LAGRANGE MULTIPLIERS IN THE PROBABILITY DISTRIBUTIONS ELICITATION PROBLEM: AN APPLICATION TO THE 2013 FIFA CONFEDERATIONS CUP

Bezerra, Diogo de Carvalho; Rêgo, Leandro Chaves

doi:10.1590/0101-7438.2015.035.03.0489

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Articles • Pesqui. Oper. 35 (3) • Sep-Dec 2015 • https://doi.org/10.1590/0101-7438.2015.035.03.0489 copy

LAGRANGE MULTIPLIERS IN THE PROBABILITY DISTRIBUTIONS ELICITATION PROBLEM: AN APPLICATION TO THE 2013 FIFA CONFEDERATIONS CUP

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Contributions from the sensitivity analysis of the parameters of the linear programming model for the elicitation of experts' beliefs are presented. The process allows for the calibration of the family of probability distributions obtained in the elicitation process. An experiment to obtain the probability distribution of a future event (Brazil vs. Spain soccer game in the 2013 FIFA Confederations Cup final game) was conducted. The proposed sensitivity analysis step may help to reduce the vagueness of the information given by the expert.

Keywords:
Elicitation; linear programming; soccer and sensitivity analysis

	max - π	min - π
W	50.00%	25.00%
D	50.00%	50.00%
L	0.00%	25.00%
E(X)	1.5	0

Result	Gain		Gain
	home team		visitor team
	max	min	max	min
W	0.50	0.00	0.33	0.50
D	0.50	1.00	0.33	0.50
L	0.00	0.00	0.33	0.00
Restrictions	λ	λ	λ	λ
π (W) ≤ π (D) + π (L)	1.00	0.00	0.00	-0.50
π (W) ≥ π (L)	0.00	-1.00	1.33	0.00
π (D) ≥ π (L)	0.00	0.00	0.33	0.00
Sum equal 1	2.00	1.00	1.33	0.50
Objective Function
E(X)	2.00	1.00	0.33	0.50

Question	First Group Statement	Second Group Statement	Third Group Statement
1	0.1429	0.0882	0.0714
2	0.3043	0.3056	0.0714
3	0.2174	0.1667	0.1429
4	0.3095	0.1176	-
5	0.1667	0.1875	-

Expert	First Group Statement				Second Group Statement				Third Group Statement
Expert	Q1	Q2	Q3	Q4	Q5	Q1	Q2	Q3	Q4	Q5	Q1	Q2	Q3	Q4	Q5
Expert1	L	D,L	D	L	L	3.00	3.00	2.00	1.00	2.00	0.10	0.10	0.10	0.10	0.10
Expert2	W,D	W	W	W	D	0.75	0.67	0.67	0.75	0.75	-	-	-	-	-
Expert3	L	D,L	D	L	L	3.00	4.00	2.00	6.00	3.00	-	-	-	-	-
Expert4	L	D,L	D	L	D	2.00	2.00	3.00	2.00	0.33	0.1	0.1	0.15	0.15	0.15
Expert5	-	D,L	D	-	D	-	-	-	-	-	-	-	-	-	-
Expert6	W,D	D,L	D	L	D	-	-	-	-	-	-	-	-	-	-
Expert7	W,D	W	W	W	-	0.50	0.33	0.50	0.50	-	0.20	0.20	0.10	0.10	-
Expert8	L	D,L	D	L	D	-	-	-	-	-	-	-	-	-	-
Expert9	W,D	W	W	L	D	0.67	0.50	0.67	0.50	0.50	0.10	0.20	0.25	0.10	0.15
Expert10	W,D	D,L	D	W	D	0.43	2.33	1.33	1.00	0.75	-	-	-	-	-
Expert11	W,D	D,L	D	W	D	0.67	1.22	1.22	0.82	0.67	-	-	-	-	-
Expert12	W,D	D,L	W	L	D	0.50	2.00	0.50	2.00	0.33	-	-	-	-	-
Expert13	W,D	D,L	D	W	D	1.00	1.00	1.00	0.50	0.50	-	-	-	-	-
Expert14	W,D	D,L	D	L	D	-	-	-	-	-	-	-	-	-	-
Expert15	W,D	D,L	D	W	D	0.50	1.00	1.00	0.50	0.50	2.00	-	-	1.00	1.00
Expert16	W,D	W	W	W	D	0.33	0.50	1.00	0.50	0.50	-	-	-	-	-
Expert17	W,D	D,L	D	W	D	2.00	0.50	0.50	2.00	2.00	-	-	-	-	-
Expert18	W,D	W	W	W	D	0.50	0.33	1.00	0.33	0.33	-	-	-	-	-
Expert19	W,D	W	W	W	D	-	-	-	-	-	-	-	-	-	-
Expert20	W,D	W	D	W	D	0.33	0.50	1.00	0.50	1.00	0.15	0.10	-	0.10	-
Expert21	W,D	D,L	D	-	D	0.75	2.00	3.00	-	0.25	-	-	-	-	-
Expert22	-	D,L	D	L	L	-	5,00	2.00	3.00	1.50	-	0	1	3	1
Expert23	W,D	W	W	W	-	0.50	0.50	0.33	0.50	-	-	-	-	-	-

