1. Two players are each dealt one card face down.
2. Remaining cards from the deck are dealt one by one, face up. After each card is revealed, either player may request that the other turn over his or her hole card.
3. If a player's hole card matches a card that has been revealed, the requestor wins 1 unit. If the hole card does not match, the requestor loses 1 unit.
For example, the cards dealt are J55K3. Player A requests to see player B's card. Player B shows a 9. Player A gives 1 chip to player B.
Now suppose the cards dealt are 2589T. Player B requests to see player A's card. Player A thinks for a bit and requests to see player B's card as well. They turn over a 5 and a T. Each player wins one bet from the other, so the game is a draw. If player B has a J instead, then player A loses two bets, one from being challenged (correctly) by B and the other for challenging B incorrectly.
What does the optimal strategy for this game look like?
Let C = the number of cards dealt, and D = the number of distinct rank showing. If your rank has not yet been dealt, the probability that your opponent's card is out is (3 * D - (C - D) ) / (52-C-1) = (4D-C)/(51-C). If your rank is out, the probability is (4D-C-1)/(51-C).
For small number of cards, D dominates: your opponent is somewhat more likely than not to have a matching card out when D >= 8, for C < 12. 9 distinct cards is sufficient for C < 20. Should you ask for a showdown as soon as your opponent is more than 50% likely to have a matching card?
If you don't have a match to the board, then why not take another card off? Your opponent might make a mistake and challenge you. And the probability that your opponent will match after the next card is usually higher. But if you only challenge when you yourself have a match then your opponent can counter-challenge and at least draw, or maybe take 2 chips away from you! On the other hand, if your opponent counter-challenges every time, you can frequently challenge without a match and earn a profit.
So we need some strategy which incorporates both deception and the risk/reward benefit of waiting for another card.