I have a couple of basic questions about NFL betting.
Are betting odds set in stone before the game begins, or do they evolve with the progression of the game?
What are "closing odds?" Are closing odds made during the game, or before it?
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It depends on the bookmaker.
Traditionally, things like the total, money line, and spread will be set before the game starts. During the game it is possible to choose these same bets for halves of the game and sometimes even quarters of the game.
The odds will change for say the second half bets as the first half plays out.
Some bookmakers may even offer a live/ever changing odds. You would be able to bet on the winner of the game or total during the game as it evolves. The only difference is that the odds will change with the result of each play. This is a very automated system.
Closing odds refer to the last odds offered before the game starts. When the odds are first chosen by the bookmaker, they are called the opening odds. You may hear something like "the spread opened at -4.5 and closed at -3". This means that during the time that the odds were made to the time they were last offered, the spread changed by 1.5 points. This can be for a number of factors (injury, weather, suspensions, etc.), but usually is changed according to where bets are being placed leading up to the game. See setting lines.
If you placed a bet at -4.5 then you will be locked in at that spread.
In the example of during the game betting the closing odds for say the second half total would be chosen before kick off of the second half.