Game theory is the study of mathematical models of strategic interaction between rational decision-makers. It has applications in all fields of social science, as well as in computer science, it addressed zero-sum games, in which one person's gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, is now an umbrella term for the science of logical decision making in humans and computers. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games 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 mathematical economics, his paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.
Game theory was developed extensively in the 1950s by many scholars. It was explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been recognized as an important tool in many fields; as of 2014, with the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole, eleven game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Early discussions of examples of two-person games occurred long before the rise of modern, mathematical game theory; the first known discussion of game theory occurred in a letter written by Charles Waldegrave, an active Jacobite, uncle to James Waldegrave, a British diplomat, in 1713. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her, the problem is now known as Waldegrave problem. In his 1838 Recherches sur les principes mathématiques de la théorie des richesses, Antoine Augustin Cournot considered a duopoly and presents a solution, a restricted version of the Nash equilibrium.
In 1913, Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. It proved that the optimal chess strategy is determined; this paved the way for more general theorems. In 1938, the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix was symmetric. Borel conjectured that non-existence of mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture, proved false. Game theory did not exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in 1928. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics, his paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern.
The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book; this foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. During the following time period, work on game theory was focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies. In 1950, the first mathematical discussion of the prisoner's dilemma appeared, an experiment was undertaken by notable mathematicians Merrill M. Flood and Melvin Dresher, as part of the RAND Corporation's investigations into game theory. RAND pursued the studies because of possible applications to global nuclear strategy. Around this same time, John Nash developed a criterion for mutual consistency of players' strategies, known as Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern.
Nash proved that every n-player, non-zero-sum non-cooperative game has what is now known as a Nash equilibrium. Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious play, repeated games, the Shapley value were developed. In addition, the first applications of game theory to philosophy and political science occurred during this time. In 1979 Robert Axelrod tried setting up computer programs as players and found that in tournaments between them the winner was a simple "tit-for-tat" program that cooperates on the first step on subsequent steps just does whatever its opponent did on the previous step; the same winner was often obtained by natural selection. In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium. In 1994 Nash and Harsanyi became Economics Nobel Laureates for their contributi
In economics and other social sciences, preference is the ordering of alternatives based on their relative utility, a process which results in an optimal "choice". The character of the individual preferences is determined purely by taste factors, independent of considerations of prices, income, or availability of goods. With the help of the scientific method many practical decisions of life can be modelled, resulting in testable predictions about human behavior. Although economists are not interested in the underlying causes of the preferences in themselves, they are interested in the theory of choice because it serves as a background for empirical demand analysis. In 1926 Ragnar Frisch developed for the first time a mathematical model of preferences in the context of economic demand and utility functions. Up to economists had developed an elaborated theory of demand that omitted primitive characteristics of people; this omission ceased when, at the end of the 19th and the beginning of the 20th century, logical positivism predicated the need of theoretical concepts to be related with observables.
Whereas economists in the 18th and 19th centuries felt comfortable theorizing about utility, with the advent of logical positivism in the 20th century, they felt that it needed more of an empirical structure. Because binary choices are directly observable, it appealed to economists; the search for observables in microeconomics is taken further by revealed preference theory. Since the pioneer efforts of Frisch in the 1920s, one of the major issues which has pervaded the theory of preferences is the representability of a preference structure with a real-valued function; this has been achieved by mapping it to the mathematical index called utility. Von Neumann and Morgenstern 1944 book "Games and Economic Behaviour" treated preferences as a formal relation whose properties can be stated axiomatically; these type of axiomatic handling of preferences soon began to influence other economists: Marschak adopted it by 1950, Houthakker employed it in a 1950 paper, Kenneth Arrow perfected it in his 1951 book "Social Choice and Individual Values".
Gérard Debreu, influenced by the ideas of the Bourbaki group, championed the axiomatization of consumer theory in the 1950s, the tools he borrowed from the mathematical field of binary relations have become mainstream since then. Though the economics of choice can be examined either at the level of utility functions or at the level of preferences, to move from one to the other can be useful. For example, shifting the conceptual basis from an abstract preference relation to an abstract utility scale results in a new mathematical framework, allowing new kinds of conditions on the structure of preference to be formulated and investigated. Another historical turnpoint can be traced back to 1895, when Georg Cantor proved in a theorem that if a binary relation is linearly ordered it is isomorphically embeddable in the ordered real numbers; this notion would become influential for the theory of preferences in economics: by the 1940s prominent authors such as Paul Samuelson, would theorize about people having weakly ordered preferences.
