I have two doubts about the Al-Najjar and Weinstein paper—I’d be curious to hear your (or others’) thoughts on these.
First, I’m having trouble seeing where the information aversion comes in. A simpler example than the one used in the paper seems to be enough to communicate what I’m confused about: let’s say an urn has 100 balls that are each red or yellow, and you don’t know their distribution. Someone averse to ambiguity would (I think) be willing to pay up to $1 for a bet that pays off $1 if a randomly selected ball is red or yellow. But if they’re offered that bet as two separate decisions (first betting on a ball being red, and then betting on the same ball being yellow), then they’d be willing to pay less than $0.50 for each bet. So it looks like preference inconsistency comes from the choice being spread out over time, rather than from information (which would mean there’s no incentive to avoid information). What am I missing here?
(Maybe the following is how the authors were thinking about this? If you (as a hypothetical ambiguity-averse person) know that you’ll get a chance to take both bets separately, then you’ll take them both as long as you’re not immediately informed of the outcome of the first bet, because you evaluate acts, not by their own uncertainty, but by the uncertainty of your sequence of acts as a whole (considering all acts whose outcomes you remain unaware of). This seems like an odd interpretation, so I don’t think this is it.)
[edit: I now think the previous paragraph’s interpretation was correct, because otherwise agents would have no way to make ambiguity averse choices that are spread out over time and consistent, in situations like the ones presented in the paper. The ‘oddness’ of the interpretation seems to reflect the oddness of ambiguity aversion: rather than only paying attention to what might happen differently if you choose one action or another, ambiguity aversion involves paying attention to possible outcomes that will not be affected by your action, since they might influence the uncertainty of your action.]
Second, assuming that ambiguity aversion does lead to information aversion, what do you think of the response that “this phenomenon simply reflects a [rational] trade-off between the intrinsic value of information, which is positive even in the presence of ambiguity, and the value of commitment”?
On the first point, I think your intuition does capture the information aversion here, but I still think information aversion is an accurate description. Offered a bet that pays $X if I pick a color and then see if a random ball matches that color, you’ll pay more than for a bet that pays $X if a random ball is red. The only difference between these situations is that you have more information in the latter: you know the color to match is red. That makes you less willing to pay. And there’s no obvious reason why this information aversion would be something like a useful heuristic.
I don’t quite get the second point. Commitment doesn’t seem very relevant here since it’s really just a difference in what you would pay for each situation. If one comes first, I don’t see any reason why it would make sense to commit, so I don’t think that strengthens the case for ambiguity aversion in any way. But I think I might be confused here.
Offered a bet that pays $X if I pick a color and then see if a random ball matches that color, you’ll pay more
I’m not sure I follow. If I were to take this bet, it seems that the prior according to which my utility would be lowest is: you’ll pick a color to match that gives me a 0% chance of winning. So if I’m ambiguity averse in this way, wouldn’t I think this bet is worthless?
(The second point you bring up would make sense to me if this first point did, although then I’d also be confused about the papers’ emphasis on commitment.)
Sorry—you’re right that this doesn’t work. To clarify, I was thinking that the method of picking the color should be fixed ex-ante (e.g. “I pick red as the color with 50% probability”), but that doesn’t do the trick because you need to pool the colors for ambiguity to arise.
The issue is that the problem the paper identifies does not come up in your example. If I’m offered the two bets simultaneously, then an ambiguity averse decision maker, like an EU decision maker, will take both bets. If I’m offered the bets sequentially without knowing I’ll be offered both when I’m offered the first one, then neither an ambiguity-averse nor a risk-averse EU decision-maker will take them. The reason is that the first one offers the EU decision-maker a 50% chance of winning, so given risk-aversion its value is less than 50% of $1. So your example doesn’t distinguish a risk-averse EU decision-maker from an ambiguity-averse one.
So I think unfortunately we need to go with the more complicated examples in the paper. They are obviously very theoretical. I think it could be a valuable project for someone to translate these into more practical settings to show how these problems can come up in a real-world sense.
