Stake-Based Consensus for Utility Scoring

Francois Luus / Jacob Steeves / Ala Shaabana / Yuqian Hu / Sin Tai Liu

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00/ Abstract

We formulate a stake-based consensus problem in terms of a two-player game and propose a protagonist consensus policy to optimize a Nash equilibrium via a weight reduction algorithm with a guarantee of minority stake deterioration. We generalize this to a two-team game and propose a smooth density evolution algorithm that outperforms coarser estimates. We perform a full-scale Monte Carlo analysis and confirm the accuracy of our theoretical results, and show the possibility of a 40% stake + 25% utility attack. The result is a variable-expense consensus algorithm that can be fit to blockchain compute constraints to reach accurate consensus in adversarial settings.

01/ Stake-based weight consensus

1.1 Problem definition

We consider a two-player game between (protagonist) honest stake 0.5<sH1\ 0.5< s_{H} \le 1 \: and (adversarial) cabal stake ( 1sH\ 1 - s_{H} \:) competing for total fixed reward  eH+eC=1\ e_H + e_C = 1 \:with honest emission  eH\ e_{H} and cabal emission  eC\ e_{C}, respectively, followed by stake updates  sH=sH+eH2\ s^{\prime}_{H}= \frac{{s_{H} + e_{H}}}{2} \: and  sC=1sH+eC2\ s^{\prime}_{C}= \frac{{1 - s_{H}+e_{C}}}{2} \:. The honest objective  sHeH\ s_{H} \le e_{H} \: at least retains scoring power  sH\ s_{H} over all action transitions in the game, otherwise when  eHsH\ e_{H} \le s_{H} \: honest emission will erode to 0 over time, despite a starting condition of  0.5<sH\ 0.5 < s_{H}.

We assume honest stake sets objectively correct weights  wH\ w_{H} \: on itself, and  1wH\ 1 - w_{H} \: on the cabal, where honest weight wH\ w_{H} \:represents an ongoing expense of the honest player, sustained throughout the game. However, cabal stake has an action policy that freely sets weight  wC\ w_{C} \: on itself, and  1wC\ 1 - w_{C} \: on the honest player, at no cost to the cabal player, with the objective to maximize the required honest self-weight expense  wH\ w_{H} \: via wC=argmaxwCE[wH  sH=eH(sH,wH,wC)]\ w_{C}^{*}=\arg \max_{w_{C}}E[w_H| \;s_{H}=e_{H}(s_{H},w_{H},w_{C})] \:.

We then assume the honest majority  sH>0.5\ s_{H}>0.5 \: can counter with a consensus policy  π\ \pi \:allowed to modify all weights modulo player labels, so it is purely based on the anonymous weight distribution itself, optimizing the Nash equilibrium  minπmaxwCE[wH    sH=eH(sH,π(w))]\ min_{\pi}\max_{w_{C}}E[w_{H}\;|\;s_{H}=e_{H}(s_{H},\pi(\mathbf{w}))]\:.

The majority stake enforces an independent and anonymous consensus policy  π\ \pi \:(e.g. through a blockchain solution) that modifies the weights to minimize the expense  wH\ w_{H} \:, which has been maximized by the cabal applying an objectively incorrect gratis self-weight  wC\ w_{C} \:. Consensus aims to produce  π(w)(wH,wC)\ \pi( \mathbf{w}) \rightarrow (w^{ \prime}_{H}, w^{\prime}_{C}) \: so that  wC=1wH\ w^{\prime}_{C}=1-w^{\prime}_{H} \:, by correcting the error  ϵ=wC+wH1>0\ \epsilon=w^{\prime}_C+w^{\prime}_H-1>0. Note that the input cost  wH \ w_{H} \: remains fully expensed, and that  wH \ w^{\prime}_{H} \: merely modifies the reward distribution that follows, but not knowing which players are honest or cabal (anonymous property).

Emission of newly minted token vector E through subnet incentive mechanisms.

Figure 1 / Selfish weighting problem: Minority cabal sets  wC=1\ w_{C}=1 \: self-weight to maximally grow its relative stake, e.g. at (`1) honest majority stake of  sH=0.6\ s_{H}=0.6 \: and honest utility of  wH=0.75\ w_{H}=0.75 \: would require cabal to report self-weight  wC<0.62\ w_{C}<0.62 \: for honest stake to be retained. (b) Consensus solution: Stake-based consensus  η=3\ \eta=3 \: corrects excessive self-weight of minority stake, e.g. at (`2)  sH=0.6\ s_{H}=0.6 \: ,  wH=0.75\ w_{H}=0.75 \: no selfish cabal weight can prevent honest stake retention, even  wC=1\ w_{C}=1 \: results in honest stake ratio gain. Zero-weight problem: Minority cabal is virtually the only scoring incentive recipient of the cabal utility reward when its actual utility is near-zero, e.g. at (`3) where honest stake deteriorates. (c) Weight trust solution: Require the majority stake to agree that a weight is non-zero, otherwise smoothly nullify the associated reward to the degree of mistrust, which then removes the honest stake deterioration region when  wH>0.95\ w_{H}>0.95 \:. Consensus guarantee: Honest majority stake is retained when  sH0.6\ s_{H}\ge 0.6 \: and  wH0.75\ w_{H}\ge 0.75 \:, despite strategic cabal weight setting.

