4. A related natural occurrence
Consider the scenario where some user broadcasts a transaction with an anomalously large fee, U₁. Let's say that the chance of winning the next block is α, your proportion of the total power (in hashes/second), and that the typical reward for a block is U₀. Then the expected utility of cooperation (strategy C₀) is (something like):
E[C₀] = αU₀
In order to earn the enormous reward U₁ (strategy C₁) you'd have to win TWO blocks before the rest of the network wins one:
E[C₁] = α²U₁
So, in the case that U₁/U₀ ≥ α⁻¹, a rational miner with hashpower α will defect, even if all others cooperate. The larger your share of the hash power, the lower the breakpoint for a prize be to be worth contending over.
Example: Suppose the (fictional) RationalMiningPool accounts for about 33% of the hashpower. A software glitch at MyPHPWebWallet.com results in a submitted transaction with a fee exceeding 75 btc (the typical block reward is only 25 btc). RationalMiningPool does not find the first winning block to include this fee; however, it decides to keep fighting for the big prize, since the chance of finding the next two consecutive solutions is decent and the high value makes it worthwhile.
One funny consequence of this example is that if you offer a really large fee... instead of having your transaction confirm more quickly, it may actually end up taking longer as it keeps ending up in orphaned chains as miners keep fighting over it!