Hi Zach, thanks for this!
I have two doubts about the Al-Najjar and Weinstein paper—I’d be curious to hear your (or others’) thoughts on these.
First, I’m having trouble seeing where the information aversion comes in. A simpler example than the one used in the paper seems to be enough to communicate what I’m confused about: let’s say an urn has 100 balls that are each red or yellow, and you don’t know their distribution. Someone averse to ambiguity would (I think) be willing to pay up to $1 for a bet that pays off $1 if a randomly selected ball is red or yellow. But if they’re offered that bet as two separate decisions (first betting on a ball being red, and then betting on the same ball being yellow), then they’d be willing to pay less than $0.50 for each bet. So it looks like preference inconsistency comes from the choice being spread out over time, rather than from information (which would mean there’s no incentive to avoid information). What am I missing here?
(Maybe the following is how the authors were thinking about this? If you (as a hypothetical ambiguity-averse person) know that you’ll get a chance to take both bets separately, then you’ll take them both as long as you’re not immediately informed of the outcome of the first bet, because you evaluate acts, not by their own uncertainty, but by the uncertainty of your sequence of acts as a whole (considering all acts whose outcomes you remain unaware of). This seems like an odd interpretation, so I don’t think this is it.)
[edit: I now think the previous paragraph’s interpretation was correct, because otherwise agents would have no way to make ambiguity averse choices that are spread out over time and consistent, in situations like the ones presented in the paper. The ‘oddness’ of the interpretation seems to reflect the oddness of ambiguity aversion: rather than only paying attention to what might happen differently if you choose one action or another, ambiguity aversion involves paying attention to possible outcomes that will not be affected by your action, since they might influence the uncertainty of your action.]
Second, assuming that ambiguity aversion does lead to information aversion, what do you think of the response that “this phenomenon simply reflects a [rational] trade-off between the intrinsic value of information, which is positive even in the presence of ambiguity, and the value of commitment”?
Thanks! Helpful follow-ups.
On the first point, I think your intuition does capture the information aversion here, but I still think information aversion is an accurate description. Offered a bet that pays $X if I pick a color and then see if a random ball matches that color, you’ll pay more than for a bet that pays $X if a random ball is red. The only difference between these situations is that you have more information in the latter: you know the color to match is red. That makes you less willing to pay. And there’s no obvious reason why this information aversion would be something like a useful heuristic.
I don’t quite get the second point. Commitment doesn’t seem very relevant here since it’s really just a difference in what you would pay for each situation. If one comes first, I don’t see any reason why it would make sense to commit, so I don’t think that strengthens the case for ambiguity aversion in any way. But I think I might be confused here.
Thanks!
I’m not sure I follow. If I were to take this bet, it seems that the prior according to which my utility would be lowest is: you’ll pick a color to match that gives me a 0% chance of winning. So if I’m ambiguity averse in this way, wouldn’t I think this bet is worthless?
(The second point you bring up would make sense to me if this first point did, although then I’d also be confused about the papers’ emphasis on commitment.)
Sorry—you’re right that this doesn’t work. To clarify, I was thinking that the method of picking the color should be fixed ex-ante (e.g. “I pick red as the color with 50% probability”), but that doesn’t do the trick because you need to pool the colors for ambiguity to arise.
The issue is that the problem the paper identifies does not come up in your example. If I’m offered the two bets simultaneously, then an ambiguity averse decision maker, like an EU decision maker, will take both bets. If I’m offered the bets sequentially without knowing I’ll be offered both when I’m offered the first one, then neither an ambiguity-averse nor a risk-averse EU decision-maker will take them. The reason is that the first one offers the EU decision-maker a 50% chance of winning, so given risk-aversion its value is less than 50% of $1. So your example doesn’t distinguish a risk-averse EU decision-maker from an ambiguity-averse one.
So I think unfortunately we need to go with the more complicated examples in the paper. They are obviously very theoretical. I think it could be a valuable project for someone to translate these into more practical settings to show how these problems can come up in a real-world sense.