How do you feel about this example, which gives a setup where you have an incentive to bid more for a coin you think has a lower expected value?
Suppose there’s a conditional prediction market for two coins. After a week of bidding, the markets will close, whichever coin had contracts trading for more money will be flipped and $1 paid to contract-holders for head. The other market is cancelled.
Suppose you’re sure that coin A, has a bias of 60%. If you flip it lots of times, 60% of the flips will be heads. But you’re convinced coin B, is a trick coin. You think there’s a 59% chance it always lands heads, and a 41% chance it always lands tails. You’re just not sure which.
We want you to pay more for a contract for coin A, since that’s the coin you think is more likely to be heads (60% vs 59%). But if you like money, you’ll pay more for a contract on coin B. You’ll do that because other people might figure out if it’s an always-heads coin or an always-tails coin. If it’s always heads, great, they’ll bid up the market, it will activate, and you’ll make money. If it’s always tails, they’ll bid down the market, and you’ll get your money back.
You’ll pay more for coin B contracts, even though you think coin A is better in expectation. Order is not preserved. Things do not work out.
More generally, what’s the argument that the market will always select the decision that leads to he higher expected payout?
We want you to pay more for a contract for coin A, since that’s the coin you think is more likely to be heads (60% vs 59%). But if you like money, you’ll pay more for a contract on coin B. You’ll do that because other people might figure out if it’s an always-heads coin or an always-tails coin. If it’s always heads, great, they’ll bid up the market, it will activate, and you’ll make money. If it’s always tails, they’ll bid down the market, and you’ll get your money back.
Let’s call “Bidding on B, hoping that other people will figure out if B is an always-head or always-tails coin” strategy X, and call “Figure out if B is an always-head or always-tail myself & bid accordingly, or if I can’t, bid on A because it’s better in expectation” strategy Y.
If I believe that sufficient number of people in the market are using strategy Y, then it’s beneficial for me to use strategy X, and insofar as my beliefs about the market are accurate, this is okay, because sufficient number of people using strategy Y means the market will actually figure out if B is always-head or always-tail, then bid accordingly. So the market selects the right decision, insofar as my beliefs about the market is correct (Note that I’m never incentivized to place a bid on B so large that it causes B to activate, since I don’t actually know if B is always-head).
On the other hand, if I believe that the vast majority of people in the market are using strategy X instead of strategy Y, then it’s no longer beneficial for me to use strategy X myself, I should instead use strategy Y because the market doesn’t actually do the work of finding out if coin B is always-head for me. Other traders who have accurate beliefs about the market will switch to strategy Y as well, until there is a sufficient number of trader to push the market towards the right decision.
So insofar as people have accurate beliefs about the market, the market will end up selecting the right decision (either sufficient number of people use strategy Y, in which case it’s robust for me to use strategy X, or not enough people are using strategy Y, in which case people are incentivized to switch to Y)
More generally, what’s the argument that the market will always select the decision that leads to he higher expected payout?
“Always” might be too strong, but very informally:
Suppose that we have we have decision d1 d2, with outcome/payoff u & conditional market prices p1 (corresponds to d1) p2 (corresponds to d2)
if p1>E[u|d1], then traders are incentivized to sell & drive down p1. Similarly they will be incentivized to bid up p1 if p1<E[u|d1]. So p1 will tend toward E[u|d1]. We can argue similar for p2 tending towards E[u|d2]
Since we choose the decision with the higher price, and prices tend towards the expected payoff given that decision, the market end up choosing the decision that leads to the higher expected payoff.
Let’s call “Bidding on B, hoping that other people will figure out if B is an always-head or always-tails coin” strategy X, and call “Figure out if B is an always-head or always-tail myself & bid accordingly, or if I can’t, bid on A because it’s better in expectation” strategy Y.
Regarding this, I’ll note that my logic is not that different traders are following different strategies. I assume that all traders are rational agents and will maximize their expected return given their beliefs. My intended setup is that you believe coin A and coin B could have the biases stated, but you also believe that if you were to aggregate your beliefs with the beliefs of other people, the result would be more accurate than your beliefs alone.
I think this feeds into my objection to this proof:
Suppose that we have we have decision d1 d2, with outcome/payoff u & conditional market prices p1 (corresponds to d1) p2 (corresponds to d2)
if p1>E[u|d1], then traders are incentivized to sell & drive down p1. Similarly they will be incentivized to bid up p1 if p1<E[u|d1]. So p1 will tend toward E[u|d1]. We can argue similar for p2 tending towards E[u|d2]
Since we choose the decision with the higher price, and prices tend towards the expected payoff given that decision, the market end up choosing the decision that leads to the higher expected payoff.