Expert	π_max			π_min			E(X)			λ_s	λ₁	λ₂	λ₃	λ₄	λ₅		λ_s	λ₁	λ₂	λ₃
Expert	W	D	L	W	D	L	max	min	max	max	max	max	max	max	min	min	min	min	min	min
Expert1	0.25	0.25	0.50	0.00	0.00	1.00	-0.75	-3.00	-0.75	2.25	0.00	1.50	0.00	0.00	0.00	0.00	0.00	-3.00	0.00	0.00
Expert2	1.00	0.00	0.00	0.50	0.25	0.25	3.00	0.75	3.00	0.00	0.00	0.00	0.00	0.00	0.75	-3.00	-2.25	0.00	0.00	-1.50
Expert3	0.25	0.25	0.50	0.00	0.00	1.00	-0.75	-3.00	-0.75	2.25	0.00	1.50	0.00	0.00	-3.00	0.00	0.00	-3.00	0.00	0.00
Expert4	0.00	0.50	0.50	0.00	0.50	0.50	-1.50	-1.50	1.00	0.00	1.00	0.50	0.50	0.00	-1.50	0.00	0.00	0.00	0.00	-1.50
Expert5	0.50	0.50	0.00	0.00	0.00	1.00	1.50	-3.00	1.50	na	1.50	0.00	na	0.00	-1.50	na	0.00	0.00	na	-1.50
Expert6	0.28	0.44	0.28	0.00	0.50	0.50	0.00	-1.50	0.00	0.00	0.00	0.00	3.00	0.00	-1.50	-1.50	0.00	0.00	0.00	0.00
Expert7	1.00	0.00	0.00	0.50	0.00	0.50	3.00	0.00	3.00	0.00	0.00	0.00	3.00	na	0.00	0.00	0.00	0.00	-3.00	na
Expert8	0.25	0.25	0.50	0.00	0.00	1.00	-0.75	-3.00	-0.75	2.25	0.00	1.50	0.00	0.00	-3.00	0.00	0.00	-3.00	0.00	0.00
Expert9	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø
Expert10	0.50	0.50	0.00	0.33	0.33	0.33	1.50	0.00	1.50	0.00	1.50	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-3.00	0.00
Expert11	0.50	0.50	0.00	0.33	0.33	0.33	1.50	0.00	1.50	0.00	1.50	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-3.00	0.00
Expert12	0.33	0.33	0.33	0.33	0.33	0.33	0.00	0.00	0.00	0.00	0.00	0.00	3.00	0.00	0.00	0.00	0.00	-3.00	0.00	-3.00
Expert13	0.50	0.50	0.00	0.33	0.33	0.33	1.50	0.00	1.50	0.00	1.50	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-3.00	0.00
Expert14	0.28	0.44	0.28	0.00	0.50	0.50	0.00	-1.50	0.00	0.00	0.00	0.00	3.00	0.00	-1.50	-1.50	0.00	0.00	0.00	0.00
Expert15	0.50	0.50	0.00	0.33	0.33	0.33	1.50	0.00	1.50	0.00	1.50	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-3.00	0.00
Expert16	1.00	0.00	0.00	0.50	0.25	0.25	3.00	0.75	3.00	0.00	0.00	0.00	0.00	0.00	0.75	0.00	-2.25	0.00	0.00	-1.50
Expert17	0.50	0.50	0.00	0.33	0.33	0.33	1.50	0.00	1.50	0.00	1.50	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-3.00	0.00
Expert18	1.00	0.00	0.00	0.50	0.25	0.25	3.00	0.75	3.00	0.00	0.00	0.00	0.00	0.00	0.75	0.00	-2.25	0.00	0.00	-1.50
Expert19	1.00	0.00	0.00	0.50	0.25	0.25	3.00	0.75	3.00	0.00	0.00	0.00	0.00	0.00	0.75	0.00	-2.25	0.00	0.00	-1.50
Expert20	0.50	0.50	0.00	0.50	0.50	0.00	1.50	1.50	1.50	0.00	0.00	1.50	0.00	0.00	1.50	0.00	-4.50	-3.00	0.00	0.00
Expert21	0.50	0.50	0.00	0.00	0.50	0.50	1.50	-1.50	1.50	0.00	1.50	0.00	na	0.00	-1.50	-1.50	0.00	0.00	na	0.00
Expert22	0.33	0.33	0.33	0.00	0.00	1.00	0.00	-3.00	0.00	na	0.00	0.00	3.00	0.00	-3.00	na	-3.00	0.00	0.00	0.00
Expert23	1.00	0.00	0.00	0.50	0.00	0.50	3.00	0.00	3.00	0.00	0.00	0.00	0.00	na	0.00	0.00	0.00	0.00	-3.00	na
Mean	0.54	0.31	0.15	0.25	0.25	0.50	1.19	-0.75	1.31	0.31	0.52	0.30	0.78	0.00	-0.68	-0.21	-0.75	-0.68	-1.11	-0.57

