"A game in strategic form is just a function with one input for each player (a strategy) and one output for each player (a payoff). More formally, a game in strategic form is a vector function and its do-main, the strategy space. The strategy space is just the set of all possible combinations of strategies, and therefore incorporates both the player and strategy sets." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"A game with no pure strategy equilibrium can be two minimal steps from a game with two equilibria. The significant observation is that for some games the equilibrium can be quite fragile. The topological approach provides a way to examine the robustness of payoffs and strategies by seeing how they vary for neighbouring games." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"A solution set is simply a subset of the possible outcomes that either predicts how a particular game will turn out or prescribes how it should turn out. A solution concept is a rationale for picking a solution based on the information specified in the form. No other information can be used." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"A symmetric game is often the convenient representative of a collection of related games, most of which are asymmetric. [...] The symmetric games are used so often, especially in introductions to game theory, that it is easy to forget they represent a very special case. For each strategy of the row player in a symmetric game, there must be an equivalent strategy for the column player." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Any game in one of the classes can be converted into any other in the same region by some strictly monotonic transformation. Since a monotonic transform conserves order, all the games in an equivalence class are ordinally equivalent. These equivalence classes par-tition the 8-dimensional payoff space for the 2 × 2 games into 144 regions. An ordinal 2×2 game is a 2×2 game with a payoff function that maps from the strategy space to these equivalence classes." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Every game is related to every other in the sense that there is a transformation that converts the payoff structure for one into the payoff structure for the other." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Figures that plot the payoff vectors are common in the literature, as are figures that show the convex hull of the payoffs. The latter are used for discussing mixed strategies and bargaining games, and require real values. A complete representation of the strategic form requires that the strategic choices represented in the matrix be recoverable, and that is the reason for using different line styles to represent each player’s inducement correspondences." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Games of pure conflict, pure common-interest and mixed motives can be described in terms of the slopes of the inducement correspondences. In a game of pure conflict every inducement correspon-dence is negatively sloped. Any action that improves the outcome for one player must make the outcome worse for the other. In a game of pure common-interest, every inducement correspondence is positively sloped." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Game theory is a formal approach to analysing social situations employing highly stylized and parsimo-nious descriptions." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"How often even ordinal symmetry appears in the real world is an open question. A great many situations are approximately symmetric, and symmetric payoffs provide a useful starting point for analysis. On the other hand, symmetric games present a problem that human beings are equipped to evade. Symmetry theoretically erases distinctions between players, but real people are capable of exploiting very subtle distinctions." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Inducement correspondences provide a particularly easy way to explore the Nash equilibrium. Because payoffs are strictly ordered there will always be a single best response in a given inducement correspondence. The inducement correspondence can also be used to describe a number of other solution concepts, including maxi-min and solutions based on dominance." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"One important and standardized block of information in the formal descriptions used by game theorists is called a game form. A form specifies the payoffs associated with every possible combination of decisions. There are several widely used forms, including the strate-gic form, typically presented in a matrix, the extensive form, which is usually represented as a tree, and the characteristic function form, expressed as a function on subsets of players." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Order graphs allow us to describe games easily, and our indexing system lets us lay out the games in a systematic and revealing way. Symmetries in the order graph repre-sentation shed some light on the nature of symmetric and reflected games, and on the structure of the space of the 2 × 2 games. They are not directly useful for analysing behaviour." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Simplicity gives the 2 × 2 games their power: they provide re-markable diversity with the absolute minimum of machinery. The strategic situation involves only two players, each with only two al-ternatives. There are only four possible outcomes and each outcome is described by a single payoff for each player. A game is therefore fully described by just 8 numbers." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Since games are characterized by the payoff function, similar games must have similar payoff functions. To define meaningful neighbourhoods, we need to characterize the smallest significant change in the payoff function. Obviously a change affecting the payoffs of one player is smaller than a change affecting two players. The closest neighbouring games are therefore those games that differ only by a small change in the ordering of the outcomes for one player." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"Starting with a torus with the graph of a layer embedded in it and a second torus with the the graph of a pipe, imagine gluing the two toruses together so that the four points of a shared tile coincide. Now imagine puncturing the two tiles that are glued together to make a door out of the pipe-torus and into the layer-torus. We now have a figure-eight, a two-holed torus." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"The beauty of a topological approach to the 2×2 games is that every topological feature yields some surprising insight into the relationships among the games. Even ignoring features can be productive." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"The discrete order topology of the ordinal games [25] is important to understanding the relationship among 2 × 2 games but is insufficient for describing and pre-dicting patterns of behaviour." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"The graphical feature that ensures that none of these games have equilibria is that at every position, negatively sloped inducement correspondences lead away in the same direction, ensuring that if one player likes the position, the other has an improving move. The sequence of best response choices cycles around the order graph, and the games are sometimes called cycle games." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"The matrix representation illustrates how a formal deep structure can be captured in an apparently simple surface structure. The fact that the rows and columns are at right angles to each other, for example, reflects the idea that the strategies available to the two players are independent. Independence means that it is possible to speak of changing Row’s strategy without changing Column’s strategy. An assumption about the nature of the world is displayed spatially in the matrix." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"The Nash equilibrium is often rationalized using a story about how people think and how their behaviour is related to their thoughts. Economists generally assume that, from a set of alternatives, a player will actively choose the one he likes best. This is the assumption of economic rationality, one of the core assumptions of standard game theory. Rationality alone will not predict behaviour in a game, but it leads us to single out the member of any inducement correspondence that yields the greatest payoff for the player that is making the choice. The resulting behaviour is sometimes described as 'myopic' because it fails to take into account how other players might respond to a given choice." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"To get beyond the typological approach requires a notion of what it means for games to be related. It turns out that preferences pro-vide the appropriate notion of closeness. The topology induced by preferences is beautiful and it makes the systematic treatment of the 2×2 games possible." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table",2005)
"Topology is the mathematical study of properties of objects which are preserved through deformations, twistings, and stretchings but not through breaks or cuts." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table",2005)
"When the graph is embedded within a surface it is called a map. The nature of the surface needed to embed the graph without cross-ing edges is a topological feature, and the topological structure of this payoff space is not only useful, but also beautiful." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"With four outcomes and two players, a 2 × 2 game is completely described by eight numbers. An array with eight numbers is just an address in an 8-dimensional Cartesian payoff space, and there are uncountably many 2 × 2 games, each fully described by an 8-number address." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
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