# The maximin connection

December 28, 2011 Leave a comment

Wald’s famous maximin model (circa 1940) is based on a **worst case** approach to uncertainty/variability. The **Maximin Rule** argues as follows:

Rank alternatives according to their worst outcome. Hence, select the alternative whose worst outcome is at least as good as the worst outcome of any other alternative.

For our purposes, the most suitable formulation of the maximin is that of the following generic model:

where

- = set of available alternatives.
- = state of Nature.
- = set of states associated with alternative .
- = list of constraints on pairs.
- = outcome generated by alternative and state .

We shall refer to this formulation as the **full Monty model.**

This model seeks robustness with respect to both the outcomes — via the operation — and the constraints — via the clause .

Now, consider the rather simple case where robustness is sought only with respect to the constraints, namely the case where the outcome is independent of the state . In this case the **full Monty model** is simplified to

observing that since the outcome is independent of the state , the iconic operation is superfluous. We shall refer to this maximin model as the **C model**.

With this as background, it is straightforward to show that info-gap’s robustness model, namely

is a simple maximin model, by demonstrating that it is an instance of the **C model**.

Info-gap’s robustness model is a simple instance of Wald’s maximin model, say the **C model**.

**Proof.**

Consider the instance of the **C model** that is based on the correspondence , namely let

In this case, the **C model** — for given values of and — is as follows:

Now consider the correspondence and the specification

The corresponding instance of the **C model** is then as follows:

Observe that this is none other than info-gap’s decision model for robustness. Hence,

Info-gap’s decision model for robustness is an instance of Wald’s maximin model.