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50/50 Voting What Does It Mean?

(First published 2008-09-17, last modified 2008-09-18)

1. Introduction
2. Explanation of two 50/50 voting formats
3. Hypothetical example
4. Simulation: traditional format
5. Simulation: Borda count
6. Modifications of Borda count
7. Simulation: Truncated Borda count
8. Conclusions


1. Introduction

For years many fans of the Eurovision Song Contest have asked for a voting system where 50 % of the voting input comes from televoting and the other 50 % from national juries. Whereas this concept is seemingly simple, there are many ways to implement it and most suggestions have lacked details. Details are important because different implementations may lead to surprisingly different results. This article demonstrates and analyses these differences.

EBU has already announced a mix of televoting and juries will indeed be used in the final of Eurovision Song Contest 2009 which is my main motive to lay out my concerns now. No details about the system to be used have yet been confirmed, not even whether juries and televoting will carry an equal weight. Even if the weighting is different, however, points made in this article remain true to the appropriate extent.


2. Explanation of two 50/50 voting formats

Whenever the question of details is discussed in the fansites, the most typical conception of how 50/50 voting should be done is:

  1. Convert top 10 of jury voting and top 10 of televoting separately into the 'douze points' system (12 for the best, 10 for the second best, then 8, 7, 6, 5, 4, 3, 2 and 1)
  2. Sum up the points.
  3. Convert the new ranking into 'douze points' format again. These points are given out.
I call this format the traditional format from here on. Ties are common in this format. The tie-breaker could be anything, none is inherently "better" than others. In this article I've assumed televoting takes precedence in case of ties.

In my opinion, one system is even more obvious: Borda count. In Borda count the songs are compared by the sum of their placings in televoting and jury voting. If a song finishes 5th in televoting and 11th in jury voting (5+11 = 16), it's considered better than a song that finishes 3rd in televoting and 18th in jury voting (3+18 = 21).

The two formats have similarities because both only use ranking information (i.e. the actual numbers of televotes and the grades the jurors give are discarded) in a fairly straightforward fashion. Their differences are that Borda count uses complete ranking instead of top 10 only and doesn't emphasize the top three by using a greater points difference between consecutive placings there.


3. Hypothetical example

An example is the easiest way to show how the two voting formats work. Few back-up jury and televoting results have been published in their entirety, that is, with results outside top 10 included. I don't know a single case where we'd know the full ranking of both for the same country in the same contest. Thus, I went for a hypothetical example.

I use votes given by France in the final of 2007 as my example. I use their real televoting result for the top 10 placings. The lower televoting placings and the entire jury ranking is made up to fit the purpose.

The rankings are:
Televoting:Jury:
1. Turkey1. Serbia
2. Armenia2. Finland
3. Serbia3. Moldova
4. Romania4. Georgia
5. Spain5. Bulgaria
6. Bulgaria6. Slovenia
7. Ukraine7. Bosnia & H.
8. Greece8. Hungary
9. Russia9. Germany
10. Bosnia & H.10. Greece
11. Hungary...
12. Germany12. Ukraine
13. Moldova16. Russia
14. Georgia17. Spain
...19. Romania
18. Slovenia20. Armenia
19. Finland23. Turkey

If we combine these using the traditional format, televoting taking precedence in case of ties, we get the following result.
CountryTJSumPts
Serbia8122012
Turkey12-1210
Bulgaria56118
Armenia10-107
Finland-10106
Moldova-885
Romania7-74
Georgia-773
Spain6-62
Bosnia & H.1451
Slovenia-55-
Ukraine4-4-
Greece314-
Hungary-33-
Russia2-2-
Germany-22-

But if we do it using Borda count, using televoting results to break the ties here too, the result is different.
CountryTJSumPts
Serbia31412
Bulgaria651110
Moldova133168
Bosnia & H.107177
Greece810186
Georgia144185
Ukraine712194
Hungary118193
Germany129212
Finland192211
Armenia22022-
Spain51722-
Romania41923-
Turkey12324-
Slovenia18624-
Russia91625-

Here we have two wildly different results even though both are calculated from the same (imaginary) sets of votes by a system that is clearly 50/50. The choice between the two formats makes a difference of three points or more for no fewer than nine countries.

If the opinions of televoters and juries are sufficiently different, as they are here, the traditional format essentially rewards songs in the top 5 or 6 of either ranking and the lower ranks have little relevance. Turkey and Armenia, top two of the French televoting results largely because of their diaspora, would almost retain their placings in the combined vote even though they have no jury support. If the idea of the voting revision is to do something to the diaspora voting, the traditional format doesn't seem to be the way to go based on this example.