Suppose the set of all states of the world is X and an agent has a preference relation on X. It is common to mark the weak preference relation by ⪯, so that x ⪯ y means "the agent wants y at least as much as x" or "the agent weakly prefers y to x"; the symbol ∼ is used as a shorthand to the indifference relation: x ∼ y ⟺, which reads "the agent is indifferent between y and x". The symbol ≺ is used as a shorthand to the strong preference relation: x ≺ y ⟺, which reads "the agent prefers y to x". In everyday speech, the statement "x is preferred to y" is understood to mean that someone chooses x over y. However, decision theory rests on more precise definitions of preferences given that there are many experimental conditions influencing people's choices in many directions. Suppose a person is confronted with a mental experiment that she must solve with the aid of introspection, she is offered apples and oranges, is asked to verbally choose one of the two. A decision scientist observing this single event would be inclined to say that whichever is chosen is the preferred alternative.
Under several repetitions of this experiment, if the scientist observes that apples are chosen 51% of the time it would mean that x ≻ y. If half of the time oranges are chosen x ∼ y. If 51% of the time she chooses oranges it means that y ≻ x. Preference is here being identified with a greater frequency of choice; this experiment implicitly assumes. Otherwise, out of 100 repetitions, some of them will give as a result that neither apples, oranges or ties are chosen; these few cases of uncertainty will ruin any preference information resulting from the frequency attributes of the other valid cases. However, this example was used
The meerkat or suricate is a small carnivoran belonging to the mongoose family. It is the only member of the genus Suricata. Meerkats live in all parts of the Kalahari Desert in Botswana, in much of the Namib Desert in Namibia and southwestern Angola, in South Africa. A group of meerkats is called a "mob", "gang" or "clan". A meerkat clan contains about 20 meerkats, but some super-families have 50 or more members. In captivity, meerkats have an average life span of 12–14 years, about half this in the wild. "Meerkat" is a loanword from Afrikaans. The name by misidentification. In Dutch, meerkat means a monkey of the genus Cercopithecus; the word meerkat is Dutch for "lake cat", but although the suricata is a feliform, it is not of the cat family. In early literature, suricates were referred as mierkat. In colloquial Afrikaans, mier means termite, kat means cat, it has been speculated that the name comes from their frequent association with termite mounds or the termites they eat. Three subspecies are recognized: The meerkat is a small diurnal herpestid weighing on average about 0.5 to 2.5 kilograms.
Its long slender body and limbs give it a body length of 35 to 50 centimetres and an added tail length of around 25 centimetres. The meerkat uses its tail to balance, as well as for signaling, its face tapers, coming to a point at the nose, brown. The eyes always have black patches around them, they have small black crescent-shaped ears. Like cats, meerkats have binocular vision. At the end of each of a meerkat's "fingers" is a claw used for digging burrows and digging for prey. Claws are used with muscular hindlegs to help climb trees. Meerkats have four toes on long slender limbs; the coat is peppered gray, tan, or brown with silver. They have short parallel stripes across their backs, extending from the base of the tail to the shoulders; the patterns of stripes are unique to each meerkat. The underside of the meerkat has no markings, but the belly has a patch, only sparsely covered with hair and shows the black skin underneath; the meerkat uses this area to absorb heat while standing on its rear legs early in the morning after cold desert nights.
Meerkats are insectivores, but eat other animals and fungi. Meerkats are immune to certain types of venom, including the strong venom of the scorpions of the Kalahari Desert. Baby meerkats do not start foraging for food until they are about 1 month old, do so by following an older member of the group who acts as the pup's tutor. Meerkats forage in a group with one "sentry" on guard watching for predators while the others search for food. Sentry duty is approximately an hour long; the meerkat standing guard makes. A meerkat has the ability to dig through a quantity of sand equal to its own weight in just seconds. Digging is done to create burrows, to get food and to create dust clouds to distract predators. Martial eagles, tawny eagles and jackals are the main predators of meerkats. Meerkats sometimes die of snakebite in confrontations with snakes. Meerkats become sexually mature at about two years of age and can have one to four pups in a litter, with three pups being the most common litter size.
