Why do you think this? Are you trying to account for uncertainty about what the most efficient charity is? Otherwise, I don’t understand this particular argument for “multiple charities” over “one charity”.
But look for instance at this or this picture and assume it shows utility z as function of the budget of two charities x and y. For almost every point (x, y), the steepest slope is in neither in direction (0, 1) nor in direction (1, 0) but in a combination of both direction. In other words, to optimize utility, I should give part of my money to charity x, part to charity y.
OK, I realised the flaw in my argumentation. If I have 1000 GBP to give away, I could either ‘walk’ 1000 GBP in direction of charity x or 1000 GBP in direction of charity y but only sqrt(x^2 + y^2) in a combination of x and y, e.g. the maximal gradient. The optimal allocation (x, y) of money is what maximises the scalar product of gradient (dU/dx, dU/dy) * (x, y) under the restriction that x + y = 1000. If dU/dx = dU/dy a 50⁄50 allocation as good as an allocation of all money to the most effective charity. Otherwise giving all money to the most effective charity maximises utility. Sorry for the confusion and thanks for the discussion.
Why do you think this? Are you trying to account for uncertainty about what the most efficient charity is? Otherwise, I don’t understand this particular argument for “multiple charities” over “one charity”.
No, I am not trying to account for uncertainty.
But look for instance at this or this picture and assume it shows utility z as function of the budget of two charities x and y. For almost every point (x, y), the steepest slope is in neither in direction (0, 1) nor in direction (1, 0) but in a combination of both direction. In other words, to optimize utility, I should give part of my money to charity x, part to charity y.
OK, I realised the flaw in my argumentation. If I have 1000 GBP to give away, I could either ‘walk’ 1000 GBP in direction of charity x or 1000 GBP in direction of charity y but only sqrt(x^2 + y^2) in a combination of x and y, e.g. the maximal gradient. The optimal allocation (x, y) of money is what maximises the scalar product of gradient (dU/dx, dU/dy) * (x, y) under the restriction that x + y = 1000. If dU/dx = dU/dy a 50⁄50 allocation as good as an allocation of all money to the most effective charity. Otherwise giving all money to the most effective charity maximises utility. Sorry for the confusion and thanks for the discussion.