Wow, this is quite a piece of work. These guys know game theory and aren't afraid to use it. Although the models are simple and the assumptions crude, there is some interesting territory opened up here.
The book is not for the faint of heart. The full game theory was overwhelming for me personally. I just looked at the assumptions and the model. The models are crude but provide some interesting gross assumptions and boundary conditions for the emergence, and sometimes submergence of democracy.
The general consensus is that Democracy is stable when it "consolidates", the general thesis being that consolidated democracy occurs when a an enfranchised "middle class" can fully take part.
The history of ever larger participation in democracy as it occurred in the US and the UK is quite interesting. The history of democracy emerging and then failing is also and important lesson. Pre-mature democracy if it can be called such occurs when the governing system either doesn't enfranchise or create the perception of enfranchisement among enough "active" participants.
The book has an interesting thesis and is an interesting read for econometricians and economists.
Some of the shortcomings of the book are actually more a function of the models themselves and ground breaking work they have performed. Social modeling involves crude assumptions. One of the largest assumptions in the model is the concept of "state", state is a belief system combining people and a geography. Some of the largest assumptions to be left out of the modeling involve people and geography. Large assumptions were that "state" wealth was land or industrially based.
This unfortunately leaves out easily extractable natural resources which unduly influence the ability to capture wealth and power, but also can lead to more frequent coups due to the ease of appropriation. This "wealth" variable is an important function of enfranchisement and belief in participation in the democratic or ruling process, ask any Nigerian or Suadi Arabian citizen.
The second large concept left out of the "state" model is group belief systems within the concept of the state. Religion, clan, tribe, or other self identifying groups can deeply inhibit the ability for democracy to consolidate. The democratic system, even a "partial democracy" is a belief system and requires a concept of equal enfranchisement among some social strata. The more heterogeneous, a peoples' self identifications are the more difficult effective democratic consolidation becomes.
These 2 macro variables involving peoples self belief and the ability to distribute or expropriate wealth are important components in the concept of state, participation and the ability to effectively "consolidate" democracy. It will be interesting to see others build out models that factor in these components.
I do like the consolidation concept put forth in the book, which ties into political and economic stability. It also applies to many other forms of government. One could argue that Putin is consolidating government in Russia and the region. This consolidation unfortunately is less than democratic. It would be quite interesting to study consolidation of power under various forms of government and then to study the ability for phase changes to lead to steady state "consolidation" and therefore stability.