An analytics approach to the FIFA ranking procedure and the World Cup final draw

Abstract

This paper analyzes the procedure used by FIFA up until 2018 to rank national football teams and define by random draw the groups for the initial phase of the World Cup finals. A predictive model is calibrated to form a reference ranking to evaluate the performance of a series of simple changes to that procedure. These proposed modifications are guided by a qualitative and statistical analysis of the FIFA ranking. We then analyze the use of this ranking to determine the groups for the World Cup finals. After enumerating a series of deficiencies in the group assignments for the 2014 World Cup, a mixed integer linear programming model is developed and used to balance the difficulty levels of the groups.

Appendices

Appendix A: Mixed integer linear programming model

Sets

• G: groups.

• C: confederations.

• I: teams.

• S: seeded teams ($S \subset I$).

• $J_c$: teams in confederation c ($J_c \subset I, c \in C$).

Parameters

• $R_{i}$: ranking of team i ($i \in I$).

• $L_c$: minimum number of teams of confederation c in each group ($c \in C$).

• $U_c$: maximum number of teams of confederation c in each group ($c \in C$).

• SR: minimum accepted sum ranking difference.

• $\hat{R}$: upper bound on the ranking of any team.

Decision variables

$$\begin{aligned} x_{ig}= & {} \left\{ \begin{array}{ll} 1 \text { if team }i\text { is assigned to group }g\\ 0 \text { otherwise } \end{array} \right. \\ x^{\max }_{ig}= & {} \left\{ \begin{array}{ll} 1 \text { if team }i\text { is assigned to group }g\text { has the highest ranked team}\\ 0 \text { otherwise } \end{array} \right. \\ x^{\min }_{ig}= & {} \left\{ \begin{array}{ll} 1 \text { if team }i\text { is assigned to group }g\text { has the lowest ranked team}\\ 0 \text { otherwise } \end{array} \right. \end{aligned}$$

• $w_{min}$: ranking sum of the group with the lowest ranking sum value.

• $w_{max}$: ranking sum of the group with the highest ranking sum value.

• $z_{min}$: range of the group with the lowest range.

• $z_{max}$: range of the group with the highest range.

Objective function

$$\begin{aligned} \min f = z_{max}-z_{min} \end{aligned}$$

(A.1)

Constraints

$$\begin{aligned} \sum _{g \in G}x_{ig}= & {} 1 \ \ \ \ \forall \ i \in \ I \end{aligned}$$

(A.2)

$$\begin{aligned} \sum _{i \in I}x_{ig}= & {} 4 \ \ \ \ \forall \ g \in \ G \end{aligned}$$

(A.3)

$$\begin{aligned} \sum _{i \in S}x_{ig}= & {} 1 \ \ \ \ \forall \ g \in \ G \end{aligned}$$

(A.4)

$$\begin{aligned} \sum _{i \in J_c}x_{ig}\ge & {} L_c \ \ \ \ \forall \ g \in \ G, c \in C \end{aligned}$$

(A.5)

$$\begin{aligned} \sum _{i \in J_c}x_{ig}\le & {} U_c \ \ \ \ \forall \ g \in \ G, c \in C \end{aligned}$$

(A.6)

$$\begin{aligned} w_{min}\le & {} \sum _{i \in I}R_{i}x_{ig} \end{aligned}$$

(A.7)

$$\begin{aligned} w_{max}\ge & {} \sum _{i \in I}R_{i}x_{ig} \end{aligned}$$

(A.8)

$$\begin{aligned} w_{max}-w_{min}\le & {} SR \end{aligned}$$

(A.9)

$$\begin{aligned} x^{\min }_{ig} + x^{\max }_{ig}\le & {} x_{ig} \ \ \ \ \forall i \in I, g \in G \end{aligned}$$

(A.10)

$$\begin{aligned} \sum _{j} R_j x^{\max }_{jg}\ge & {} R_i~x_{ig} \ \ \ \ \forall i \in I, g \in G \end{aligned}$$

(A.11)

$$\begin{aligned} \sum _{j} (\bar{R}-R_j)~x^{\min }_{jg}\ge & {} (\bar{R}-R_i)~x_{ig} \ \ \ \ \forall i \in I, g \in G \end{aligned}$$

(A.12)

$$\begin{aligned} \sum _{i \in I}x^{\max }_{ig}= & {} 1 \ \ \ \ \forall \ g \in \ G \end{aligned}$$

(A.13)

$$\begin{aligned} \sum _{i \in I}x^{\min }_{ig}= & {} 1 \ \ \ \ \forall \ g \in \ G \end{aligned}$$

(A.14)

$$\begin{aligned} z_{min}\le & {} \sum _{i \in I}R_{i}(x^{\max }_{ig} - x^{\min }_{ig}) \end{aligned}$$

(A.15)

$$\begin{aligned} z_{max}\ge & {} \sum _{i \in I}R_{i}(x^{\max }_{ig} - x^{\min }_{ig}) \end{aligned}$$

(A.16)

$$\begin{aligned} x_{ig},\, x^{\max }_{ig},\, x^{\min }_{ig}\in & {} \ \{0,1\} \quad \forall i \in I, g \in G \end{aligned}$$

(A.17)

$$\begin{aligned} w_{min},\,w_{max},\,z_{min},\, z_{max}\ge & {} 0 \ \ \ \ \end{aligned}$$

(A.18)

Constraints (A.2) ensure that every team is assigned to exactly one group. Constraints (A.3) specify that each group contains exactly four teams while constraints (A.4) require that one of the four teams is a seed. Constraints (A.5) and (A.6) impose upper and lower bounds on the number of teams from a single confederation assigned to a single group (in the 2014 World Cup, only one team from each confederation was allowed in a group except for the European confederation, in which case the limit was two; and at least one UEFA team per group is necessary). Constraints (A.7)–(A.8) aid in calculating the group’s minimum and maximum ranking sums, while constraint (A.9) imposes a bound on the difference of such values. Constraints (A.10)–(A.14) compute the range of each group, and constraints (A.15)–(A.16) aid in calculating the group’s minimum and maximum ranges. Constraints (A.18) defines the nature of the variables, and finally, the objective function (A.1) minimizes the difference between the maximum and minimum ranges.

Appendix B: Estimates for the predictive model of Section 2

See Table 17.

Table 17 Poisson regression outcome for predictive model in Sect. 2