
…em valid? It checks all the previous transactions that are correlated to the wallet you use to send Bitcoins via the references that each one has as inputs. To simplify and speed up the verification process a special record of unspent transactions is kept by the network nodes. Thanks to this security check, it is not possible to double-spend received Bitcoins.
We can involve a neutral third party. Each of us gives $50 to a third party, she then will give the total amount to the winner. But hey, she could also run away with all our money. So we end up in one of the first two options: trust or contract.