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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

THE DUCKWORTH-LEWIS METHOD

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

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


One day cricket
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?


Possibilities
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


A solution
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


A solution via mathematical models
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


Mathematical models
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


Mathematical models1
MATHEMATICAL MODELS

b) World Cup 1996

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


Mathematical models2
MATHEMATICAL MODELS

c) Clark Curves

  • Too complicated

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


Mathematical models3
MATHEMATICAL MODELS

d) Duckworth-Lewis

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


Duckworth lewis
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’


Duckworth lewis1
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)



Duckworth lewis expression1
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


Example application
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


Example application1
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)]}


A real example england vs new zealand 1983
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 19831
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
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.


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