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If someone increase the transaction fee then it would get priority and will confirmed first but normal people will try to keep the fee as low as possible some if someone or few trying to increase it won't even last long for a day.
It depends on the amount you want to transfer. No one wants to pay $4(the highest fee currently) transferring $20, but if it's $20,000 and you want it fast, why not? Apparently, not so many people are craving for their transaction to be included in the very first block. They decide to wait additional 10-15 minutes but save around $3 on tx fee, and thus we don't have fees race, when transaction fees go higher and higher. The above mentioned race can be bad not only for regular users, but for miners as well, because if BTC price goes down consequently, they lose a lot of USD on each block reward.