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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Properties of discrete cake-cutting procedures that use a minimal number of cuts (n A¢â‚¬â€œ 1 if there are n players) are analyzed. None is always envy-free or efficient, but divide-and-conquer (D&C) minimizes the maximum number of players that any single player may envy. It works by asking n ? 2 players successively to place marks on a cake that divide it into equal or approximately equal halves, then halves of these halves, and so on. Among other properties, D&C (i) ensures players of more than 1 n shares if their marks are different and (ii) is strategyproof for risk-averse players. However, D&C may not allow players to obtain proportional, connected pieces if they have unequal entitlements. Possible applications of D&C to land division are briefly discussed. One key question in cooperative game theory is that of coalitional stability. A coalition in such games is stable when no subset of the agents in it has a rational incentive to leave the coalition. Finding a division of the gains of the coalition (an imputation) lies at the heart of many cooperative game theory solution concepts, the most prominent of which is the core. However, some coalitional games have empty cores, and any imputation in such a game is unstable. We investigate the possibility of stabilizing the coalitional structure using external payments. In this case, a supplemental payment is offered to the grand coalition by an external party which is interested in having the members of the coalition work together. The sum of this payment plus the gains of the coalition, called the coalition's "adjusted gains", may be divided among the members of the coalition in a stable manner. We call a division of the adjusted gains a super-imputation, and define the cost of stability (CoS) as the minimal sum of payments that stabilizes the coalition. We examine the cost of stability in weighted voting games, where each agent has a weight, and a coalition is successful if the sum of its weights exceeds a given threshold. Such games offer a simple model of decision making in political bodies, and of cooperation in multiagent settings. We show that it is coNP-complete to test whether a super-imputation is stable, but show that if either the weights or payments of agents are bounded then there exists a polynomial algorithm for this problem. We provide a polynomial approximation algorithm for computing the cost of stability. 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. 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. Cutting a cake, dividing up the property in an estate, determining the borders in an international dispute - such problems of fair division are ubiquitous. Fair Division treats all these problems and many more through a rigorous analysis of a variety of procedures for allocating goods (or 'bads' like chores), or deciding who wins on what issues, when there are disputes. Starting with an analysis of the well-known cake-cutting procedure, 'I cut, you choose', the authors show how it has been adapted in a number of fields and then analyze fair-division procedures applicable to situations in which there are more than two parties, or there is more than one good to be divided. In particular they focus on procedures which provide 'envy-free' allocations, in which everybody thinks he or she has received the largest portion and hence does not envy anybody else. They also discuss the fairness of different auction and election procedures. Coalition formation has been a very active area of research in multiagent systems. Most of this research has concentrated on decentralized procedures that allow self-interested agents to negotiate the formation of coalitions and division of coalition payoffs. A different line of research has addressed the problem of finding the optimal division of agents into coalitions such that the sum total of the the payoffs to all the coalitions is maximized (Larson and Sandholm, 1999). This is the optimal coalition structure identification problem. Deterministic search algorithms have been proposed and evaluated under the assumption that the performance of a coalition is independent of other coalitions. We use an order-based genetic algorithm (OBGA) as a stochastic search process to identify the optimal coalition structure. We compare the performance of the OBGA with a representative deterministic algorithm presented in the literature. Though the OBGA has no performance guarantees, it is found to dominate the deterministic algorithm in a significant number of problem settings. An additional advantage of the OBGA is its scalability to larger problem sizes and to problems where performance of a coalition depends on other coalitions in the environment. 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. Coalition formation is a key problem in automated negotiation among self-interested agents. In order for coalition formation to be successful, a key question that must be answered is how the gains from cooperation are to be distributed. Various solution concepts have been proposed, but the computational questions around these solution concepts have received little attention. We study a concise representation of characteristic functions which allows for the agents to be concerned with a number of independent issues that each coalition of agents can address. For example, there may be a set of tasks that the capacity-unconstrained agents could undertake, where accomplishing a task generates a certain amount of value (possibly depending on how well the task is accomplished). Given this representation, we show how to quickly compute the Shapley value--a seminal value division scheme that distributes the gains from cooperation fairly in a certain sense. We then show that in (distributed) marginal-contribution based value division schemes, which are known to be vulnerable to manipulation of the order in which the agents are added to the coalition, this manipulation is NP-complete. Thus, computational complexity serves as a barrier to manipulating the joining order. Finally, we show that given a value division, determining whether some subcoalition has an incentive to break away (in which case we say the division is not in the core) is NP-complete. So, computational complexity serves to increase the stability of the coalition. Task execution