This paper extends the classic cake-cutting problem to a situation in which the "cake" is divided among families. Each piece of cake is owned and used simultaneously by all members of the family. A typical example of such a cake is land. We examine three ways to assess the fairness of such a division, based on the classic no-envy criterion: (a) Average envy-freeness means that for each family, the average value of its share (averaged over all family members) is weakly larger than the average value of any other share; (b) Unanimous envy-freeness means that in each family, each member values the family's share weakly more than any other share; (c) Democratic envy-freeness means that in each family, at least half the members value the family's share weakly more than any other share. We study each of these definitions from both an existential and a computational perspective.
@cite_17 @cite_27 study the concept of cake-cutting. Their model is the standard cake-cutting model in which the cake is divided among (and not among families as in our model). They define a group-envy-free division as a division in which no coalition of individuals can take the pieces allocated to another coalition with the same number of individuals and re-divide the pieces among its members such that all members are weakly better-off. Coalitions are also studied by @cite_31 @cite_7 . In our setting, the families are pre-determined and the agents do not form coalitions on-the-fly. In an alternative model, in which agents allowed to form coalitions based on their preferences, the family-cake-cutting problem becomes easier. For instance, it is easy to achieve a division with connected pieces between two coalitions: ask each agent to mark its median line, find the median of all medians, then divide the agents to two coalitions according to whether their median line is to the left or to the right of the median-of-medians.
@cite_27 @cite_31 @cite_7 @cite_17
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Cake cutting is a common metaphor for the division of a heterogeneous divisible good. There are numerous papers that study the problem of fairly dividing a cake; a small number of them also take into account self-interested agents and consequent strategic issues, but these papers focus on fairness and consider a strikingly weak notion of truthfulness. In this paper we investigate the problem of cutting a cake in a way that is truthful and fair, where for the first time our notion of dominant strategy truthfulness is the ubiquitous one in social choice and computer science. We design both deterministic and randomized cake cutting algorithms that are truthful and fair under different assumptions with respect to the valuation functions of the agents. Cake cutting has been recognized as a fundamental model in fair division and several envy-free cake cutting algorithms have been proposed Recent works from the computer science field proposed novel mechanisms for cake cutting, whose approaches are based on the theory of mechanism design; these mechanisms are strategy-proof, i.e., no agent has any incentive to misrepresent her utility function, as well as envy-free. We consider a different type of manipulations; each agent might create fake identities to cheat the mechanism. Such manipulation have been called Sybils or false-name manipulations, and designing robust mechanisms against them, i.e., false-name-proof, is a challenging problem in mechanism design literature. We first show that no randomized false-name-proof cake cutting mechanism simultaneously satisfies ex-post envy-freeness and Pareto efficiency We then propose a new randomized mechanism that is optimal in terms of worst-case loss among those that satisfy false-name-proofness, ex-post envy-freeness, and a new weaker efficiency property. However, it reduces the amount of allocations for an agent exponentially with respect to the number of agents. To overcome this negative result, we provide another new cake cutting mechanism that satisfies a weaker notion of false-name-proofness, as well as ex-post envy freeness and Pareto efficiency. We propose an online form of the cake cutting problem. This models situations where agents arrive and depart during the process of dividing a resource. We show that well known fair division procedures like cut-and-choose and the Dubins-Spanier moving knife procedure can be adapted to apply to such online problems. We propose some fairness properties that online cake cutting procedures can possess like online forms of proportionality and envy-freeness. We also consider the impact of collusion between agents. Finally, we study theoretically and empirically the competitive ratio of these online cake cutting procedures. Based on its resistance to collusion, and its good performance in practice, our results favour the online version of the cut-and-choose procedure over the online version of the moving knife procedure. We study cake cutting on a graph, where agents can only evaluate their shares relative to their neighbors. This is an extension of the classical problem of fair division to incorporate the notion of social comparison from the social sciences. We say an allocation is locally envy-free if no agent envies a neighbor's allocation, and locally proportional if each agent values its own allocation as much as the average value of its neighbors' allocations. We generalize the classical Cut and Choose" protocol for two agents to this setting, by fully characterizing the set of graphs for which an oblivious single-cutter protocol can give locally envy-free (thus also locally-proportional) allocations. We study the price of envy-freeness , which compares the total value of an optimal allocation with that of an optimal, locally envy-free allocation. Surprisingly, a lower bound of @math on the price of envy-freeness for global allocations also holds for local envy-freeness in any connected graph, so sparse graphs do not provide more flexibility asymptotically with respect to the quality of envy-free allocations.