Who is the winner in the Low-High card game?

Question

Recently, I do watch a Low-Card game in Korean Game show called Running Man. Here is the link. Basically, the game is like this.

1. There are 20 cards in the table, numbered from 1 to 20.
2. There are 4 players and each player need to choose two cards.
3. Then, each player need to give hint the cards they have. 1 - 10 called as Low and 11 - 20 is High. So, if player A get 11 and 15, he need to shout "High, High"
4. Then, each player need to discard one of their card and can only use one remaining card
5. So, bigger number beats smaller number (19 beats 18, 17, 16 ... 1) but card number 1 can beat card number 20 only.

At first, I though that this was a simple game but the more I think, the game can come a bit confusing.

Consider that:

1. Player 1 have card 19
2. Player 2 have card 20
3. Player 3 have card 1
4. Player 4 have card 2

Who will win the bet? Well;

1. Player 1 can beat Player 3 and Player 4
2. Player 2 can beat Player 1 and Player 4
3. Player 3 can beat Player 2
4. Player 4 can beat Player 3

Of course, I also took a consideration that the confusing can happen if there are more than two players in the game. If there only two players, then it will be a simple game.

Answer 1

The game is not well defined in that case since you have a collection 3 people that beat one of the others and lose to one of the others, meaning there is no "highest" card.

I think the way it's meant to be played in the video is that 1 only wins if there are only 20's and 1's on the table. That's just a guess, but otherwise the rules just don't cover that case.