Info-gap Decision Theory - Basic Example: Budget

Basic Example: Budget

As a simple example, consider a worker with uncertain income. They expect to make $100 per week, while if they make under $60 they will be unable to afford lodging and will sleep in the street, and if they make over $150 they will be able to afford a night's entertainment.

Using the info-gap absolute error model:

where one would conclude that the worker's robustness function is $40, and their opportuneness function is $50: if they are certain that they will make $100, they will neither sleep in the street nor feast, and likewise if they make within $40 of $100. However, if they erred in their estimate by more than $40, they may find themselves on the street, while if they erred by more than $50, they may find themselves in clover.

As stated, this example is only descriptive, and does not enable any decision making – in applications, one considers alternative decision rules, and often situations with more complex uncertainty.

Consider now the worker thinking of moving to a different town, where the work pays less but lodgings are cheaper. Say that here they estimate that they will earn $80 per week, but lodging only costs $44, while entertainment still costs $150. In that case the robustness function will be $36, while the opportuneness function will be $70. If they make the same errors in both cases, the second case (moving) is both less robust and less opportune.

On the other hand, if one measures uncertainty by relative error, using the fractional error model:

in the first case robustness is 40% and opportuneness is 50%, while in the second case robustness is 45% and opportuneness is 87.5%, so moving is more robust and less opportune.

This example demonstrates the sensitivity of analysis to the model of uncertainty.