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I was wondering if some one could explain to me, in this example what the benefit of the website owner(s) is? It seems to me they do not benefit at all from this contract. There are obviously benefits to the user, namely they cannot possibly lose their bitcoins from what I can tell. In this scenario, what is the 'payment' that the user making besides not having access (as far as I can tell no one has access) to their coins for 6 months?

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In the example, the user is demonstrating that they are willing to lose access to some amount of Bitcoin for a period of time. The example is based on an assumption that a rational user wouldn't do that unless they were serious about something. The implied benefit is that casual users or bots won't want to put a lock on their funds.

However, as Tadje Dryja pointed out in a recent talk, time value of money is not necessarily equal among users. In the example, the contract is for 10 BTC. For some folks 10 BTC may be a lot to part with for 6 months. For others it may be no big deal. So it's questionable if the site gets the benefit they think they are.

Imagine an exclusive club where you had to deposit $1,000 just to get in the door, but you were guaranteed the return of your money when you left. It does act as a selection filter for who is able and willing to go through with this arrangement. Whether or not it really selects for the clientele the club is looking for is another question.

Also, at the risk of pointing out the obvious, I would add a step 7 to that example that the web site verify that Tx1 is confirmed on the block chain and is as expected before proceeding. The web site already knows the transaction ID for Tx1 since that was given in step 3.

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