We revisit the issue of the emergence of fair behavior in the framework of the spatial Ultimatum game, adding many important results and insights to the pioneering work by Page et al. [2000. The spatial Ultimatum game. Proc. R. Soc. London B 267, 2177], who showed in a specific example that on a two-dimensional setup evolution may lead to strategies with some degree of fairness. Within this spatial framework, we carry out a thorough simulation study and show that the emergence of altruism is a very generic phenomenon whose details depend on the dynamics considered. A very frequent feature is the spontaneous emergence and fixation of quasiempathetic individuals, whose offers are very close to their acceptance thresholds. We present analytical arguments that allow an understanding of our results and give insights on the manner in which local effects in evolution may lead to such non-rational or apparently maladaptive behaviors.