THE DUCKWORTH-LEWIS METHOD

1. THE DUCKWORTH-LEWIS METHOD (to decide a result in interrupted one-day cricket) http://www.flickr.com/photos/elkinator/3624920915

2. ONE-DAY CRICKET • Match restricted to one day • Fixed number, N, overs for each team • Draw is unacceptable if match is not finished THE PROBLEM • How can a result be decided if rain stops play?

3. POSSIBILITIES a) Team 1 completes Team 2 interrupted b) Team 1 completes late Team 2 left short of overs c) No of overs reduced for both teams d) Both teams interrupted

4. A SOLUTION • Team 1 had all N overs • Suppose Team 2 interrupted after u overs • Compare average runs per over • Compare Team 2 total with u overs of Team 1 (First u, Last u, Best u?) • Compare best u’ < u overs of each – still questions DIFFICULTIES • All these solutions can cause bias. We could • Use c) with Team 1’s best overs scaled

5. A SOLUTIONVIAMATHEMATICAL MODELS • Formulate and quantify • Team 2’s expected score allowing for the remaining N-u overs – compare • A target that Team 2 needs to win

6. MATHEMATICAL MODELS a) Parabola No of runs, Z(u), in u overs  Z(u)=7.46 u – 0.059 u2 (1) • 225 runs in 50 overs – assumed typical • Allows for team getting tired • Anomalous maximum at u = 63. Negative for u > 126

7. MATHEMATICAL MODELS b) World Cup 1996 • Identical to parabola with Z(u) expressed as a percentage of 225, i.e. 100 Z(u)/225

8. MATHEMATICAL MODELS c) Clark Curves • Too complicated • Allows for different kinds of stoppage and adjusts for the number of wickets, w, fallen

9. MATHEMATICAL MODELS d) Duckworth-Lewis • Includes explicitly the number of wickets, w, fallen. (w < 10)

10. DUCKWORTH-LEWIS 1) Starting point is w-independent Z(u)= Z0[1-exp(-bu)] (2) • b accounts for the team getting tired • If b small Eq. (2) is essentially Eq. (1) • DL call Z0 ‘asymptotic’

11. DUCKWORTH-LEWIS 2) Influence of w • If many overs, N-u, and few wickets, 10-w, are left or vice versa Eq. (2) needs to be changed • DL modified it to include w-dependence Z(u,w)= Z0(w){1-exp[-b(w)u]} (3)

12. DUCKWORTH-LEWIS EXPRESSION

13. DUCKWORTH-LEWIS EXPRESSION ~ 260 runs maximum for 80 overs ~ 225 runs for maximum 50 overs DL formula (3) for 0 wickets is roughly parabola or World Cup 1996

14. EXAMPLE APPLICATION Proportion of runs still to be scored with u overs left and w wickets down is P(u,w)=Z(u,w)/ Z(N,0) (4) Wickets lost w Overs left u

15. EXAMPLE APPLICATION • Team 1 scores S runs, Team 2 stopped at u1 overs left w wickets down, play resumes but time only for u2 overs • Overs lost = u1-u2. • Resource lost = P(u1,w)-P(u2,w) • Score to win = S{1-[P(u1,w)-P(u2,w)]}

16. A REAL EXAMPLE:ENGLAND VS NEW ZEALAND 1983 • 50 overs expected. • England batted first, scored 45 for 3 in 17.3 overs, were stopped for 27 overs and scored 43 in 5.7 overs i.e. 88 in 23 overs. • New Zealand were given 23 overs to score a target of 89 to win, which they did easily.

17. A REAL EXAMPLE:ENGLAND VS NEW ZEALAND 1983 • In the DL method England’s score is altered and the calculation gives New Zealand a target of 112 to win. • England were disadvantaged by the unexpected shortening of their innings. New Zealand knew in advance that they had a maximum 23 overs and planned accordingly. • DL claim that their method avoids this.

18. A REAL EXAMPLE:SOUTH AFRICA VS SRI LANKA 2003 • 50 overs expected. • Sri Lanka batted first, scored 268 for 9 • South Africa were 229 for 6 when rain stopped play after 45 overs. The DL target was 229, so the game was a draw.