Unfair coin tossing

An ideal coin tossing resource for two parties outputs the same random bit to both parties. We introduce the notion of an unfair coin tossing resource by relaxing both the fairness and the non-influenceability guarantees that an ideal coin toss would provide. The presence of this non-ideal behavior is necessary in order to understand what coin tossing protocols really achieve in the setting of two distrustful parties, since it is known that such an ideal coin tossing resource cannot be constructed whenever a majority of players is dishonest.