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@@ -68,8 +68,10 @@ the probability that a party holding all the stake will be selected to
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be a leader. Stakeholder is selected as leader for slot j with
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probability $\phi_f(\alpha_i)$, $\alpha_i$ is $U_i$ stake.
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+### absolute stake aggregation dependent leader selection functions
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+
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+#### linear functions
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-### linear aggregation dependent leader selection
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in the previous leader selection function, it has the unique property of
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independent aggregation of the stakes, meaning the property of a leader winning leadership with stakes $\sigma$ is independent of whether the stakeholder would act as a pool of stakes, or distributed stakes on competing coins.
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"one minus the probability" of winning leadership with aggregated stakes is
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@@ -80,13 +82,15 @@ $\prod_{i}^{n}(1-\phi(\sigma_i))=-(1-f)^{\sum_i(\sigma_i)}$
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thus:
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$$ 1-\phi(\sum_{i}\sigma_i) =\prod_{i}^{n}(1-\phi(\sigma_i)) $$
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-#### linear leader selection
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+##### linear leader selection
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+
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$$y < T $$
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$$y = 2^lk \mid 0 \le k \le 1$$
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$$T = 2^l\phi(v)$$
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$$ \phi(v)=\frac{1}{v_{max}}v $$
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-#### dependent aggregation
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+##### dependent aggregation
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+
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linear leader selection has the dependent aggregation property, meaning it's favorable to compete in pools with sum of the stakes over aggregated stakes of distributed stakes:
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$$\phi(\sum_{i}{\sigma_i})>\prod_{i}^{n}{\sigma_i}$$
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@@ -95,17 +99,49 @@ let's assume the stakes are divided to stakes of value $\sigma_i=1$ for $\Sigma>
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$$V>(\frac{1}{v_{max}})^{n-1}$$
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note that $(\frac{1}{v_{max}})^{n-1} < 1, V>1$, thus competing with single coin of the sum of stakes held by the stakeholder is favourable.
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-### scalar linear aggregation dependent leader selection
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+##### scalar linear aggregation dependent leader selection
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+
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a target function T with scalar coefficients can be formalized as
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$$T=2^lk\phi(\Sigma)=2^l(\frac{1}{v_{max}})\Sigma$$
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let's assume $v_{max}=2^v$, then:
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$$T=2^lk\phi(\Sigma)=2^{l-v}\Sigma$$
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then the lead statement is
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- $$y<2^{l-v}\Sigma$$ for example for a group order or l=24 bits, and maximum value of $v_{max}=2^{10}$, then lead statement:
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+ $$y<2^{l-v}\Sigma$$ for example for a group order or l= 24 bits, and maximum value of $v_{max}=2^{10}$, then lead statement:
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$$y<2^{14}\Sigma$$
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-### pairing leader selection independent aggregation function
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-TODO
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+##### competing max value coins
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+
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+for a stakeholder with $nv_{max}$ absolute stake, $\mid n \in \mathbb{Z}$ it's advantageous for the stakeholder to
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+distribute stakes on $n$ competing coins.
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+
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+
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+#### inverse functions
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+
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+inverse lead selection functions doesn't require maximum stake, most suitable for absolute stake,
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+it has the disadvantage that it's inflating with increasing rate as time goes on , but it can be function of the inverse of the slot
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+to control the increasing frequency of winning leadership.
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+
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+##### leader selection without maximum stake upper limit
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+
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+the inverse leader selection without maximum stake value can be $\phi(v)=\frac{v}{v+c}$ where c is $ > 1$
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+and inversely proportional with probability of winning leadership, let it be called leadership coefficient.
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+
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+
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+##### decaying linear leader selection
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+
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+as the time goes one, and stakes increase, this means the combined stakes of all stakeholders increases the probability
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+of winning leadership in next slots leading to more leaders at a single slot, to maintain, or to be more general to control this frequency of leaders per slot,
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+c (the leadership coefficient) need to be function of the slot $sl$, i.e $c(sl) = \frac{sl}{R}$ where $R$ is epoch size (number of slots in epoch).
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+
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+##### pairing leader selection independent aggregation function
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+
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+the only family of functions that are isomorphic to summation on multiplication (having the independent aggregation property) is the exponential function,
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+and since it's impossible to implement in plonk,
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+
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+###### TODO (proof)
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+
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+a re-formalization of the lead statement using pairing that is isomorphic to summation on multiplication is also an options.
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+
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## Leaky non-resettable beacon
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mohab