It appears that running the tip selection algorithm on the tangle does not only require seeing a large part of it but also involves some computation effort that is not insignificant. This leads to the question if light nodes can delegate this effort to full nodes. Is this so and if yes, how are full nodes rewarded for performing this task.
Currently the tip selection is only performed on full nodes, and cannot be performed on light nodes. This is not only true for the tip selection, but also for broadcasting transactions and for searching the balance or inclusion (confirmation) states.
You run a full node because either you are an individual who wants to help the network or have influence on its trust finding (similar to people who run Bitcoin/Ethereum full nodes but are not mining), or because you are an organization who uses IOTA and depends on having full nodes available at a certain service level agreement.
For both these cases, there is no need to reward full nodes (similar to Bitcoin/Ethereum).