# Does Nash Envy Immunity

## TL;DR

This paper introduces a new stability concept called envy-proofness, which enhances robustness against irrational behaviors in games, and explores its relation to Nash equilibrium and immunity, including existence and computational aspects.

## Contribution

It proposes the envy-proof stability notion, analyzes its relationship with existing concepts, and studies existence and computation in two-player and multi-player games.

## Key findings

- Envy-proof profiles can be characterized and related to Nash equilibria.
- Existence of envy-proof strategies depends on game type and parameters.
- Algorithms for finding envy-proof strategies are developed for certain classes of games.

## Abstract

The most popular stability notion in games should be Nash equilibrium under the rationality of players who maximize their own payoff individually. In contrast, in many scenarios, players can be (partly) irrational with some unpredictable factors. Hence a strategy profile can be more robust if it is resilient against certain irrational behaviors. In this paper, we propose a stability notion that is resilient against envy. A strategy profile is said to be envy-proof if each player cannot gain a competitive edge with respect to the change in utility over the other players by deviation. Together with Nash equilibrium and another stability notion called immunity, we show how these separate notions are related to each other, whether they exist in games, and whether and when a strategy profile satisfying these notions can be efficiently found. We answer these questions by starting with the general two player game and extend the discussion for the approximate stability and for the corresponding fault-tolerance notions in multi-player games.