Minimax, Maximin and Safety-First Play

Expected-value thinking asks "what's my average outcome if I play this strategy?" — a fine question when nature is indifferent to your choice, like a coin flip. But against an opponent who is actively trying to minimize your payoff (a genuinely adversarial game, not a random one), maximizing your average outcome can walk straight into a trap, because a smart opponent will steer toward whichever branch hurts you most. Minimax reasoning instead asks: for each of my strategies, what's the worst the opponent can do to me, and which strategy makes that worst case least bad?

The two names, one idea

Maximin is the player's own perspective: among your available strategies, find the one whose worst-case payoff is the highest — you're maximizing the minimum you're guaranteed. Minimax is the same idea stated from the perspective of bounding the opponent: they're trying to minimize the maximum payoff they'll let you have. In a two-player zero-sum game (one player's gain is exactly the other's loss), these two calculations converge to the same value and the same optimal strategy — this convergence is the content of von Neumann's minimax theorem, one of the founding results of game theory.

Worked example

You and an opponent each secretly choose Rock, Paper, or a rigged third option: you get a payoff table (your gain, in dollars) where your choices are Left or Right, and the opponent's choices are Up or Down:

	Opponent: Up	Opponent: Down
You: Left	+10	−8
You: Right	+1	+2

If you play Left hoping the opponent picks Up, you could win +10 — but a rational adversary, seeing you'll play Left, plays Down and hands you −8. The worst case for Left is −8. If you play Right, the worst case is +1 (the opponent's best response, Down, only limits you to +2, and Up gives you +1 — either way you're never worse than +1). Maximin compares the worst cases: −8 for Left versus +1 for Right, and picks Right, guaranteeing at least +1 regardless of what the opponent does — even though Left has a tempting +10 sitting in it, that number is only available if the opponent cooperates with your hope, which a genuine adversary never will.

What this means in practice

Minimax reasoning is the right lens whenever you're facing a genuinely adversarial counterparty who profits directly from your mistakes — a market maker being picked off by informed flow, or negotiating a price with someone who knows your walk-away point. It's the wrong lens against a passive, non-adversarial environment, where expected-value thinking recovers more of the upside a hostile opponent would never let you keep.