Jeopardy!: Most of their tournaments work the same way.
Week 1: 5 independent quarter-final games. The winners and the four top-scoring non-winners advance to the semifinals.
Week 2, Monday-Wednesday: The nine players appear in three semifinal games. Only the winners advance.
Thursday-Friday: The three finalists play two games, and their totals for the two days are added together to determine the winner.
The $100,000 Pyramid: The three players who win the $10,000 or $25,000 in the shortest amount of time over the course of six to eight weeks, i.e., since the previous tournament, come back for the $100,000 tournament. On any given day, two of them play two games. If one of them wins the $100,000, the tournament is over. If not, the player who wins more money in the Winner's Circle comes back the next day to face the player who was left out. (On Day 1, the players are the two with the fastest qualifying times.) This was the set-up for both Clark and Davidson's shows. Osmond's Pyramid has only had three tournaments, two of them with the same structure which doesn't have a guaranteed prize anyway, so I won't go into it.
The Joker's Wild: Generally, a variation on this ... 16 players are randomly paired off. Whoever wins the game (or best of 3 or best of 5) advances to the next level. In the second and third tournaments, the ones where Frank Dillon was the previous year's defending champion, he was excluded from all levels of play and faced the last person remaining for the title. When he lost, they made his successor start from the bottom with everyone else.
Celebrity Sweepstakes: I'm aware of one Tournament; they may have had more. The one I saw was pretty straightforward: Ten players appeared in one game each over the course of the week. The player with the biggest total in his game was the winner. I don't know whether the players were aware of the totals put up prior to their own games. (This is the only one where I'll state an opinion plainly: I don't think this made for interesting game play, especially if there was no prize for those who won their day but not the top prize. If you only want to do one week, bring back eight players and have the top two players come back Friday.)
