# Info-gap’s decision model for robustness

Info-gap decision theory puts forward two types of decision-making models: one for decision-makers who are risk-averse and thus seek robustness, and one for decision-makers who are risk-takers and thus seek opportuneness.

Recall that according to info-gap decision theory, the robustness of decision $x\in X$ is defined as follows: $\displaystyle \rho(x,\tilde{u}):= \max_{\rho\ge 0}\ \{\rho: r^{*}\le r(x,u),\forall u\in N(\rho,\tilde{u})\}$

where

• $X$ = decision space.
• $u$ = parameter of interest whose true value is subject to severe uncertainty.
• $\tilde{u}$ = point estimate of the true value of $u$.
• $N(\rho,\tilde{u})$ = neighborhood of size $\rho$ around $\tilde{u}$.
• $r^{*}$ = critical performance level.
• $r(x,u)$ = performance level of decision $x$ associated with the given value of $u$.

The greater the value of $\rho(x,\tilde{u})$, the more robust the decision.

Hence, insofar as decision-makers pursuing robustness are concerned, the best (optimal) decision is that whose robustness is the highest (largest). Therefore, info-gap’s decision model for robustness is as follows: $\begin{array}{rl} \displaystyle \rho(\tilde{u}):= & \displaystyle \max_{x\in X} \rho(x,\tilde{u})\\ \displaystyle \equiv & \displaystyle \max_{x\in X} \max_{\rho\ge 0}\ \{\rho: r^{*}\le r(x,u),\forall u\in N(\rho,\tilde{u})\}\\ \displaystyle \equiv & \displaystyle \max_{\substack{x\in X\\\rho\ge 0}}\ \{\rho: r^{*}\le r(x,u),\forall u\in N(\rho,\tilde{u})\} \end{array}$

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