n players, you need two identical decks of cards, each with cards numbered
n-1. Keep the first deck ordered from
1 at the bottom, to
n-1 at the top, and add a random card from the second deck to the top of the first deck. Set the rest of the second deck aside. Each player will get a single card from the first deck, telling them the relative position of the (alleged) saboteur. For example, if you get a
2 that means the saboteur is the 2nd person to your left (or yourself). Because the cards are ordered, all the cards from the first deck will point to the same person. The one card from the other deck points to another player at random.
To distribute the cards, the order must be maintained but the starting point can be anywhere in the deck. So, the deck is "shuffled" by repeatedly cutting it, which preserves the order but randomizes the start. If the cards are dealt clockwise (to the left), then the relative positions will be also be clockwise.
Everyone has equal probability of being the saboteur, and the civilian the saboteur is mistakenly suspicious of is chosen with equal probability. Equipment is relatively simple, you only need
2*(n-1) cards where
n is the maximum number of players. Two decks of playing cards with the same backs could be used in a pinch. (With a bit of preparation, a single deck could be used. Separate out
n-1 in two suits, and pick one of each rank at random to separate the two "decks".) The only drawback I see is that mis-shuffling will mess things up, but if players are careful it shouldn't be a problem.
For a worked example of this scheme see TTTPPP's answer.
I'll save my previous solution attempts below, none work completely as the one above does.
Almost Solution: Lots of ranked cards
Have many more uniquely ranked cards than players, e.g. cards numbered 1 to 100. Every player is dealt one card, and whoever has the lowest number is the saboteur. Players reveal their cards, as you suggest, to everyone but themselves (maybe Indian Poker-style, card on forehead). It generalizes: need more than one saboteur? Use the two lowest cards. Need a unknowing "citizen leader"? Whoever has the highest card.
You could even do it with a regular deck of playing cards. Just determine a rank order (make sure everyone knows if Aces are high or low) and a suit order to breaks ties in rank (such as Clubs, Spades, Hearts, Diamonds, as in Sheepshead--which is where I got this idea).
Everyone has a guess to who the saboteur is. Everyone except the saboteur is right about their guess. It can be implemented with standard supplies.
The non-saboteurs can't be 100% sure that they aren't the saboteur--but I really don't see a way around this. It requires a large number of cards that probably won't have another use in the game. During the reveal, there's a danger of people accidentally giving it away by asking questions clarifying the ranking of cards/staring/asking another player to confirm who they think the saboteur is.
It also enables people to make some educated guesses--if no one else has a particularly low card, you might get suspicious that you're the saboteur. My guess is people will overthink this and it will actually throw people off more often than it will give a good hint, which might end up making things more exciting.
Other Brainstorming: Cards numbered from
This has a slightly different result, but it might be on to something so I'm recording it here in case it inspires someone else: If you have
n players, have
n+1 cards labeled from
n+1. Deal each player one card, with the remaining card remaining hidden (call it the hole card). The saboteur is whoever holds the card one above the hole card, with the
1 card considered one above the
n+1 card. As above, everyone reveals their card to all but themselves. Let's say we have
n=5 players, so 6 cards, and the
3 card is the hole card. The two people adjacent to the hole card (including the saboteur, who is 1 above the hole card) see a "big hole", so they know that their card is either 1 above or 1 below the hole card, so they know they have a 50% probability of being the saboteur, and that whoever is above the big hole is the only other possibility. The people who are not adjacent the hole card see two small holes, so they can also narrow the saboteur down to two possibilities, but they know for sure that they themselves are not the saboteur.