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The least of the worries that conform the basis on a Loyalty Reward Program is whether Blockchain could be part of it. Corporations use custom tailored loyalty programs based on their strategical and tactical needs. Companies tend to be zealous of their customers and data, and when they offer rewards that can be redeemed on third-party stores or businesses other than their own, it is because there is a defined win-win situation between the corporations involved (i.e. co-marketing of products).
Typically, a loyalty program may redirect its customers to a different site if the site is complementary and does not compete with them in any way. There are specific agreements placed on the volume of customers that redeem in those referenced stores, managing on the one hand discounts for the customers, and potentially some return for the corporation behind the loyalty program itself.
Blockchain is not the fastest of technologies, and the key feature should be transparency alongside immutability (i.e. read-only data). There is no benefit to creating a Loyalty Program based on the blockchain whatsoever in general terms, although some niche ideas could conform a side benefit if dealt with properly. For example, the Loyaly Program could state that it matches a percentage of the value redeemed by its customers to donate it to certain OGNs. They (the corporation) could use blockchain there in order to be transparent with their donations (although they can also do that currently with a simple webpage and a bit of trust on our behalf).