import matplotlib.pyplot as plt head = 1 source = 1 target = 1 delay = 7/8 proposer = 1/8 fig, (ax1,ax2) = plt.subplots(1,2,figsize=(16,10)) ax1.pie( [head, source, target, delay, proposer], labels=['head', 'source', 'target', 'delay', 'proposer'], autopct='%1.1f%%' ) ax1.set_title( "Pre-Altair Reward Weights for Validator Duties" ) HEAD_WEIGHT = 14 SOURCE_WEIGHT = 14 TARGET_WEIGHT = 26 SYNC_WEIGHT = 2 PROPOSER_WEIGHT = 8 WEIGHT_DENOMINATOR = 64 ax2.pie( [HEAD_WEIGHT, SOURCE_WEIGHT, TARGET_WEIGHT, SYNC_WEIGHT, PROPOSER_WEIGHT], labels=['head', 'source', 'target', 'sync', 'proposer'], autopct='%1.1f%%' ) ax2.set_title( "Altair Reward Weights for Validator Duties" ) plt.show()

SECONDS_PER_SLOT = 12 SLOTS_PER_EPOCH = 32 EPOCHS_PER_COMMITTEE = 256 COMMITTEE_SIZE = 512 SECONDS_PER_YEAR = 31556952 seconds_per_committee = SECONDS_PER_SLOT * SLOTS_PER_EPOCH * EPOCHS_PER_COMMITTEE committees_per_year = SECONDS_PER_YEAR / seconds_per_committee print(f"{committees_per_year:.1f} sync committees are selected each year") NUM_VALIDATORS = 200000 expected_committee_selections_per_year = committees_per_year * COMMITTEE_SIZE / NUM_VALIDATORS print(f"with a validator set of {NUM_VALIDATORS} validators, on average each validator will be assigned " f"to a sync committee {expected_committee_selections_per_year:.2f} times per year")

from scipy.stats import binom SECONDS_PER_YEAR = 31556952 SECONDS_PER_SLOT = 12 SLOTS_PER_EPOCH = 32 COMMITTEE_EPOCHS = 256 NUM_VALIDATORS = 200000 COMMITTEE_VALIDATORS = 512 epochs_per_year = SECONDS_PER_YEAR / (SECONDS_PER_SLOT * SLOTS_PER_EPOCH) x = [el for el in range(7)] p_selection = COMMITTEE_VALIDATORS / NUM_VALIDATORS committees_per_year = epochs_per_year / COMMITTEE_EPOCHS y = binom.pmf(x, committees_per_year, p_selection) fig, ax = plt.subplots(figsize=(12, 8)) ax.bar(x, y) ax.set_xlim(xmin=-0.5) ax.set_ylim(ymin=0) ax.set_title('Probability mass function (200,000 validators) — number of sync committees per year') ax.set_xlabel('Number of sync committee selections in a year') ax.set_ylabel('Proportion of validators') print(y[0])

