Incentives for early arrival (I4EA) was recently proposed for studying online cooperative games. In an online cooperative game, players arrive in an unknown order, and the value increase after each player arrived should be distributed immediately among all the arrived players. Although there is only one arriving order in the game, we also hope that the value distribution is equal to their Shapley value in expectation. To achieve these goals, the early solutions ignored the fairness in each single arriving order. More specifically, an important player may receive nothing in a game, which seems unfair in reality. To combat this, we propose refined fairness in this paper and design new solutions in 0-1 value games. Specifically, we compute the distance of the distribution in each order to the Shapley value and aim to minimize it. We propose a new mechanism called Egalitarian Value-Sharing (EVS) to do so. We also show that the mechanism can maximize the egalitarian welfare among all the players who made contributions.

Diffusion auction design is a new trend in mechanism design which extends the original incentive compatibility property to include buyers' private connection report. Reporting connections is equivalent to inviting their neighbors to join the auction in practice. Then, the social welfare is collectively accumulated by all participants: reporting high valuations or inviting high-valuation neighbors. Hence, we can measure each participant's contribution by the marginal social welfare increase due to her participation. Therefore, in this paper, we introduce a new property called Shapley fairness to capture participants' social welfare contribution and use it as a benchmark to guide our auction design for a fairer utility allocation. Not surprisingly, none of the existing diffusion auctions has ever approximated the fairness, because Shapley fairness depends on each buyer's own valuation and this dependence can easily violate incentive compatibility. Thus, we combat this challenge by proposing a new diffusion auction called Permutation Diffusion Auction (PDA) for selling k homogeneous items, which is the first diffusion auction satisfying 1/(k+1)-Shapley fairness, incentive compatibility and individual rationality. Moreover, PDA can be extended to the general combinatorial auction setting where the literature did not discover meaningful diffusion auctions yet.

We study cooperative games where players join sequentially, and the value generated by those who have joined at any point must be irrevocably divided among these players. We introduce two desiderata for the value division mechanism: that the players should have incentives to join as early as possible, and that the division should be considered fair. For the latter, we require that each player's expected share in the mechanism should equal her Shapley value if the players' arrival order is uniformly at random. When the value generation function is submodular, allocating the marginal value to the player satisfies these properties. This is no longer true for more general functions. Our main technical contribution is a complete characterization of 0-1 value games for which desired mechanisms exist. We show that a natural mechanism, Rewarding First Critical Player (RFC), is complete, in that a 0-1 value function admits a mechanism with the properties above if and only if RFC satisfies them; we analytically characterize all such value functions. Moreover, we give an algorithm that decomposes, in an online fashion, any value function into 0-1 value functions, on each of which RFC can be run. In this way, we design an extension of RFC for general monotone games, and the properties are proved to be maintained.