Expert	π_max			π_min			E(X)			λ_s	λ₁	λ₂	λ₃	λ₄	λ₅		λ_s	λ₁	λ₂	λ₃
Expert	W	D	L	W	D	L	max	min	max	max	max	max	max	max	min	min	min	min	min	min
Expert1	0.08	0.17	0.75	0.00	0.00	1.00	-2.00	-3.00	-2.00	1.00	0.00	1.00	0.00	0.00	-3.00	0.00	0.00	-3.00	0.00	0.00
Expert2	1.00	0.00	0.00	0.60	0.23	0.17	3.00	1.29	3.00	0.00	0.00	0.00	0.00	0.00	1.29	0.00	-0.86	0.00	0.00	-0.43
Expert3	0.08	0.17	0.75	0.00	0.00	1.00	-2.00	-3.00	-2.00	1.00	0.00	1.00	0.00	0.00	-3.00	0.00	0.00	-3.00	0.00	0.00
Expert4	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø
Expert5	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na
Expert6	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na
Expert7	1.00	0.00	0.00	0.75	0.00	0.25	3.00	1.50	3.00	0.00	0.00	0.00	0.00	na	1.50	0.00	-1.50	0.00	0.00	na
Expert8	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na
Expert9	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø
Expert10	0.30	0.70	0.00	0.30	0.41	0.30	0.90	0.00	0.90	0.00	0.30	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-1.00	0.00
Expert11	0.45	0.55	0.00	0.00	1.00	0.00	1.35	0.00	1.35	0.00	0.30	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-0.55	0.00
Expert12	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø
Expert13	0.50	0.50	0.00	0.00	1.00	0.00	1.50	0.00	1.50	0.00	1.50	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-1.50	0.00
Expert14	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na
Expert15	0.50	0.50	0.00	0.00	1.00	0.00	1.50	0.00	1.50	0.00	1.50	0.00	0.00	0.00	0.00	0.00	0.00	0.00	-1.50	0.00
Expert16	1.00	0.00	0.00	0.67	0.22	0.11	3.00	1.67	3.00	0.00	0.00	0.00	0.00	0.00	1.67	0.00	-1.33	0.00	0.00	-1.00
Expert17	1.00	0.00	0.00	0.67	0.11	0.22	3.00	1.33	3.00	0.00	0.00	0.00	0.00	0.00	1.33	0.00	-1.67	0.00	0.00	-1.00
Expert18	1.00	0.00	0.00	0.75	0.19	0.06	3.00	2.06	3.00	0.00	0.00	0.00	0.00	0.00	2.06	0.00	-0.94	0.00	0.00	-0.75
Expert19	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na	na
Expert20	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø
Expert21	0.50	0.50	0.00	0.00	0.80	0.20	1.50	-0.60	1.50	0.00	1.50	0.00	na	0.00	-0.60	0.00	0.00	0.00	na	-0.60
Expert22	0.11	0.22	0.67	0.00	0.00	1.00	-1.67	-3.00	-1.67	na	0.00	2.33	0.00	1.33	-3.00	na	0.00	-3.00	0.00	0.00
Expert23	1.00	0.00	0.00	0.67	0.00	0.33	3.00	1.00	3.00	0.00	0.00	0.00	0.00	na	1.00	-1.00	-0.50	0.00	0.00	na
Mean	0.63	0.24	0.13	0.32	0.37	0.31	1.35	0.02	1.47	0.13	0.41	0.27	0.00	0.10	0.02	-0.13	-0.46	-0.56	-0.40	-0.29

Expert	π_max			π_min			E(X)			λ_s	λ₁	λ₂	λ₃	λ₄	λ₅		λ_s	λ₁	λ₂	λ₃
Expert	W	D	L	W	D	L	max	min	max	max	max	max	max	max	min	min	min	min	min	min
Expert1	0.01	0.22	0.78	0.00	0.20	0.80	-2.30	-2.40	-2.00	1.00	0.00	1.00	0.00	0.00	-3.00	0.00	0.00	-3.00	0.00	0.00
Expert4	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø
Expert7	1.00	0.00	0.00	0.80	0.00	0.20	3.00	1.80	3.00	0.00	0.00	0.00	0.00	na	1.50	0.00	-1.50	0.00	0.00	na
Expert9	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø
Expert15	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø
Expert20	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø
Expert22	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø	Ø
Mean	0.50	0.11	0.39	0.40	0.10	0.50	0.35	-0.30	0.50	0.50	0.00	0.50	0.00	-	-0.75	0.00	-0.75	-1.50	0.00	-

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