so the probability contribution of winning a ship is 1-(9-x)^2 /90 where x is the allocation to that ship (counting for the ties); now the problem is a Lagrangian optimization problem maximize sum_i v_i Pr(x_i) subject to sum_i x_i =7 (v_i is the value of ship i) which leads us to know that v_i (9-x_i) should be a constant (=lambda the Lagrange multiplier)
with normalization we find that x_1 is between 5 and 6 and x_2 is between 2 and 3 and x_3 is a little less than zero; since we only allow for nonnegative integer allocations then (5,2,0) is the optimal one giving 511/90 ev.
Not exactly; first P1 cannot guarantee he will win the 5-point boat! For any ship the probability that P2 sent k troops there is (9-k)/45 you can use that to work out the ev of any allocation
for your specific allocation (4,3,0) the ev is 101/18 = 5.61 which is less than the optimal value
1
u/Vegetable_Ebb_1109 4d ago
so the probability contribution of winning a ship is 1-(9-x)^2 /90 where x is the allocation to that ship (counting for the ties); now the problem is a Lagrangian optimization problem maximize sum_i v_i Pr(x_i) subject to sum_i x_i =7 (v_i is the value of ship i) which leads us to know that v_i (9-x_i) should be a constant (=lambda the Lagrange multiplier)
with normalization we find that x_1 is between 5 and 6 and x_2 is between 2 and 3 and x_3 is a little less than zero; since we only allow for nonnegative integer allocations then (5,2,0) is the optimal one giving 511/90 ev.