Emission of newly minted token vector E through subnet incentive mechanisms.

Figure 2 / Retention line interpretation: (a) Honest incentive share contour plot for  sH=0.6\ s_{H}=0.6 \:only, highlighting where the emission is  eH=0.6\ e_{H}=0.6 \:, e.g. at (`1). However, at (`2) the contour recedes again due to the zero-weight problem. (b) Similarly, the specific emission contour plot for  sH=0.7\ s_{H}=0.7 \:, highlighting the contour where the emission is  eH=0.7\ e_{H}=0.7 \:, which means with inflation the honest share ratio of  sH=0.7\ s_{H}=0.7 \: can be retained if honest utility is at least  wH>0.75\ w_{H}>0.75 \: like at (`3). (c) Retention lines: A compound plot combines all the highlighted  sH=eH\ s_{H}=e_{H} \:contours from individual contour plots (e.g.  sH=0.6 and sH=0.7\ s_{H}=0.6 \ and \ s_{H}=0.7), to show the overall retention profile. Generally, the higher the honest stake, the higher the honest utility requirement to retain stake proportion under adversarial weight setting.

Emission of newly minted token vector E through subnet incentive mechanisms.

Figure 3 / Scoring incentive: A percentage share of the utility rewards equal to stake in a score, as incentive to encourage honest scoring. (a) No scoring incentive leads to extreme selfish weight setting, since the cabal does not share in the honest rewards. (b)-(f) Higher scoring incentive reduces selfishness evidenced by receding honest self-weight requirement for stake retention. However, the zero-weight problem at (`1) increases as well, since only the cabal can claim reward share from both honest and cabal subsets while honest sets zero weight on the cabal. The weight trust solution with smooth edge coverage  wH>0.95)\ w_{H}>0.95) \: can only be extended so far before legitimate low-utility is also nullified, which practically limits scoring incentive around 50%.

1.2 Reward emission

In the two-player characterization of the game, there are two bimodal weight distributions of  (wH,  1wC)\ (w_{H},\; 1-w_{C}) \: and  (1wH,wC)\ (1-w_{H}, w_{C}) \: on the honest and cabal players, respectively. The stake proportions behind the bimodal distributions are  (bHH, bCH)=(wHsH, (1wC)(1sH))\ (b_{HH}, \ b_{CH})=(w_Hs_H, \ (1-w_{C})(1-s_{H})) \: and (bHC, bCC)=((1wH)sH, wC(1sH))\ (b_{HC}, \ b_{CC})=((1-w_{H})s_{H}, \ w_{C}(1-s_{H})) \: , respectively.

 bHH=wHsH   bCH=(1wC)(1sH)\ b_{HH}=w_{H}s_{H} \ \ \ b_{CH}=(1-w_{C})(1-s_{H}) \:
 bHC=(1wH)sH   bCC=wC(1sH)\ b_{HC}=(1-w_{H})s_{H} \ \ \ b_{CC}=w_{C}(1-s_{H}) \:

Primary incentive ii is the normalized sum of stake proportions, where honest rank  rH=wHsH+(1wC)(1sH)\ r_{H}=w_{H} s_{H}+(1-w_{C})(1-s_{H}) \: and cabal rank  rC=(1wH)sH+wC(1sH)\ r_{C}=(1-w_{H})s_{H}+w_{C}(1-s_{H}) \: are normalized to give  iH=rHrH+rC\ i_{H}=\frac{r_{H}}{r_{H}+r_{C}} \: and  iC=rCrH+rC\ i_{C}= \frac{r_{C}}{r_{H}+r_{C}} \: . An additional reward dd is the scoring share of incentive, i.e.  dH=wHsHrHiH+(1wH)sHrCiC\ d_{H}=\frac{w_{H} s_{H}}{r_{H}}i_{H}+\frac{(1-w_{H})s_{H}}{r_{C}}i_{C} \: and dC=(1wC)sCrHiH+wCsCrCiC\ d_{C}=\frac{(1-w_{C})s_{C}}{r_{H}}i_{H}+\frac{w_{C}s_{C}}{r_{C}}i_{C} \:. Finally, the complete reward emissions are eH=iH+dH2\ e_{H}=\frac{i_{H}+d_{H}}{2} and  eC=iC+dC2\ e_{C}=\frac{i_{C}+d_{C}}{2} \:, such that  eH+eC=1\ e_{H}+e_{C}=1 \: .

