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# THE DUCKWORTH-LEWIS METHOD - PowerPoint PPT Presentation

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

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### THE DUCKWORTH-LEWIS METHOD

(to decide a result in interrupted one-day cricket)

http://www.flickr.com/photos/elkinator/3624920915

• 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?

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

• 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

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

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

b) World Cup 1996

• Identical to parabola with Z(u) expressed as a percentage of 225, i.e. 100 Z(u)/225

c) Clark Curves

• Too complicated

• Allows for different kinds of stoppage and adjusts for the number of wickets, w, fallen

d) Duckworth-Lewis

• Includes explicitly the number of wickets, w, fallen. (w < 10)

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’

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)

~ 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

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

• 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)]}

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.

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.

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.