Optimal strategies for a game on amenable semigroups

The semigroup game is a two-person zero-sum game defined on a semigroup S as follows: Players 1 and 2 choose elements x and y in S, respectively, and player 1 receives a payoff f(xy) defined by a function f from S to [-1,1]. If the semigroup is amenable in the sense of Day and von Neumann, one can extend the set of classical strategies, namely countably additive probability measures on S, to include some finitely additive measures in a natural way. This extended game has a value and the players have optimal strategies. This theorem extends previous results for the multiplication game on a compact group or on the positive integers with a specific payoff. We also prove that the procedure of extending the set of allowed strategies preserves classical solutions: if a semigroup game has a classical solution, this solution solves also the extended game.Comment: 17 pages. To appear in International Journal of Game Theor

Achieving Good Angular Resolution in 3D Arc Diagrams

We study a three-dimensional analogue to the well-known graph visualization approach known as arc diagrams. We provide several algorithms that achieve good angular resolution for 3D arc diagrams, even for cases when the arcs must project to a given 2D straight-line drawing of the input graph. Our methods make use of various graph coloring algorithms, including an algorithm for a new coloring problem, which we call localized edge coloring.Comment: 12 pages, 5 figures; to appear at the 21st International Symposium on Graph Drawing (GD 2013

Reproducing in cities

Reproducing in cities has always been costly, leading to lower fertility (that is, lower birth rates) in urban than in rural areas. Historically, although cities provided job opportunities, initially residents incurred the penalty of higher infant mortality, but as mortality rates fell at the end of the 19th century, European birth rates began to plummet. Fertility decline in Africa only started recently and has been dramatic in some cities. Here it is argued that both historical and evolutionary demographers are interpreting fertility declines across the globe in terms of the relative costs of child rearing, which increase to allow children to outcompete their peers. Now largely free from the fear of early death, postindustrial societies may create an environment that generates runaway parental investment, which will continue to drive fertility ever lower