Research On The Validity Of Fair Distribution

Fair division, especially the problem of trade-off between fairness and efficiency, has always been followed with high interest. In this thesis, we study the loss of social welfare when a division has to be meeting some fairness criterion. The model we discuss in this thesis is as following. A unit interval should be allocated to some agents, each of whom should get a connected subinterval, and whose utility is determined by their utility functions. The social welfare is to maximize the total utility of all agents. The fairness criterions we mainly consider in this thesis are envy-freeness and proportionality.There are 3 Chapters in this thesis. In Chapter 1, we introduce the fair division problem with the basic conception, theory and results included and the description of the main model we study. In Chapter 2, we study the efficiency of envy-freeness by giving the tight bound of price of envy-freeness when considering the social welfare as the total utility and when the number of agents is small, and by discussing the upper bound of the price of envy-freeness when the number of agents is an arbitrary positive integer. In Chapter 3 we study the efficiency of proportionality by giving a lower bound of the price of proportionality when there are 3 agents.