Meerkats can reproduce any time of the year. The pups are allowed to leave the burrow at two to three weeks old. There is no precopulatory display. Gestation lasts 11 weeks and the young are born within the underground burrow and are altricial; the young's ears open at about 10 days of age, their eyes at 10–14 days. They are weaned around 49 to 63 days; the alpha pair reserves the right to mate and kills any young not its own, to ensure that its offspring have the best chance of survival. The dominant couple may evict, or kick out the mothers of the offending offspring. New meerkat groups are formed by evicted females joining a group of males. Females appear to be able to discriminate the odour of their kin from the odour of their non-kin. Kin recognition is a useful ability that facilitates cooperation among relatives and the avoidance of inbreeding; when mating does occur between meerkat relatives, it results in negative fitness consequences or inbreeding depression. Inbreeding depression was evident for a variety of traits: pup mass at emergence from the natal burrow, hind-foot length, growth until independence and juvenile survival.
These negative effects are due to the increased homozygosity that arises from inbreeding and the consequent expression of deleterious recessive mutations. The avoidance of inbreeding and the promotion of outcrossing allow the masking of deleterious recessive mutations. Meerkats are small burrowing animals, living in large underground networks with multiple entrances which they leave only during the day, except to avoid the heat of the afternoon, they are social creatures and they live in colonies together. Animals in the same group groom each other regularly; the alpha pair scent-mark subordinates of the group to express their authority. There may be up to 30 meerkats in a group. To look
An extensive-form game is a specification of a game in game theory, allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when they make a decision, their payoffs for all possible game outcomes. Extensive-form games allow for the representation of incomplete information in the form of chance events modeled as "moves by nature"; some authors in introductory textbooks define the extensive-form game as being just a game tree with payoffs, add the other elements in subsequent chapters as refinements. Whereas the rest of this article follows this gentle approach with motivating examples, we present upfront the finite extensive-form games as constructed here; this general definition was introduced by Harold W. Kuhn in 1953, who extended an earlier definition of von Neumann from 1928. Following the presentation from Hart, an n-player extensive-form game thus consists of the following: A finite set of n players A rooted tree, called the game tree Each terminal node of the game tree has an n-tuple of payoffs, meaning there is one payoff for each player at the end of every possible play A partition of the non-terminal nodes of the game tree in n+1 subsets, one for each player, with a special subset for a fictitious player called Chance.
Each player's subset of nodes is referred to as the "nodes of the player". Each node of the Chance player has a probability distribution over its outgoing edges; each set of nodes of a rational player is further partitioned in information sets, which make certain choices indistinguishable for the player when making a move, in the sense that: there is a one-to-one correspondence between outgoing edges of any two nodes of the same information set—thus the set of all outgoing edges of an information set is partitioned in equivalence classes, each class representing a possible choice for a player's move at some point—, every path in the tree from the root to a terminal node can cross each information set at most once the complete description of the game specified by the above parameters is common knowledge among the playersA play is thus a path through the tree from the root to a terminal node. At any given non-terminal node belonging to Chance, an outgoing branch is chosen according to the probability distribution.
At any rational player's node, the player must choose one of the equivalence classes for the edges, which determines one outgoing edge except the player doesn't know which one is being followed. A pure strategy for a player thus consists of a selection—choosing one class of outgoing edges for every information set. In a game of perfect information, the information sets are singletons. It's less evident, it is assumed that each player has a von Neumann–Morgenstern utility function defined for every game outcome. The above presentation, while defining the mathematical structure over which the game is played, elides however the more technical discussion of formalizing statements about how the game is played like "a player cannot distinguish between nodes in the same information set when making a decision"; these can be made precise using epistemic modal logic. A perfect information two-player game over a game tree can be represented as an extensive form game with outcomes. Examples of such games include tic-tac-toe and infinite chess.
A game over an expectminimax tree, like that of backgammon, has no imperfect information but has moves of chance. For example, poker has both moves of imperfect information. A complete extensive-form representation specifies: the players of a game for every player every opportunity they have to move what each player can do at each of their moves what each player knows for every move the payoffs received by every player for every possible combination of moves The game on the right has two players: 1 and 2; the numbers by every non-terminal node indicate. The numbers by every terminal node represent the payoffs to the players; the labels by every edge of the graph are the name of the action. The initial node belongs to player 1. Play according to the tree is as follows: player 1 chooses between U and D; the payoffs are as specified in the tree. There are four outcomes represented by the four terminal nodes of the tree:, and; the payoffs associated with each outcome are as follows, and. If player 1 plays D, player 2 will play U' to maximise their payoff and so player 1 will only receive 1.