in multi-agent environments may require cooperation among agents. Given a set of agents and a set of tasks which they have to satisfy, we consider situations where each task should be attached to a group of agents that will perform the task. Task allocation to groups of agents is necessary when tasks cannot be performed by a single agent. However it may also be beneficial when groups perform more efficiently with respect to the single agents' performance. In this paper we present several solutions to the problem of task allocation among autonomous agents, and suggest that the agents form coalitions in order to perform tasks or improve the efficiency of their performance. We present efficient distributed algorithms with low ratio bounds and with low computational complexities. These properties are proven theoretically and supported by simulations and an implementation in an agent system. Our methods are based on both the algorithmic aspects of combinatorics and approximation algorithms for NP-hard problems. We first present an approach to agent coalition formation where each agent must be a member of only one coalition. Next, we present the domain of overlapping coalitions. We proceed with a discussion of the domain where tasks may have a precedence order. Finally, we discuss the case of implementation in an open, dynamic agent system. For each case we provide an algorithm that will lead agents to the formation of coalitions, where each coalition is assigned a task. Our algorithms are any-time algorithms, they are simple, efficient and easy to implement. We consider the well-known cake cutting problem in which a protocol wants to divide a cake among n ≥ 2 players in such a way that each player believes that they got a fair share. The standard Robertson-Webb model allows the protocol to make two types of queries, Evaluation and Cut, to the players. A deterministic divide-and-conquer protocol with complexity O(n log n) is known. We provide the first an Ω(n log n) lower bound on the complexity of any deterministic protocol in the standard model. This improves previous lower bounds, in that the protocol is allowed to assign to a player a piece that is a union of intervals and only guarantee approximate fairness. We accomplish this by lower bounding the complexity to find, for a single player, a piece of cake that is both rich in value, and thin in width. We then introduce a version of cake cutting in which the players are able to cut with only finite precision. In this case, we can extend the Ω(n log n) lower bound to include randomized protocols. Coalition formation is a key problem in automated negotiation among self-interested agents, and other multiagent applications. A coalition of agents can sometimes accomplish things that the individual agents cannot, or can accomplish them more efficiently. Motivating the agents to abide by a solution requires careful analysis: only some of the solutions are stable in the sense that no group of agents is motivated to break off and form a new coalition. This constraint has been studied extensively in cooperative game theory: the set of solutions that satisfy it is known as the core. The computational questions around the core have received less attention. When it comes to coalition formation among software agents (that represent real-world parties), these questions become increasingly explicit. In this paper we define a concise, natural, general representation for games in characteristic form that relies on superadditivity. In our representation, individual agents' values are given as well as values for those coalitions that introduce synergies. We show that this representation allows for efficient checking of whether a given outcome is in the core. We then show that determining whether the core is nonempty is NP-complete both with and without transferable utility. We demonstrate that what makes the problem hard in both cases is determining the collaborative possibilities (the set of outcomes possible for the grand coalition); we do so by showing that if these are given, the problem becomes solvable in time polynomial in the size of the representation in both cases. However, we then demonstrate that for a hybrid version of the problem, where utility transfer is possible only within the grand coalition, the problem remains NP-complete even when the collaborative possibilities are given. Finally, we show that for convex characteristic functions, a solution in the core can be computed efficiently (in O(nl^2) time, where n is the number of agents and l is the number of synergies), even when the collaborative possibilities are not given in advance. Coalition formation is a desirable behavior in a multiagent system, when a group of agents can perform a task more efficiently than any single agent can. Computational and communications complexity of traditional approaches to coalition formation, e.g., through negotiation, make them impractical for large systems. We propose an alternative, physics-motivated mechanism for coalition formation that treats agents as randomly moving, locally interacting entities. A new coalition may form when two agents encounter one another and it may grow when a single agent encounters it. Such agent-level behavior leads to a macroscopic model that describes how the number and distribution of coalitions change with time. We increase the generality and complexity of the model by letting the agents leave coalitions with some probability. The model is expressed mathematically as a series of differential equations. These equations have steady state solutions that describe the equilibrium distribution of coalitions. Within a context of a specific multi-agent application, we analyze and discuss the connection between the global system utility the parameters of the model. A cake is a metaphor for a heterogeneous, divisible good. When two players divide such a good, there is always a perfect division — one that is efficient (Pareto-optimal), envy-free, and equitable — which can be effected with a finite number of cuts under certain mild conditions; this is not always the case when there are more than two players (Brams, Jones, and Klamler, 2011b). We not only establish the existence of such a division but also provide an algorithm for determining where and how many cuts must be made, relating it to an algorithm, “Adjusted Winner” (Brams and Taylor, 1996, 1999), that yields