On each day, the reasoner receives 50¢ from T, but after day t, the reasoner must pay $1 every day thereafter.
Hm. It’s a bit complicated and there are several possible ways to set things up. Reading that paragraph, I’m not sure about this sentence either.
In the version I was trying to explain, where traders are “forced to sell” every morning before the day of trading begins, the reasoner would receive 50¢ from the trader every day, but would return that money next morning. Also, in the version I was describing, the reasoner is forced to set the price to $1 rather than 50¢ as soon as the deductive process proves 1+1=2. So, that morning, the reasoner has to return $1 rather than 50¢. That’s where the reasoner loses money to the trader. After that, the price is $1 forever, so the trader would just be paying $1 every day and getting that $1 back the next morning.
I would then define exploitation as “the trader’s total wealth (across different times) has no upper bound”. (It doesn’t necessarily escape to infinity—it might oscillate up and down, but with higher and higher peaks.)
Now, the LI paper uses a different definition of exploitation, which involves how much money a trader has within a world (which basically means we imagine the deductive process decides all the sentences, and we ask how much money the trader would have; and, we consider all the different ways the deductive process could do this). This is not equivalent to my definition of exploitation in general; according to the LI paper, a trader ‘exploits’ the market even if its wealth is unbounded only in some very specific world (eg, where a specific sequence of in-fact-undecidable sentences gets proved).
However, I do have an unpublished proof that the two definitions of exploitation are equivalent for the logical induction algorithm and for a larger class of “reasonable” logical inductors. This is a non-trivial result, but, justifies using my definition of exploitation (which I personally find a lot more intuitive). My basic intuition for the result is: if you don’t know the future, the only way to ensure you don’t lose unbounded money in reality is to ensure you don’t lose unbounded money in any world. (“If you don’t know the future” is a significant constraint on logical inductors.)
Also, when those definitions do differ, I’m personally not convinced that the definition in the logical induction paper is better… it is stronger, in the sense that it gives us a more stringent logical induction criterion, but the “irrational” behaviors which it helps rule out don’t seem particularly irrational to me. Simply put, I am only convinced that I should care about actually losing unbounded money, as opposed to losing unbounded money in some hypothetical world.
In the version that I have in my head, theres a market with PCH and LCH in it that assigns constant price to the unactualised bet, so neither of them gain or lose anything with their trades on it, and LCH exploits PCH on the actualised ones.
Why is the price of the un-actualized bet constant? My argument in the OP was to suppose that PCH is the dominant hypothesis, so, mostly controls market prices. PCH thinks it gains important information when it sees which action we actually took, so it updates the expectation for the un-actualized action. So the price moves. Similarly, if PCH and LCH had similar probability, we would expect the price to move.
Why is the price of the un-actualized bet constant? My argument in the OP was to suppose that PCH is the dominant hypothesis, so, mostly controls market prices.
Thinking about this in detail, it seems like what influence traders have on the market price depends on a lot more of their inner workings than just their beliefs. I was thinking in a way where each trader only had one price for the bet, below which they bought and above which they sold, no matter how many units they traded (this might contradict “continuous trading strategies” because of finite wealth), in which case there would be a range of prices that could be the “market” price, and it could stay constant even with one end of that range shifting. But there could also be an outcome like yours, if the agents demand better and better prices to trade one more unit of the bet.
I think its still possible to have a scenario like this. Lets say each trader would buy or sell a certain amount when the price is below/above what they think it to be, but the transition being very steep instead of instant. Then you could still have long price intervalls where the amounts bought and sold remain constant, and then every point in there could be the market price.
I’m not sure if this is significant. I see no reason to set the traders up this way other than the result in the particular scenario that kicked this off, and adding traders who don’t follow this pattern breaks it. Still, its a bit worrying that trading strategies seem to matter in addition to beliefs, because what do they represent? A traders initial wealth is supposed to be our confidence in its heuristics—but if a trader is mathematical heuristics and trading strategy packaged, then what does confidence in the trading strategy mean epistemically? Two things to think about:
Is it possible to consistently define the set of traders with the same beliefs as trader X?
