Approximate Pareto efficiency with ordinal preferences

Question

Pareto-efficiency is a very important property, but in some cases it cannot be attained (for example, when agents play a game such as the Prisonner's Dilemma). In such cases, it can be useful to measure how far the outcome is from being Pareto-efficient.

A simple measure of approximate Pareto-efficiency (which I found e.g. here and here) uses a multiplicative factor: an outcome is $r$-approximately-efficient if there is no alternate outcome in which each agent’s utility increases by a factor of at least $r$. However, this measure assumes that agents have cardinal utility functions. In economics, it is usually assumed that agents' utility functions are just representations of preference relations. In particular, applying a monotone transformation to a utility function leads to a different function that represents the same preferences. If we apply a monotone transformation to a utility function (even a simple one such as adding or subtracting 1), the multiplicative approximation ratio changes.

MY QUESTION: what is a notion of approximate Pareto-efficiency, which is meaningful even when agents have ordinal preferences?

Answer 1

Consider a game:

• Set of Players $= \{1,2\}$
• Action Sets of the players: $\triangle_1\subseteq\mathbb{R}^n$ and $\triangle_2\subseteq\mathbb{R}^m$. For Eg: In case of Prisoner's Dilemma, $\triangle_1=\{(p,1-p)|0\leq p\leq 1\}$, where $p$ is the probability of co-operation.
• Payoffs: $u_i:\triangle_1\times \triangle_2\rightarrow\mathbb{R}$ where $i\in\{1,2\}$

We assume $u_i$s are continuous.

Definition 1. We can define an outcome $(p,q)\in \triangle_1\times \triangle_2$ to be $\epsilon -$ approximate Pareto efficient if there exists a Pareto efficient outcome $(p',q')$ in its $\epsilon-$ neighbourhood i.e. $(p',q')\in \triangle_1\times \triangle_2 \cap \mathcal{N}_{\epsilon}((p,q))$, where $\mathcal{N}_{\epsilon}((p,q))$ is the $\epsilon-$ neighbourhood of $(p,q)$.

Alternatively,

Definition 2. We can define an outcome $(p,q)\in \triangle_1\times \triangle_2$ to be $\delta -$ approximate Pareto efficient if $\delta=\inf \{d((p,q),(p',q'))|(p',q')\in \text{Set of Pareto efficient Outcomes}\}$, where $d$ is a metric on $\triangle_1\times \triangle_2$ and the Set of Pareto efficient Outcomes $\neq\emptyset$.

You can refine the definition(s) to suit your needs.