Minimax and Worst-Case Optimization

Most optimization so far assumes you know the objective function precisely and are just trying to find its minimum. But many real decisions face genuine uncertainty about what the "objective" even looks like — a market maker doesn't know which direction an informed trader will pick off her quote, and a portfolio manager doesn't know exactly how future returns or correlations will behave. Optimizing for the average case can leave you badly exposed if reality turns out to be worse than average. Minimax optimization instead asks: across every strategy I could choose, and every scenario an adversary (or an uncooperative world) could choose in response, what's the best guarantee I can make about my worst possible outcome?

An analogy: choosing an umbrella, not betting on the weather

Suppose you're picking a coat for a trip with unknown weather. Betting on the average forecast might mean packing a light jacket, which is fine most days but leaves you soaked if it rains hard. A minimax approach instead asks: for each possible coat, what's the worst weather could do to me while wearing it? A heavy raincoat's worst case (mild sunburn on a hot day) is much better than a light jacket's worst case (soaked and cold in a storm) — so minimax picks the raincoat, even though on an average day the light jacket might have been more comfortable. It optimizes against the worst scenario, not the typical one.

The framework, one symbol at a time

Given a decision variable $x$ (your strategy) chosen from a feasible set, and an adversarial or uncertain parameter $y$ chosen from its own feasible set, with a cost function $f(x, y)$, the minimax problem is

$$\min_x \; \max_y \; f(x, y) .$$

In plain English: for every candidate $x$ you might choose, first imagine the adversary picks the $y$ that makes things worst for that particular $x$ (the inner $\max$), then choose the $x$ that makes this worst case as small as possible (the outer $\min$). This is fundamentally different from $\max_y \min_x f(x,y)$ (the adversary committing first) — in general these two quantities differ, with $\min_x\max_y f \geq \max_y\min_x f$ always holding (this gap is closely related to weak duality), and only equal under special conditions such as convexity in $x$, concavity in $y$, and a compact feasible set (the minimax theorem).

Worked example 1: a simple 2x2 payoff table

A market maker must set a spread as either "wide" or "narrow," and the market can turn out either "calm" or "volatile." Costs (in expected $ lost, lower is better) are: wide+calm = 3, wide+volatile = 4, narrow+calm = 1, narrow+volatile = 9. Choosing "narrow" has worst case $\max(1,9) = 9$. Choosing "wide" has worst case $\max(3,4)=4$. The minimax choice is "wide," with guaranteed worst-case cost of 4 — even though "narrow" is much better ($1 vs $3) if the market happens to be calm, minimax refuses to bet on that, and picks the option that limits maximum possible damage.

Worked example 2: robust portfolio weight under return uncertainty

A quant doesn't trust a single return estimate for an asset and instead considers it could be anywhere in $[-2\%, 6\%]$ (worst to best case) depending on unmodeled risk. For a simple one-asset allocation of weight $w \in [0,1]$, portfolio return under the worst case is $w \times (-2\%) + (1-w)\times 0\%$ (cash return zero), i.e., the worst case is always $-2w\%$, monotonically worsening as $w$ increases. Minimizing the worst case alone would pick $w=0$ (no exposure at all) — a valid but maximally conservative answer, illustrating why pure minimax is often blended with expected-value considerations (a portfolio manager rarely uses raw worst-case optimization alone, since it can be needlessly pessimistic; it's more often used to set risk limits rather than the entire allocation).

What this means in practice

Minimax reasoning underlies robust portfolio optimization (protecting against the worst plausible set of returns rather than trusting a single point estimate), adversarial machine learning (training models to resist worst-case perturbations), and market-making spread-setting when the counterparty's information advantage is unknown. It's also the backbone of two-player zero-sum game theory, where the minimax value is provably the rational outcome both sides converge to.