My main objection to this logic is that there doesn’t seem to be any reflection of the idea that different traders will have different beliefs. (It’s possible that the market does give causal estimates with that assumption, but it’s definitely not an assumption I’d be willing to make, since I think the central purpose of prediction markets is to aggregate diverse beliefs.) All my logic is based on a setup where different traders have different beliefs.
So I don’t think the condition “p1>E[u|d1]” really makes sense? I think a given trader will drive down that market iff their estimate of the utility conditioned on that market activating is higher than p1, i.e. if p1>E_i[u|d1, market 1 activates]. I’m claiming that for trader i, E_i[u|d1, market 1 activates] != E_i[u|d1], basically because the event that market 1 activates contains extra information, and this makes it unlikely that the market will converge to E[u|d1].
My main objection to this logic is that there doesn’t seem to be any reflection of the idea that different traders will have different beliefs.[...] All my logic is based on a setup where different traders have different beliefs.
Over time, traders who have more accurate beliefs (& act rationally according to those beliefs) will accumulate more money in expectation (& vice versa), so in the limit we can think of futarchy as aggregating the beliefs of different traders weighted by how accurate their beliefs were in the past
So I don’t think the condition “p1>E[u|d1]” really makes sense? [...]and this makes it unlikely that the market will converge to E[u|d1].
If I pay p1 for a contract in market 1, my expected payoff is:
(E[u|d1]−p1)P(d1)+0×P(d2) (since I get my money back if d2/market 2 is activated)
this is negative iff p1>E[u|d1] and positive iff p1<E[u|d1]
and if we commit to using futarchy to choose the decision, then d1 is chosen iff market 1 activates, so E_i[u|d1, market 1 activates] should equal E_i[u|d1]
If I pay p1 for a contract in market 1, my expected payoff is:
(E[u|d1]−p1)P(d1)+0×P(d2) (since I get my money back if d2/market 2 is activated)
this is negative iff p1>E[u|d1] and positive iff p1<E[u|d1]
This is incorrect. There are two errors here:
The first expectation needs to be conditioned on the market activating. (That is not conditionally independent of u given d1 in general.)
Different people have different beliefs, so the expectations are different for different traders. You can’t write “E” without specifying for which trader.
I agree that if you assume u is conditionally independent of market activation given d1 and that all traders have the same beliefs then the result seems to hold. But those assumptions are basically always false.
The first expectation needs to be conditioned on the market activating. (That is not conditionally independent of u given d1 in general.)
If we commit to using futarchy to choose decision, then market 1 activating will have exactly the same truth conditions as executing d1, so “market activating and d1” would be the exact same thing as “d1“ itself (commiting to use futarchy to choose decision means we assign 0 probability to “first market activating & execute d2” or “Second market activating & execute d1”)
Different people have different beliefs, so the expectations are different for different traders. You can’t write “E” without specifying for which trader.
Yes, we can replace with E_i, and then argue that traders with accurate beliefs will accumulate more money over time, making market estimates more accurate in the limit
Yes, we can replace with E_i, and then argue that traders with accurate beliefs will accumulate more money over time, making market estimates more accurate in the limit
There’s a chicken-and-egg problem here. You’re assuming that markets are causal (meaning traders that are better at estimating causal probabilities) and then using that assumption to prove that markets are causal.
There’s a chicken-and-egg problem here[...] and then using that assumption to prove that markets are causal.
That argument was more about accomodating “different traders with different beliefs”, but here’s an independent argument for market being causal:
When I cause a particular effect/outcome, that means I mediate the influence between the cause of my action and the effect/outcome of my action, the cause of my action is conditionally independent of the effect of my action given me
Futarchy is a similar case: There may be many causes that influence market prices, which in turn determines the decision chosen, & market prices mediate the influence between the cause of market prices (e.g. different traders’ beliefs) and the decision chosen. Any information can only influence what decision will be chosen through influencing the market prices. This seems like what it means for market to be causal (In a bayesnet, the decision chosen will literally only have market prices as the parent, assuming we commit to using futarchy to choose decisions).
How do you feel about this example, which gives a setup where you have an incentive to bid more for a coin you think has a lower expected value?
More generally, what’s the argument that the market will always select the decision that leads to he higher expected payout?
Let’s call “Bidding on B, hoping that other people will figure out if B is an always-head or always-tails coin” strategy X, and call “Figure out if B is an always-head or always-tail myself & bid accordingly, or if I can’t, bid on A because it’s better in expectation” strategy Y.
If I believe that sufficient number of people in the market are using strategy Y, then it’s beneficial for me to use strategy X, and insofar as my beliefs about the market are accurate, this is okay, because sufficient number of people using strategy Y means the market will actually figure out if B is always-head or always-tail, then bid accordingly. So the market selects the right decision, insofar as my beliefs about the market is correct (Note that I’m never incentivized to place a bid on B so large that it causes B to activate, since I don’t actually know if B is always-head).