Borda count, on the other hand, throws Turkey and Armenia all the way out of the top 10. Instead of requiring a very high placing in either list, Borda count rewards songs that score decently with both televoters and juries. A disastrous placing in either eliminates the chance of scoring high points. This has an obvious downside too, however. Whereas Borda count may be a better system assuming jurors are honest, it's prone to tactical voting. By ranking a song last jurors can make it nearly certain it's outside the top 10 in the combined ranking even if it wins televoting. That said, tactical voting always was a possibility with juries (more so than with televoters who can't realistically coordinate their votes to get a desired effect).

Of course, this example was hypothetical. It may be argued it's carefully constructed to produce seemingly absurd outcomes and that anything this chaotic is unlikely to occur in reality. I admit I constructed this example specifically to point out the peculiarities of different 50/50 systems. I have no idea how often results look like this in reality or how often juries and televoters find a better agreement which leads to a "neater" combined outcome probably expected by the proponents of 50/50 voting. The point is: nobody knows. Few jury votes have been made public at all which means no one without inside information knows what the situation have been this far. Moreover, because the way jury voting is conducted will also be revised, the state of affairs could be different next year.

Luckily, we do not have to rely on one hypothetical example. With few assumptions we can simulate the performance of the two 50/50 voting formats and see whether these concerns are relevant.


4. Simulation: traditional format

For the simulations, let us assume:

  • there are 24 songs to be voted for (There are 25 finalists in 2009 but finalist countries are in a majority and can't vote for themselves);
  • there are no ties within either televoting results or jury results;
  • televoting takes precedence in case of ties in the combined result; and
  • results of televoting and jury voting have no correlation.

    The first three assumptions are unproblematic. The third is more questionable. In normal circumstances we could expect jury and televoting results to correlate positively, possibly quite strongly, even. In current circumstances, however, I wouldn't be surprised if there was a negative correlation in many countries. Juries might very well vote tactically against countries which are bound to get a hefty amount of televotes from their diaspora anyway.

    Even if there was no tactical voting, zero correlation is not all that unlikely. In many countries the televoting result is heavily affected by diaspora, so much so that it proves little about the preferences of the native population. If the televoting result has little to do with songs and jurors vote "fairly" based on songs alone, near-zero correlation is what we should logically expect.

    My justification for the assumption isn't some divine knowledge of how the results will correlate, however. The justification is far more simple: the case of clear positive correlation is uninteresting. In such a case all 50/50 voting systems would perform quite similarly. Furthermore, they would show little difference to the plain televoting result (which would beg the question why use juries at all?). What's interesting is how the voting formats will perform with a near-zero correlation between the sources of votes or with a negative correlation.

    With these assumptions I simulated 100 000 combined votings under the traditional format. Here are the results with each row corresponding to a placing in televoting and each column corresponding to an amount of points in the combined vote. Each cell contains the probability of that combination. For example, under these assumptions, the televoting winner has a probability of 39,8 % to get the douze of the combined vote. The probability of the televoting winner getting exactly six points is as low as 0,6 %. Rows 1124 are virtually identical (and would be exactly identical if the simulation was run infinitely many times) because the traditional format ignores placings outside top 10.

    Some further explanations:

  • 0,0 % means the combination did occur in the simulation but less than 0,05 % of the time.
  • 0 means the combination didn't occur in the simulation but theoretically could.
  • - means the combination is mathematically impossible under these rules.
  • Expected votes means the expected value of combined votes.
  • Expected gain means the expected amount of points gained by gaining a placing in the televoting ranking.

    Percentages above 10 % are highlighted.

    Traditional format televoters' perspective
     Combined votesExpected
    votes
    Expected
    gain
    1210876543210
    T
    e
    l
    e
    v
    o
    t
    i
    n
    g

