First, the scope and assumptions of the question of learning in games.<\/p>\n
Second, several learning models.<\/p>\n
\\[S^{i} = BR^{i}\\left(S^{-i}\\right)\\]
\nand
\n\\[S^{i}\\left(t+1\\right) = BR^{i}\\left(S^{-i}\\left(t\\right)\\right)\\]
\nWhere \\(S^{i}\\) is the pure strategy of player \\(i\\) and \\(S^{-i}\\) is the pure strategy state of players other than the player \\(i\\)<\/p>\n
\\[\\rho^{i} = BR^{i}\\left(\\rho^{-i}\\right)\\]
\nand
\n\\[\\rho^{i}\\left(t+1\\right) = BR^{i}\\left(\\rho^{-i}\\left(t\\right)\\right)\\]
\nWhere \\(\\rho^{i}\\) is the mixed strategy of player \\(i\\) and \\(\\rho^{-i}\\) is the pure strategy state of players other than the player \\(i\\)<\/p>\n
\\[S^{i}\\left(t+1\\right) = BR^{i}\\left(\\rho^{-i, E}\\left(t\\right)\\right)\\]
\nWhere \\(\\rho^{-i,E}\\) is the empirical distribution of strategies of players other than the player \\(i\\) from the whole history, or certain length of the previous actions<\/p>\n
\\[Prob\\left(S^{i}\\left(t+1\\right)=S^{j}\\left(t\\right)\\right) = \\delta_{E^j\\left(t\\right), Max\\left(E^{1}\\left(t\\right), \\cdots, E^{i}\\left(t\\right), \\cdots, E^{N}\\left(t\\right)\\right)}\\]
\nor
\n\\[Prob\\left(S^{i}\\left(t+1\\right)=S^{j}\\left(t\\right)\\right) \\propto e^{\\beta\\left(E^{j}\\left(t\\right)-E^{i}\\left(t\\right)\\right)}\\]<\/p>\n
\\[S^{i} = \\bar{BR}^{i}\\left(S^{-i}\\right)\\]
\nand
\n\\[S^{i}\\left(t+1\\right) = \\bar{BR}^{i}\\left(S^{-i}\\left(t\\right)\\right)\\]
\nwhere
\n\\[\\bar{BR}^{i}\\left(\\rho^{-i}\\right)\\propto e^{\\beta E\\left(s^{i},\\rho^{-i}\\right)}\\]
\nis a probability distribution of player \\(i\\)’s strategies and \\(S^{i}\\) takes one sample from this probability distribution at a time.<\/p>\n
\\[S^{i}\\left(t+1\\right) = \\bar{BR}^{i}\\left(\\rho^{-i, E}\\left(t\\right)\\right)\\]
\nAgain \\(S^{i}\\) takes one sample from this probability distribution at a time.<\/p>\n
\\[\\rho^{i} = \\bar{BR}^{i}\\left(\\rho^{-i}\\right)\\]
\nand
\n\\[\\rho^{i}\\left(t+1\\right) = \\bar{BR}^{i}\\left(\\rho^{-i}\\left(t\\right)\\right)\\]
\nWhat it does is to simply replace the static\/dynamical mixed best response by static\/dynamical mixed smoothed best response. This is what we have done in this field: Dynamical QRE and its stability.<\/p>\n
\\[\\rho^{i}\\left(t+1\\right) = \\bar{BR}^{i}\\left(\\rho^{-i,E}\\left(t\\right)\\right)\\]
\nwhere \\(\\rho^{-i,E}\\left(t\\right)\\) is some kind of empirical distribution of strategies of players other than player \\(i\\). For example, one approach can be taking average of all historical \\(\\rho^{j}\\left(\\tau<t+1\\right)\\)s,
\n\\[\\rho^{j,E}\\left(t\\right) = \\sum_{\\tau<t+1}\\frac{\\rho^{j}\\left(\\tau\\right)}{t}.\\]
\nNot sure this has been discussed by others or not.\n<\/ol>\n
All the above models can be simultaneously updated or alternatively updated.<\/p>\n","protected":false},"excerpt":{"rendered":"