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Strategic Bankruptcy Problem
A strategic bankruptcy problem is a variant of a bankruptcy problem (also called ''claims problem'') in which claimants may act strategically, that is, they may manipulate their claims or their behavior. There are various kinds of strategic bankruptcy problems, differing in the assumptions about the possible ways in which claimants may manipulate. Definitions There is a divisible resource, denoted by ''E'' (=Estate or Endowment). There are ''n'' people who claim this resource or parts of it; they are called ''claimants''. The amount claimed by each claimant ''i'' is denoted by ''c_i''. Usually, \sum_^n c_i > E, that is, the estate is insufficient to satisfy all the claims. The goal is to allocate to each claimant an amount ''x_i'' such that \sum_^n x_i = E. Unit-selection game O'Neill describes the following game. * The estate is divided to small units (for example, if all claims are integers, then the estate can be divided into ''E'' units of size 1). * Each claimant ''i'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bankruptcy Problem
A bankruptcy problem, also called a claims problem, is a problem of distributing a homogeneous divisible good (such as money) among people with different claims. The focus is on the case where the amount is insufficient to satisfy all the claims. The canonical application is a bankrupt firm that is to be liquidated. The firm owes different amounts of money to different creditors, but the total worth of the company's assets is smaller than its total debt. The problem is how to divide the scarce existing money among the creditors. Another application would be the division of an estate amongst several heirs, particularly when the estate cannot meet all the deceased's commitments. A third application is ''tax assessment''. One can consider the claimants as taxpayers, the claims as the incomes, and the endowment as the total after-tax income. Determining the allocation of total after-tax income is equivalent to determining the allocation of tax payments. Definitions The amount ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Bargaining Solution
Cooperative bargaining is a process in which two people decide how to share a surplus that they can jointly generate. In many cases, the surplus created by the two players can be shared in many ways, forcing the players to negotiate which division of payoffs to choose. Such surplus-sharing problems (also called bargaining problem) are faced by management and labor in the division of a firm's profit, by trade partners in the specification of the terms of trade, and more. The present article focuses on the ''normative'' approach to bargaining. It studies how the surplus ''should'' be shared, by formulating appealing axioms that the solution to a bargaining problem should satisfy. It is useful when both parties are willing to cooperate in implementing the fair solution. The five axioms, any Nash Bargaining Solution should satisfy are Pareto Optimality (PAR), Individual Rationality (IR), Independent of Expected Utility Representations (INV), Independence of Irrelevant Alternatives (IIA) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constrained Equal Losses
Constrained equal losses (CEL) is a division rule for solving bankruptcy problems. According to this rule, each claimant should lose an equal amount from his or her claim, except that no claimant should receive a negative amount. In the context of taxation, it is known as poll tax. Formal definition There is a certain amount of money to divide, denoted by ''E'' (=Estate or Endowment). There are ''n'' ''claimants''. Each claimant ''i'' has a ''claim'' denoted by ''c_i''. Usually, \sum_^n c_i > E, that is, the estate is insufficient to satisfy all the claims. The CEL rule says that each claimant ''i'' should receive \max(0, c_i-r), where ''r'' is a constant chosen such that \sum_^n \max(0, c_i-r) = E. The rule can also be described algorithmically as follows: * Initially, all agents are active, and each agent gets his full claim. * While the total allocation is larger than the estate: ** Remove one unit equally from all active agents. ** Each agent whose total allocation dr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truthful Mechanism
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] |
Subgame Perfect Equilibrium
In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that at any point in the game, the players' behavior from that point onward should represent a Nash equilibrium of the continuation game (i.e. of the subgame), no matter what happened before. Every finite extensive game with perfect recall has a subgame perfect equilibrium. Perfect recall is a term introduced by Harold W. Kuhn in 1953 and ''"equivalent to the assertion that each player is allowed by the rules of the game to remember everything he knew at previous moves and all of his choices at those moves"''. A common method for determining subgame perfect equilibria in the case of a finite game is backward induction. Here one first considers the last actions of the game and determ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherence (fairness)
Coherence, also called uniformity or consistency, is a criterion for evaluating rules for fair division. Coherence requires that the outcome of a fairness rule is fair not only for the overall problem, but also for each sub-problem. Every part of a fair division should be fair. The coherence requirement was first studied in the context of apportionment. In this context, failure to satisfy coherence is called the new states paradox: when a new state enters the union, and the house size is enlarged to accommodate the number of seats allocated to this new state, some other unrelated states are affected. Coherence is also relevant to other fair division problems, such as bankruptcy problems. Definition There is a ''resource'' to allocate, denoted by h. For example, it can be an integer representing the number of seats in a ''h''ouse of representatives. The resource should be allocated between some n ''agents''. For example, these can be federal states or political parties. The agent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominant Strategy
In game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play. Terminology When a player tries to choose the "best" strategy among a multitude of options, that player may compare two strategies A and B to see which one is better. The result of the comparison is one of: * B is equivalent to A: choosing B always gives the same outcome as choosing A, no matter what the other players do. * B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other players do. * B weakly dominates A: choosing B always gives at least as good an outcome as choosing A, no matter what the other players do, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgame Perfect Equilibrium
In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that at any point in the game, the players' behavior from that point onward should represent a Nash equilibrium of the continuation game (i.e. of the subgame), no matter what happened before. Every finite extensive game with perfect recall has a subgame perfect equilibrium. Perfect recall is a term introduced by Harold W. Kuhn in 1953 and ''"equivalent to the assertion that each player is allowed by the rules of the game to remember everything he knew at previous moves and all of his choices at those moves"''. A common method for determining subgame perfect equilibria in the case of a finite game is backward induction. Here one first considers the last actions of the game and determ ... [...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] |
Supermodular Function
In mathematics, a function :f\colon \mathbb^k \to \mathbb is supermodular if : f(x \uparrow y) + f(x \downarrow y) \geq f(x) + f(y) for all x, y \isin \mathbb^, where x \uparrow y denotes the componentwise maximum and x \downarrow y the componentwise minimum of x and y. If −''f'' is supermodular then ''f'' is called submodular, and if the inequality is changed to an equality the function is modular. If ''f'' is twice continuously differentiable, then supermodularity is equivalent to the condition : \frac \geq 0 \mbox i \neq j. Supermodularity in economics and game theory The concept of supermodularity is used in the social sciences to analyze how one Agent (economics), agent's decision affects the incentives of others. Consider a symmetric game with a smooth payoff function \,f defined over actions \,z_i of two or more players i \in . Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: z_i \in [a,b]. In this context, sup ... [...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] |
