The Max-Min Inequality

Harish Guda


One of my favorite results in optimization is the max-min inequality. This is taught in just about every introductory course in optimization, math. for econ, and/or allied courses. The result is as follows.

The Result

Consider a bivariate function \(f(z, w)\), \(z \in Z\), \(w \in W\). It’s easier to think of the sets \(Z\) and \(W\) as finite sets, or a convex subset of \(\mathbb{R}\) (although the result is more general). Then, \[ \max_{z \in Z} \min_{w \in W} f(z, w) \le \min_{w \in W}\max_{z \in Z}f(z, w). \]

An Interpretation

While it’s easy to algebraically think/vizualize this result in a 3D space, one intuitive way to think of this result is a game with an adversary (a zero-sum game). Suppose your (resp., adversary’s) action set is \(Z\) (resp., \(W\)). Your payoff from an action \(z\) and adversary’s action \(w\) is \(f(z,w)\), while the adversary’s payoff is \(-f(z, w)\). The result states that your best worst-case payoff is inferior to the worst best-case payoff. Stated differently, you are better off moving second than moving first in a game with an adversary. I find it intuitive to remember this result as the second-mover result.

Edit: I was informed recently that this interpretation is also present in the textbook Convex Optimization by Boyd and Vandenberghe (see Chapter 5.4.3).

Where This (Can) Show Up

This result has a lot of applications, but one place I’ve seen this result often is in robust optimization, which is essentially a game with an adversary. This result provides a simple bound on the worst payoff a decision-maker can get if they were to play an adversary.

A (Simple) Proof

I’ve seen at least three different proof ideas for this result. My favorite one is the proof by contradiction. I share that below.

Suppose the result does not hold. That is, \(\exists\) an \(f(\cdot)\) and \(z_0\) s.t. \[ \min_{w \in W} f(z_0, w) > \min_{w \in W} \max_{z \in Z} f(z,w). \] In words, the above inequality states that there exists an action \(z_0\) that guarantees you a strictly higher payoff than what would happen if you move second.

But this is equivalent to \[ \min_{w \in W}\max_{z \in \{z_0\}}f(z, w) > \min_{w \in W} \max_{z \in Z} f(z,w) \] That is, you do strictly better if your action set is the singleton set \(\{z_0\}\) than a superset of \(z_0\). But this is impossible! Since your action set (in the RHS) is a superset of \(z_0\), you must do at least as well. We have a contradiction, and this completes the proof.