it is important to judge the saddle points when we analyze the Jacobian matrix in evolutionary game models. however, i find that some people use the detJ
Dear Jimin,
first we have to set dimensionality of your problem. Probably you are talking about 2 dimensions. By definition a saddle point is when 2 real eigenvalues of characteristic equation have opposite signs. In this case there exists classification based on the values of determinant det J=\delta and track trJ=\tau; see http://www.scholarpedia.org/article/File:Equilibrium_figure_summary_2d.gif .
Indeed, det J
Yuri's comment may be misleading.
Most 2-dimensional EvGames that you might encounter have 3 alternative strategies whose shares sum to 1.0. To ensure that the sum remains constant, there is an eigenvalue of 0 associated with the system of 3 ODEs specifying the dynamics.
Yuri's comment is true if it is interpreted to apply to the projection onto the space orthogonal to the eigenvector (1,1,1) corresponding to that 0 eigenvalue. See Sandholm's book, or my forthcoming book with Sinervo for details.
Dear Daniel,
I think that I gave a correct answer to mathematical question of Yimin.
There was a question in another branch where we also discussed how to model corruption in China: https://www.researchgate.net/post/what_are_the_causes_of_collusive_corruption/1
As far as modelling of evolutionary games is concerned, he wanted to model it using EGT and was asking for literature: https://www.researchgate.net/post/could_someone_recommand_me_some_good_books_papers_or_syllabus_about_evolutionary_game_theory
I think that Yimin is just a beginner. So perhaps it will be too complex mathematically for him to start from 3-strategy models (may be better to use just two but without any restrictions). It may be much more important to focus on setting interaction between agents in a correct way that it grasps peculiarities of Chinese corruption. I was also suggesting network approach and if you can recommend how to model networks for this case this would be helpful.
Dear Yimin,
It is important to create a good model of interaction between those who give bribes and who receive them. It is good if you will be able to explain some stylized facts by a simple mathematical model with just 2 types of players. Then you can move to 3-dimensional game using the book of Daniel where mathematics might be much more complex. But you may start reading this and other books first just to understand how those simple models may look like. If you just start from a complex story, you might be able to miss important assumptions. Note that there are many approaches starting from no rationality at all: just play game according to your type (give bribe or not), then get payoff, replicate yourself at the next generation depending on your fitness, play again, etc.
If the population of honest people will increase over time, fitness of bribers will go down, and the phenomenon will disappear. Look why this does not happen.
As I already mentioned in previous remarks, a good model of network structure supporting corruption might give a clue to its persistence. But this may require more sophisticated math. If you look at Duncan Watts' works, you can see that there might be free scale or small world networks, having very different topology. In a "small world" everybody in a cluster has a link with almost everybody, but very few with an outer world. If bribing mafia would have such structure, the probability to detect it from outside will be lower, and thus even severe laws on corruption punishment may not work since the risk to be discovered is small. But another reason may be in low probability of network participants to inform police; here we can have a story with incentives.
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