For many systems characterized as "complex" the patterns exhibited on different scales differ markedly from one another. For example the biomass distribution in a human body "looks very different" depending on the scale at which one examines it. Conversely, the patterns at different scales in "simple" systems (e.g., gases, mountains, crystals) vary little from one scale to another. Accordingly, the degrees of self-dissimilarity between the patterns of a system at various scales constitute a complexity "signature" of that system.
Here I present a novel quantification of self-dissimilarity. This quantification can be measured for many kinds of real-world data. This allows comparisons of the complexity signatures of wholly different kinds of systems (e.g., systems involving information density in a digital computer vs. species densities in a rain-forest vs. capital density in an economy, etc.). Moreover, in contrast to many other suggested complexity measures, evaluating the self-dissimilarity of a system does not require one to already have a model of the system.
These facts may allow self-dissimilarity signatures to be used as the underlying observational variables of an eventual overarching theory relating all complex systems. To illustrate self-dissimilarity I present several numerical experiments. In particular, I show that underlying structure of the logistic map is picked out by the self-dissimilarity signature of time series' produced by that map.
In many cultures a major factor affecting a person's happiness is the difference between their income and that of their neighbors, independent of their own income. This effect is strongest when the neighbor has higher income. In addition a person's lifetime happiness tends to follow a "U" shape, with a minimum in the 40's. Previous models have separately explained some of these phenomena, typically by assuming the person has cognitive limitations, e.g., their happiness has a finite number of possible values. Here I present a model which explains all of the phenomena, and does not assume any cognitive limitations. In this model moderately greater income of your neighbor is statistical data that, if carefully analyzed, would recommend that you explore for a new income-generating strategy. This explains unhappiness that your neighbor has moderately greater income, as an emotional "prod'' that induces you to explore, exactly as a detailed statistical analysis of the income difference would recommend. It explains the "U" shape of happiness in a similar manner.
The Ultimatum Game is a prototypical example used in experimental economics to show that humans are ``irrational'' by making choices other than equilibrium solutions. Using the frame work of persona games, we introduce the fair persona and show that using this persona in the Ultimatum Game is perfectly rational.
Many statistics problems involve predicting the joint strategy of players in a noncooperative game. Conventional game theory predicts the joint strategy will satisfy an "equilibrium concept". Relative probabilities of the joint strategies satisfying the equilibrium concept are unspecified, and all joint strategies not satisfying it are assigned probability zero.
As an alternative, I cast the prediction problem as one of statistical inference. In this alternative the "data" is the game specification, which induces a posterior probability distribution over all joint strategies. I show that this alternative provides a unique best prediction for any noncooperative game, thereby solving a long-standing problem of conventional game theory. I also present an application of this alternative to predicting the behavior of a set of airlines during a weather disruption. In particular I show how to sample from the posterior distribution of airline joint strategies, and how to estimate associated quantities like covariances in airline behavior.
