I don't understand these reward schemes. Surely they are being funded by higher merchant fees, which means merchants charge more, which just means that the rewards are a transfer scheme between people who pay by cash/unrewarded cards and those with rewards cards.

I think you are neglecting the fact that, even if some people don't carry a balance, most credit card users do, and the rewards scheme acts as an incentive (perhaps not a rational one, but people demonstrably don't behave rationally) to additional use of the card, all other things being equal. So, credit card issuers probably make up the additional costs of rewards cards in interest and fees from those cardholders, on average, even though perfectly disciplined use provides a net win to the cardholder.

They're mostly being funded by the people who don't pay off their balance in full every month. Accidentally miss one or two payments and all of a sudden all your 'profits' from using the card have been wiped out.

I realize I'm losing money by paying with cash, because I'm paying the markup for credit card swipes (assuming a retailer that charges the same for credit cards and cash). Even if I can only get back 2% (there are several no-free credit cards that give 2% back on all purchases), that's more than with the cash where I'm paying higher prices and not able to get any of that back with cashback.

Depending on agreement, card type, and other factors, the swipe fee is sometimes higher than 2%, so I don't always get all the difference back.

There's a lot of truth to that (cash paying customers pay the same, even though the cost to serve them is less).

But unrewarded cards aren't really any different. Most merchants still have to pay the same high fee for them. Since the bank/processor is going to take the maximum amount of money they can regardless, and most merchants will have to pay that full fee regardless, you might as well use a rewards card over an unrewarded one.

Handling cash isn't free for a merchant. It has to be stored, counted, transported, banked, insured and it's easily stolen. Even though the fees aren't the same cash still has a fee.

(In the UK)

American Express cards offer the best rewards, but the cost to the merchant is much greater. Many shops or restaurants don't accept AmEx, and sometimes those that do will say they prefer you to use an alternative card.

But within one network, I think it's correct that the cost is the same regardless of the card's promotion.

That's exactly right. It's a game you are forced to play, even if you don't want to use any cards at all.

I've had the same qualms, my guess was always that it is subsidized by those that use the rewards cards in a suboptimal way, such as carrying a balance and paying interest. Either way, the end result has been me having a desire to be on the rewards card side and using it for a net gain.

Merchants can't always charge more. The amount of the charge that's passed on to the consumer depends on the good/service being sold and its price elasticity of demand.

That's the prisoner dilemma, and leads to the expected results.

You can also get Amazon gift cards at grocery stores that provide gas points. For example, at Safeway $500 Amazon gift card purchase will give you $30 AMEX cashback + $25 discount on gas if you dont mind bringing a couple of gas cans with you to the station.