We study classic cake-cutting problems, but in discrete models rather than using infiniteprecision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision protocols (Robertson-Webb and movingknife), we roughly partition the various fair-allocation problems into 3 classes: "easy" (constant number of rounds of logarithmic many bits), "medium" (poly-logarithmic total communication), and "hard". Our main technical result concerns two of the "medium" problems (perfect allocation for 2 players and equitable allocation for any number of players) which we prove are not in the "easy" class. Our main open problem is to separate the "hard" from the "medium" classes.Moving beyond Cut-and-Choose, protocols for more players can get very complex and, in fact, a central question in the cake cutting literature is that of computing fair allocations despite the informational challenge of private preferences. The goal of a mediator (or center) is to help the parties reach a fair solution, which in turn requires it to learn enough valuation information. The existing cake cutting protocols are broadly classified in three categories as follows. The largest class contains so called "discrete" protocols, which operate in a query model (due to Robertson and Webb [RW98, WS07]), where the center can repeatedly ask the players to make a cut or evaluate an existing piece. Despite the fact that the Robertson-Webb (RW) query model is viewed as discrete in the cake cutting literature, it is an infinite precision model, which is unavoidable for the purpose of computing exact solutions (or even approximate ones for arbitrarily complex instances.) Continuous (moving knife) procedures generally involve multiple knives sliding over the cake, with stopping conditions that signal the presence of a fair allocation. In a companion paper [BN17] we formalize a general class of moving knife procedures and prove that they can be approximately simulated by logarithmically many queries in the RW model. Finally, direct revelation protocols were studied also in the context of mechanism design [MT10, MN12, CLPP13, BM15], where the players directly submit their entire valuation to the center, whose goal is to compute a fair allocation despite the strategic behavior of the players. The positive results in the direct revelation model are either only for very restricted preferences, such as piecewise uniform functions [CLPP13], or rely again on infinite precision for general preferences [MT10].In this paper we study the communication complexity [KN96, LS09] of cake cutting: we assume that each player knows its own valuation function, and study the amount of communication that the players need to exchange (between themselves or with a mediator) in order to find an ǫ-fair allocation, for various notions of fairness.There are several motivations for studying the communication complexity of fair allocation problems. First, it is natural to use discrete protocols for studying notions of comp...
We study the problem of allocating divisible resources to agents with different preferences. We analyze a market game known as Trading Post, first considered by Shapley and Shubik, where each agent gets a budget of virtual currency to bid on goods: after bids are placed, goods are allocated to players in proportion to their bids. In this setting, the agents choose their bids strategically, aiming to maximize their utility, and this gives rise to a game. We study the equilibrium allocations of this game, measuring the quality of an allocation via the Nash social welfare, the geometric mean of utilities (a measure of aggregate welfare that respects individual needs). We show that any Nash equilibrium of Trading Post approximates the optimal Nash welfare within a factor of two for all concave valuations, and the mechanism is essentially optimal for Leontief valuations.
The Fisher market model is one of the most fundamental resource allocation models in economics. In a Fisher market, the prices and allocations of goods are determined according to the preferences and budgets of buyers to clear the market. In a Fisher market game, however, buyers are strategic and report their preferences over goods; the market-clearing prices and allocations are then determined based on their reported preferences rather than their real preferences. We show that the Fisher market game always has a pure Nash equilibrium, for buyers with linear, Leontief, and Cobb-Douglas utility functions, which are three representative classes of utility functions in the important Constant Elasticity of Substitution (CES) family. Furthermore, to quantify the social efficiency, we prove Price of Anarchy bounds for the game when the utility functions of buyers fall into these three classes respectively.
It is well known that strategic behavior in elections is essentially unavoidable; we therefore ask: how bad can the rational outcome be? We answer this question via the notion of the price of anarchy, using the scores of alternatives as a proxy for their quality and bounding the ratio between the score of the optimal alternative and the score of the winning alternative in Nash equilibrium. Specifically, we are interested in Nash equilibria that are obtained via sequences of rational strategic moves. Focusing on three common voting rules — plurality, veto, and Borda — we provide very positive results for plurality and very negative results for Borda, and place veto in the middle of this spectrum.
Off-chain transaction channels represent one of the leading techniques to scale the transaction throughput in cryptocurrencies. However, the economic effect of transaction channels on the system has not been explored much until now.We study the economics of Bitcoin transaction channels, and present a framework for an economic analysis of the lightning network and its effect on transaction fees on the blockchain. Our framework allows us to reason about different patterns of demand for transactions and different topologies of the lightning network, and to derive the resulting fees for transacting both on and off the blockchain.Our initial results indicate that while the lightning network does allow for a substantially higher number of transactions to pass through the system, it does not necessarily provide higher fees to miners, and as a result may in fact lead to lower participation in mining within the system.
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