    r
    a
    n
    k
    139,8 %32,8 %20,4 %6,3 %0,8 %0,0 %0,0 %0---10,171,91
    220,0 %11,6 %18,7 %27,7 %17,2 %4,4 %0,5 %0,0 %00-8,261,60
    312,2 %10,1 %6,5 %7,0 %19,3 %26,5 %14,7 %3,3 %0,3 %0,0 %06,661,17
    49,3 %8,7 %6,1 %5,0 %5,4 %11,2 %24,2 %21,7 %7,3 %1,0 %0,0 %5,491,18
    56,9 %7,3 %5,7 %5,0 %4,8 %4,7 %6,6 %17,3 %25,4 %13,5 %2,8 %4,311,15
    64,8 %5,9 %5,0 %4,8 %4,7 %4,4 %3,6 %4,7 %12,2 %23,4 %26,5 %3,160,78
    73,1 %4,4 %4,1 %4,6 %4,9 %4,3 %3,7 %3,2 %3,8 %9,0 %54,9 %2,380,45
    82,3 %3,6 %2,9 %3,5 %4,4 %4,5 %3,5 %3,1 %3,1 %3,6 %65,6 %1,930,39
    91,1 %2,7 %2,8 %2,6 %3,4 %4,2 %3,6 %3,1 %2,9 %3,1 %70,5 %1,540,31
    100,7 %1,7 %2,2 %2,4 %2,5 %3,3 %3,4 %3,1 %3,1 %3,0 %74,6 %1,230,31
    11-0,8 %1,8 %2,2 %2,3 %2,4 %2,7 %2,8 %3,0 %3,0 %79,0 %0,92 
    12-0,8 %1,9 %2,3 %2,3 %2,3 %2,6 %2,8 %3,0 %3,1 %78,8 %0,93 
    13-0,9 %1,7 %2,3 %2,3 %2,3 %2,6 %3,0 %3,0 %3,1 %78,8 %0,92 
    14-0,8 %1,8 %2,2 %2,3 %2,4 %2,5 %2,9 %3,0 %3,0 %79,1 %0,91 
    15-0,8 %1,8 %2,2 %2,3 %2,3 %2,6 %2,9 %2,9 %3,2 %79,0 %0,92 
    16-0,8 %1,8 %2,2 %2,3 %2,3 %2,6 %3,0 %2,9 %3,1 %78,9 %0,92 
    17-0,8 %1,9 %2,2 %2,3 %2,3 %2,6 %2,8 %3,0 %3,1 %79,0 %0,92 
    18-0,8 %1,9 %2,2 %2,3 %2,3 %2,6 %2,9 %2,9 %3,1 %79,0 %0,92 
    19-0,8 %1,8 %2,2 %2,4 %2,3 %2,5 %2,9 %3,0 %3,0 %79,2 %0,91 
    20-0,8 %1,7 %2,2 %2,4 %2,4 %2,6 %2,9 %3,1 %3,0 %78,9 %0,92 
    21-0,8 %1,8 %2,2 %2,3 %2,3 %2,6 %2,9 %3,0 %3,1 %78,9 %0,92 
    22-0,8 %1,8 %2,2 %2,4 %2,3 %2,5 %2,9 %3,0 %3,1 %78,9 %0,92 
    23-0,8 %1,8 %2,2 %2,4 %2,3 %2,6 %2,9 %3,1 %3,3 %78,7 %0,92 
    24-0,8 %1,8 %2,2 %2,3 %2,3 %2,6 %3,0 %3,0 %3,1 %78,8 %0,92 

    Now, now, where to start. The table looks pretty much as I expected it to look like before running the simulation. The televoting favourites retain their high placings with a high probability. The intuitively understandable idea that the first and second places of televoting are worth about six and five points (i.e. half of their original value) respectively if the songs have no jury support is clearly wrong. With the assumption of no correlation any song received nul points from the jury (placing out of top 10) 58 % (14/24) of the time. Thus, in such cases the televoting winner usually got 8 or 10 points in the combined ranking; the televoting runner-up, 6 to 8 points.

    The two right-hand columns hold the most interesting piece of information here. As expected, televoting placings in lower top 10 are worth little. (If the idea was to make them more important because that's where the non-diaspora favourites are expected to be, this is a severe problem). Gaining a single placing in the televoting ranking on those positions is theoretically worth 0,3 to 0,4 points, less than half of what it is now.

    Gaining placings in the top 5 of televoting, however, whoa! This came out of the blue. By leaping from sixth place to third (or a smaller number of placings in this range) in the televoting results you gain, theoretically, more points than you do under full televoting! Each placing gained is worth more than one full point! This is why I highlighted this part of the table.

    First I thought this had to be a mistake because it didn't make sense to me (and I thought I had a pretty good intuitive grasp of things like these). After checking and re-checking and finding no mistakes, I had to accept this as a correct result and find an explanation for it. Luckily, the explanation is actually rather simple. Ties are very common in the traditional format with assumption of no correlation. Results much like the hypothetical example I constructed earlier are common. By gaining a point in televoting, you are likely to gain one position in the combined ranking but could very well gain two or three. This is true for lower positions too but the risk of being outside the top 10 of the combined list is so high that the gain is smaller.

    Another funny-looking and also highlighted aspect of the table is a related thing: the high 'peaks' in the first six rows of the table. There is an explanation to the peaks. As I mentioned before, each song received nul points from the jury for 58 % of the time. The placings they usually reclaim if they get no jury support is where the peak occurs.

    The same table from the juries' perspective looks quite similar but more subdued. The aforementioned 'peaks' are also lower for any given position. With televoting taking precedence in case of ties, juries actually have less than 50 % of the voting power. Of course, the precedence could be given to juries in which case these two tables are interchangeable. Other tie-breakers are also possible.