Emission of newly minted token vector E through subnet incentive mechanisms.

Figure 4 / Larger minority excess: The excess weight above consensus is larger for minority-stake when  wH<wC\ w_{H}<w_{C} \:. (a) Positive contours when  wH<wC\ w_{H}<w_{C} \: at (`1) indicate regions of cabal error-correction potential. (b) Cabal error-correction region grows as majority stake increases  sH=0.550.95\ s_{H}=0.55\rightarrow 0.95 \:. (c) However at (`2), larger majority excess appears on the right-side when  wC<wH\ w_{C}<w_{H} \:(to be avoided), which negatively impacts the majority weight more than the minority.

1.3 Consensus deviation

The weight consensus is the stake-proportion weight average wj=i(siwij)wij/k(skwkj)\ \overline{w_{j}} = \sum_i(s_{i} w_{ij})w_{ij} / \sum_k(s_{k} w_{kj})\:, and accordingly the consensus weights for the honest and cabal players are
 wH=sHwH2+(1sH)(1wC)2sHwH+(1sH)(1wC)\ \overline{w_{H}}=\frac{s_{H} w_{H}^2 + (1-s_{H})(1-w_{C})^2}{s_{H} w_{H} + (1-s_{H})(1-w_{C})} \: and ,
 wC=sH(1wH)2+(1sH)wC2sH(1wH)+(1sH)wC\ \overline{w_{C}}=\frac{s_{H} (1-w_{H})^2 + (1-s_{H})w_{C}^2}{s_{H} (1-w_{H}) + (1-s_{H})w_{C}} \: respectively

Under typical adversarial play with  1wH<wC\ 1-w_{H}<w_{C} \: , the upper modes  wH>wH\ w_{H}> \overline{w_{H}} \: and  wC>wC\ w_{C}> \overline{w_{C}} \: of the honest and cabal weight distributions, respectively, will exceed the consensus. Honest excess  wH<wH\ \overline{w_{H}}<w_{H} \: is present when  1wC<wH: \ 1-w_{C}<w_{H}: \:

 wH<wH\ \overline{w_{H}}<w_{H} \: sHwH2+(1sH)(1wC)2sHwH+(1sH)(1wC)<wH\ \frac{s_{H} w_{H}^{2} + (1-s_{H})(1-w_{C})^2}{s_{H} w_{H} + (1-s_{H})(1-w_{C})}<w_{H} \: sHwH2+(1sH)(1wC)2<\ s_{H} w_{H}^{2} + (1-s_{H})(1-w_{C})^{2} < \: sHwH2+(1sH)(1wC)wH\ \qquad\qquad\qquad\qquad s_{H} w_{H}^{2} + (1-s_{H})(1-w_{C})w_{H} \: (1sH)(1wC)2<(1sH)(1wC)wH\ \qquad\qquad\qquad\qquad (1-s_{H})(1-w_{C})^2<(1-s_{H})(1-w_{C})w_{H} \: 1wC<wH\ \qquad\qquad\qquad\qquad 1-w_{C}<w_{H} \:

Similarly,  1wH<wC\ 1-w_{H}<w_{C} \: produces cabal excess  wC<wC\ \overline{w_{C}}<w_{C} \: :

 wC<wC\ \overline{w_{C}}<w_{C} \: sH(1wH)2+(1sH)wC2sH(1wH)+(1sH)wC<wC\ \frac{s_{H} (1-w_{H})^2 + (1-s_{H})w_{C}^{2}}{s_{H} (1-w_{H}) + (1-s_{H})w_{C}}<w_{C} \: sH(1wH)2+(1sH)wC2<\ s_{H} (1-w_{H})^{2} + (1-s_{H})w_{C}^{2} < \: sH(1wH)wC+(1sH)wC2\ \qquad\qquad\qquad\qquad s_{H} (1-w_{H})w_{C} + (1-s_{H})w_{C}^{2} \: sH(1wH)2<sH(1wH)wC\ \qquad\qquad\qquad\qquad s_{H} (1-w_{H})^{2} < s_{H} (1-w_{H})w_{C} \: 1wH<wC\ \qquad\qquad\qquad\qquad 1-w_{H}<w_{C} \:

Lemma 1 (Larger Minority Excess). Minority-stake excess weight is larger than majority-stake excess weight, i.e. wCwC>wHwH, when wH<wC\ w_{C} - \overline{w_{C}} > w_{H} - \overline{w_{H}}, \ when \ w_{H} < w_{C} \:

We use a symbolic solver (Wolfram) to show that the cabal excess is larger when  wH<wC,\ w_{H} < w_{C}, \: i.e.
 dwCdwH=wCwCwHwH>1\ \frac{dw_{C}}{dw_{H}}=\frac{w_{C}-\overline{w_{C}}}{w_{H}-\overline{w_{H}}} > 1 \:
The cabal excess is larger with majority honest stake when