import math import pandas as pd def get_quantile(pmf, quantile): cumulative = 0 for x, prob in sorted(pmf.items()): cumulative += prob if cumulative >= quantile: return x slots_per_year = SECONDS_PER_YEAR / SECONDS_PER_SLOT GWEI_PER_ETH = int(1e9) gwei_per_validator = 32 * GWEI_PER_ETH BASE_REWARD_FACTOR = 64 PREALTAIR_NUM_BASE_REWARDS = 4 PREALTAIR_PROPOSER_REWARD_QUOTIENT = 8 base_reward = gwei_per_validator * BASE_REWARD_FACTOR // math.isqrt(NUM_VALIDATORS * gwei_per_validator) total_reward = base_reward * NUM_VALIDATORS prior_proposer_share = base_reward // PREALTAIR_NUM_BASE_REWARDS // PREALTAIR_PROPOSER_REWARD_QUOTIENT prior_proposer_reward = prior_proposer_share * NUM_VALIDATORS // SLOTS_PER_EPOCH prior_att_reward = base_reward - prior_proposer_share altair_proposer_reward = total_reward * PROPOSER_WEIGHT // SLOTS_PER_EPOCH // WEIGHT_DENOMINATOR altair_att_reward = base_reward * (HEAD_WEIGHT + SOURCE_WEIGHT + TARGET_WEIGHT) // WEIGHT_DENOMINATOR sync_reward = total_reward * COMMITTEE_EPOCHS * SYNC_WEIGHT // COMMITTEE_VALIDATORS // WEIGHT_DENOMINATOR # distribution of committee selections per year n_committees = [el for el in range(11)] pmf_committees = binom.pmf(n_committees, committees_per_year, COMMITTEE_VALIDATORS / NUM_VALIDATORS) # distribution of block proposal opportunities per year n_proposals = [el for el in range(51)] pmf_proposals = binom.pmf(n_proposals, slots_per_year, 1 / NUM_VALIDATORS) n_bins = 32 bins = [1.7 + i / 40 for i in range(n_bins)] altair_hist = [0] * n_bins prior_hist = [0] * n_bins # calculate all possible reward levels (up to 50 block proposals) assuming perfect participation prior_pmf = {} for props in n_proposals: reward = props * prior_proposer_reward + epochs_per_year * prior_att_reward prior_pmf[reward] = pmf_proposals[props] # bin the rewards to generate histogram for reward_gwei, prob in prior_pmf.items(): reward = reward_gwei / GWEI_PER_ETH for i, edge in enumerate(bins[1:]): if reward < edge: prior_hist[i] += prob break prior_mean = sum([p * r / GWEI_PER_ETH for r, p in prior_pmf.items()]) prior_sigma = math.sqrt(sum([p * (r / GWEI_PER_ETH)**2 for r, p in prior_pmf.items()]) - prior_mean**2) prior_lq = get_quantile(prior_pmf, 0.25) / GWEI_PER_ETH prior_median = get_quantile(prior_pmf, 0.5) / GWEI_PER_ETH prior_uq = get_quantile(prior_pmf, 0.75) / GWEI_PER_ETH prior_iqr = prior_uq - prior_lq print('Pre-Altair annual reward statistics (ETH)') print('-----------------------------------------') print(f' median: {prior_median:.4f}') print(f' mean: {prior_mean:.4f}') print(f' standard deviation: {prior_sigma:.4f}') print(f'interquartile range: {prior_iqr:.4f}') #print(sum(prior_hist)) # check histogram sums to unity # calculate all possible reward levels (up to 50 block proposals and 10 committee selections) altair_pmf = {} for comms in n_committees: for props in n_proposals: reward = comms * sync_reward + props * altair_proposer_reward + epochs_per_year * altair_att_reward prob = pmf_committees[comms] * pmf_proposals[props] if reward in altair_pmf: altair_pmf[reward] += prob else: altair_pmf[reward] = prob # bin the rewards to generate histogram for reward_gwei, prob in altair_pmf.items(): reward = reward_gwei / GWEI_PER_ETH for i, edge in enumerate(bins[1:]): if reward < edge: altair_hist[i] += prob break altair_mean = sum([p * r / GWEI_PER_ETH for r, p in altair_pmf.items()]) altair_sigma = math.sqrt(sum([p * (r / GWEI_PER_ETH)**2 for r, p in altair_pmf.items()]) - altair_mean**2) altair_lq = get_quantile(altair_pmf, 0.25) / GWEI_PER_ETH altair_median = get_quantile(altair_pmf, 0.5) / GWEI_PER_ETH altair_uq = get_quantile(altair_pmf, 0.75) / GWEI_PER_ETH altair_iqr = altair_uq - altair_lq print('\nAltair annual reward statistics (ETH)') print('-------------------------------------') print(f' median: {altair_median:.4f}') print(f' mean: {altair_mean:.4f}') print(f' standard deviation: {altair_sigma:.4f}') print(f'interquartile range: {altair_iqr:.4f}') #print(sum(altair_hist)) # check histogram sums to unity print(f'\nrelative spread: {altair_sigma / prior_sigma:.1f} (standard deviation) / ' f'{altair_iqr / prior_iqr:.1f} (interquartile range)') fig, (ax1,ax2) = plt.subplots(2, 1, figsize=(12,10)) ax1.bar(bins, prior_hist, 1 / n_bins, align='edge') ax1.set_title('Pre-Altair Annual Rewards Distribution (200,000 validators, perfect participation)') ax1.set_ylabel('Proportion of validators') ax2.bar(bins, altair_hist, 1 / n_bins, align='edge') ax2.set_title('Altair Annual Rewards Distribution (200,000 validators, perfect participation)') ax2.set_xlabel('Annual reward (ETH)') ax2.set_ylabel('Proportion of validators') ax.set_title('Distribution of annual rewards assuming perfect performance') ax.set_xlabel('Annual reward (ETH)') ax.set_ylabel('Proportion of validators');