a perfect division of multiple homogenous goods. We consider optimizing the coalition structure in Coalitional Skill Games (CSGs), a succinct representation of coalitional games (Bachrach and Rosenschein 2008). In CSGs, the value of a coalition depends on the tasks its members can achieve. The tasks require various skills to complete them, and agents may have different skill sets. The optimal coalition structure is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that CSGs can represent any characteristic function, and consider optimal coalition structure generation in this representation. We provide hardness results, showing that in general CSGs, as well as in very restricted versions of them, computing the optimal coalition structure is hard. On the positive side, we show that the problem can be reformulated as constraint satisfaction on a hyper graph, and present an algorithm that finds the optimal coalition structure in polynomial time for instances with bounded tree-width and number of tasks. Procuring multiple agents with different ability levels to independently solve the same task is common in labor markets, crowdsourcing environments and research and development projects due to two reasons: some agents may fail to provide a satisfactory solution, and the redundancy increases the quality of the best solution found. However, incentivizing large number of agents to compete for one task is difficult; agents need fair ex-ante guaranteed payoffs that consider their ability levels and failure rates to exert efforts. We model such domains as a cooperative game called the Max-Game, where each agent has a weight representing its ability level, and the value of an agent coalition is the maximal weight of the agents in the coalition. When agents may fail, we redefine the value of a coalition as the expected maximal weight of its surviving members. We analyze the core, the Shapley value, and the Banzhaf index as methods of payoff division. Surprisingly, the latter two, which are usually computationally hard, can be computed in polynomial time. Finally, we initiate the study of a new form of sabotage where agents may be incentivized to influence the failure probabilities of their peers, and show that no such incentive is present in a restricted case of Max-Games. We consider the partitioning of a society into coalitions in purely hedonic settings, i.e., where each player's payoff is completely determined by the identity of other members of her coalition. We first discuss how hedonic and nonhedonic settings differ and some sufficient conditions for the existence of core stable coalition partitions in hedonic settings. We then focus on a weaker stability condition: individual stability, where no player can benefit from moving to another coalition while not hurting the members of that new coalition. We show that if coalitions can be ordered according to some characteristic over which players have single-peaked preferences, or where players have symmetric and additively separable preferences, then there exists an individually stable coalition partition. Examples show that without these conditions, individually stable coalition partitions may not exist. We also discuss some other stability concepts, and the incompatibility of stability with other normative properties. Journal of Economic Literature Classification Numbers: C71, A14, D20. 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. A key problem when forming effective coalitions of autonomous agents is determining the best groupings, or the optimal coalition structure, to select to achieve some goal. To this end, we present a novel, anytime algorithm for this task that is significantly faster than current solutions. Specifically, we empirically show that we are able to find solutions that are optimal in 0.082 of the time taken by the state of the art dynamic programming algorithm (for 27 agents), using much less memory (O(2n) instead of O(3n) for n agents). Moreover, our algorithm is the first to be able to find solutions for more than 17 agents in reasonable time (less than 90 minutes for 27 agents, as opposed to around 2 months for the best previous solution). A natural requirement of a resource allocation system is to guarantee fairness to its participants. Fair allocation can be achieved either by distributed protocols known as cake-cutting algorithms or by centralized approaches, which first collect the agents’ preferences and then decide on the allocation. Compared with cake-cutting algorithms, centralized approaches ageless restricted and can therefore achieve more favorable allocations. Our work uses as a starting point a recent centralized algorithm that achieves an envy-free (i.e., fair) and Pareto optimal (i.e., efficient) allocation of multiple divisible goods. In fair allocation algorithms, agents who do not follow the protocol cannot prevent other agents from being allocated a fair share, but in certain situations, agents can increase their own allocation by submitting untruthful preferences. A recent article has shown that the only mechanisms that do not allow such manipulations(i.e., the only incentive-compatible mechanisms) are dictatorial. Nevertheless, we present a method that reduces possible gains from untruthful manipulation. Our mechanism uses a heuristic to approximate optimal manipulations, and compensates agents who submitted suboptimal preferences by increasing their allocation. We empirically demonstrate that when our method is used, the additional benefit that agents can achieve by untruthful manipulation is insignificant, hence they have insignificant incentives to lie. Weighted threshold games are coalitional games in which each player has a weight (intuitively corresponding to its voting power), and a coalition is successful if the sum of its weights exceeds a given threshold. Key questions in coalitional games include finding coalitions that are stable (in the sense that no member of the coalition has any rational incentive to leave it), and finding a division of payoffs to coalition members (an imputation) that is fair. We investigate the computational complexity of such questions for weighted threshold games. We study the core, the least core, and the nucleolus, distinguishing those problems that are polynomial-time computable from those that are NP-hard, and providing pseudopolynomial and approximation algorithms for the NP-hard problems.