Card game with arduino and nfc

Hi, i would like to create a physical version of Gwent using NFC chips and Arduino, if you don't know what Gwent is it's basically a card game where you have some cards, any card has a score and at the end of each round the sum of all the card of a player get summed and who has the higher score wins.
What I would like to make is a board with NFC (I guess they're called active chips but I'm not sure) chips that can read the information in the (also guessing passive chips) card's chips.
Once the card is on the table it should send the information to the Arduino so that it can sum all the points (for each player) and show them on a little screen.
Is there a way to do that?
I saw a friend using some passive chips for promoting something so i guess that the writing part, on the card's chip, should be kinda easy but i can't find any active chip. There are those things that can read/write on chips (like this one: Lettore codificatore da tavolo 13.56MHzMHz » but buying a lot of them would be too much expensive. Is there something that would let me do this cheaply (i would like to stay under 100€ for the whole thing) with Arduino?