However, if player 1 plays U, player 2 maximises their payoff by playing D' and player 1 receives 2. Player 1 prefers 2 to 1 and s
Tragedy of the commons
The tragedy of the commons is a term used in environmental science to describe a situation in a shared-resource system where individual users acting independently according to their own self-interest behave contrary to the common good of all users by depleting or spoiling that resource through their collective action. The concept originated in an essay written in 1833 by the British economist William Forster Lloyd, who used a hypothetical example of the effects of unregulated grazing on common land in Great Britain and Ireland; the concept became known as the "tragedy of the commons" over a century due to an article written by the American ecologist and philosopher Garrett Hardin in 1968. In this modern economic context, commons is taken to mean any shared and unregulated resource such as atmosphere, rivers, fish stocks and highways, or an office refrigerator, it has been argued that the term "tragedy of the commons" is a misnomer since "the commons" referred to land resources with rights jointly owned by members of a community, no individual outside the community had any access to the resource.
However, the term is now used in social science and economics when describing a problem where all individuals have equal and open access to a resource. Hence, "tragedy of open access regimes" or "the open access problem" are more apt terms; the "tragedy of the commons" is cited in connection with sustainable development, meshing economic growth and environmental protection, as well as in the debate over global warming. It has been used in analyzing behavior in the fields of economics, evolutionary psychology, game theory, politics and sociology. Although common resource systems have been known to collapse due to overuse, many examples have existed and still do exist where members of a community with access to a common resource co-operate or regulate to exploit those resources prudently without collapse. Elinor Ostrom was awarded the Nobel Prize in economics for demonstrating this concept in her book Governing the Commons, which included examples of how local communities were able to do this without top-down regulations.
In 1833, the English economist William Forster Lloyd published a pamphlet which included a hypothetical example of over-use of a common resource. This was the situation of cattle herders sharing a common parcel of land on which they are each entitled to let their cows graze, as was the custom in English villages, he postulated that if a herder put more than his allotted number of cattle on the common, overgrazing could result. For each additional animal, a herder could receive additional benefits, but the whole group shared damage to the commons. If all herders made this individually rational economic decision, the common could be depleted or destroyed, to the detriment of all. In 1968, ecologist Garrett Hardin explored this social dilemma in his article "The Tragedy of the Commons", published in the journal Science; the essay derived its title from the pamphlet by Lloyd, which he cites, on the over-grazing of common land. Hardin discussed problems that cannot be solved by technical means, as distinct from those with solutions that require "a change only in the techniques of the natural sciences, demanding little or nothing in the way of change in human values or ideas of morality".
Hardin focused on human population growth, the use of the Earth's natural resources, the welfare state. Hardin argued that if individuals relied on themselves alone, not on the relationship of society and man the number of children had by each family would not be of public concern. Parents breeding excessively would leave fewer descendants because they would be unable to provide for each child adequately; such negative feedback is found in the animal kingdom. Hardin said that if the children of improvident parents starved to death, if overbreeding was its own punishment there would be no public interest in controlling the breeding of families. Hardin blamed the welfare state for allowing the tragedy of the commons. In his article, Hardin lamented the following proposal from the United Nations: The Universal Declaration of Human Rights describes the family as the natural and fundamental unit of society, it follows that any choice and decision with regard to the size of the family must irrevocably rest with the family itself, cannot be made by anyone else.
In addition, Hardin pointed out the problem of individuals acting in rational self-interest by claiming that if all members in a group used common resources for their own gain and with no regard for others, all resources would still be depleted. Overall, Hardin argued against relying on conscience as a means of policing commons, suggesting that this favors selfish individuals – known as free riders – over those who are more altruistic. In the context of avoiding over-exploitation of common resources, Hardin concluded by restating Hegel's maxim, "freedom is the recognition of necessity", he suggested. By recognizing resources as commons in the first place, by recognizing that, as such, they require management, Hardin believed that humans "can preserve and nurture other and more precious freedoms". Hardin's article was the start of the modern use of "Commons" as a term connoting a shared resource; as Frank van Laerhoven & Elinor Ostrom have stated: "Prior to the publication of Hardin’s article on the tragedy of the commons, titles containing the words'the commons','common pool resources,' or'common property' were
In game theory, a sequential game is a game where one player chooses their action before the others choose theirs. The players must have some information of the first's choice, otherwise the difference in time would have no strategic effect. Sequential games hence are governed by the time axis, represented in the form of decision trees. Unlike sequential games, simultaneous games do not have a time axis as players choose their moves without being sure of the other's, are represented in the form of payoff matrices. Extensive form representations are used for sequential games, since they explicitly illustrate the sequential aspects of a game. Combinatorial games are sequential games. Games such as chess, infinite chess, tic-tac-toe and Go are examples of sequential games; the size of the decision trees can vary according to game complexity, ranging from the small game tree of tic-tac-toe, to an immensely complex game tree of chess so large that computers cannot map it completely. In sequential games with perfect information, a subgame perfect equilibrium can be found by backward induction.
