user1082: Let's say Alice has two CPUs, and Bob has only one. Let's assume Alice and Bob are the only miners in the Bitcoin network.
What will happen if Alice did the following:

Calculate the nonce for the next block and send it occasionally.
Whenever Bob solves and transmits a block, DON'T accept his block, but instead try to solve another block at the same height, and a successive block.
When done calculating, send both blocks to the network. They should be accepted instead of Bob's transaction, and since you're 2 times speedier than Bob, in average you'll be able to prevent Bob from mining coins.

While this is a somewhat artificial example, I think that it might represent a real problem. A group of strong participants can prevent weaker players from mining, even if this group does not control the majority of CPU resources in the network.

user4101: It appears that some strongest players - e.g. manufacturers of ASIC devices - purposely give up their chance at monopoly in exchange for immediate profit (sell their devices to multiple customers) - precisely to increase the reliability of the network.
While the network is vulnerable to the "51% attack", its result will be an immediate, catastrophic and permanent crash of the bitcoin economy. All of the investment in hardware that let whoever did it achieve the 51% hashpower will be irrevocably lost - as they will be able to dominate the market for a worthless resource, compromised bitcoin being deprived of any value.
The big players understand it and willingly give up a part of their hashing power, simply to keep Bitcoin strong, reliable, and desired - keeping the prices up. That way they produce less but earn more simply because that unit is more valuable.
In other words, it is in best interest of anyone producing or owning a lot of bitcoin to keep the bitcoin network running smoothly = keep value of bitcoins they produce/have high. Bitcoin is a network of trust; break that trust and it loses all value.
Such an attack would be possible e.g. by a financial institution with a grudge against Bitcoin, or some government. It would be costly but its only returns would be in whatever profit destroying bitcoin entirely would bring. Any immediate profits of corrupting the transaction process would be relatively small, and very short-lived compared to the damages (including own hardware becoming worthless) - so it is in best interest of anyone holding any power capable of doing it not to do it - a powerful economic incentive against this.

user575: This isn't something that can happen unintentionally. If everyone plays by the rules, someone who has x% of the hashrate will find x% of the blocks, no matter how small or large x is (of course, for very small x the variance will be substantial, but without any change in the expectation).
You can attempt a double spend even without having >50% hashrate, but the probability of success is small. The chance improves as your hashrate approaches 50%. What's special about >50% is that you are guaranteed eventual success, no matter how many blocks the receiver waits.
There's a little-known hashrate amplification attack (discussed here and here) that allows you to find more blocks than your hashrate normally allows, which doesn't require >50%. It still requires a relatively large hashrate, about 41%. For example, with 46% of the hashrate you can get 51.29% of the total blocks. For >50% hashrate you get 100% of the blocks.

user1061: That is essentially a 51% attack. It is a very real, but unlikely threat to the Bitcoin network. Historically as it turns out, it is more lucrative to simply add more processing power to the network than fight it.

1082: <p>Let's say Alice has two CPUs, and Bob has only one. Let's assume Alice and Bob are the only miners in the Bitcoin network.</p>

<p>What will happen if Alice did the following:</p>

<ol>
<li>Calculate the nonce for the next block and send it occasionally.</li>
<li>Whenever Bob solves and transmits a block, DON'T accept his block, but instead try to solve another block at the same height, and a successive block.</li>
<li>When done calculating, send both blocks to the network. They should be accepted instead of Bob's transaction, and since you're 2 times speedier than Bob, in average you'll be able to prevent Bob from mining coins.</li>
</ol>

<p>While this is a somewhat artificial example, I think that it might represent a real problem. A group of strong participants can prevent weaker players from mining, even if this group does not control the majority of CPU resources in the network.</p>
 4101: <p>It appears that some strongest players - e.g. manufacturers of ASIC devices - purposely give up their chance at monopoly in exchange for immediate profit (sell their devices to multiple customers) - precisely to increase the reliability of the network.</p>

<p>While the network is vulnerable to the "51% attack", its result will be an immediate, catastrophic and permanent crash of the bitcoin economy. All of the investment in hardware that let whoever did it achieve the 51% hashpower will be irrevocably lost - as they will be able to dominate the market for a worthless resource, compromised bitcoin being deprived of any value.</p>

<p>The big players understand it and willingly give up a part of their hashing power, simply to keep Bitcoin strong, reliable, and desired - keeping the prices up. That way they produce less but earn more simply because that unit is more valuable.</p>

<p>In other words, it is in best interest of anyone producing or owning a lot of bitcoin to keep the bitcoin network running smoothly = keep value of bitcoins they produce/have high. Bitcoin is a network of trust; break that trust and it loses all value.</p>

<p>Such an attack would be possible e.g. by a financial institution with a grudge against Bitcoin, or some government. It would be costly but its only returns would be in whatever profit destroying bitcoin entirely would bring. Any immediate profits of corrupting the transaction process would be relatively small, and very short-lived compared to the damages (including own hardware becoming worthless) - so it is in best interest of anyone holding any power capable of doing it not to do it - a powerful economic incentive against this.</p>
 575: <p>This isn't something that can happen unintentionally. If everyone plays by the rules, someone who has x% of the hashrate will find x% of the blocks, no matter how small or large x is (of course, for very small x the variance will be substantial, but without any change in the expectation).</p>

<p>You can attempt a double spend even without having >50% hashrate, but the probability of success is small. The chance improves as your hashrate approaches 50%. What's special about >50% is that you are guaranteed eventual success, no matter how many blocks the receiver waits.</p>

<p>There's a little-known hashrate amplification attack (discussed <a href="https://bitcointalk.org/index.php?topic=2227.msg30064#msg30064" rel="nofollow">here</a> and <a href="https://bitcointalk.org/index.php?topic=2227.msg30083#msg30083" rel="nofollow">here</a>) that allows you to find more blocks than your hashrate normally allows, which doesn't require >50%. It still requires a relatively large hashrate, about 41%. For example, with 46% of the hashrate you can get 51.29% of the total blocks. For >50% hashrate you get 100% of the blocks.</p>
 1061: <p>That is essentially a <a href="https://bitcoin.stackexchange.com/q/658/659">51% attack</a>. It is a very real, but unlikely threat to the Bitcoin network. Historically as it turns out, it is more lucrative to simply add more processing power to the network than fight it.</p>
