Minimax means planning for the worst your opponent can do
Suppose one of your options could pay 5 but might cost you 100, while another guarantees at least 2. Minimax thinking picks the safe one. The rule, born in game theory, asks a pessimist's question: whatever I choose, what is the worst that can happen, and how do I make that worst as good as possible?
The idea is used in artificial intelligence, decision theory, statistics and philosophy. Framed as losses, it minimises the largest possible loss; framed as gains, it is called maximin, maximising the smallest gain. It began with zero-sum games, in which one side's win is the other's loss, whether players move in turn or at the same moment, and was later stretched to general decisions under uncertainty.
A small example shows the logic. A row player chooses T, M or B and a column player chooses L or R. Picking M could bring 5 but could also mean losing 10; B risks a loss of 100 in exchange for a possible 4. T, by contrast, pays at least 2 whatever the other side does, so T is the maximin choice: the best outcome you can be sure of without knowing your rival's move. The column player, reasoning the same way, can lock in at least 0 by playing L.
Minimax flips the order. Instead of maximising before you know what others will do, it describes the smallest value opponents can hold you to, or equivalently the most you can secure once you know their moves. That puts the player in a much stronger position, so the two numbers generally differ.
Simple maximin choices can be unstable. In one three-by-three zero-sum game, each side's cautious pick invites the other to switch for a gain, which provokes another switch, round and round until both realise choosing is hard. The escape is to randomise. In any zero-sum contest between two players with a limited menu of options, both players have randomised strategies and a number V exist so that the first can guarantee V and the second can guarantee minus V. Each side minimises the maximum the other can win, which in a zero-sum game also minimises its own maximum loss.
Source: Minimax
