I'm confused about defection becoming a dominant strategy.. Because the existence of a dominant strategy suggests to me that there should exist a unique Nash equilibrium here, which is not the case. Everyone defecting is a Nash equilibrium, but 50 people cooperating and 49 defecting is a Nash equilibrium as well, and a better one at that. Something (quite likely my intuition regarding Nash equilibria in games with more than 2 players) is off here. Also, it is of course possible to calculate the optimal probability that we should defect and I agree with FeepingCreature that this should be 0.5-e, where e depends on the size of the player base and goes to 0 when the player base becomes infinite. But I highly doubt that there's an elegant formula for it. It seems (in my head at least) that already for, say, n=5 you have to do quite a bit of calculation, let alone n=99.
Nice. If we analyze the game using Vitalik's 2x2 payoff matrix, defection is a dominant strategy. But now I see that's not how game theorists would use this phrase. They would work with the full 99-dimensional matrix, and there defection is not a dominant strategy, because as you say, it's a bad strategy if we know that 49 other people are cooperating, and 49 other people are defecting.
There's a sleight of hands going on in Vitalik's analysis, and it is located at the phrase "regardless of one’s epistemic beliefs [one is better off defecting]". I...
Vitalik Buterin has a new post about an interesting theoretical attack against Bitcoin. The idea relies on the assumption that the attacker can credibly commit to something quite crazy. The crazy thing is this: paying out 25.01 BTC to all the people who help him in his attack to steal 25 BTC from everyone, but only if the attack fails. This leads to a weird payoff matrix where the dominant strategy is to help him in the attack. The attack succeeds, and no payoff is made.
Of course, smart contracts make such crazy commitments perfectly possible, so this is a bit less theoretical than it sounds. But even as an abstract though experiment about decision theories, it looks pretty interesting.
By the way, Vitalik Buterin is really on a roll. Just a week ago he had a thought-provoking blog post about how Decentralized Autonomous Organizations could possibly utilize a concept often discussed here: decision theory in a setup where agents can inspect each others' source code. It was shared on LW Discussion, but earned less exposure than I think it deserved.
EDIT 1: One smart commenter of the original post spotted that an isomorphic, extremely cool game was already proposed by billionaire Warren Buffett. Does this thing already have a name in game theory maybe?
EDIT 2: I wrote the game up in detail for some old-school game theorist friends:
The attacker orchestrates a game with 99 players. The attacker himself does not participate in the game.
Rules:
Each of the players can either defect or cooperate, in the usual game theoretic setup where they do announce their decisions simultaneously, without side channels. We call "aggregate outcome" the decision that was made by the majority of the players. If the aggregate outcome is defection, we say that the attack succeeds. A player's payoff consists of two components:
1. If her decision coincides with the aggregate outcome, the player gets 10 utilons.
and simultaneously:
2. if the attack succeeds, the attacker gets 1 utilons from each of the 99 players, regardless of their own decision.
There are two equilibria, but the second payoff component breaks the symmetry, and everyone will cooperate.
Now the attacker spices things up, by making a credible commitment before the game. ("Credible" simply means that somehow they make sure that the promise can not be broken. The classic way to achieve such things is an escrow, but so called smart contracts are emerging as a method for making fully unbreakable commitments.)
The attacker's commitment is quite counterintuitive: he promises that he will pay 11 utilons to each of the defecting players, but only if the attack fails.
Now the payoff looks like this:
Defection became a dominant strategy. The clever thing, of course, is that if everyone defects, then the attacker reaches his goal without paying out anything.