Bingo works better as a metaphor for death than as a fun party game - you know your number's going to be called eventually, but you don't know when and there's nothing you can do about it. Though there is at least one online guide to bingo strategy, it's not strategy in the conventional sense (since tips are things like "use auto-daub" and "buy more cards"). While this has its advantages - primarily in lowering the bar to playing about as far as it will go - here are some ideas for a way to add actual strategy to Bingo. As a downside, note that most of these make the game less suitable for vast numbers of players. (If you need a thinking game for that situation, try Streams.)
The basic idea: only the first square is picked randomly, in the usual manner; after that, one player is chosen randomly from those who did not mark a square, and they choose the next square to mark. Control then passes to one player randomly chosen from those who didn't mark a square and so on until the game ends.
This means that any move that gives you an advantage (by marking one of your squares) also reduces your control over subsequent outcomes (by passing play to someone else). Since it's typical to play with multiple cards, count each "card" as a player - while this makes it easier to monopolize the game, you are necessarily giving your enemies a square for every one you give yourself, which means that as long as no one player holds a majority of the cards everyone should have a fighting chance. (Really, though, it might be better to limit everyone to the same number of cards rather than allow power imbalances.)
One potentially unclear situation is what to do when everyone (or nobody) marks a square. Two obvious options are 1) pick a random square in the usual manner, or 2) pass control to a random player. The first option adds a strong element of chance to the game as non-intentional moves can change strategies, while the second gambles on getting control again directly and skipping the normal limit on passing control.
The above rules are all that's needed to make the game strategic, but here are some other complications that might make it more interesting:
- Each card has a list of numbers it prevents the holder from calling (these may be public or private)
- Each round only certain numbers can be called (for example alternate rows, columns, or even and odd)
- At the start of a round, certain squares are (publicly or secretly) marked as causing various effects such as letting the current player go again, marking another square randomly, or transferring control randomly (without regard to who marked the square)
I'm not sure how well any of this would work, but I'd like to give it a try sometime. If you give it a try, be sure to let me know. Ψ