We know that Felix is an incredibly hard to make potion, that only a few people can make it correctly, that it has a very long preparation time and that the ingredients are hard to acquire.

This would make Felix a very valuable —and probably utterly expensive— potion. So why would Slughorn give it to 6th year students only for the best attempt at making the Draught of Living Death? while that potion looks difficult to brew, it seems like a very big reward for a comparatively easy task.

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Slughorn is portrayed as a compulsive networker; someone who makes connections with his students at an early age and parlays those relationships into favourable outcomes for himself. Dumbledore describes him as someone who operates behind the scenes, influencing the rich and powerful through sheer force of persuasion.

‘Horace,’ said Dumbledore, relieving Harry of the responsibility to say any of this, ‘likes his comfort. He also likes the company of the famous, the successful and the powerful. He enjoys the feeling that he influences these people. He has never wanted to occupy the throne himself; he prefers the back seat – more room to spread out, you see. He used to handpick favourites at Hogwarts, sometimes for their ambition or their brains, sometimes for their charm or their talent, and he had an uncanny knack for choosing those who would go on to become outstanding in their various fields. Horace formed a kind of club of his favourites with himself at the centre, making introductions, forging useful contacts between members, and always reaping some kind of benefit in return, whether a free box of his favourite crystallised pineapple or the chance to recommend the next junior member of the Goblin Liaison Office.’

It makes perfect sense to create a sense of obligation with the top potion-maker in his senior class, someone who's very likely to go on to become an important and influential figure in the wizarding world.

What better way to create that obligation that by offering them the sort of lifelong memory of a "perfect day" that only Felix can provide, intimately connecting Slughorn to their happiest memory?

On top of that, because of Harry's heritage (and Slughorn's feelings about Harry's mother) Slughorn strongly indicates that he was expecting Harry to win, little realising that the only reason he succeeds is through dumb luck. He would view the bottle of Felix as less of a prize that could be won by anyone and more of a downpayment toward "collecting" Harry to add to his inner circle of confidants.

‘The clear winner!’ he cried to the dungeon. ‘Excellent, excellent, Harry! Good Lord, it’s clear you’ve inherited your mother’s talent, she was a dab hand at Potions, Lily was! Here you are, then, here you are – one bottle of Felix Felicis, as promised, and use it well!’

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