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Fair Division Of A Single Homogeneous Resource
Fair division of a single homogeneous resource is one of the simplest settings in fair division problems. There is a single resource that should be divided between several people. The challenge is that each person derives a different utility from each amount of the resource. Hence, there are several conflicting principles for deciding how the resource should be divided. A primary conflict is between efficiency and equality. Efficiency is represented by the ''utilitarian'' rule, which maximizes the sum of utilities; equality is represented by the ''egalitarian'' rule, which maximizes the minimum utility. Setting In a certain society, there are: * t units of some divisible resource. * n agents with different "utilities". * The utility of agent i is represented by a function u_i; when agent i receives y_i units of resource, he derives from it a utility of u_i(y_i). This setting can have various interpretations. For example: * The resource is wood, the agents are builders, and the ut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Division
Fair division is the problem in game theory of dividing a set of resources among several people who have an Entitlement (fair division), entitlement to them so that each person receives their due share. The central tenet of fair division is that such a division should be performed by the players themselves, without the need for external arbitration, as only the players themselves really know how they value the goods. There are many different kinds of fair division problems, depending on the nature of goods to divide, the criteria for fairness, the nature of the players and their preferences, and other criteria for evaluating the quality of the division. The archetypal fair division algorithm is divide and choose. The research in fair division can be seen as an extension of this procedure to various more complex settings. Description In game theory, fair division is the problem of dividing a set of resources among several people who have an Entitlement (fair division), entitlem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utilitarian Rule
In social choice theory, social choice and operations research, the utilitarian rule (also called the max-sum rule) is a Social welfare function, rule saying that, among all possible alternatives, society should pick the alternative which maximizes the ''sum of the utilities'' of all individuals in society. It is a formal mathematical representation of the Utilitarianism, utilitarian philosophy, and is often justified by reference to Harsanyi's utilitarian theorem or the Von Neumann–Morgenstern utility theorem, Von Neumann–Morgenstern theorem. Definition Let X be a set of possible "states of the world" or "alternatives". Society wishes to choose a single state from X. For example, in a single-winner election, X may represent the set of candidates; in a resource allocation setting, X may represent all possible allocations of the resource. Let I be a finite set, representing a collection of individuals. For each i \in I, let u_i:X\longrightarrow\mathbb be a ''utility, utilit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-freeness
Envy-freeness, also known as no-envy, is a criterion for fair division. It says that, when resources are allocated among people with equal rights, each person should receive a share that is, in their eyes, at least as good as the share received by any other agent. In other words, no person should feel envy. General definitions Suppose a certain resource is divided among several agents, such that every agent i receives a share X_i. Every agent i has a personal preference (economics), preference relation \succeq_i over different possible shares. The division is called envy-free (EF) if for all i and j: :::X_i \succeq_i X_j Another term for envy-freeness is no-envy (NE). If the preference of the agents are represented by a value functions V_i, then this definition is equivalent to: :::V_i(X_i) \geq V_i(X_j) Put another way: we say that agent i ''envies'' agent j if i prefers the piece of j over his own piece, i.e.: :::X_i \prec_i X_j :::V_i(X_i) 2 the problem is much harder. See e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egalitarian Rule
In social choice and operations research, the egalitarian rule (also called the max-min rule or the Rawlsian rule) is a rule saying that, among all possible alternatives, society should pick the alternative which maximizes the ''minimum utility'' of all individuals in society. It is a formal mathematical representation of the egalitarian philosophy. It also corresponds to John Rawls' principle of maximizing the welfare of the worst-off individual. Definition Let X be a set of possible `states of the world' or `alternatives'. Society wishes to choose a single state from X. For example, in a single-winner election, X may represent the set of candidates; in a resource allocation setting, X may represent all possible allocations. Let I be a finite set, representing a collection of individuals. For each i \in I, let u_i:X\longrightarrow\mathbb be a ''utility function'', describing the amount of happiness an individual ''i'' derives from each possible state. A '' social choice rule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concave Function
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex. Definition A real-valued function f on an interval (or, more generally, a convex set in vector space) is said to be ''concave'' if, for any x and y in the interval and for any \alpha \in ,1/math>, :f((1-\alpha )x+\alpha y)\geq (1-\alpha ) f(x)+\alpha f(y) A function is called ''strictly concave'' if :f((1-\alpha )x+\alpha y) > (1-\alpha ) f(x)+\alpha f(y) for any \alpha \in (0,1) and x \neq y. For a function f: \mathbb \to \mathbb, this second definition merely states that for ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diminishing Returns
