an adversary argument
Imagine playing a guessing game where the answer is not fixed in advance: a mischievous opponent watches your questions and quietly arranges the secret to be as inconvenient as possible, while staying consistent with every answer already given. If, no matter how cleverly you ask, the opponent can keep the truth ambiguous until you have asked many questions, then many questions are genuinely necessary. That is an adversary argument: a lower-bound proof in which a malicious 'adversary' answers the algorithm's probes adaptively to force lots of work.
Concretely, the adversary does not pick the input first. Instead it answers each comparison (or query) on the fly, choosing the reply that keeps the most possible inputs alive, subject only to staying consistent with all earlier replies. The algorithm cannot stop until only one valid answer remains. To prove a lower bound of k, you exhibit an adversary strategy guaranteeing that after fewer than k queries at least two consistent inputs with different correct answers still survive — so the algorithm cannot yet be certain and must ask more. A clean bookkeeping trick is to track 'state' the adversary maintains (for example, which elements are still known to be the maximum candidate) and show each query can retire only a bounded amount of uncertainty.
Adversary arguments are powerful because they are adaptive: the adversary tailors its answers to whatever the algorithm just did, which often yields tighter bounds than a static counting argument. They prove some of the prettiest exact bounds in the field — n-1 comparisons to find the maximum, ceil(3n/2) - 2 to find max and min together. The honesty caveats: the adversary must NEVER contradict an earlier answer (the input it implicitly maintains must always be consistent), or the proof is invalid; and like all these techniques the bound is worst-case and model-specific. A correct adversary argument shows the lower bound holds against every deterministic algorithm in the model, since the adversary can react to any of them.
Finding the maximum of n numbers: the adversary insists every element is still 'possibly the max' until it has lost a comparison. Each comparison can knock out at most one candidate, and to be sure of the winner all n-1 losers must be eliminated, so at least n-1 comparisons are forced — matching the obvious algorithm exactly.
The adversary answers adaptively to keep the truth ambiguous, forcing the algorithm to keep asking.
The adversary must stay consistent: there must always exist at least one real input matching every answer it has given, or the argument proves nothing. It is a proof device, not an actual opponent — it shows what every algorithm must face in the worst case.