Preliminary Thoughts on Numbers

Some preliminary thoughts about the pennant points system (source post here, a useful translation here), and its wider implications:

• From a high level, it seems to have encompassed the three major types of "processes" that form the crux of the 48 Group's popularity: theater performances, handshakes and elections. From a completeness point of view, it looks reasonable.
• One point that I stress that has been emphasised in the following detailed discussion, is that if you look beyond the maths involved, there are no new practical implications arising for fans. The objective is once again, as is always in the idol fandom, to maximise your spending on these members across all three areas. However, what this has done is to slightly shift the focus away from the individual to the team, and incentivised the shift accordingly. 
• Some of my understanding, and the comments/implications arising from it, may change depending on what further information (if any) AKS decides to release - as well as the obvious language barrier. 

1) Theater Performances

• My understanding is that each team gets 2 pts for every stage performance, plus an additional 5 if they exceed a prescribed application ratio. 
• The base application ratio ("BAR") that they described in technical detail seems to suggest that it's based on a historical set of member appearance data that they then chuck into a linear regression model, which spits out a basic metric that is then used throughout the entire pennant race applicable period. When the actual application ratio ("AAR") > BAR, then the 5 pts would be awarded for that particular show. 
• How the AAR is calculated is not disclosed, but it would probably be along the lines of a simple AAR = n / N ratio (where n = number of applications, N = number of tickets available to the stated categories, where N is fixed). 
• Furthermore, the BAR appears to be an aggregate measure, as indicated by the "pi" in the regression equation. Also, based purely on the BAR equation alone, there's no weighting of each "a_k", which on a mathematical level eliminates the differences in popularity between each individual member - in practical terms, Jurina's absences in Team K shows would be balanced out by KK's frequent appearances. So from a team basis, the effect of individual popularity would be equalised. 
• Mathematically speaking, y increases if a_k increases, where a_k = determined via historical appearances for each member, irrespective of individual member popularity.
• However, there's no discussion as to the range of k or n, nor what the exact data set is for the regression equation (apart from the broad "historical member performance data from the previous year"), or whether they are employing multiple regression models, one for each team. In light of this, and based on what I think is a reasonable assumption, y would be based off the entire 48G, not just tied to a particular team (and therefore having multiple y's). Otherwise you'd have different thresholds, which may lead to unfair disparity. 
• That's not to say individual popularity isn't factored in entirely; however, its effect on the mathematical equation is rather indirect. That is to say, if Jurina appears in a Team K show, then it is more likely for that particular show, AAR > BAR. 
• Because this threshold applies only for those who have a mobile FC or Hashira no Kai subscription, it appears AKS have incentivised the need for subscriptions. Then again, most of the domestic fans would likely already have a subscription anyway, but for the remaining group of fans, this may give them an incentive to sign up. 
• The implication of all this is, to maximise the number of points, the team's fans should ideally all be set up with a particular FC subscription if they haven't done so already and a full house team lineup be present. None of this is anything new, but now the incentivisation to do so has been quantified. 
• In fact, seeing that the AAR allocation is not scaled according to the degree that AAR > BAR, and that the assumption is of a static BAR (apart from weightings for special shows), an optimal maxima may actually exist as to the number of applications required in order for the AAR to be > BAR. Of course, fans may not actually get access to that information for fear of gaming the result, but one should keep that in mind. 
• On a note of interest, those members with concurrencies may find that the latter more difficult to achieve due to logistics and scheduling. Again, that is not a new point but the quantitative effect is now in force. It appears that only Team E will have an issue here with only one concurrency counting towards the scheme, as most of the other teams have an even spread of 2-4 concurrencies each. So unless you're in Team E, none of the other teams are heavily biased or significantly disadvantaged in this aspect. 