    Traditional format juries' perspective
     Combined votesExpected
    votes
    Expected
    gain
    1210876543210
    J
    u
    r
    y

    r
    a
    n
    k
    126,4 %22,1 %27,8 %18,1 %5,0 %0,6 %0,0 %00--9,201,56
    218,9 %10,7 %7,3 %17,7 %26,0 %15,4 %3,7 %0,3 %0,0 %007,651,56
    312,5 %9,3 %6,2 %5,0 %7,5 %18,7 %24,2 %13,3 %3,1 %0,3 %0,0 %6,081,13
    49,7 %8,3 %5,7 %4,7 %4,7 %5,5 %11,9 %23,2 %19,2 %6,2 %0,9 %4,951,11
    57,6 %6,9 %5,3 %4,8 %4,6 %4,4 %4,0 %7,7 %18,5 %22,9 %13,4 %3,840,93
    65,6 %5,6 %4,7 %4,6 %4,6 %4,3 %3,5 %3,2 %5,5 %14,5 %44,0 %2,910,54
    73,9 %4,5 %4,0 %4,5 %4,5 %4,3 %3,5 %2,9 %3,0 %4,6 %60,4 %2,370,37
    83,1 %3,6 %2,7 %3,7 %4,4 %4,2 %3,3 %3,0 %2,9 %3,0 %66,0 %1,990,36
    91,6 %3,0 %2,8 %2,6 %3,7 %4,0 %3,3 %2,9 %3,0 %3,0 %70,0 %1,630,33
    101,1 %1,9 %2,3 %2,3 %2,6 %3,4 %3,2 %2,9 %2,9 %3,1 %74,2 %1,300,15
    110,7 %1,7 %2,3 %2,3 %2,3 %2,6 %2,8 %2,9 %3,0 %3,0 %76,4 %1,16 
    120,7 %1,7 %2,2 %2,3 %2,4 %2,5 %2,8 %2,9 %3,1 %3,0 %76,5 %1,14 
    130,7 %1,7 %2,2 %2,3 %2,3 %2,5 %2,8 %3,0 %3,0 %3,0 %76,5 %1,15 
    140,7 %1,7 %2,2 %2,3 %2,3 %2,5 %2,9 %2,9 %3,0 %3,0 %76,4 %1,15 
    150,7 %1,7 %2,1 %2,3 %2,3 %2,5 %2,8 %2,9 %3,0 %3,1 %76,6 %1,13 
    160,7 %1,7 %2,2 %2,2 %2,3 %2,5 %2,8 %2,9 %3,0 %3,1 %76,6 %1,14 
    170,7 %1,8 %2,3 %2,3 %2,3 %2,5 %2,8 %2,9 %2,9 %3,0 %76,4 %1,16 
    180,6 %1,8 %2,2 %2,3 %2,3 %2,5 %2,8 %2,9 %3,0 %3,0 %76,5 %1,14 
    190,7 %1,8 %2,2 %2,3 %2,3 %2,5 %2,8 %2,9 %3,0 %3,0 %76,7 %1,14 
    200,7 %1,7 %2,3 %2,3 %2,3 %2,5 %2,8 %2,9 %3,1 %3,1 %76,3 %1,16 
    210,7 %1,8 %2,2 %2,3 %2,3 %2,6 %2,7 %2,8 %2,9 %3,1 %76,5 %1,15 
    220,7 %1,7 %2,2 %2,3 %2,3 %2,6 %2,9 %2,9 %2,9 %3,0 %76,5 %1,15 
    230,7 %1,6 %2,3 %2,3 %2,4 %2,6 %2,8 %2,9 %2,9 %3,0 %76,5 %1,15 
    240,7 %1,7 %2,2 %2,3 %2,4 %2,4 %2,8 %3,0 %2,9 %3,0 %76,7 %1,14 


    5. Simulation: Borda count

    Onto Borda count then. The assumptions remain the same, the format of the table is the same and this simulation was run 100 000 times too. Without further ado, let's jump to the results.

    Borda count televoters' perspective
     Combined votesExpected
    votes
    Expected
    gain
    1210876543210
    T
    e
    l
    e
    v
    o
    t
    i
    n
    g

