Example: multi-party coin toss P1, P2 and P3 are playing a game: each of them outputs a random bit(0 or 1), then the final outcome is the XOR of the three bits. P1 prefers the outcome to be 0, and P2, P3 prefer 1. The players who can guess the right outcome could get a payoff of 1. otherwise, losers get nothing from this game. Notice: if they all prefer the same number, the outcome will be the globally preferred number.

What is multi-party coin toss In a game played by n parties(honest or corrupt), each party wants to get 0 or 1. Preference profile: each party has a preference. The vector of all parties’ preference, denoted Coin-toss protocol: a protocol is a system of interactions between players. Those n parties can jointly decide an outcome of the protocol between 0 and1. Outcome of the protocol: Payoff: Winners(prefer the outcome) can get a payoff of 1, and losers get 0.

Strong Fairness and Weak Fairness Strong Fairness: A corrupt party can’t bias the outcome. e.g. the outcome of 0 or 1 are still both be [½ -negl(k), ½ + negl(k)]. k is the security parameter of the protocol. Weak Fairness: A corrupt party can bias the outcome of the remaining honest party, but the bias must not be in the corrupt party’s favor. e.g. If the corrupt party prefers 0, then the Pr(outcome=1) is at least ½-negl(k).

However, Strong Fairness is too strong to satisfy, now, we are trying to extend the notion of weak fairness.

Here comes. Maximin Fairness. Cooperative-Strategy-Proof Fairness Here comes Maximin Fairness Cooperative-Strategy-Proof Fairness Strong Nash Equilibrium

Maximin Fairness Maximin fairness requires that no honest party should be harmed by any corrupt coalition. In other words, a corrupt coalition should not be able to (non-negligibly) decrease the expected payoff for any honest party relative to an all-honest execution. In an execution, with an adversary, the expected reward for any honest party is at least 1 −negl(κ).

Cooperative-strategy-proof Fairness Cooperative-strategy-proof (CSP) fairness requires that no deviation by a corrupt coalition of size up to n − 1 can noticeably improve the coalition’s total expected reward relative to an honest execution. It is not difficult to see that CSP fairness is equivalent to maximin fairness for zero- sum cases: when exactly half prefer 0 and half prefer 1. However, the two notions are incomparable in general.

Strong Nash Equilibrium SNE fairness requires that no deviation by a coalition can improve every coalition member’s expected reward. SNE fairness is strictly weaker than CSP fairness in general.

Nash Equilibrium Two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

Corruption Model

Maximin Fairness Unanimous preference profile: all players prefer 0 or 1. Almost unanimous preference profile: everyone agrees in preference except one party. Amply divided preference profile: there are at least 2 0-supporters and at least 2 1- supporters.

Maximin Fairness: the case for amply divided P

Maximin Fairness: the case for almost unanimous P Fail-stop adversaries Possibility of maximin fairness for fail-stop adversaries

Maximin Fairness: the case for almost unanimous P Malicious adversaries Impossibility of maximin fairness for malicious adversaries, even under computational assumptions.