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How much do your eyes betray you?
Have two subjects play matching pennies. They will face each other but separated by a one-way mirror. Only one subject will be able to see the other’s face. He can only see the face, not anything below the chin.
Each subject selects his action by touching a screen. Touch the screen to the West to play Heads, touch the screen on the East to play Tails. (East-West rather than left-right so that my Tails screen is on the same side as your Tails screen. This makes it easier to keep track.)
You have to touch a lighted region of the screen in order to have your move registered and the lighted region is moving around the screen. This is going to require you to look at the screen you want to touch. But you can look in one direction and then the other and touch only the screen you want. Your hands are not visible to the other subject.
How much more money is earned by the player who can see the other’s eyes?
(Conversation with Adriana Lleras-Muney)
My brother-in-law wanted to sell something with an auction but first he wanted to assemble as many interested buyers as he could. His problem is that while he knew there were many interested buyers in the world he didn’t know who they were or how to find them. But he had a good idea: people who are interested in his product probably know other people who are also interested. He asked me for advice on how to use finders’ fees to incentivize the buyers he already know about to introduce him to any new potential buyers they know.
This is a very interesting problem because it interacts two different incentive issues. First, to get someone to refer you to someone they know you have to confront a traditional bilateral monopoly problem. You are a monopoly seller of your product but your referrer is a monopoly provider of access to his friend because only he knows which and how many of his friends are interested. If your finder’s fee is going to work it’s going to have to give him his monopoly rents.
The interesting twist is that your referrer has an especially strong incentive not to give you any references. Because anybody he introduces to you is just going to wind up being competition for him in the auction for your product. So your finder’s fee has to be even more generous in order to compensate your referrer for the inevitable reduction in the consumer’s surplus he was expecting from the auction.
I told my brother-in-law not to use finder’s fees. That can’t be the optimal way to solve his problem. Because there is another instrument he has at his disposal which must be the more efficient way to deal with this compound incentive problem.
Here’s the problem with finder’s fees. Every dollar of encouragement I give to my buyers is going to cost me a full dollar. But I have a way to give him a dollar’s worth of encouragement at a cost to me of strictly less than a dollar. I leverage my monopoly power and I use the object I am selling as the carrot.
In fact there is a basic principle here which explains not only why finder’s fees are bad incentive devices but also why employers give compensation in the form of employee discounts, why airlines use frequent flier miles as kickbacks and why a retailer would always prefer to give you store credit rather than cash refunds. It costs them less than a dollar to provide you with a dollar’s value.
Why is that? Because any agent with market power inefficiently under-provides his product. By setting high prices, he creates a wedge between his cost of supplying the good and your value for receiving it. If he wants to do you a favor he could either give you cash or he could give you the cash value in product. It’s always cheaper to do the latter.
So what does this say about incentivizing referrals to an auction? How do you “use the object” in place of a finder’s fee? The optimal way to do that is the following. You tell your potential referrer that you will give him an advantage in the auction if he brings to you a new potential buyer. Because you are a monopoly auctioneer there is always a wedge that you can capitalize on to do this at minimal cost to yourself.
In this particular example the wedge is your reserve price. Your referrer knows that you are going to extract your profits by setting a high reserve price and thereby committing not to sell the object if he is not willing to pay at least that much. You will induce your referrer to bring in new competition by offering to lower his reserve price when he does.
Now of course you have to deal with the problem of collusion and shills. Of course that’s a problem in any auction and even more of a problem with monetary finder’s fees but that’s a whole nuther post.
(Ongoing collaboration with Ahmad Peivandi)
Subjects video chat with each other. In one treatment subject A sees her own image in a small window in the corner of the chat, and in the other treatment (the control) there is no small window and she sees only the chat partner.
Subject B is not told about the two treatments and is simply asked to report how attractive subject A is. We want to know whether attractiveness is higher in the self-image treatment versus the control treatment.
This gets at a few different issues but the one I am curious about is this: do people know what it is about them that makes them attractive to others?
Also, we would want to track eye movements during the chat.
Suppose you and I are playing a series of squash matches and we are playing best 2 out of 3. If I win the first match I have an advantage for two reasons. First is the obvious direct reason that I am only one match short of wrapping up the series while you need to win the next two. Second is the more subtle strategic reason, the discouragement effect. If I fight hard to win the next match my reward is that my job is done for the day, I can rest and of course bask in the glow of victory. As for you, your effort to win the second match is rewarded by even more hard work to do in the third match.