    r
    a
    n
    k
    127,0 %12,5 %9,6 %7,8 %7,1 %6,3 %5,6 %5,4 %5,1 %4,6 %9,0 %7,090,55
    221,6 %13,3 %9,6 %8,1 %6,9 %6,2 %5,8 %5,3 %4,9 %4,8 %13,3 %6,530,54
    316,6 %13,4 %10,0 %8,1 %6,9 %6,4 %6,0 %5,3 %5,1 %4,6 %17,6 %5,990,52
    412,4 %13,1 %9,9 %8,3 %7,1 %6,4 %5,9 %5,5 %4,9 %4,7 %21,8 %5,460,53
    58,7 %12,2 %10,0 %8,2 %7,1 %6,4 %5,8 %5,4 %5,1 %4,7 %26,3 %4,930,53
    65,7 %10,4 %9,9 %8,3 %7,2 %6,4 %5,9 %5,4 %5,1 %4,7 %30,9 %4,400,47
    73,6 %8,5 %9,6 %8,3 %7,3 %6,4 %5,8 %5,4 %5,1 %4,9 %35,1 %3,940,44
    82,1 %6,5 %8,6 %8,3 %7,4 %6,5 %6,0 %5,4 %5,2 %4,9 %39,1 %3,500,47
    91,1 %4,4 %7,3 %8,0 %7,3 %6,4 %6,1 %5,4 %5,1 %4,8 %44,2 %3,020,38
    100,5 %2,8 %5,8 %7,4 %7,3 %6,6 %5,8 %5,5 %5,1 %4,7 %48,4 %2,640,37
    110,2 %1,6 %4,2 %6,4 %6,8 %6,6 %5,8 %5,6 %5,1 %4,9 %52,7 %2,270,35
    120,1 %0,8 %2,7 %5,0 %6,3 %6,5 %6,0 %5,6 %5,1 %4,8 %57,1 %1,910,32
    130,0 %0,3 %1,5 %3,6 %5,4 %6,1 %6,0 %5,4 %5,0 %4,9 %61,7 %1,590,27
    140,0 %0,2 %0,8 %2,2 %4,2 %5,7 %5,8 %5,6 %5,3 %4,9 %65,4 %1,330,28
    150,0 %0,0 %0,3 %1,2 %3,0 %4,5 %5,4 %5,5 %5,2 %5,0 %70,0 %1,040,25
    1600,0 %0,1 %0,5 %1,6 %3,3 %4,6 %5,3 %5,3 %4,9 %74,5 %0,800,21
    1700,0 %0,0 %0,2 %0,7 %1,9 %3,6 %4,7 %5,0 %4,9 %78,9 %0,590,17
    18000,0 %0,0 %0,3 %1,0 %2,3 %3,8 %4,6 %4,8 %83,2 %0,410,15
    19000,0 %0,0 %0,1 %0,4 %1,2 %2,5 %3,8 %4,5 %87,5 %0,270,11
    2000000,0 %0,1 %0,4 %1,4 %2,6 %3,8 %91,6 %0,160,08
    2100000,0 %0,0 %0,1 %0,5 %1,5 %2,8 %95,1 %0,080,05
    220000000,0 %0,1 %0,6 %1,6 %97,7 %0,030,02
    230000000,0 %0,0 %0,1 %0,6 %99,3 %0,010,01
    24-0000000,0 %0,0 %0,2 %99,8 %0,00 

    Well, well. Here we have a far more balanced outlook than in the traditional format. A high placing in the televoting is no guarantee of high points (or, indeed, any points) and ranks outside top 10 have a real meaning, too. A placing gained in the televoting results helps in a straightforward fashion; the oddities of the traditional format aren't present.

    I'd like to be able to find something to be surprised about or something to criticise in this table but I'm not. It looks beautiful and reasonable.

    Borda count isn't that different from the juries' perspective but let's show that table, too, for sake of completeness.

    Borda count juries' perspective
     Combined votesExpected
    votes
    Expected
    gain
    1210876543210
    J
    u
    r
    y

