coordination mechanism in place then multiple validators B_1, B_2, …, B_n may enter simultaneously. It is unclear to me why all but one will just sit there and watch.

Furthermore, if in reality only one validator does the job and everyone else just sit-and-watch, we may wonder who will monitor the incumbent and how the challenge mechanism can be effectively triggered.

While none of such questions may seem urgent as long as the allowlist on validators is still on, I feel they may become relevant once the allowlist is removed (or even further relaxed).