Dynamic programming in cricket: Protecting the weaker batsman

A simple Dynamic Programming model of cricket is presented. The state is the facing batsmen and the number of runs on offer. The decision is whether to run or not, with the objective to maximise the chance the better batsman is on strike at the start of the next over. The model is solved analytically to find the optimal policy and the value of the objective function. The simple initial model is extended to a more realistic one requiring no further calculations and a numerical example is given. An alternative optimality criterion is investigated and we demonstrate that trying to put the better batsman on strike at the start of the over does not necessarily maximise the expected duration of the partnership. This alternative objective function is investigated numerically, and it is shown that the better batsman should generally run if possible off the second last or last ball of the over.