{0<wC0.50.5<wH1wC0.5<sH<1} or {0.5<wC10.5<wHwC0.5<sH<1}\begin{Bmatrix} 0<w_{C} \leq 0.5 \\ 0.5<w_{H} \leq 1-w_{C} \\ 0.5<s_{H}<1 \end{Bmatrix} \ or \ \begin{Bmatrix} 0.5<w_{C} \leq 1 \\ 0.5<w_{H} \le w_{C} \\ 0.5<s_{H}<1 \end{Bmatrix}

Otherwise, for the following conditions

{0<wC0.51wC<wH} or {0.5<wC1wC<wH<1}\begin{Bmatrix} 0<w_{C} \leq 0.5 \\ 1 - w_{C} < w_{H} \end{Bmatrix} \ or \ \begin{Bmatrix} 0.5<w_{C} \leq 1 \\ w_{C} < w_{H} < 1 \end{Bmatrix}

Otherwise, for the following conditions

 wC2wCwC2wCwH2+wHwC2wH2wC2wHwCwH2+wCwH(wC2wCwH2+wH)2<sH<1.\ \frac{w_{C}^{2}-w_{C}}{w_{C}^{2}-w_{C}-w_{H}^{2}+w_{H}}- \\ \sqrt{\frac{w_{C}^{2} w_{H}^{2}-w_{C}^{2} w_{H}-w_{C} w_{H}^{2}+w_{C} w_{H}}{\left(w_{C}^{2}-w_{C}-w_{H}^{2}+w_{H}\right)^{2}}}<s_{H}<1. \:

1.4 Excess weight reduction

Stake-proportional consensus advantages the honest player with  sH>0.5\ s_{H}>0.5 \: , since it biases the consensus weight toward the honest vote and exposes the cabal excess self-weight wC>wC\ w_{C} > \overline{w_{C}} \: where  dwC>dwH\ dw_{C}>dw_{H} \: . Consequently, a consensus policy  π(w)=min(w,w)\ \pi(\mathbf{w})=\min(\overline{\mathbf{w}}, \mathbf{w}) \: can reduce excess weight above the consensus w\ \overline{\mathbf{w}} \: , where cabal weight should decrease more than honest weight. The weight reductions normally only happen in the upper modes  wH\ w_{H} \: and  wC\ w_{C} \: of the honest and cabal weights, respectively.

Emission of newly minted token vector E through subnet incentive mechanisms.

Figure 5 / Larger minority excess: The excess weight above consensus is larger for minority-stake when  wH<wC\ w_{H}<w_{C} \:. (a) Positive contours when  wH<wC\ w_{H}<w_{C} \: at (`1) indicate regions of cabal error-correction potential. (b) Cabal error-correction region grows as majority stake increases  sH=0.550.95\ s_{H}=0.55\rightarrow 0.95 \:. (c) However at (`2), larger majority excess appears on the right-side when  wC<wH\ w_{C}<w_{H} \:(to be avoided), which negatively impacts the majority weight more than the minority.

The consensus policy  π(w)(wH,wC)\ \pi(\mathbf{w})\rightarrow (w^{\prime}_{H}, w^{\prime}_{C}) \: attempts to correct the error  ϵ=wH+wC1\ \epsilon=w_{H}+w_{C}-1 \: so that

 wC=1wHwCΔwC=1(wHΔwH)ΔwH+ΔwC=wH+wC1η(dwH+dwC)=wH+wC1η=wH+wC1dwH+dwC. \ w^{\prime}_{C} = 1 - w^{\prime}_{H} \\ w_{C} - \Delta w_{C} = 1 - (w_{H} - \Delta w_{H}) \\ \Delta w_{H} + \Delta w_{C} = w_{H} + w_{C} - 1 \\ \eta(d w_{H} + d w_{C}) = w_{H} + w_{C} - 1 \\ \eta = \frac{{w_{H} + w_{C} - 1}}{d w_{H} + d w_{C}}.\

The approximate number of weight reduction steps is  η\ \eta \:, and the consensus policy is thus converted to an iterated function  π=fη\ \pi=f^{\eta} \: , where the function is repeated  η\ \eta \: times  f3(w)=f(f(f(w)))\ f^3(\mathbf{w})=f(f(f(\mathbf{w}))) \: . Note that f(w)=min(w,w)\ f(\mathbf{w})=\min(\overline{\mathbf{w}}, \mathbf{w}) \: recomputes the consensus weight  w\ \overline{\mathbf{w}} \: each time.

Emission of newly minted token vector E through subnet incentive mechanisms.

