By Lucien Guillou, Alexis Marin
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Extra info for A la recherche de la topologie perdue: I Du côté de chez Rohlin, II Le côté de Casson
It is an unnecessarily special approach. One can instead generate a topological space from an arbitrary set of abstract elements, called points, by imposing a topology on the set . In our case the set of points is the set of 2 × 2 games. Any set for which a topology has been specified is called a topological space (, p. 76). Modern topology in fact takes no account of the individual nature of the elements, but merely of their mutual relationships. In an approach based on points, however, what is meant by the expression neighbourhood of a point must be defined axiomatically.
Row comes to care only for Column’s payoffs. Perhaps it’s love. 10 In fact, the pattern on the bottom row accounts for one third of all games with three asymmetric quadrilaterals. 16 has an R reflection in the lower left of the same layer. Each 36-game layer can be folded along its negative diagonal so that the game in the upper right coincides with its R reflection in the lower left. The 24 games on the negative diagonals of the layers are invariant under R . 11. Although R does not change the order graph, and the graphs are obviously symmetric, these games are not symmetric in the game-theoretic sense.
Weise and Noack, “Aspects of topology” ( p. 593) In Chapter 2 we described the 2 × 2 games in terms of the payoff function. Payoff functions provide a complete description of a game in strategic form. We now introduce enough additional structure to induce a topology on the set of games as a whole. The topology allows us to relate the 144 2 × 2 games in a new and systematic way. 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.
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