Because you are behind, you have less incentive than me to win the second match and so you are not going to fight as hard to win it. This is the discouragement effect. Many people are skeptical that it has any measurable effect on real competition. Well I found a new paper that demonstrates an interesting new empirical implication that could be used to test it.
Go back to our squash match and now lets suppose instead that it’s a team competition. We have three players on our teams and we will match them up according to strength and play a best two out of three team competition. Same competition as before but now each subsequent game is played by a different pair of players.
A new paper by Fu, Lu, and Pan called “Team Contests With Multiple Pairwise Battles” analyzes this kind of competition and shows that they exhibit no discouragement effect. The intuition is straightforward: if I win the second match, the additional effort that would have to be spent to win the third match will be spent not by me, but by my teammate. I internalize the benefits of winning because it increases the chance that my team wins the overall series but I do not internalize the costs of my teammate’s effort in the third match. This negative externality is actually good for team incentives.
The implied empirical prediction is the following. Comparing individual matches versus team matches, the probability of a comeback victory conditional on losing the first match will be larger in the team competition. A second prediction is about the very first match. Without the discouragement effect, the benefit from winning the first match is smaller. So there will be less effort in the first match in the team versus individual competition.
My son and I went to see the Cubs last week as we do every Spring.
The Cubs won 8-0 and Matt Garza was one out away from throwing a complete game shutout, a rarity for a Cub. The crowd was on its feet with full count to the would-be final batter who rolled the ball back to the mound for Garza to scoop up and throw him out. We were all ready to give a big congratulatory cheer and then this happened. This is a guy who was throwing flawless pitches to the plate for nine innings and here with all the pressure gone and an easy lob to first he made what could be the worst throw in the history of baseball and then headed for the showers. Cubs win!
But this Spring we weren’t so interested in the baseball out on the field as we were in the strategery down in the toilet. Remember a while back when I wrote about the urinal game? It seems like it was just last week (fuzzy vertical lines pixellating then unpixellating the screen to reveal the flashback:)
Consider a wall lined with 5 urinals. The subgame perfect equilibrium has the first gentleman take urinal 2 and the second caballero take urinal 5. These strategies are pre-emptive moves that induce subsequent monsieurs to opt for a stall instead out of privacy concerns. Thus urinals 1, 3, and 4 go unused.
So naturally we turn our attention to The Trough.
A continuous action space. Will the trough induce a more efficient outcome in equilibrium than the fixed array of separate urinals? This is what you come Cheap Talk to find out.
Let’s maintain the same basic parameters. Assume that the distance between the center of two adjacent urinals is d and let’s consider a trough of length 5d, i.e. the same length as a 5 side-by-side urinals (now with invincible pink mystery ice located invitingly at positions d/2 + kd for k = 1, 2, 3, 4.) The assumption in the original problem was that a gentleman pees if and only if there is nobody in a urinal adjacent to him. We need to parametrize that assumption for the continuos trough. It means that there is a constant r such that he refuses to pee in a spot in which someone is currently peeing less than a distance r from him. The assumption from before implies that d < r < 2d. Moreover the greater the distance to the nearest reliever the better.
The first thing to notice is that the equilibrium spacing from the original urinal game is no longer a subgame-perfect equilibrium. In our continuous trough model that spacing corresponds to gentlemen 1 and 2 locating themselves at positions d/2 and 7d/2 measured from the left boundary of the trough. Suppose r <= 3d/2. Then the third man can now utilize the convex action space and locate himself at position 2d where he will be a comfortable distance 3d/2>= r away from the other two. If instead r > 3d/2, then the third man is strictly deterred from intervening but this means that gentleman number 2 would increase his personal space by locating slightly farther to the right whilst still maintaining that deterrence.
So what does happen in equilibrium? I’ve got good news and bad news. The good news first. Suppose that r < 5d/4. Then in equilibrium 3 guys use the trough whereas only 2 of the arrayed urinals were used in the original equilibrium. In equilibrium the first guy parks at d/2 (to be consistent with the original setup we assume that he cannot squeeze himself any closer than that to the left edge of the trough without risking a splash on the shoes) the second guy at 9d/2 and the third guy right in the middle at 5d/2. They are a distance of 2d> r from one another, and there is no room for anybody else because anybody who came next would have to be standing at most a distance d< r from two of the incumbents. This is a subgame perfect equilibrium because the second guy knows that the third guy will pick the midpoint and so to keep a maximal distance he should move to the right edge. And foreseeing all of this the first guy moves to the left edge.
Note well that this is not a Pareto improvement. The increased usage is offset by reduced privacy.They are only 2d away from each other whereas the two urinal users were 3d away from each other.