Figure 6 / Emission slashing: Iterative weight correction reduces effective scoring weight and incentive share. (a) Initial scoring weight is always 1, but weight correction reduces this whenever  w1\ \sum w\neq 1 \:, with the largest reduction seen at (`1). (b) Emission is the average of scoring incentive and utility reward and cabal emission is slashed, i.e.  e(η=3)<e(η=0)\ e^{(\eta=3)} < e^{(\eta=0)} \: particularly in the  wH<wC\ w_{H}<w_{C} \: and  0.5<sH\ 0.5<s_{H} \: region around (`2). (c) Consequently, honest emission is boosted in region (`2), but the zero weight vulnerability at (`3) slashes honest emission, although comparatively little.

Emission of newly minted token vector E through subnet incentive mechanisms.

Figure 7 / Cabal slash (two-team). (a) Statistical analysis simplifies the two-player result. (b) The two-team generalization fully enacts the weight distribution, and a Monte Carlo analysis reveals the worst-case cabal slash closely following the two-player result. (c) Cabal incentive is boosted at the zero weight vulnerability of (`1), but weight trust solution ensures active cabal slash at (`2).

We compute  η=wH+wC1dwH+dwC\ \eta = \frac{w_{H} + w_{C} - 1}{d w_{H} + d w_{C}} \: and compare with the correction factor dwC/dwH\ dw_{C}/dw_{H} \: to identify the optimal  η\ \eta \: avoiding over-correction in the detrimental wC<wH\ w_{C}<w_{H} \: region where  dwC/dwH<1\ dw_{C}/dw_{H}<1 \:

We observe that higher  η>3\ \eta>3 \: values extend the correction further into the detrimental wC<wH\ w_{C}<w_{H} \: region where  dwCdwH<0\ dw_{C} - dw_{H} < 0 \: , hence an optimal  η3\ \eta\approx 3 \: is identified, which is large enough to provide sufficient correction when  wH<wC\ w_{H}<w_{C} \: .

Application of the consensus policy  π(w)=fη3(w)\ \pi(\mathbf{w})=f^{\eta\approx 3}(\mathbf{w}) \: can partially correct the error  ϵ=wC+wH1>0\ \epsilon=w^{\prime}_{C}+w^{\prime}_{H}-1>0 \: , in particular the previous expense  wH=1\ w_{H}=1 \: is reduced to  wH<0.75\ w_{H}<0.75 \: for  sH=0.6\ s_{H}=0.6 \: , even at Nash equilibrium with  wC=0.8\ w_C^{*}=0.8 \: . Importantly, the consensus policy  π(w)\ \pi(\mathbf{w}) \: operates on anonymized weights and do not assume the player identities, thus behaves impartially in terms of a stake-based consensus.

1.5 Smoothed weight reduction

The correction function  f(w)=min(w,w)\ f(\mathbf{w})=\min(\overline{\mathbf{w}}, \mathbf{w}) \: should be smoothed to ensure limdw0fη(w+dw)fη(w)<ε\ \lim_{dw\rightarrow 0}f^{\eta}(w+dw) - f^{\eta}(w)<\varepsilon \: where adjacent weights are corrected to a similar degree. The correction factor should also depend on the magnitude of deviation from consensus, in terms of a standard deviation σ\sigma. We opt for a stake-weighted mean absolute deviation, since it does not make the normal assumption as strongly as mean square deviation, as follows

σ(w)=isiwijwijwkskwkj.\sigma(\mathbf{w})=\frac{\sum_{i}s_{i}w_{ij}|w_{ij} - \overline{\mathbf{w}}|}{\sum_{k}s_{k}w_{kj}}.

The standard correction  0α<1\ 0 \le \alpha<1 \: fully applies when ww=σ(w)\ w - \overline{\mathbf{w}}=\sigma(\mathbf{w}) \:, and amplifies when ww>σ(w)\ w - \overline{\mathbf{w}}>\sigma(\mathbf{w}) \: to a maximum correction at w\ \overline{\mathbf{w}} \: with a proposed smoothed function iterate

 f(ww>w)=w+(ww)αwwσ(w)=w+(ww)(1δ)=w+wwδ(ww)=wδ(ww)=wδdw \ f(\mathbf{w}|w>\overline{\mathbf{w}})=\overline{\mathbf{w}} + (w - \overline{\mathbf{w}})\alpha^{\frac{w - \overline{\mathbf{w}}}{\sigma(\mathbf{w})}} \\ =\overline{\mathbf{w}} + (w - \overline{\mathbf{w}})(1-\delta)\\ =\overline{\mathbf{w}} + w - \overline{\mathbf{w}} -\delta (w - \overline{\mathbf{w}})\\ =w -\delta (w - \overline{\mathbf{w}})\\ =w -\delta \cdot dw\ \:
Emission of newly minted token vector E through subnet incentive mechanisms.

Figure 8 / Weight trust: Majority stake should set non-zero weight to a player, otherwise its reward is nullified. (a) Two-team honest emission with just the consensus policy, with zero weight vulnerability at (`1). (b) Applying weight trust protects the  0.95<wH\ 0.95<w_{H} \: region and removes the exploit. (c) Honest retention is now monotonically possible at increasing honest self-weight.

