By: Sandra Johnson, Kerrie Mengersen, Anders L Madsen
WWW: Anders L Madsen and Sandra Johnson
Created: 2023-12-08
Latest update: 2023-12-17
Modelling Proposer Selection with increased MaxEB (EIP-7251)
This research has been funded by Consensys Software Inc and EF Academic Grant FY23-1030 (Sandra Johnson, Kerrie Mengersen, Patrick O'Callaghan). We acknowledge the valuable reviews, discussions and insights given by Barnabé Monnot, Ben Edgington, Mike Neuder, Mikhail Kalinin and Roberto Saltini.
EIP-7251 proposes increasing the MAX_EFFECTIVE_BALANCE constant from 32 ETH to 2,048 ETH. The EIP aims to reduce the validator set size by facilitating consolidation, and auto-compounding of multiple validators into larger validators, up to a maximum consolidation of an effective balance of 2,048 ETH.
The active validator set comprises stakers that can roughly be categorised into five groups. For each of these staker categories, we use an example consolidation strategy across validators consisting of 32 ETH, 64 ETH, 160 ETH, 320 ETH, 960 ETH and 2,048 ETH.
The blog post on ethresear.ch describes the model in more detail.
Below is a set of HUGIN widgets to interact with the model:
Validator Economics
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Scenario 1: Running the model with no evidence provided.
The result of running the BN model is shown in the interface above (no evidence entered). Therefore, assuming the active validator set consists of the staking categories as described along with their respective example consolidation strategies, the validator set will be reduced to 329,810 from 716,800. This adjusted validator set has different proportions of consolidated validators as shown in node Consolidated validator types. Across this validator set, the proportion of validators that will pass the proposer test is 20.8% and the marginal probability, across the various validator types in the validator set, of being selected as the first candidate is 26.4%, and of being the proposer of the next block is 0.69%.
The probability of being the next proposer depends on being selected as the next candidate and then passing the test. It appears counter-intuitive that the probability is so much smaller than the individual probabilities in the BN. The reason for this becomes clearer when we look at the conditional probability table of this node:
Scenario 2: Running the model with all validators having 32 ETH.
If we enter evidence (shown as a state of a node being red) that the validator set consists entirely of unconsolidated validators, and therefore there is no reduction in the size of the validator set. The probability of selecting an unconsolidated validator type is understandably 100%, with the probability of passing the proposer check being 1.56%, and therefore probability of an unconsolidated proposer being the proposer of the next block is simply its probability of passing the proposer check, which is 1.56%.
References and Further Reading
[1] U. B. Kjærulff and A. L. Madsen, Bayesian Networks and Influence Diagrams, Second Edi. Springer, 2013.
[2] R.E. Neapolitan, Learning Bayesian Networks, Pearson Education, Inc., 2004
[3] F.V. Jensen, Bayesian Networks and Decision Graphs, Springer-Verlag, Inc., 2001
[4] K.B. Korb and A.E. Nicholson, Bayesian Artificial Intelligence, Chapman & Hall/CRC, Second Edition, 2011
[5] J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann Publishers, 1988
[6] D. Koller and N. Friedman, Probabilistic Graphical Models: Principles and Techniques, MIT Press, 2009
[7] D. N. Barton et al., Bayesian networks in environmental and resource management, Integr. Environ. Assess. Manag., vol. 8, no. 3, 2012.
[8] S. Johnson et al., Modeling the viability of the free-ranging cheetah population in Namibia: an object-oriented Bayesian network approach, Ecosphere, vol. 4, no. 7, p. art90, Jul. 2013.
[9] J. Holt et al., Bayesian Networks to Compare Pest Control Interventions on Commodities Along Agricultural Production Chains, Risk Anal., vol. 38, no. 2, 2018.
[10] F. Taroni, Bayesian Networks for Probabilistic Inference and Decision Analysis in Forensic Science, 2nd Edition, 2nd edition. John Wiley & Sons, 2014.
[11] B. G. Marcot et al, Guidelines for developing and updating Bayesian belief networks applied to ecological modeling and conservation, Can. J. For. Res., 2007.
[12] B. G. Marcot and T. D. Penman, Advances in Bayesian network modelling: Integration of modelling technologies, Environmental Modelling and Software. 2019.
[13] V. T. Hoang, B. Morris, and P. Rogaway, “An Enciphering Scheme Based on a Card Shuffle,” 2014.
Contact information
For further details on the study and Bayesian network model: Sandra Johnson (sandra(dot)johnson(at)consensys(dot)net)
For further details on the use of Bayesian networks and web deployment of models contact: Anders L Madsen (alm(at)hugin(dot)com)