import json with open('aggregate_rewards.json') as f: rewards = json.load(f) with open('check.json') as f: check = json.load(f) # we're using the "reduced genesis set" of validators as in the previous article with open('reduced_genesis_set.json') as f: reduced_genesis_set = json.load(f) altair_rewards = [] prior_rewards = [] for validator_index in reduced_genesis_set: altair_rewards += [rewards[str(validator_index)] / 1e9] prior_rewards += [check[str(validator_index)] / 1e9] df = pd.DataFrame({'altair_rewards': altair_rewards, 'prior_rewards': prior_rewards}) prior_iqr = df['prior_rewards'].quantile(0.75) - df['prior_rewards'].quantile(0.25) altair_iqr = df['altair_rewards'].quantile(0.75) - df['altair_rewards'].quantile(0.25) prior_mean = df["prior_rewards"].mean() altair_mean = df["altair_rewards"].mean() mean_diff = prior_mean - altair_mean print('Statistics — first 62,000 epochs (pre-Altair rewards scheme, measured in ETH)') print('-----------------------------------------------------------------------------') print(f' median: {df["prior_rewards"].quantile(0.5):.4f}') print(f' mean: {prior_mean:.4f}') print(f' standard deviation: {df["prior_rewards"].std():.4f}') print(f'interquartile range: {prior_iqr:.4f}') print('\nStatistics — first 62,000 epochs (Altair rewards scheme, measured in ETH)') print('-------------------------------------------------------------------------') print(f' median: {df["altair_rewards"].quantile(0.5):.4f}') print(f' mean: {altair_mean:.4f}') print(f' standard deviation: {df["altair_rewards"].std():.4f}') print(f'interquartile range: {altair_iqr:.4f}') print('\nComaprison') print('-----------') print(f'The mean per-validator reward under Altair changed by {100*(altair_mean / prior_mean - 1):.1f}%') print(f'The interquartile range (spread) of rewards under Altair was {altair_iqr / prior_iqr:.1f} times greater') fig, (ax1,ax2) = plt.subplots(2, 1, figsize=(12,10)) bins = [b/20 for b in range(50)] df['prior_rewards'].plot.hist(ax=ax1, bins=bins) df['altair_rewards'].plot.hist(ax=ax2, bins=bins) ax1.set_title("Net rewards — pre-Altair reward scheme (first 62,000 epochs)") ax1.set_ylabel("Number of validators") ax2.set_title("Net rewards — Altair reward scheme (first 62,000 epochs)") ax2.set_xlabel("Net reward (ETH)") ax2.set_ylabel("Number of validators");

with open('issuance.json') as f: issuance = json.load(f) delta = [] sum_old = sum_new = 0 for (old, new) in zip(issuance['old_issuance'], issuance['alt_issuance']): delta.append(100 * (new / old - 1)) print(f"Mean change in issuance: {sum(delta) / len(delta):.2f}%") print(f"Greatest per-epoch drop in issuance from Altair: {min(delta):.2f}%, greatest increase: {max(delta):.2f}%") fig, ax = plt.subplots(figsize=(12, 8)) ax.plot(pd.Series(delta).rolling(16).mean()) ax.set_title('Percentage Change in Issuance after Applying Altair Rewards Scheme (16-epoch moving average)') ax.set_xlabel('Epoch') ax.set_ylabel('% change');