Authority Distribution
The solution concept authority distribution was formulated by Lloyd Shapley and his student X. Hu in 2003 to measure the authority power of players in a well-contracted organization. The index generates the Shapley-Shubik power index and can be used in ranking, planning and organizational choice. Definition The organization contracts each individual by boss and approval relation with others. So each individual has its own authority structure, called command game. The Shapley-Shubik power index for these command games are collectively denoted by a power transit matrix Ρ. The authority distribution π is defined as the solution to the counterbalance equation π=πΡ. The basic idea for the counterbalance equation is that a person's power comes from his critical roles in others' command game; on the other hand, his power could also be redistributed to those who sit in his command game as vital players. For a simple legislative body, π is simply the Shapley-Shubik power index, b ...
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Lloyd Shapley
Lloyd Stowell Shapley (; June 2, 1923 – March 12, 2016) was an American mathematician and Nobel Prize-winning economist. He contributed to the fields of mathematical economics and especially game theory. Shapley is generally considered one of the most important contributors to the development of game theory since the work of von Neumann and Morgenstern. With Alvin E. Roth, Shapley won the 2012 Nobel Memorial Prize in Economic Sciences "for the theory of stable allocations and the practice of market design." Life and career Lloyd Shapley was born on June 2, 1923, in Cambridge, Massachusetts, one of the sons of astronomers Harlow Shapley and Martha Betz Shapley, both from Missouri. He attended Phillips Exeter Academy and was a student at Harvard when he was drafted in 1943. He served in the United States Army Air Corps in Chengdu, China and received the Bronze Star decoration for breaking the Soviet weather code. After the war, Shapley returned to Harvard and graduated wi ...
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Shapley Value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a unique distribution (among the players) of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable properties. Hart (1989) provides a survey of the subject. The setup is as follows: a coalition of players cooperates, and obtains a certain overall gain from that cooperation. Since some players may contribute more to the coalition than others or may possess different bargaining power (for example threatening to destroy the whole surplus), what final distribution of generated surplus among the players should arise in any particular game? Or phrased differently: how important is each player to the overall cooperation, and what payoff can he or she reasonably expect? The Shapley val ...
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Banzhaf Power Index
The Banzhaf power index, named after John F. Banzhaf III (originally invented by Lionel Penrose in 1946 and sometimes called Penrose–Banzhaf index; also known as the Banzhaf–Coleman index after James Samuel Coleman), is a power index defined by the probability of changing an outcome of a vote where voting rights are not necessarily equally divided among the voters or shareholders. To calculate the power of a voter using the Banzhaf index, list all the winning coalitions, then count the critical voters. A ''critical voter'' is a voter who, if he changed his vote from yes to no, would cause the measure to fail. A voter's power is measured as the fraction of all swing votes that he could cast. There are some algorithms for calculating the power index, e.g., dynamic programming techniques, enumeration methods and Monte Carlo methods. Examples Voting game Simple voting game A simple voting game, taken from ''Game Theory and Strategy'' by Philip D. Straffin: ; 4, 3, 2, 1 T ...
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Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathema ...
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Cooperative Games
Cooperative game may refer to: * Cooperative board game, board games in which players work together to achieve a common goal * Cooperative game theory, in game theory, a game with competition between groups of players and the possibility of cooperative behavior * Cooperative video game A cooperative (also known as co-operative, co-op, or coop) is "an autonomous association of persons united voluntarily to meet their common economic, social and cultural needs and aspirations through a jointly owned and democratically-contro ...