I might be misunderstanding, but it looks to me like your proposed extension is essentially just the Elo model with some degrees of freedom that don’t yet appear to matter?
The dot product has the property that <theta_A-theta_B,w> = <theta_A,w> - <theta_B,w>, so the only thing that matters is the <theta_P,w> for each player P, which is just a single scalar. So we are on a one-dimensional scale again where predictions are based on taking a sigmoid of the difference between a single scalar associated with each player.
As far as I can tell, the way that such a model could still be a nontrivial extension of Elo would be if you posited w could vary between games, whether randomly from some distribution or whether via additional parameters associated to players that influence what w is in the games they are involved in, or other things like that. But it seems you would need something like that, or else some source of nonlinearity, because if w is constant then every dimension orthogonal to that fixed w can never have any effect on predictions by the model.
I might be misunderstanding, but it looks to me like your proposed extension is essentially just the Elo model with some degrees of freedom that don’t yet appear to matter?
The dot product has the property that <theta_A-theta_B,w> = <theta_A,w> - <theta_B,w>, so the only thing that matters is the <theta_P,w> for each player P, which is just a single scalar. So we are on a one-dimensional scale again where predictions are based on taking a sigmoid of the difference between a single scalar associated with each player.
As far as I can tell, the way that such a model could still be a nontrivial extension of Elo would be if you posited w could vary between games, whether randomly from some distribution or whether via additional parameters associated to players that influence what w is in the games they are involved in, or other things like that. But it seems you would need something like that, or else some source of nonlinearity, because if w is constant then every dimension orthogonal to that fixed w can never have any effect on predictions by the model.