Value of Holding Money

Conceptual Framework: Epistemic Uncertainty and Placeholder Opportunities

Research prioritisation faces a fundamental epistemic uncertainty: there are funding opportunities that we don't know of yet, and that we may discover (or that may start to exist) later on. This uncertainty adds some potential value to holding money rather than allocating all available funds to known funding opportunities.

To account for this epistemic uncertainty, we use the concept of placeholder opportunities - hypothetical funding opportunities that represent the expected value of newly discovered funding opportunities we don't know of yet. By including placeholder opportunities in our modelling, we aim to represent the expected value of holding money for future opportunities.

There are two main parameters that control placeholder opportunity creation:

VirtualOrgsSamplingStrategy: Configures how placeholder opportunities are sampled from existing opportunity expected values
VirtualOrgsPerYear: Specifies how many placeholder opportunities to add each year

Both parameters are located in the meta tab of the input spreadsheet.

Technical Implementation: Placeholder Opportunity Creation Process

The placeholder opportunities sampling process is repeated for each year starting from year 2 (future years), with placeholder opportunities being prolonged to subsequent years using median discounting. Year 1 represents the present year with only real opportunities.

Step 1: Extract Expected Values for Year i

The system starts with the Expected Value (EV) matrix for year i. Each "world" or "simulation" is considered independently. For simulation_1, we extract opportunity EVs: \(EV_{i,sim1}\)

Year i EV Matrix:

Simulation	Opportunity A	Opportunity B	Opportunity C	Opportunity D	...
simulation_1	0.8	0.3	0.9	0.5	...
simulation_2	0.6	0.7	0.2	0.9	...
simulation_3	0.9	0.4	0.8	0.3	...
...	...	...	...	...	...

Step 2: Sample Placeholder Opportunities for Year i

We sample from \(EV_i\) using the chosen sampling strategy for as many opportunities as specified by VirtualOrgsPerYear.

The sampling methods are detailed in the section Sampling Strategies.

Step 3: Add Placeholder Opportunities to Year i

Placeholder opportunities are added to the EV matrix for year i with newly generated values:

Year i EV Matrix (with Placeholder Opportunities):

Simulation	Opportunity A	Opportunity B	Opportunity C	Opportunity D	VirtualOrg1	VirtualOrg2	...
simulation_1	0.8	0.3	0.9	0.5	0.7	0.4	...
simulation_2	0.6	0.7	0.2	0.9	0.8	0.6	...
simulation_3	0.9	0.4	0.8	0.3	0.5	0.9	...
...	...	...	...	...	...	...	...

Step 4: Prolong Placeholder Opportunities to Subsequent Years

Placeholder opportunities created in year i are prolonged to subsequent years (i+1, i+2, etc.) using the median discount rate. The median discount rate is calculated from all opportunities' discount rates in the system.

Discount Rate Calculation:

Opportunity	Discount Rate
Opportunity A	0.05
Opportunity B	0.03
Opportunity C	0.07
Opportunity D	0.04
Median	0.045

Note: In typical cases, all opportunities will have similar discount rates, so the median provides a representative rate for all opportunities.

Year i+1 EV Matrix (with Prolonged Placeholder Opportunities):

Simulation	Opportunity A	Opportunity B	Opportunity C	Opportunity D	VirtualOrg1	VirtualOrg2	...
simulation_1	0.76	0.29	0.86	0.48	0.67	0.38	...
simulation_2	0.57	0.67	0.19	0.86	0.76	0.57	...
simulation_3	0.86	0.38	0.76	0.29	0.48	0.86	...
...	...	...	...	...	...	...	...

Note: All opportunity values (real and placeholder) are discounted by the median discount rate (0.045) from year i to year i+1

Step 5: Repeat Process for Each Year

This entire process (Steps 1-4) is repeated for each year starting from year 2. Year 1 represents the present year with only real opportunities.

Year 1 (Present): Only real opportunities exist
Year 2: Extract EVs → Sample placeholder opportunities → Add to matrix → Prolong to Year 3
Year 3: Extract EVs → Sample placeholder opportunities → Add to matrix → Prolong to Year 4
Year 4: Extract EVs → Sample placeholder opportunities → Add to matrix → Prolong to Year 5
...and so on for all future years

Multi-Year Placeholder Opportunity Timeline:

Year	Description	Placeholder Opportunities Added	Placeholder Opportunities from Previous Years	Total Placeholder Opportunities
1	Present	0	0	0
2	Future	2	0	2
3	Future	2	2 (discounted)	4
4	Future	2	4 (discounted)	6
5	Future	2	6 (discounted)	8
...	...	...	...	...

Step 6: Final Multi-Year EV Matrix

The final EV matrix contains all real opportunities plus all placeholder opportunities across all years, with placeholder opportunities properly discounted for each year they appear in.

Sampling Strategies

Each strategy combines two dimensions: how values are generated (sample vs percentile) and whether we fit a distribution or use empirical data; plus whether weights are used.

Strategy	Mode	Fit	Weighted
`fit_sample`	sample	Yes	No
`fit_percentile`	percentile	Yes	No
`percentile`	percentile	No	No
`weighted_fit_sample`	sample	Yes	Yes
`weighted_fit_percentile`	percentile	Yes	Yes
`weighted_percentile`	percentile	No	Yes

Fit Sample (`"fit_sample"`)

Fits candidate distributions (lognormal, gamma, normal) to \(EV_{i,sim}\)
Uses Kolmogorov-Smirnov to select the best fit
Draws n random samples from the fitted distribution

Fit Percentile (`"fit_percentile"`)

Fits as above, then returns n evenly spaced interior percentiles from the fitted distribution
Deterministic for given inputs; useful for stable, representative points

Percentile (`"percentile"`)

Takes actual percentiles from the existing opportunity data (empirical)
For n=1: 50th percentile; n=2: ~33rd/66th; n=3: 25th/50th/75th
Fast and preserves the empirical shape

Weighted Fit Sample (`"weighted_fit_sample"`)

Like fit_sample, but uses room for funding as weights when fitting
Emphasises opportunities with higher funding capacity

Weighted Fit Percentile (`"weighted_fit_percentile"`)

Like fit_percentile, but uses room for funding as weights when fitting
Returns percentiles of the weighted fitted distribution

Weighted Percentile (`"weighted_percentile"`)

Like percentile, but computes percentiles using room for funding as weights
With equal weights, matches percentile

Implementation

The sampling strategies are implemented in draw_samples.R.