The smoothed function iterate now requires more steps, where a larger  α1δ\ \alpha\approx 1-\delta \:results in a larger  η\ \eta \: , such that  ηδ3\ \eta\delta\approx 3 \: , according to

 wC=1wHwCΔwC=1(wHΔwH)ΔwH+ΔwC=wH+wC1ηδ(dwH+dwC)=wH+wC1η=wH+wC1δ(dwH+dwC).\ w^{\prime}_{C} = 1 - w^{\prime}_{H}\\ w_{C} - \Delta w_{C} = 1 - (w_{H} - \Delta w_{H})\\ \Delta w_{H} + \Delta w_{C} = w_{H} + w_{C} - 1\\ \eta\delta(d w_{H} + d w_{C}) = w_{H} + w_{C} - 1\\ \eta = \frac{w_{H} + w_{C} - 1}{\delta(d w_{H} + d w_{C})}.\\ \:

We compare the previous retention graph  (η=3,α=0)\ (\eta=3, \alpha=0) \:against  (η=3/(10.95),α=0.95)\ (\eta=3/(1-0.95), \alpha=0.95) \: and observe a reduced cost  wH=0.7<0.75\ w_{H}=0.7<0.75 \:with the smoothed iterate with  α>0\ \alpha>0 \:.

We ensure monotonicity of the consensus policy by choosing the minimum  ε>0\ \varepsilon>0 \:added to the deviation  σ(w)\ \sigma(\mathbf{w}) \:.

 f(ww>w)<f(w+dww>w)w+(ww)αwwσ(w)+ε<w+(w+dww)αw+dwwσ(w)+ε(ww)αwwσ(w)+ε<(w+dww)αw+dwwσ(w)+εwww+dww<αw+dwwσ(w)+εwwσ(w)+εlogwww+dww<logαdwσ(w)+εdwlogαlogwww+dwwσ(w)<ε\ f(\mathbf{w}|w>\overline{\mathbf{w}})<f(\mathbf{w} + dw\|w>\overline{\mathbf{w}})\\ \overline{\mathbf{w}} + (w - \overline{\mathbf{w}})\alpha^{\frac{w - \overline{\mathbf{w}}}{\sigma(\mathbf{w})+\varepsilon}}<\overline{\mathbf{w}} + (w + dw - \overline{\mathbf{w}})\alpha^{\frac{w + dw - \overline{\mathbf{w}}}{\sigma(\mathbf{w})+\varepsilon}}\\ (w - \overline{\mathbf{w}})\alpha^{\frac{w - \overline{\mathbf{w}}}{\sigma(\mathbf{w})+\varepsilon}}< (w + dw - \overline{\mathbf{w}})\alpha^{\frac{w + dw - \overline{\mathbf{w}}}{\sigma(\mathbf{w})+\varepsilon}}\\ \frac{w - \overline{\mathbf{w}}}{w + dw - \overline{\mathbf{w}}}< \alpha^{\frac{w + dw - \overline{\mathbf{w}}}{\sigma(\mathbf{w})+\varepsilon} - \frac{w - \overline{\mathbf{w}}}{\sigma(\mathbf{w})+\varepsilon}}\\ \log\frac{w - \overline{\mathbf{w}}}{w + dw - \overline{\mathbf{w}}}< \log\alpha^{\frac{dw}{\sigma(\mathbf{w})+\varepsilon}}\\ \frac{dw\log\alpha}{\log\frac{w - \overline{\mathbf{w}}}{w + dw - \overline{\mathbf{w}}}} - \sigma(\mathbf{w})<\varepsilon\\ \:

1.6 Weight trust

Weight trust  T=(W>0)S\ T=(W>0)S \: is the sum of stake assigning a non-zero weight to a player, and a consensus C=(1+exp(ρ(Tκ)))1\ C=(1 + \exp(-\rho(T-\kappa)))^{-1} \: provides a smooth threshold at κ\kappa where exceedingκ\kappa ratio of stake quickly allows for high trust. A modified rank r=rc\ r^{\prime}=rc \: multiplies rank with the weight trust consensus, which influences the emission so that zero cabal weight  wC=1wH0\ w^{\prime}_{C}=1-w_{H}\approx 0 \: receives low consensus thereby penalizing cabal emissions.

The vulnerable region of  wH=1\ w_{H}=1 \: and  0.8<wC<0.95\ 0.8<w_{C}<0.95 \:allows for cabal stake gain when  sH=0.6\ s_{H}=0.6 \:, but the weight trust consensus smoothly pads the region around wH=1\ w_{H}=1 \: and removes the vulnerability. The cabal can thus not claim reward when the honest majority deems cabal utility to be zero, despite the non-zero self-weight reported by the minority cabal.

Emission of newly minted token vector E through subnet incentive mechanisms.