On the other hand, if I believe that the vast majority of people in the market are using strategy X instead of strategy Y, then it’s no longer beneficial for me to use strategy X myself, I should instead use strategy Y because the market doesn’t actually do the work of finding out if coin B is always-head for me. Other traders who have accurate beliefs about the market will switch to strategy Y as well, until there is a sufficient number of trader to push the market towards the right decision.
So insofar as people have accurate beliefs about the market, the market will end up selecting the right decision (either sufficient number of people use strategy Y, in which case it’s robust for me to use strategy X, or not enough people are using strategy Y, in which case people are incentivized to switch to Y)
“Always” might be too strong, but very informally:
Suppose that we have we have decision d1 d2, with outcome/payoff u & conditional market prices p1 (corresponds to d1) p2 (corresponds to d2)
if p1>E[u|d1], then traders are incentivized to sell & drive down p1. Similarly they will be incentivized to bid up p1 if p1<E[u|d1]. So p1 will tend toward E[u|d1]. We can argue similar for p2 tending towards E[u|d2]
Since we choose the decision with the higher price, and prices tend towards the expected payoff given that decision, the market end up choosing the decision that leads to the higher expected payoff.
Regarding this, I’ll note that my logic is not that different traders are following different strategies. I assume that all traders are rational agents and will maximize their expected return given their beliefs. My intended setup is that you believe coin A and coin B could have the biases stated, but you also believe that if you were to aggregate your beliefs with the beliefs of other people, the result would be more accurate than your beliefs alone.
I think this feeds into my objection to this proof:
My main objection to this logic is that there doesn’t seem to be any reflection of the idea that different traders will have different beliefs. (It’s possible that the market does give causal estimates with that assumption, but it’s definitely not an assumption I’d be willing to make, since I think the central purpose of prediction markets is to aggregate diverse beliefs.) All my logic is based on a setup where different traders have different beliefs.
So I don’t think the condition “p1>E[u|d1]” really makes sense? I think a given trader will drive down that market iff their estimate of the utility conditioned on that market activating is higher than p1, i.e. if p1>E_i[u|d1, market 1 activates]. I’m claiming that for trader i, E_i[u|d1, market 1 activates] != E_i[u|d1], basically because the event that market 1 activates contains extra information, and this makes it unlikely that the market will converge to E[u|d1].
Over time, traders who have more accurate beliefs (& act rationally according to those beliefs) will accumulate more money in expectation (& vice versa), so in the limit we can think of futarchy as aggregating the beliefs of different traders weighted by how accurate their beliefs were in the past
If I pay p1 for a contract in market 1, my expected payoff is:
(E[u|d1]−p1)P(d1)+0×P(d2) (since I get my money back if d2/market 2 is activated)
this is negative iff p1>E[u|d1] and positive iff p1<E[u|d1]
and if we commit to using futarchy to choose the decision, then d1 is chosen iff market 1 activates, so E_i[u|d1, market 1 activates] should equal E_i[u|d1]
This is incorrect. There are two errors here:
The first expectation needs to be conditioned on the market activating. (That is not conditionally independent of u given d1 in general.)
Different people have different beliefs, so the expectations are different for different traders. You can’t write “E” without specifying for which trader.
I agree that if you assume u is conditionally independent of market activation given d1 and that all traders have the same beliefs then the result seems to hold. But those assumptions are basically always false.
If we commit to using futarchy to choose decision, then market 1 activating will have exactly the same truth conditions as executing d1, so “market activating and d1” would be the exact same thing as “d1“ itself (commiting to use futarchy to choose decision means we assign 0 probability to “first market activating & execute d2” or “Second market activating & execute d1”)
Yes, we can replace with E_i, and then argue that traders with accurate beliefs will accumulate more money over time, making market estimates more accurate in the limit
There’s a chicken-and-egg problem here. You’re assuming that markets are causal (meaning traders that are better at estimating causal probabilities) and then using that assumption to prove that markets are causal.
That argument was more about accomodating “different traders with different beliefs”, but here’s an independent argument for market being causal:
When I cause a particular effect/outcome, that means I mediate the influence between the cause of my action and the effect/outcome of my action, the cause of my action is conditionally independent of the effect of my action given me
Futarchy is a similar case: There may be many causes that influence market prices, which in turn determines the decision chosen, & market prices mediate the influence between the cause of market prices (e.g. different traders’ beliefs) and the decision chosen. Any information can only influence what decision will be chosen through influencing the market prices. This seems like what it means for market to be causal (In a bayesnet, the decision chosen will literally only have market prices as the parent, assuming we commit to using futarchy to choose decisions).