abstract
probabilistic shapley attribution (psa) is a scalable method for fairly rewarding transactions in a focus flow computation (ffc) network. it approximates shapley values, which measure each transaction’s marginal contribution to the final equilibrium distribution (p^*), using sampling and local influence metrics. psa enables superintelligence-scale attribution with millions of transactions and billions of edges per epoch.
background
- focus flow computation minimises free energy (\mathcal{F}) to compute a global equilibrium distribution (p^*).
- transactions introduce new cyberlinks and provide proofs of computation.
- rewards should reflect how much each transaction contributed to lowering free energy and improving the network’s focus.
exact shapley values are computationally infeasible for large networks ((O(n!)) complexity).
key idea
psa combines:
- local marginal contributions – approximate each transaction’s individual effect on free energy.
- monte carlo sampling – randomly sample subsets of transactions to approximate interactions.
- hierarchical batching – cluster transactions by the nodes/edges they affect to further reduce complexity.
algorithm outline
step 1 – local marginal computation
- for each transaction (tx_i), compute the local change in free energy: [ \Delta \mathcal{F}_i = \mathcal{F}(p) - \mathcal{F}(p + tx_i) ]
- this gives a first-order estimate of contribution.
step 2 – monte carlo sampling
- randomly sample (k) subsets of transactions.
- compute each transaction’s average marginal contribution within the sampled permutations.
step 3 – cluster correction (hierarchical batching)
- cluster transactions that touch the same nodes/edges.
- compute shapley-like contributions at the cluster level.
- distribute rewards within clusters proportionally to local (\Delta \mathcal{F}).
step 4 – final reward
[ R_i = \alpha \Delta \mathcal{F}_i + (1-\alpha) \hat{S}_i ]
- (\Delta \mathcal{F}_i): local marginal effect.
- (\hat{S}_i): sampled shapley estimate.
- (\alpha): weight controlling the trade-off between local and sampled contributions.
complexity
- local marginals: (O(n)).
- monte carlo shapley: (O(k \cdot n)), with (k \ll n) (e.g., 50–500 samples).
- hierarchical batching: reduces cost by grouping transactions.
feasible for 10^6–10^7 transactions per epoch and 10^9+ edges on distributed hardware.
advantages
- fairness: approximates true shapley values, rewarding contributions based on marginal effect.
- scalability: complexity grows linearly with transactions for practical settings.
- robustness: combining local and sampled estimates reduces gaming opportunities.
integration with focus flow incentives
- psa is used after each epoch to compute rewards for all transactions.
- transactions include:
- new cyberlinks (edges)
- proof-of-computation (partial focus flow updates)
- base fee (like eip-1559) prevents spam.
- rewards are distributed after delayed settlement to account for long-term network effects.
significance
psa provides a scalable, fair attribution mechanism for proof-of-intelligence in focus flow networks. it ensures:
- meaningful transactions become profitable.
- spam or low-impact transactions lose money.
- rewards correlate with long-term contributions to global focus and negentropy.
this creates a self-optimising economy for intelligence, where agents are incentivised to submit useful cyberlinks and computation proofs that improve the network’s equilibrium.