Figure 9 / Evolution smoothness: Evolving through more smaller steps with η=59\ \eta=59 \: reduces the zero weight exploit at black (`3), compared to  η=3\ \eta=3 \: at (`1), since fine-grained correction steps more accurately track changes in consensus. The two-team honest emissions (Monte Carlo worst-case analysis) tend to further reduce the zero weight exploits at (`2), (`4) compared to the two-player case at (`1), (`3).

02/ Density generalization

2.1 Overview

We generalize the two-player game to a two-team game with  H\ |H| \: honest and  C\ |C| \: cabal players, that have  iHsi=sH\ \sum_{i\in H}s_{i}=s_{H} \: honest stake and  iCsi=1sH\ \sum_{i\in C}s_{i}=1-s_{H} \: cabal stake. Honest players  iH\ i\in H \: set  jHwij=wH\ \sum_{j\in H}w_{ij}=w_{H} \: self-weight and  jCwij=1wH\ \sum_{j\in C}w_{ij}=1-w_{H} \: weight on cabal players, while cabal players  iC\ i\in C \: set  jCwij=wC\ \sum_{j\in C}w_{ij}=w_{C} \: self-weight and  jHwij=1wC\ \sum_{j\in H}w_{ij}=1-w_{C} \:weight on honest players. The rank components result in the same aggregates in the two-player game.

 bHH=jHwijiHsi=wHsHbCH=jHwijiCsi=(1wC)(1sH)bHC=jCwijiHsi=(1wH)sHbCC=jCwijiCsi=wC(1sH)\ b_{HH}=\sum_{j\in H}w_{ij}\sum_{i\in H}s_{i}=w_{H}s_{H}\\ b_{CH}=\sum_{j\in H}w_{ij}\sum_{i\in C}s_{i}=(1-w_{C})(1-s_{H})\\ b_{HC}=\sum_{j\in C}w_{ij}\sum_{i\in H}s_{i}=(1-w_{H})s_{H}\\ b_{CC}=\sum_{j\in C}w_{ij}\sum_{i\in C}s_{i}=w_{C}(1-s_{H}) \:

In particular, the weight consensus of an individual honest player is shown to be  wh=wHH\ \overline{w_{h}}=\frac{\overline{w_{H}}}{|H|} \: as follows (and similarly for cabal players  wc=wCC\ \overline{w_{c}}=\frac{\overline{w_{C}}}{|C|} \:)

 wh=isiwih2/kskwkh=iHsiwih2+iCsiwih2kHskwkh+kCskwkhiHsiwH2/H2+iCsi(1wC)2/H2kHskwH/H+kCsk(1wC)/H=1HsHwH2+(1sH)(1wC)2sHwH+(1sH)(1wC)=wHH \ \overline{w_{h}}=\sum_{i} s_{i} w^{2}_{ih}\left/ \sum_k s_{k} w_{kh}\right.\\ =\frac{\sum_{i\in H} s_{i} w^{2}_{ih} + \sum_{i\in C} s_{i} w^{2}_{ih}}{\sum_{k\in H} s_{k} w_{kh} + \sum_{k\in C} s_{k} w_{kh}}\\ \approx\frac{\sum_{i\in H} s_{i} w_{H}^{2}/|H|^{2} + \sum_{i\in C} s_{i} (1 - w_{C})^{2}/|H|^{2}}{\sum_{k\in H} s_{k} w_{H}/|H| + \sum_{k\in C} s_{k} (1 - w_{C})/|H|}\\ =\frac{1}{|H|}\frac{s_{H} w_{H}^{2} + (1-s_{H})(1-w_{C})^{2}}{s_{H} w_{H} + (1-s_{H})(1-w_{C})}=\frac{\overline{w_{H}}}{|H|}\ \:

Under the minimal assumption that the average self-weights set on honest and cabal players are wHH\ \frac{w_{H}}{|H|} \: and  wCC\ \frac{w_{C}}{|C|} \: we can construct weight densities ph(w)=phh(w)+pch(w)\ p_h(w) = p_{hh}(w) + p_{ch}(w) \: and  pc(w)=phc(w)+pcc(w)\ p_{c}(w) = p_{hc}(w) + p_{cc}(w) \:, here according to the normal assumption (other densities with a similar first moment could possibly also be valid)

 phh(w)=sHwN(wHH,wHHσ)pch(w)=(1sH)wN(1wCH,1wCHσ)phc(w)=sHwN(1wHC,1wHCσ)pcc(w)=(1sH)wN(wCC,wCCσ)\ p_{hh}(w) = s_{H} w \mathcal{N}\left(\frac{w_{H}}{|H|}, \frac{w_{H}}{|H|}\sigma\right) \\ p_{ch}(w) = (1-s_{H}) w \mathcal{N}\left(\frac{1-w_{C}}{|H|}, \frac{1-w_{C}} {|H|}\sigma\right)\\ p_{hc}(w) = s_{H} w \mathcal{N}\left(\frac{1-w_{H}}{|C|}, \frac{1-w_{H}}{|C|}\sigma\right)\\ p_{cc}(w) = (1-s_{H}) w \mathcal{N}\left(\frac{w_{C}}{|C|}, \frac{w_{C}}{|C|}\sigma\right) \:

