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- Type of Document: Ph.D. Dissertation
- Language: Farsi
- Document No: 51705 (19)
- University: Sharif University of Technology
- Department: Computer Engineering
- Advisor(s): Ghodsi, Mohammad
- Abstract:
- In this study, we consider the fair division problem. In this problem, a heterogeneous resource must be fairly divided among a set of agents with different preferences.The resource can be either a divisible good (i.e., time, land), or a set of indivisible goods. When the resource is a single divisible good, the problem is commonly known as cake cutting. To measure fairness, several notions are defined, i.e., envy-freeness, proportionality, equability, maximin-share. In this thesis, we first give a formal definition of these notions. Next, we present our results for envy-freeness and maximin-share.First, we prove the existence of an allocation that guarantees each agent a factor 3/4 of his maximin-share. This improves upon the work of Procaccia and Wang [65] wherein the authors prove the existence of 2/3 approximation guarantee. We also extend our results to the case of submodular and fractionally subadditive valuations.More precisely, we give a 1/3 approximation guarantees for submodular agents and a 1/5 approximation guarantee for XOS agents. Next, we extend the definition of maximin-share to the case of the agents with unequal entitlements. We show that, in some cases with n agents, no allocation can guarantee better than 1/n approximation of maximin-share when the entitlements are not necessarily equal. Furthermore, we consider a restricted version of the problem where the valuation of every agent for each good is bounded by the total value he wishes to receive in a fair allocation. We show that this assumption enables us to find guarantee 1/2 approximation of maximin-share.In the last part of this study, we consider the problem of envy-free division of a cake among strategic players. Roughly, we wish to divide the cake among n players in a way that meets the following criteria: (I) the allocation satisfies envy-freeness, (II) the allocation is strategy-proof (truthful), and (III) the number of cuts made on the cake is minimal. We provide methods, namely expansion process and expansion process with unlocking, for dividing the cake under different assumptions on the valuation
- Keywords:
- Fair Division ; Maximin Share ; Envy-Free Division ; Competitive Equilibrium ; Approximate Algorithm ; Mechanism Design
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