It seems that logical induction is using a trick, where it avoids inconsistent discrete traders, but includes an infinite sequence of continuous traders with ever steeper transitions to get some of the effects. This could lead to unexpected differences between behaviour “at all finite steps” vs “at the limit”. What can we say about logical induction if trading strategies need to be lipschitz-continuous with a shared upper limit on the lipschitz constant?
Hm. It’s a bit complicated and there are several possible ways to set things up. Reading that paragraph, I’m not sure about this sentence either.
In the version I was trying to explain, where traders are “forced to sell” every morning before the day of trading begins, the reasoner would receive 50¢ from the trader every day, but would return that money next morning. Also, in the version I was describing, the reasoner is forced to set the price to $1 rather than 50¢ as soon as the deductive process proves 1+1=2. So, that morning, the reasoner has to return $1 rather than 50¢. That’s where the reasoner loses money to the trader. After that, the price is $1 forever, so the trader would just be paying $1 every day and getting that $1 back the next morning.
I would then define exploitation as “the trader’s total wealth (across different times) has no upper bound”. (It doesn’t necessarily escape to infinity—it might oscillate up and down, but with higher and higher peaks.)
Now, the LI paper uses a different definition of exploitation, which involves how much money a trader has within a world (which basically means we imagine the deductive process decides all the sentences, and we ask how much money the trader would have; and, we consider all the different ways the deductive process could do this). This is not equivalent to my definition of exploitation in general; according to the LI paper, a trader ‘exploits’ the market even if its wealth is unbounded only in some very specific world (eg, where a specific sequence of in-fact-undecidable sentences gets proved).
However, I do have an unpublished proof that the two definitions of exploitation are equivalent for the logical induction algorithm and for a larger class of “reasonable” logical inductors. This is a non-trivial result, but, justifies using my definition of exploitation (which I personally find a lot more intuitive). My basic intuition for the result is: if you don’t know the future, the only way to ensure you don’t lose unbounded money in reality is to ensure you don’t lose unbounded money in any world. (“If you don’t know the future” is a significant constraint on logical inductors.)
Also, when those definitions do differ, I’m personally not convinced that the definition in the logical induction paper is better… it is stronger, in the sense that it gives us a more stringent logical induction criterion, but the “irrational” behaviors which it helps rule out don’t seem particularly irrational to me. Simply put, I am only convinced that I should care about actually losing unbounded money, as opposed to losing unbounded money in some hypothetical world.
Why is the price of the un-actualized bet constant? My argument in the OP was to suppose that PCH is the dominant hypothesis, so, mostly controls market prices. PCH thinks it gains important information when it sees which action we actually took, so it updates the expectation for the un-actualized action. So the price moves. Similarly, if PCH and LCH had similar probability, we would expect the price to move.
Thinking about this in detail, it seems like what influence traders have on the market price depends on a lot more of their inner workings than just their beliefs. I was thinking in a way where each trader only had one price for the bet, below which they bought and above which they sold, no matter how many units they traded (this might contradict “continuous trading strategies” because of finite wealth), in which case there would be a range of prices that could be the “market” price, and it could stay constant even with one end of that range shifting. But there could also be an outcome like yours, if the agents demand better and better prices to trade one more unit of the bet.
The continuity property is really important.
I think its still possible to have a scenario like this. Lets say each trader would buy or sell a certain amount when the price is below/above what they think it to be, but the transition being very steep instead of instant. Then you could still have long price intervalls where the amounts bought and sold remain constant, and then every point in there could be the market price.
I’m not sure if this is significant. I see no reason to set the traders up this way other than the result in the particular scenario that kicked this off, and adding traders who don’t follow this pattern breaks it. Still, its a bit worrying that trading strategies seem to matter in addition to beliefs, because what do they represent? A traders initial wealth is supposed to be our confidence in its heuristics—but if a trader is mathematical heuristics and trading strategy packaged, then what does confidence in the trading strategy mean epistemically? Two things to think about:
Is it possible to consistently define the set of traders with the same beliefs as trader X?
It seems that logical induction is using a trick, where it avoids inconsistent discrete traders, but includes an infinite sequence of continuous traders with ever steeper transitions to get some of the effects. This could lead to unexpected differences between behaviour “at all finite steps” vs “at the limit”. What can we say about logical induction if trading strategies need to be lipschitz-continuous with a shared upper limit on the lipschitz constant?