Your second option still implicitly assumes that you’re the only decider. In fact each of the possible deciders in each branch of the simulation would be making an evaluation of expected payoff—and there are nine times as many in the “tails” branches.
There are twenty branches of the simulation, ten with nine deciders and ten with one decider. In the one-decider branches, the result of saying “yea” is a guaranteed $100; in the nine-decider branches, it’s $1000 in the single case where everyone agrees, $700 in the single case where everyone disagrees, and $0 otherwise. The chance of defection depends on the strength of everyone’s precommitments, and the problem quickly goes recursive if you start taking expected behavior into account, but in the simplified case where everyone has a well-defined probability of defection the expected payoffs are much higher if your precommitments are strong.
Your second option still implicitly assumes that you’re the only decider. In fact each of the possible deciders in each branch of the simulation would be making an evaluation of expected payoff—and there are nine times as many in the “tails” branches.
There are twenty branches of the simulation, ten with nine deciders and ten with one decider. In the one-decider branches, the result of saying “yea” is a guaranteed $100; in the nine-decider branches, it’s $1000 in the single case where everyone agrees, $700 in the single case where everyone disagrees, and $0 otherwise. The chance of defection depends on the strength of everyone’s precommitments, and the problem quickly goes recursive if you start taking expected behavior into account, but in the simplified case where everyone has a well-defined probability of defection the expected payoffs are much higher if your precommitments are strong.