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Simultaneous Eating Algorithm
A simultaneous eating algorithm (SE) is an algorithm for allocating divisible objects among agents with ordinal preferences. "Ordinal preferences" means that each agent can rank the items from best to worst, but cannot (or does not want to) specify a numeric value for each item. The SE allocation satisfies SD-efficiency - a weak ordinal variant of Pareto-efficiency (it means that the allocation is Pareto-efficient for ''at least one'' vector of additive utility functions consistent with the agents' item rankings). SE is parametrized by the "eating speed" of each agent. If all agents are given the same eating speed, then the SE allocation satisfies SD-envy-freeness - a strong ordinal variant of envy-freeness (it means that the allocation is envy-free for ''all'' vectors of additive utility functions consistent with the agents' item rankings). This particular variant of SE is called the Probabilistic Serial rule (PS). SE was developed by Hervé Moulin and Anna Bogomolnaia as a sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Preferences
In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask ''how much'' better it is or how good it is. All of the theory of consumer decision-making under conditions of certainty can be, and typically is, expressed in terms of ordinal utility. For example, suppose George tells us that "I prefer A to B and B to C". George's preferences can be represented by a function ''u'' such that: :u(A)=9, u(B)=8, u(C)=1 But critics of cardinal utility claim the only meaningful message of this function is the order u(A)>u(B)>u(C); the actual numbers are meaningless. Hence, George's preferences can also be represented by the following function ''v'': :v(A)=9, v(B)=2, v(C)=1 The functions ''u'' and ''v'' are ordinally equivalent – they represent George's preferences equally well. Ordinal utility contrasts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strategyproofness
In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information about what the others do, you fare best or at least not worse by being truthful. SP is also called truthful or dominant-strategy-incentive-compatible (DSIC), to distinguish it from other kinds of incentive compatibility. An SP game is not always immune to collusion, but its robust variants are; with group strategyproofness no group of people can collude to misreport their preferences in a way that makes every member better off, and with strong group strategyproofness no group of people can collude to misreport their preferences in a way that makes at least one member of the group better off without making any of the remaining members worse off. Examples Typical examples of SP mechanisms are majority voting between two alternatives, second- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Envy-freeness
Group envy-freeness (also called: coalition fairness) is a criterion for fair division. A group-envy-free division is a division of a resource among several partners such that every group of partners feel that their allocated share is at least as good as the share of any other group with the same size. The term is used particularly in problems such as fair resource allocation, fair cake-cutting and fair item allocation. Group-envy-freeness is a very strong fairness requirement: a group-envy-free allocation is both envy-free and Pareto efficient, but the opposite is not true. Definitions Consider a set of ''n'' agents. Each agent ''i'' receives a certain allocation ''Xi'' (e.g. a piece of cake or a bundle of resources). Each agent ''i'' has a certain subjective preference relation <''i'' over pieces/bundles (i.e. means that agent ''i'' prefers piece ''X'' to piece ''Y''). Consider a group ''G'' of the agents, with its current allocation |
Separation Oracle
A separation oracle (also called a cutting-plane oracle) is a concept in the mathematical theory of convex optimization. It is a method to describe a convex set that is given as an input to an optimization algorithm. Separation oracles are used as input to ellipsoid methods. Definition Let ''K'' be a convex and compact set in R''n''. A strong separation oracle for ''K'' is an oracle (black box) that, given a vector ''y'' in R''n'', returns one of the following: M. Grötschel, L. Lovász, A. Schrijver: ''Geometric Algorithms and Combinatorial Optimization'', Springer, 1988. *Assert that ''y'' is in ''K''. * Find a hyperplane that separates ''y'' from ''K'': a vector a in R''n'', such that a\cdot y > a\cdot x for all ''x'' in ''K''. A strong separation oracle is completely accurate, and thus may be hard to construct. For practical reasons, a weaker version is considered, which allows for small errors in the boundary of ''K'' and the inequalities. Given a small error toleran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Social Choice Theory
Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.Amartya Sen (2008). "Social Choice,". ''The New Palgrave Dictionary of Economics'', 2nd EditionAbstract & TOC./ref> Whereas choice theory is concerned with individuals making choices based on their preferences, social choice theory is concerned with how to translate the preferences of individuals into the preferences of a group. A non-theoretical example of a collective decision is enacting a law or set of laws under a constitution. Another example is voting, where individual preferences over candidates are collected to elect a person that best represents the group's preferences. Social choice blends elements of welfare economics and public choice theory. It is methodologically individualistic, in that it aggregates preferences and behaviors of individual member ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leximin Order
