# Approximate Pareto efficiency with ordinal preferences

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?

• I doubt that there is a useful context-independent generalization. For an approach that uses the structure of commodity spaces, take a look at Debreu's 1951 paper "The Coefficient of Resource Utilization." Commented May 31, 2023 at 17:08

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.

• Sounds good, thanks! Is there any reference where these or similar definitions are used? Commented Jun 1, 2023 at 9:08
• No, I don't think so, and if there are some, I am not aware of it. Let me know if you want to discuss these further; I'll be happy to discuss them.
– Amit
Commented Jun 1, 2023 at 9:17
• So, intuitively, an outcome is approximately-Pareto-optimal if it is near a Pareto-optimal outcome. What do you think of the following alternative definition: "an outcome is approximately-Pareto-optimal if it is not Pareto-dominated by a faraway outcome". Formally: "an outcome $(p,q)$ is $\epsilon$-Pareto-optimal if there is no outcome outside its $\epsilon$-neighborhood which Pareto-dominates $(p,q)$." Is it stronger/weaker than your definition? Commented Jun 1, 2023 at 15:05
• Yes, that's right. An outcome is approximately-Pareto-optimal if it is near a Pareto-optimal outcome and continuity of $u_i$s guarantees that.
– Amit
Commented Jun 2, 2023 at 2:25
• For comparing the two definitions, let me call them: Def - old: An outcome $(p,q)$ is $\epsilon-$Pareto-optimal if there is exists a Pareto optimal allocation in its $\epsilon-$ neighbourhood. Def - new: An outcome $(p,q)$ is $\epsilon-$Pareto-optimal if there is no outcome outside its $\epsilon-$ neighbourhood which Pareto-dominates $(p,q)$.
– Amit
Commented Jun 2, 2023 at 2:34