    r
    a
    n
    k
    122,7 %12,7 %9,5 %8,1 %6,9 %6,3 %5,9 %5,4 %5,0 %4,6 %13,0 %6,590,45
    219,4 %12,3 %9,5 %7,9 %6,9 %6,2 %5,7 %5,4 %4,9 %4,9 %16,9 %6,140,44
    316,2 %12,0 %9,4 %7,7 %6,8 %6,2 %5,8 %5,3 %4,9 %4,7 %20,9 %5,700,46
    413,1 %11,4 %9,2 %7,8 %6,7 %6,2 %5,7 %5,2 %4,9 %4,7 %25,1 %5,230,45
    59,9 %11,0 %9,0 %7,6 %6,9 %6,1 %5,6 %5,2 %5,1 %4,7 %28,9 %4,790,45
    67,0 %10,0 %9,0 %7,6 %6,7 %6,2 %5,7 %5,3 %5,0 %4,8 %32,8 %4,340,42
    74,9 %8,8 %8,6 %7,6 %6,8 %6,1 %5,6 %5,1 %4,9 %4,6 %37,1 %3,920,40
    83,1 %7,4 %8,2 %7,4 %6,7 %6,1 %5,6 %5,2 %5,0 %4,7 %40,7 %3,510,39
    91,8 %5,6 %7,6 %7,5 %6,6 %6,0 %5,5 %5,2 %4,9 %4,5 %44,8 %3,120,39
    101,0 %3,8 %6,3 %7,1 %6,6 %6,1 %5,5 %5,2 %5,0 %4,6 %48,7 %2,730,36
    110,5 %2,4 %5,2 %6,5 %6,5 %6,0 %5,5 %5,2 %4,9 %4,6 %52,8 %2,380,33
    120,2 %1,4 %3,6 %5,7 %6,3 %5,9 %5,5 %5,2 %4,8 %4,7 %56,7 %2,050,30
    130,1 %0,7 %2,4 %4,4 %5,8 %5,9 %5,6 %5,2 %4,8 %4,6 %60,5 %1,740,29
    140,0 %0,3 %1,3 %3,2 %5,0 %5,6 %5,4 %5,1 %4,8 %4,6 %64,7 %1,460,27
    150,0 %0,1 %0,6 %2,1 %3,7 %5,0 %5,3 %5,1 %4,8 %4,6 %68,7 %1,190,23
    160,0 %0,0 %0,3 %1,1 %2,6 %4,1 %5,0 %5,0 %4,8 %4,6 %72,5 %0,950,22
    170,0 %0,0 %0,1 %0,5 %1,5 %3,0 %4,3 %4,6 %4,8 %4,4 %76,8 %0,730,19
    1800,0 %0,0 %0,2 %0,7 %1,9 %3,3 %4,3 %4,7 %4,5 %80,5 %0,550,16
    19000,0 %0,0 %0,2 %0,9 %2,2 %3,4 %4,2 %4,4 %84,6 %0,380,14
    200000,0 %0,1 %0,3 %1,1 %2,4 %3,5 %4,1 %88,6 %0,240,10
    210000,0 %0,0 %0,1 %0,4 %1,3 %2,5 %3,5 %92,1 %0,150,07
    220000,0 %0,0 %0,0 %0,1 %0,6 %1,4 %2,6 %95,3 %0,080,05
    23000000,0 %0,0 %0,2 %0,6 %1,4 %97,8 %0,030,02
    240000000,0 %0,0 %0,1 %0,6 %99,3 %0,01 


    6. Modifications of Borda count

    After I first published this article, it was perceived by many that the chances of the televoting winner getting no points at all in Borda count is too high. Whereas I believe this is a realistic risk mostly in cases where the televoting result is heavily affected by diaspora voting, it's nevertheless a concern that could be addressed better. There is a middle ground between traditional Eurovision scoring system and pure Borda count.

    Chances of televoting winner missing out on top 10 in the combined vote can be greatly decreased or altogether eliminated by any of the following modifications to pure Borda count.

  • Truncation: Set a maximum amount of points a song can receive from either ranking. Ranks worse than that are replaced by the maximum. That is, if the maximum is set at 19, a song that finishes 1st in televoting and 23rd in jury voting will be placed as if its Borda count was 20 (1 + 19) rather than 24 (1 + 23).
  • Ad hoc rule: Winner of the televoting gets one point if it doesn't reach the top 9 of Borda count (could also be minimums of 3, 2 and 1 for songs finishing first, second and third respectively in televoting).
  • Discontinuity: More general version of the previous; the last point is (or the last two or three point-scoring positions are) given out based on televoting alone. If the televoting winner fails to reach top 9, this works exactly like the 'ad hoc rule'. However, if eg. all the top 5 songs of the televoting reach the top 9 in Borda count but the 6th-placed song doesn't, the 6th-placed song of the televoting gets the last point no matter where the jury has rated it. This is partially modelled after the current semifinal qualification system where the first nine qualifiers are decided by televoting but the last one by juries.

    I strongly dislike the 'discontinuity' format. It's too arbitrary and potentially confusing and produces an artificial gap between two songs that might have finished close to each other with pure Borda count. The artificial gap is even greater if more than one point-scoring position is decided by televoting only. The 'ad hoc rule' is somewhat more acceptable because it would only have an impact in a small number of cases, at least if only the winner of televoting has this kind of a safety net.

    Truncated Borda count is a bit arbitrary, too, but not much more so than the traditional format. (Traditional Eurovision scoring system is a truncated Borda count, too, with the extra modification that gaps between the first three placings are doubled.) If Borda count is truncated to separate the first 20 placings, the probability of televoting winner getting no points decreases from 9,0 % to 1,5 %. If only 19 placings are separated, the probability drops to slightly below 0,5 %. With 43 participants (as in 2008), we should expect this to happen roughly once every five years or once every 200 years in any particular country.

    If even that tiny chance is unacceptable, truncated Borda count more or less has to be coupled with the 'ad hoc rule'. Truncated Borda count guarantees a top 10 place for the televoting winner only if 11 placings or fewer are separated. That, of course, would be reinventing the traditional Eurovision scoring format.

