One experiment randomly assigned participants either to the trust game or a coin-flip
game, in which one participant was told that if he or she handed over the 5 dollars, the other participant would then flip a coin to determine whether to return 10 dollars.
Whoever jumps best will probably come out on top, so it's a coin-flip
on that score' however, Il Duce has a better turn of foot than Paul Nicholls's gelding and, on a speed-favouring track, that just sways me his direction.
The main difference between their results and ours are: (a) they require fair Boolean-valued local coins, where we allow arbitrary distributions and ranges on the local coins; (b) they allow the adversary to replace a coin-flip with a new value of its choosing, where we assume a weaker adversary that can only hide a coin-flip by replacing it with a fixed value [perpendicular to]; and (c) they obtain a tight result that shows that combining the local coins with the majority function (or, in general, any threshold function) minimizes the adversary's influence over the global coin.
Since Pr[1|[Alpha], A] is not 0 or 1, A must eventually cause the protocol to execute a coin-flip after some deterministic extension [Alpha]' or [Alpha].
The lemma above implies that we can always reach an execution that either is null-valent or permits a coin-flip.