(N.B. I am the author of the linked post, not Tarn Fletcher.
He offered to link it because I was wary of logging in via Facebook. I’ve done so now, though.)
My central claim is that the desiderata are too weak, not that they are not pragmatically achievable. Specifically, I think that there is something fundamentally unsatisfactory about the definition of “inexploitability” (AKA “dominance” in your paper). I hadn’t seen your paper before and I haven’t looked at it in very much detail, but as far as I can tell, my central claim applies to it as well. (I.e. it isn’t specific to formal logic as opposed to prediction)
In the tumblr post, I complained that this definition was “asymptotic,” which I realize was not very clear. What I meant was that the definition captures only qualitative convergence or lack thereof, throwing away any rate information. Dominance just tells us whether the trader can or can’t make an infinite profit, not how quickly the inductor “plugs the hole” which would allow the trader to make an infinite profit if left unplugged forever.
This means that any further convergence results we get from dominance considerations across multiple traders must be pointwise with respect to the individual traders. For a uniform convergence result, we would need access to convergence rate information about each of the individual traders, so we could ensure that they have all gotten within ϵ at the same time. But the dominance statements for the traders lack this information.
Consider, for instance, procedures like TradingFirm in the LI paper or eq. 31 in your paper, where a set of traders is used to construct a single trader that is “strictly better” than every trader in the set with respect to dominance. For instance, your eq. 31 reads
Gξn:=n∑k=0ξ(k)Gkn
Although dominance carries over from each trader in the set, convergence rates and other finite-time properties do not. Concretely: at any time n, all but finitely many of the Gk from which Gξ is inheriting dominance have not yet appeared in the sum, so they are not yet affecting the behavior of Gξ.
For this reason, telling me only that an inductor dominates a set of traders does not tell me anything about the relation of that inductor to any of those traders at any given time. (After all, for any given trader and time n, it might be using a procedure like the above in which the trader is not considered until some time N>n.)
Say I’m interested in this particular set of traders because, for each trader, there is some convergence property related to it which I would like the inductor to have. (This seems quite generally true.) So for ϵ>0, I’m associating with the trader Gk some property pkn(ϵ), something like “at time n, the hole exploited by Gk has been plugged to within tolerance ϵ.” What I want to conclude here is that, at time n, some of these properties hold. But suppose the inductor dominates the set via a procedure like the one above. Then whenever I ask about some pkn(ϵ), I have no way of knowing whether n is high enough for Gk to have been included in the sum yet, according to the enumeration used to by the procedure. Thus I can’t conclude any of the facts pkn(ϵ).
In short, this framework just isn’t built for uniform convergence. If we want to get any results about how the inductor is doing “in general” at any given time, we need information that the dominance concept throws away.
Finally, about abstract asymptotic results leading to efficient practical algorithms—yes, this happens, but it’s important to think about what information beyond mere convergence is necessary for it to happen. Consider root-finding for a differentiable function f from R→R. Here’s one method that converges (given some conditions): Newton’s method. Here’s another: enumerate the rational numbers in an arbitrary order and evaluate f at one rational number per timestep. (You can approximate the root arbitrarily well with rationals, the function is continuous, blah blah.) Even though these are both convergent, there’s obviously a big difference; the former is actually converging to the result in the intuitive sense of that phrase, while the latter is just trolling you by satisfying your technical criteria but not the intuitions behind them. (Cf. the enumeration-based trader constructions.)
(N.B. I am the author of the linked post, not Tarn Fletcher. He offered to link it because I was wary of logging in via Facebook. I’ve done so now, though.)
My central claim is that the desiderata are too weak, not that they are not pragmatically achievable. Specifically, I think that there is something fundamentally unsatisfactory about the definition of “inexploitability” (AKA “dominance” in your paper). I hadn’t seen your paper before and I haven’t looked at it in very much detail, but as far as I can tell, my central claim applies to it as well. (I.e. it isn’t specific to formal logic as opposed to prediction)
In the tumblr post, I complained that this definition was “asymptotic,” which I realize was not very clear. What I meant was that the definition captures only qualitative convergence or lack thereof, throwing away any rate information. Dominance just tells us whether the trader can or can’t make an infinite profit, not how quickly the inductor “plugs the hole” which would allow the trader to make an infinite profit if left unplugged forever.
This means that any further convergence results we get from dominance considerations across multiple traders must be pointwise with respect to the individual traders. For a uniform convergence result, we would need access to convergence rate information about each of the individual traders, so we could ensure that they have all gotten within ϵ at the same time. But the dominance statements for the traders lack this information.
Consider, for instance, procedures like TradingFirm in the LI paper or eq. 31 in your paper, where a set of traders is used to construct a single trader that is “strictly better” than every trader in the set with respect to dominance. For instance, your eq. 31 reads
Gξn:=n∑k=0ξ(k)Gkn
Although dominance carries over from each trader in the set, convergence rates and other finite-time properties do not. Concretely: at any time n, all but finitely many of the Gk from which Gξ is inheriting dominance have not yet appeared in the sum, so they are not yet affecting the behavior of Gξ.
For this reason, telling me only that an inductor dominates a set of traders does not tell me anything about the relation of that inductor to any of those traders at any given time. (After all, for any given trader and time n, it might be using a procedure like the above in which the trader is not considered until some time N>n.)
Say I’m interested in this particular set of traders because, for each trader, there is some convergence property related to it which I would like the inductor to have. (This seems quite generally true.) So for ϵ>0, I’m associating with the trader Gk some property pkn(ϵ), something like “at time n, the hole exploited by Gk has been plugged to within tolerance ϵ.” What I want to conclude here is that, at time n, some of these properties hold. But suppose the inductor dominates the set via a procedure like the one above. Then whenever I ask about some pkn(ϵ), I have no way of knowing whether n is high enough for Gk to have been included in the sum yet, according to the enumeration used to by the procedure. Thus I can’t conclude any of the facts pkn(ϵ).
In short, this framework just isn’t built for uniform convergence. If we want to get any results about how the inductor is doing “in general” at any given time, we need information that the dominance concept throws away.
Finally, about abstract asymptotic results leading to efficient practical algorithms—yes, this happens, but it’s important to think about what information beyond mere convergence is necessary for it to happen. Consider root-finding for a differentiable function f from R→R. Here’s one method that converges (given some conditions): Newton’s method. Here’s another: enumerate the rational numbers in an arbitrary order and evaluate f at one rational number per timestep. (You can approximate the root arbitrarily well with rationals, the function is continuous, blah blah.) Even though these are both convergent, there’s obviously a big difference; the former is actually converging to the result in the intuitive sense of that phrase, while the latter is just trolling you by satisfying your technical criteria but not the intuitions behind them. (Cf. the enumeration-based trader constructions.)