Envy-freeness, also known as no-envy, is a criterion for fair division. It says that, when resources are allocated among people with equal rights, each person should receive a share that is, in their eyes, at least as good as the share received by any other agent. In other words, no person should feel envy.

Suppose a certain resource is divided among several agents, such that every agent ${\displaystyle i}$ receives a share ${\displaystyle X_{i}}$. Every agent ${\displaystyle i}$ has a personal preference relation ${\displaystyle \succeq _{i}}$ over different possible shares. The division is called envy-free (EF) if for all ${\displaystyle i}$ and ${\displaystyle j}$:

Another term for envy-freeness is no-envy (NE).

If the preference of the agents are represented by a value functions ${\displaystyle V_{i}}$, then this definition is equivalent to:

Put another way: we say that agent ${\displaystyle i}$ envies agent ${\displaystyle j}$ if ${\displaystyle i}$ prefers the piece of ${\displaystyle j}$ over his own piece, i.e.:

A division is called envy-free if no agent envies another agent.

The notion of envy-freeness was introduced by George Gamow and Marvin Stern in 1958. They asked whether it is always possible to divide the brandy (a heterogeneous resource) among 3 people, such that everyone gets their fair share, which is at least one-third. The difficulty arises from the fact that each person, when brandy has been divided, has their own opinion on which glass has the most brandy. For n=2 this can be done by the Divide and choose algorithm, but for n=3 the problem has been stated as unsolvable. In later works, this problem has been usually formulated as the envy-free cake-cutting, where the cake is divided among children with different tastes.

In cake-cutting, EF means that each child believes that their share is at least as large as any other share; in the chore division, EF means that each agent believes their share is at least as small as any other share (the crucial issue in both cases is that no agent would wish to swap their share with any other agent). See chore division.