In mathematics, leximin order is a total preorder on finite-dimensional vectors. A more accurate, but less common term is leximin preorder. The leximin order is particularly important in social choice theory and fair division. Definition A vector x = (''x''1, ..., ''x''''n'') is ''leximin-larger'' than a vector y = (''y''1, ..., ''y''''n'') if one of the following holds: * The smallest element of x is larger than the smallest element of y; * The smallest elements of both vectors are equal, and the second-smallest element of x is larger than the second-smallest element of y; * ... * The ''k'' smallest elements of both vectors are equal, and the (''k''+1)-smallest element of x is larger than the (''k''+1)-smallest element of y. Examples The vector (3,5,3) is leximin-larger than (4,2,4), since the smallest element in the former is 3 and in the latter is 2. The vector (4,2,4) is leximin-larger than (5,3,2), since the smallest elements in both are 2, but the second-smallest elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Network Flow Problem
In combinatorial optimization, network flow problems are a class of computational problems in which the input is a flow network (a graph with numerical capacities on its edges), and the goal is to construct a flow, numerical values on each edge that respect the capacity constraints and that have incoming flow equal to outgoing flow at all vertices except for certain designated terminals. Specific types of network flow problems include: *The maximum flow problem, in which the goal is to maximize the total amount of flow out of the source terminals and into the sink terminals *The minimum-cost flow problem, in which the edges have costs as well as capacities and the goal is to achieve a given amount of flow (or a maximum flow) that has the minimum possible cost *The multi-commodity flow problem, in which one must construct multiple flows for different commodities whose total flow amounts together respect the capacities * Nowhere-zero flow, a type of flow studied in combinatorics in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Priority Item Allocation
Random priority (RP), also called Random serial dictatorship (RSD), is a procedure for fair random assignment - dividing indivisible items fairly among people. Suppose n partners have to divide n (or fewer) different items among them. Since the items are indivisible, some partners will necessarily get the less-preferred items (or no items at all). RSD attempts to insert fairness into this situation in the following way. Draw a random permutation of the agents from the uniform distribution. Then, let them successively choose an object in that order (so the first agent in the ordering gets first pick and so on). Properties RSD is a truthful mechanism when the number of items is at most the number of agents, since you only have one opportunity to pick an item, and the obviously dominant strategy in this opportunity is to pick the best available item. RSD is ex-ante envy-free (EF), since each agent has the same chance to appear in each position in the ordering. Obviously, it is not e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximin Strategy
In game theory, a simultaneous game or static game is a game where each player chooses their action without knowledge of the actions chosen by other players. Simultaneous games contrast with sequential games, which are played by the players taking turns (moves alternate between players). In other words, both players normally act at the same time in a simultaneous game. Even if the players do not act at the same time, both players are uninformed of each other's move while making their decisions. Normal form representations are usually used for simultaneous games.Mailath, G., Samuelson, L. and Swinkels, J., 1993. Extensive Form Reasoning in Normal Form Games. Econometrica, nline61(2), pp.273-278. Available at: ccessed 30 October 2020 Given a continuous game, players will have different information sets if the game is simultaneous than if it is sequential because they have less information to act on at each step in the game. For example, in a two player continuous game that is se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk Aversion
In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more certain outcome. Risk aversion explains the inclination to agree to a situation with a more predictable, but possibly lower payoff, rather than another situation with a highly unpredictable, but possibly higher payoff. For example, a risk-averse investor might choose to put their money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chance of losing value. Example A person is given the choice between two scenarios: one with a guaranteed payoff, and one with a risky payoff with same average value. In the former scenario, the person receives $50. In the uncertain scenario, a coin is flipped to decide whether the person receives $100 or nothing. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs. If each player has chosen a strategy an action plan based on what has happened so far in the game and no one can increase one's own expected payoff by changing one's strategy while the other players keep their's unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