Simultaneous game Subgame perfection Sequential auction
In social psychology, social loafing is the phenomenon of a person exerting less effort to achieve a goal when he or she works in a group than when working alone. This is seen as one of the main reasons groups are sometimes less productive than the combined performance of their members working as individuals, but should be distinguished from the accidental coordination problems that groups sometimes experience. Research on social loafing began with rope pulling experiments by Ringelmann, who found that members of a group tended to exert less effort in pulling a rope than did individuals alone. In more recent research, studies involving modern technology, such as online and distributed groups, have shown clear evidence of social loafing. Many of the causes of social loafing stem from individual members feeling that his or her effort will not matter to the group; the first known research on the social loafing effect began in 1913 with Max Ringelmann's study. He found that, when he asked a group of men to pull on a rope, they did not pull as hard collectively as they did when each was pulling alone.
This research did not distinguish whether this was the result of the individuals in a group putting in less effort or of poor coordination within the group. In 1974, Alan Ingham, James Graves, colleagues replicated Ringelmann's experiment using two types of group: 1) Groups with real participants in groups of various sizes or 2) Pseudo-groups with only one real participant. In the pseudo-groups, the researchers' assistants only pretended to pull on the rope; the results showed a decrease in the participants' performance. Groups of participants who all exerted effort exhibited the largest declines; because the pseudo-groups were isolated from coordination effects, Ingham proved that communication alone did not account for the effort decrease, that motivational losses were the more cause of the performance decline. In contrast with Ringelmann's first findings, Bibb Latané, et al. replicated previous social loafing findings while demonstrating that the decreased performance of groups was attributable to reduced individual effort, as distinct from a deterioration due to coordination.
They showed this by blindfolding male college students while making them wear headphones that masked all noise. They asked them to shout both in actual groups and pseudogroups in which they shouted alone but believed they were shouting with others; when subjects believed one other person was shouting, they shouted 82 percent as intensely as they did alone, but with five others, their effort decreased to 74 percent. Latané, et al. concluded that increasing the number of people in a group diminished the relative social pressure on each person: "If the individual inputs are not identifiable the person may work less hard. Thus if the person is dividing up the work to be performed or the amount of reward he expects to receive, he will work less hard in groups." In a 1993 meta-analysis and Williams proposed the Collective Effort Model, used to generate predictions. The CEM integrates expectancy theories with theories of group-level social comparison and social identity to account for studies that examine individual effort in collective settings.
From a psychological state, it proposes that Expectancy multiplied by Instrumentality multiplied by Valence of Outcome produces the resulting Motivational Force. Karau, et al. concluded that social loafing occurred because there was a stronger perceived contingency between individual effort and valued outcomes when working individually. When working collectively, other factors determine performance, valued outcomes are divided among all group members. All individuals are assumed to try to maximize the expected utility of their actions; the CEM acknowledges that some valued outcomes do not depend on performance. For example, exerting strong effort when working on intrinsically meaningful tasks or with respected team members may result in self-satisfaction or approval from the group if the high effort had little to no impact on tangible performance outcomes. Notable or novel findings by Karau and Williams following their implementation of the CEM include: The magnitude of social loafing is reduced for women and individuals originating from Eastern cultures.
Individuals are more to loaf when their co-workers are expected to perform well. Individuals reduce social loafing when working with acquaintances and do not loaf at all when they work in valued groups. A 2005 study by Laku Chidambaram and Lai Lai Tung based their research model on Latané's social impact theory, hypothesized that as group size and dispersion grew, the group's work would be affected in the following areas: Members would contribute less in both quantity and quality, final group output would be of lower quality, a group's output would be affected both by individual factors and contextual factors. A sample of 240 undergraduate business students was randomly split into forty teams which were randomly assigned to either a collocated or distributed setting; the participants were to complete a task that asked them to act as a board of directors of a winery with an image problem. They were to find and discuss alternatives, at the end submit their alternative with rationale. Co-located groups worked at a table together, while distributed groups did the same task at separate computers that allowed for electronic, networked communication.
The same technology was used by distributed groups. Chidambaram and Tung found that group size mattered immensely in a group's performance; the smaller the group, the more each member