David Ding: Riddler Classic December 9, 2022

Speaking of “football,” the Riddler Football Playoff (RFP) consists of four teams. Each team is assigned a random real number between 0 and 1, representing the “quality” of the team. If team A has quality a and team B has quality b, then the probability that team A will defeat team B in a game is a/(a+b) (no ties).

In the semifinal games of the playoff, the team with the highest quality (the “1 seed”) plays the team with the lowest quality (the “4 seed”), while the other two teams play each other as well. The two teams that win their respective semifinal games then play each other in the final.

On average, what is the quality of the RFP champion?

function qualityChamp = getQuality()
    % Gets the quality of the champion

    % First, we get the seedings:
    seeds = rand(4, 1);
    arrSeeds = sort(seeds, 'descend');

    % Semifinal one
    probSf1 = arrSeeds(1)/(arrSeeds(1) + arrSeeds(4));
    if rand < probSf1
        qualF1 = arrSeeds(1);
    else
        qualF1 = arrSeeds(4);
    end

    % Semifinal two
    probSf2 = arrSeeds(2)/(arrSeeds(2) + arrSeeds(3));
    if rand < probSf2
        qualF2 = arrSeeds(2);
    else
        qualF2 = arrSeeds(3);
    end

    % Final
    probFinal = qualF1/(qualF1 + qualF2);
    if rand < probFinal
        qualityChamp = qualF1;
    else
        qualityChamp = qualF2;
    end
end

It is interesting to note that the CDF of the champion quality is convex, resulting in that the average quality, 0.674, is less than the median quality, which is about 0.71. This means that there will be, on average, more champions of "lower" quality than that of "higher" quality. Indeed, about 25% of the time, the champion will have quality of less than 0.5, which is the average of a team's quality. This is buoyed by two facts: 1) it is possible that the 1-seed is already of low quality, and 2) upsets with a lesser seed becoming the champion. With the tournament having two elimination rounds, upsets are not out of the norm.

If we map the "Riddler Football Playoff" to the "College Football Playoff (CFP)", and if we "normalize" the playoff seeds such that the third and fourth seeds have qualities at or below 0.5 (relative to the other teams in the playoff), then the historical trends actually reflect well with our Riddler case. Consider that since the inception of CFP, the one-seed have actually won just twice out of 8 times! We have the 4-seed winning twice (!), the 3-seed winning once, and the 2-seed winning three times. Now, let's evenly divide the four seeds onto the 0-1 continuum, such that the 4-seed has the quality 0.25, the 3-seed has the quality 0.5, the 2-seed 0.75 and the one seed with quality of 1, then the average quality of the champions from the historical data is: \(( 2 + 3 \times 0.75 + 0.5 + 2 \times 0.25) / 8 = 0.653\), which is not far from the average quality of champions in the Riddler case. Of course, in actuality, the College Football Playoff teams might have quality that are all very high, but this is just a fun hypothetical exercise.

Riddler Classic: December 9, 2022

Quality Champion