Rethinking the FIFA World Cup™ final draw

Appendix A: The suggested method is evenly distributed

Let us prove that our suggested procedure is evenly distributed, i.e., that under this procedure all the admissible outcomes are equally likely. Since Draw I and Draw II are independent, and follow the same rules, it is enough to prove that all the possible outcomes of Draw I are equally likely. Let d be an admissible outcome of Draw I and c denote its continental distribution. Let C denote the random continental distribution drawn at the first step, and D the final result of the random draw.

The first step of the procedure guarantees that ℙ(C=c)=1​/​NI. Besides, each d corresponds to exactly one c, say c(d), and each c allows the same number of d’s, say n_d. Indeed, for a given c, the set of d’s is in one-to-one correspondence with the set O₄×O₅×O₈, where O_i denotes the product of the sets of permutations of teams per continent for Pot i. As a consequence, n_d=m₄m₅m₈ where mi=∏j=1Jkij! where J is the number of continents represented at the World Cup and (k_i1, …, k_iJ) are the numbers of teams per continent for Pot i (k_i1+…+k_iJ=4). Now, at the second step, Pots 8, 5, and 4 are emptied randomly, uniformly, so that the order of the k_ij teams from continent j drawn from Pot i is uniform over the k_ij! possible orders. Moreover, all these orders are independent. This ensures that

ℙ(D=d)=ℙ(C=c(d))ℙ(D=d|C=c(d))=1NI×1nd

which does not depend on d: The distribution of D is uniform.

For the 2014 World Cup, the number of admissible draws would have been N_I×3!×3!×3!=1296 for Draw I, and N_II×3!×2!×2!=576 for Draw II, hence a total of 1296×576=746,496 possible outcomes for the entire draw. Each of those would have had probability 1/746,496.

Appendix B: Numbers of admissible continental distributions for previous World Cups

In this section we report the number of admissible continental distributions N_I and N_II for the 2010, 2006, 2002, and 1998 FIFA World Cups.

For the final draw of the 2010 edition, FIFA used the October 2009 FIFA ranking to seed the teams, so we use this same ranking to define the S-curve, shown in Table 5. However, for the 2006, 2002, and 1998 events, FIFA used their rankings in combination with performances of national teams in the two or three previous World Cups, resulting in the S-curves of Tables 6–8; see Wikipedia websites, and Marcuccitti (2005) for criticism. The corresponding numbers of admissible continental distributions are given in Table 9. We reported the 2014 numbers again for comparison. We consider two cases, depending on whether groups with no European teams are allowed or not. As Australia qualified to the 2006 World Cup as a member of the Oceania Football Confederation (OFC) but became a member of the Asian Football Confederation (AFC) on January 1st, 2006, we consider both possible affiliations for 2006, respectively denoted by 2006 (OFC) and 2006 (AFC). For 2014, we also show the values of N_I and N_II if we use the Elo ratings as of June 1, 2014 for seeding, instead of the FIFA rankings.

Our suggested procedure proves to be always tractable: The numbers of admissible continental distributions N_I and N_II are small, from six to a few tens or hundreds. For the 2002 and 1998 World Cups, 15 European teams were qualified, and the lower part of the S-curve was crowded by two many European teams so N_II=0. Using the S-curve rebalancing algorithm described in Section 3.4 solves the problem.