In economics, diminishing returns means the decrease in marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, holding all other factors of production equal ('' ceteris paribus''). The law of diminishing returns (also known as the law of diminishing marginal productivity) states that in a productive process, if a factor of production continues to increase, while holding all other production factors constant, at some point a further incremental unit of input will return a lower amount of output. The law of diminishing returns does not imply a decrease in overall production capabilities; rather, it defines a point on a production curve at which producing an additional unit of output will result in a lower profit. Under diminishing returns, output remains positive, but productivity and efficiency decrease. The modern understanding of the law adds the dimension of holding other outputs equal, since a giv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Function
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (mathematics), ''epigraph'' (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup \cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap \cap. A twice-differentiable function, differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain of a function, domain. Well-known examples of convex functions of a single variable include a linear function f(x) = cx (where c is a real number), a quadratic function cx^2 (c as a nonnegative real number) and an exponential function ce^x (c as a nonnegative real number). Convex functions pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Increasing Returns
In economics, diminishing returns means the decrease in marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, holding all other factors of production equal (''ceteris paribus''). The law of diminishing returns (also known as the law of diminishing marginal productivity) states that in a productive process, if a factor of production continues to increase, while holding all other production factors constant, at some point a further incremental unit of input will return a lower amount of output. The law of diminishing returns does not imply a decrease in overall production capabilities; rather, it defines a point on a production curve at which producing an additional unit of output will result in a lower profit. Under diminishing returns, output remains positive, but productivity and efficiency decrease. The modern understanding of the law adds the dimension of holding other outputs equal, since a given pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resource Monotonicity
Resource monotonicity (RM; aka aggregate monotonicity) is a principle of fair division. It says that, if there are more resources to share, then all agents should be weakly better off; no agent should lose from the increase in resources. The RM principle has been studied in various division problems. Allocating divisible resources Single homogeneous resource, general utilities Suppose society has m units of some homogeneous divisible resource, such as water or flour. The resource should be divided among n agents with different utilities. The utility of agent i is represented by a function u_i; when agent i receives y_i units of resource, he derives from it a utility of u_i(y_i). Society has to decide how to divide the resource among the agents, i.e, to find a vector y_1,\dots,y_n such that: y_1+\cdots+y_n = m. Two classic allocation rules are the egalitarian rule - aiming to equalize the utilities of all agents (equivalently: maximize the minimum utility), and the utilitari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utilitarianism
In ethical philosophy, utilitarianism is a family of normative ethical theories that prescribe actions that maximize happiness and well-being for the affected individuals. In other words, utilitarian ideas encourage actions that lead to the greatest good for the greatest number. Although different varieties of utilitarianism admit different characterizations, the basic idea that underpins them all is, in some sense, to maximize utility, which is often defined in terms of well-being or related concepts. For instance, Jeremy Bentham, the founder of utilitarianism, described ''utility'' as the capacity of actions or objects to produce benefits, such as pleasure, happiness, and good, or to prevent harm, such as pain and unhappiness, to those affected. Utilitarianism is a version of consequentialism, which states that the consequences of any action are the only standard of right and wrong. Unlike other forms of consequentialism, such as egoism and altruism, egalitarian util ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egalitarianism
Egalitarianism (; also equalitarianism) is a school of thought within political philosophy that builds on the concept of social equality, prioritizing it for all people. Egalitarian doctrines are generally characterized by the idea that all humans are equal in fundamental worth or moral status. As such, all people should be accorded Equal rights before the law, equal rights and Equality before the law, treatment under the law. Egalitarian doctrines have supported many modern social movements, including the Age of Enlightenment, Enlightenment, feminism, civil rights, and International human rights law, international human rights. Egalitarianism is the foundation of left-wing politics. One key aspect of egalitarianism is its emphasis on equal opportunities for all individuals, regardless of their background or circumstances. This means ensuring that everyone has access to the same resources, education, and opportunities to succeed in life. By promoting equal opportunities, egalita ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