    It is also possible to truncate the jury ranking only. That way a televoting winner with no jury support would almost certainly score some points but the opposite wouldn't be true; a jury winner finishing last in televotes would almost certainly miss out on points. If such an effect is wanted, it can be done. Strictly speaking, however, that would not be a 50/50 system.


    7. Simulation: Truncated Borda count

    The simulation for truncated Borda count was run similarly to others. Truncation was done to a maximum of 19 points. That is, all ranks from 19th to 24th counted as a 19th place.

    The probability of televoting winner or runner-up not scoring goes down to 0,5 % and 3,0 % respectively. The concept of peaks, very evident in the traditional format, is present here again and shows where the top-ranked songs of televoting are likely to end up if they are disliked by the jury. With this method a televoting winner with no jury support would usually get 35 points; the runner-up, 24. This looks like a fair middle ground between a realistic chance of getting no points at all in pure Borda count and having 810 (winner) or 68 points (runner-up) virtually guaranteed in the traditional format.

    Truncated Borda count televoters' perspective
     Combined votesExpected
    votes
    Expected
    gain
    1210876543210
    T
    e
    l
    e
    v
    o
    t
    i
    n
    g

    r
    a
    n
    k
    126,9 %12,8 %9,5 %8,1 %7,8 %8,8 %10,0 %8,6 %5,1 %1,9 %0,5 %7,520,73
    221,4 %13,2 %9,8 %8,1 %6,9 %6,5 %7,6 %9,0 %8,7 %5,7 %3,0 %6,790,70
    316,8 %13,3 %9,9 %8,0 %7,0 %6,1 %5,5 %6,4 %8,1 %8,8 %10,0 %6,090,67
    412,4 %13,1 %9,8 %8,1 %7,0 %6,0 %5,2 %4,8 %5,7 %7,8 %20,1 %5,420,58
    58,8 %11,9 %10,0 %8,4 %7,0 %6,1 %5,2 %4,4 %4,2 %5,3 %28,6 %4,840,51
    65,9 %10,5 %9,9 %8,3 %7,1 %6,2 %5,2 %4,4 %4,1 %4,1 %34,2 %4,340,52
    73,6 %8,5 %9,4 %8,3 %7,1 %6,3 %5,1 %4,4 %4,0 %3,9 %39,3 %3,820,43
    82,2 %6,4 %8,8 %8,2 %7,2 %6,1 %5,4 %4,5 %4,0 %3,8 %43,3 %3,390,44
    91,1 %4,5 %7,4 %8,0 %7,2 %6,2 %5,4 %4,5 %4,1 %3,9 %47,8 %2,950,39
    100,6 %2,9 %5,9 %7,3 %7,2 %6,3 %5,5 %4,6 %3,9 %3,8 %52,2 %2,560,38
    110,2 %1,6 %4,2 %6,5 %6,9 %6,3 %5,4 %4,6 %4,1 %3,8 %56,4 %2,180,37
    120,1 %0,8 %2,6 %5,1 %6,4 %6,2 %5,4 %4,7 %4,1 %3,8 %61,0 %1,810,31
    130,0 %0,3 %1,5 %3,5 %5,4 %6,0 %5,4 %4,6 %4,1 %3,9 %65,2 %1,500,29
    140,0 %0,1 %0,7 %2,2 %4,3 %5,3 %5,2 %4,7 %4,1 %3,9 %69,6 %1,210,27
    150,0 %0,0 %0,3 %1,2 %2,8 %4,4 %4,9 %4,5 %4,1 %3,8 %74,0 %0,950,24
    1600,0 %0,1 %0,5 %1,5 %3,1 %4,3 %4,4 %4,0 %3,8 %78,3 %0,710,21
    1700,0 %0,0 %0,2 %0,7 %1,9 %3,0 %3,8 %3,9 %3,8 %82,7 %0,500,17
    18000,0 %0,0 %0,2 %0,8 %1,9 %2,9 %3,5 %3,7 %86,8 %0,330,15
    19-000,0 %0,0 %0,2 %0,7 %1,7 %2,7 %3,5 %91,1 %0,18 
    20-000,0 %0,0 %0,2 %0,8 %1,7 %2,7 %3,5 %91,2 %0,18 
    21-000,0 %0,0 %0,2 %0,8 %1,7 %2,8 %3,3 %91,2 %0,18 
    22-000,0 %0,0 %0,2 %0,8 %1,7 %2,7 %3,4 %91,1 %0,18 
    23-000,0 %0,0 %0,2 %0,7 %1,7 %2,7 %3,4 %91,2 %0,18 
    24-000,0 %0,0 %0,2 %0,8 %1,7 %2,7 %3,3 %91,2 %0,18 