2) Handshakes

• The methodology seemingly employed here, or at least my understanding of it, appears to exhibit a few holes from our understanding of handshake event applications. 
• The initial allocation of points is on a total tickets sold per event basis, in which they allocate a point between 0.05 to 1 based on the member's sell out % for that event. For subsequent events, if a member increases her sales total, then that initial points allocation is scaled accordingly by the % increase over the previous sales total (for simplicity, I'll call this "SSF" or "subsequent scaling factor"). Adjustments are also noted for any member that sells out 100% event on event. 
• Some possible issues with this methodology:
• The biggest being that selling rounds are not taken into account. For example, member A sells out all her 30 slots in round 1, member B sells out all her 30 slots by round 4 whilst member C sells out all her 4 slots in round 2. Under this system, from an initial allocation perspective, all three members would earn the same points, i.e. 1.00. It would appear that the SSF would help to cover this with respect to future events for those with originally lower number of slots (such that member C would get less than 1.1, assuming she doesn't sell out) but A and B, from the original e.g., would still be treated the same, even though A is implicitly the more popular. 
• A counterpoint to this, I would suspect, is that because the purpose of the pennant race is to award points on a teams basis. Therefore, the timing would be irrelevant - total sales would then be the more accurate variable to use. From a practical viewpoint, this makes sense - if member A and member C are on the same team, it doesn't really matter how many tickets they have to sell in total: as long as they sell out, the team benefits. 
• I find the SSF formulae itself a little perplexing. Let's use the example that AKS has given, i.e. assuming member A had received 0.8 pts and following that, she sold more tickets by an SSF of 1.5 (150/100), then the subsequent points allocated is 1.2pt. Yet if we compare this to member X who consistently sold 100% of her tickets at every event with the max. available tickets possible, then the maximum pts. she get is 1.1 for the next round, 1.21 afterward and so forth (1 x 1.1^n). Which means despite selling more than member A, X will only receive less pts credited to her team than A. Unless I've misinterpreted the methodology, that presents a bit of a problem.
• To draw a concept from auditing, there's also no indication as to the sales cutoff given by AKS: is it at the closure of each round, or at the point of invoicing? If it's the latter, then no issues. The only logical implication of this is now there is less of an incentive for false orders. If it's the former however, then there is the risk of false orders will heavily influence the statistics, misstating the amount of points to be allocated. 
• From a practical perspective, the good thing about it is that assuming I've understood the methodology correctly, it would be those up-and-coming members with low number of slots that stand to gain the most benefit for the team as these members' tickets would easily sell out more than those who have more slots allocated, as they will simply get 1 pt - the same as those with 30 slots and sells them all out. Even better if you're holding a concurrency; however, as mentioned previously, the effect of concurrencies wouldn't have that large of an effect overall.
• The logical strategy would be to boost up members' sellout ratios by maxing out all of their allocated slots through increasing the number of tickets sold. Once again, I reiterate this is not a new concept for all as many have been doing this since handshake events started, but the incentive to do so has now been quantified. 

3) Elections

• The methodology employed here is actually one of the easier ones to understand. However, as the points allocated to each member here are few, and that this is entirely based off one event, the effect of election ranks won't count for much. 
• As has been explained by other fans, this is essentially a modified version of the D'Hondt method. Even if one doesn't understand how that works, it's a very simple algorithm - keep dividing the total number of votes until you hit a "cap", from which you start allocating points on that basis. Keep doing that for all 80 members and you'll get the desired points allocation for everyone.
• Strictly speaking, there's nothing wrong with the methodology itself. However, the broader implication of this is that it may introduce an element of vote-rigging in order to maximise the number of points allocation the higher ranked members can get. That is to say, while it is unlikely those lower ranked may have the possibility of their votes artificially capped (as there really is no control over voting patterns save for a mass vote rigging operation), fans may wish to increase the voting gap between those on lower ranks and those on higher ranks in order to maximise the number of "divisors" before they hit the threshold. 
• Take the official example given. Akarin, the lowest ranked in the sample, is on 43k votes, at which point the threshold is determined. Those that want to increase Sasshi's point allocation by one would necessitate an increase of 22,338* votes - a 15% increase required. 
• In fact, there are two options: spread more votes across more members of the same team, or maximise the votes on a few members in that same team (mainly the more popular ones). Both of these options have been tried before - all one needs is sufficient enough capital and/or people with the same mindset to help out. 
• Once again, there is nothing new with the broader practical implications - maximising points by purchasing more votes is an age old concept. But as I have stressed at the start, there is no real incentive as this is a one-off event unlikely to contribute significantly to the overall team tally**. 

*: If we want Sasshi to get 4 pts allocated, then it is calculated as Akarin's votes of 43,252 x 4 = 173,008, less Sasshi's actual result of 150,670, giving you a required difference of 22,338. The same logic can apply if we increase the sample to all 64 ranks (example as shown here) or simply increasing the desired point allocation to 5, and so forth. Of course, if Akarin increased her votes by 1% on last year, then the required difference would undoubtedly be greater. 

**: To take a hypothetical example, suppose we desire 100 points allocated to Sasshi. Based on prior year vote counts, and using the same methodology, this would necessitate a whopping 4,325,200 votes in total, or an additional 4,169,130 votes above last year's result. That amounts to an extra 4,169,130,000 yen (or USD$40.8 million) in spending on votes alone. As a comparison, using a rough estimate of approx. 6 shows on average per month per team, that would amount to 84 standard points alone during the applicable period of Apr-Oct., plus (assuming at least 50% of those shows exceeded the BAR) an additional 105 points, giving you a grand total of 189 points. Whilst an over 50% influence (100/189 = 53%) in the total points result is sizeable, the costs involved would be far too high at 41.69mil yen/pt; one would be better off maximising the AAR across all shows where the rate is 600 yen/pt.