Givewell models in explained maths

Stating these models as clear mathematical equations and breaking them down

All models - basics

Give Directly

Connected to Causal interface here

Note (DR): the equation below can and should be massively simplified and made clearer. Some tips on this in the fold. I made a start to this below. Note that the equation interface here is limited, at least in the interactive version (e.g., how do I do multiline equations)? ... I recommend writing equations elsewhere and moving them here.

$$value/donation =$$

$$\ln 2 \times \frac{p}{s} \times ...$$

$$\Big[\Big( PV(c, n - 1, -(\ln(b + \frac{s}{h} i r) - \ln(b))\Big) + ...$$

$$... + \ln\Big(b + (\frac{s}{h} (1 - i))\Big) - \ln(b)\Big]$$

Where

p = percent transferred, i.e., 'how much makes it to recipients after admin, leakage etc')

s = average size of donation (transfer to household?)

g = Value lost to negative spoiler DR: I don't see this anywhere

PV = Present Value function, as represented in excel

c = discount rate

n = duration of investment

h = average household size [DR: flag -- something possibly weird here -- smaller households mean greater benefits and greater cost-effectiveness ... in such a strong way, with no countervailing 'but we are helping fewer people' thing?]

b = baseline consumption (?in absence of transfer)

v = amount of investment returned (at end of investment period?)

i = percent of transfer invested

r = ???

Side discussion: how to simplify and give insight to equations:

Some useful techniques for legibility:

1. Use spacing and multiple lines carefully and cleverly

2. Use 'large parentheses' and other shapes of braces ... in latex, $\Big( x+3 \Big)$ etc for outer terms ... more examples below

3. Factor common terms out where possible (at least if it doesn't destroy intuition)

   1. Sometimes re-expressing things in negatives or reciprocals can aid this

4. Name variables in ways that are easy to connect to the thing they represent

5. Under-braces with labels and explanations can be extremely helpful (see examples below)

6. Define intuitive combinations of the elements and break these out to make the equation simpler... (but also make it very quick to see and easy to remember what these represent; so-so example below, from another context)

Examples (latex code to be shared later where absent)

Multiple sizes of and shapes of braces, brackets, parentheses

\begin{align*} \Omega(q(\gamma), q(\beta)):=\sum_{s, \sigma} P(s, \sigma)\Big[ q(\sigma)W(e, s)+(1-q(\sigma))W(ne, s)\Big]. \end{align*}

Defining and separating out terms

Defining and separating out terms

Underbracing and grouping for explanations

Combining these methods: