That isn't the same question at all. The amount of the penalty for not using the loyalty card is chosen by the merchant and the merchant naturally chooses an amount that induces a majority of their customers to use it. That doesn't mean the customers like it.
The interesting question is to what extent the customer benefits from providing the merchant with the information outside of the artificial penalty. But in all likelihood providing the information has negative value to the customer, because the merchant can use the information against the customer, e.g. to better determine when they can get away with charging higher prices.
The trouble is that it's a collective action problem. Paying the penalty to opt out doesn't prevent the merchant from figuring out how to get more of your money without providing you with any additional value, because they can do it with a statistical sample of people who are like you but didn't opt out. So everyone is collectively worse off when a critical mass of people use the loyalty cards, but taking the "discount" is in the selfish interest of each person individually. Classic tragedy of the commons.
The way we solve problems like that is with legislation, e.g. by requiring merchants to charge the same price regardless of whether the customer uses a loyalty card.