The consensus and mean absolute deviations of a weight density function  p(w)\ p(w) \: are

 p=wp(w)dw,andσ(p)=wpp(w)dw.\ \overline{p}=\int w p(w) dw,\quad {and} \quad\sigma(p)= \int |w - \overline{p}|p(w) dw. \:

We overload the iterated function  f\ f \: as a density evolution function  f(p(w))\ f(p(w)) \: that contracts a density p(w)\ p(w) \: above consensus p\ \overline{p} \: by a nominal degree of  α\ \alpha \: at a single deviation  wpσ(p)\ \frac{w-\overline{p}}{\sigma(p)} \:, in order to correct the error  ϵ=wH+wC1\ \epsilon=w_{H}+w_{C}-1 \:. The density is contracted via  g(w)=f1(w)\ g(w)=f^{-1}(w) \:involving the original iterated function  f\ f \:.

 f(p(w))=p(w  wp)+p(g(w)  p<w)wg(w)dg(w)dw\ f(p(w)) = p(w \ | \ w\le \overline{p}) + p(g(w) \ | \ \overline{p}<w) \frac{ w}{g(w)}\frac{dg(w)}{dw} \:

The final rank after applying the consensus policy  π=fη\ \pi=f^{\eta} \: is  ri=fη(pi(w))dw\ r_{i} = \int f^{\eta}(p_{i}(w))dw \:, where a single function iteration contracts the consensus by  pp\ \overline{p^{\prime}}-\overline{p} \:, which is equal to

 wp(w)dwwp(g(w)  μp<w)wg(w)dg(w)dwdw.\ \int w p(w) dw - \int w p(g(w) \ | \ \mu_{p}<w) \frac{ w}{g(w)}\frac{dg(w)}{dw}dw. \:
Emission of newly minted token vector E through subnet incentive mechanisms.

Figure 10 / Density evolution: Weight correction through density evolution reduces cabal weight consensus more than honest reduction  (sH=0.6,wH=0.7,wC=0.8)\ (s_{H}=0.6, w_{H}=0.7, w_{C}=0.8) \: . The honest weights and cabal weights have an equal starting consensus at (`1), but through density evolution the cabal reduces more to (`3), versus a higher end honest consensus at (`2). Density evolution thus succeeds in penalizing the minority cabal and allows for honest stake retention. The theoretical probability densities in (a), (c) closely match the stochastically sampled results in (b), (d). The crosshair markers indicate the consensus flanked by a standard deviation above and below.

Simulating the consensus policy  (η=3/(10.95),α=0.95)\ (\eta=3/(1-0.95), \alpha=0.95) \: on weight densities set on honest and cabal players where  sH=0.6,wH=0.7,wC=0.8\ s_{H}=0.6, w_{H}=0.7, w_{C}=0.8 \:, we see an equal starting consensus weight reduce further for the cabal players  (6.763.8)\ (6.76\rightarrow 3.8) \: vs honest players  (6.764.41)\ (6.76\rightarrow 4.41) \:. The consensus policy acts as an upper-mode resiliency test where cabal self-weight with minority stake fails comparatively to honest self-weight with majority stake.

2.2 Stochastic sampling

We move from theoretical density analysis to a stochastic sampling analysis, where the original π(w)=fη(w)\ \pi(\mathbf{w})=f^{\eta}(\mathbf{w}) \: can be applied directly to a weight sample  w\ \mathbf{w} \: for a player, gradually contracting excess weight toward the consensus until an optimal contraction volume is reached. We observe very similar density evolution results as with the theoretical density analysis.

2.3 Two-team game

We perform a worst-case Monte Carlo analysis of a full-scale two-team game by sampling from normal densities, primarily to confirm the accuracy of the preceding aggregate analysis. We run a number of Monte Carlo iterations and record the worse-case results. A blockchain-based consensus algorithm has space and compute limitations, which would favor a smaller  η\ \eta \: number of density evolution operations, each of which requires O(n2)\ O(n^{2}) \: operations. A small  η=3\ \eta=3 \: with  α=0\ \alpha=0 \: produce a full-scale result very close to the aggregate result.

Increasing the number of density evolution steps to  η=59\ \eta=59 \: with α=0.95\ \alpha=0.95 \: manages to remove the zero-utility exploit at  wH>0.98\ w_{H}>0.98 \: seen at  η=3\ \eta=3 \:. However, the aggregate result in the theoretical honest retention deviates slightly, likely due to deviation of the upper mode density below consensus not accounted for in the aggregate.

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