    Truncated Borda count juries' perspective
     Combined votesExpected
    votes
    Expected
    gain
    1210876543210
    J
    u
    r
    y

    r
    a
    n
    k
    122,8 %12,6 %9,6 %8,0 %7,0 %7,1 %8,1 %9,5 %8,4 %4,7 %2,2 %6,930,62
    219,7 %12,3 %9,4 %7,9 %6,9 %6,1 %5,9 %6,9 %8,6 %8,3 %8,1 %6,310,60
    316,1 %11,9 %9,4 %7,9 %6,8 %6,1 %5,4 %5,1 %6,0 %7,9 %17,4 %5,710,51
    413,0 %11,7 %9,2 %7,7 %6,8 %6,0 %5,2 %4,4 %4,5 %5,6 %26,0 %5,200,51
    59,7 %10,9 %9,0 %7,7 %6,8 %5,9 %5,2 %4,7 %4,0 %4,2 %31,9 %4,690,43
    67,2 %10,0 %9,0 %7,6 %6,6 %5,9 %5,1 %4,3 %3,9 %3,8 %36,5 %4,260,43
    74,9 %8,9 %8,6 %7,5 %6,7 %5,9 %5,1 %4,4 %3,8 %3,6 %40,7 %3,830,43
    83,0 %7,3 %8,2 %7,4 %6,7 %5,8 %5,1 %4,3 %3,8 %3,6 %44,8 %3,400,38
    91,9 %5,5 %7,4 %7,4 %6,6 %5,9 %5,0 %4,4 %3,8 %3,7 %48,4 %3,020,39
    100,9 %3,9 %6,4 %7,1 %6,6 %5,7 %4,9 %4,3 %3,7 %3,6 %52,8 %2,630,34
    110,5 %2,5 %5,1 %6,6 %6,3 %5,7 %5,0 %4,2 %3,8 %3,8 %56,4 %2,290,33
    120,2 %1,4 %3,7 %5,6 %6,3 %5,6 %4,9 %4,3 %3,9 %3,6 %60,4 %1,960,32
    130,1 %0,7 %2,5 %4,5 %5,6 %5,6 %4,9 %4,2 %3,8 %3,6 %64,5 %1,640,30
    140,0 %0,3 %1,4 %3,2 %4,9 %5,2 %4,8 %4,1 %3,7 %3,6 %68,8 %1,340,25
    150,0 %0,1 %0,7 %2,0 %3,8 %4,7 %4,7 %4,1 %3,6 %3,6 %72,6 %1,090,25
    160,0 %0,0 %0,3 %1,1 %2,4 %3,8 %4,2 %4,0 %3,7 %3,7 %76,7 %0,840,21
    1700,0 %0,1 %0,4 %1,4 %2,7 %3,7 %3,7 %3,6 %3,8 %80,6 %0,630,19
    1800,0 %0,0 %0,1 %0,6 %1,5 %2,5 %3,3 %3,6 %3,7 %84,6 %0,430,13
    19000,0 %0,0 %0,2 %0,7 %1,7 %2,7 %3,3 %3,6 %87,8 %0,30 
    20000,0 %0,0 %0,2 %0,8 %1,7 %2,6 %3,3 %3,6 %87,8 %0,30 
    21000,0 %0,0 %0,2 %0,8 %1,7 %2,5 %3,3 %3,6 %87,9 %0,30 
    22000,0 %0,0 %0,2 %0,8 %1,7 %2,7 %3,3 %3,7 %87,8 %0,30 
    23000,0 %0,0 %0,2 %0,7 %1,8 %2,5 %3,3 %3,6 %87,9 %0,30 
    24000,0 %0,0 %0,2 %0,8 %1,7 %2,7 %3,4 %3,6 %87,5 %0,31 


    8. Conclusions

    Of the two 50/50 systems compared in this article, Borda count is definitely more efficient in countering the biases of televoting. The traditional format emphasizes the topmost placings too heavily and essentially only rewards the top 5 or 6 jury choices and the top 5 or 6 televoters' choices (which are most likely to be immigrant-induced). Borda count utilizes both rankings in a more balanced way.

    Even though it won't be obvious to the casual viewer, Borda count actually gives "ordinary" televoters more power in countries where voting is most heavily affected by immigrants. Any placing change in the televoting list, no matter how low down, can potentially have an impact in the points the country eventually gives out.

    The balancedness of Borda count is also its potential pitfall. Measures will have to be taken by EBU to ensure tactical voting won't be rampant. This shouldn't be too difficult in the 21st century. There is no need for jurors to know each other or ever meet. If they don't, they can't coordinate their votes or choose a joint strategy. Peer pressure won't affect the results either.

    Even though I've only demonstrated two 50/50 systems, I find it difficult to believe I could have missed a system that's as simple as and better than Borda count. Thus, Borda count gets my full endorsement.

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