Now the bad news when r >5d/4. In this case it is possible for the first two to keep the third out. For example suppose that 1 is at 5d/4 and 2 is at 15d/4. Then there is no place the third guy can stand and be more than 5d/4 away hence more than r from the others. In this case the equilibrium has the first two guys positioning themselves with a distance between them equal to exactly 2r, thus maximizing their privacy subject to the constraint that the third guy is deterred. (One such equilibrium is for the first two to be an equal distance from their respective edges, but there are other equilibria.)
The really bad news is that when r is not too large, the two guys even have less privacy than with the urinals. For example if r is just above 5d/4 then they are only 10d/4 away from each other which is less than the 3d distance from before. What’s happening is that the continuous trough gives more flexibility for the third guy to squeeze between so the first two must stand closer to one another to keep him away.
Instant honors thesis for any NU undergrad who can generalize the analysis to a trough of arbitrary length.
Bicycle “sprints.” This is worth 6 minutes of your time.
Thanks to Josh Knox for the link.
In basketball the team benches are near the baskets on opposite sides of the half court line. The coaches roam their respective halves of the court shouting directions to their team.
As in other sports the teams switch sides at halftime but the benches stay where they were. That means that for half of the game the coaches are directing their defenses and for the other half they are directing their offenses.
If coaching helps then we should see more scoring in the half where the offenses are receiving direction.
This could easily be tested.
Over the course of your life you have to decide your position on a number of philosophical/social/political issues. You are open-minded so you collect as much data as you can before forming an opinion. But you are human and you can only remember so many facts.
There will come a time when the data you have collected make a very strong case for one particular position on issue A, say the right-wing position. When that happens you are pretty sure that there is never going to be enough evidence to overturn your position.
That’s not because you are closed-minded. That’s because you are very open-minded and based on the weight of all the evidence you collected and processed as objectively as a person can do, you have concluded that its very likely that this is the right position on A. And the fact that this is very likely the right position on A does not just imply but is indeed equivalent to saying that you attach very low probability to the future occurrence of strong evidence in the other direction.
Now that means that there’s not much point in collecting any more information about A. And indeed there’s not much point in remembering the detailed information that led you to this conclusion. The only reason for doing that would be to weigh it against future evidence but we’ve already established that this is unlikely to make any difference.
So what you optimally, rationally, perfectly objectively do is allow yourself to forget everything you know about A including all the reasons that justify your strongly-held views on A and to just make an indelible mental note that “The right-wing position on A is the correct one no matter what anyone else says and no matter what evidence to the contrary should come along in the future.”
The reason this is the rational thing to do is that you have scarce memory space. By allowing those memories to fade away you free up storage space for information about issues B, C, and D which you are still carefully collecting information on, forming an objective opinion about, in preparation for eventually also adopting a well-informed dogmatic opinion about.
- If you have a blog and you write about potential research questions, write the question out clearly but give a wrong answer. This solves the problem I raised here.
- When I send an email to two people I feel bad for the person whose name I address second (“Dear Joe and Jane”) so I put it twice to make it up to them (“Dear Joe and Jane and Jane.”)
- If you have a rich country and a poor country and their economies are growing at the same rate you will nevertheless have rising inequality over time simply because, as is well documented, the poor have more kids.
- Are there arguments against covering contraception under health insurance that don’t also apply to covering vaccines?
- The most interesting news is either so juicy that the source wants it kept private or so important that the source wants to make it public. This is why Facebook is an inferior form of communication: as neither private nor fully public it is an interior minimum.
Email is the superior form of communication as I have argued a few times before, but it can sure aggravate your self-control problems. I am here to help you with that.
As you sit in your office working, reading, etc., the random email arrival process is ticking along inside your computer. As time passes it becomes more and more likely that there is email waiting for you and if you can’t resist the temptation you are going to waste a lot of time checking to see what’s in your inbox. And it’s not just the time spent checking because once you set down your book and start checking you won’t be able to stop yourself from browsing the web a little, checking twitter, auto-googling, maybe even sending out an email which will eventually be replied to thereby sealing your fate for the next round of checking.
One thing you can do is activate your audible email notification so that whenever an email arrives you will be immediately alerted. Now I hear you saying “the problem is my constantly checking email, how in the world am i going to solve that by setting up a system that tells me when email arrives? Without the notification system at least I have some chance of resisting the temptation because I never know for sure that an email is waiting.”
Yes, but it cuts two ways. When the notification system is activated you are immediately informed when an email arrives and you are correct that such information is going to overwhelm your resistance and you will wind up checking. But, what you get in return is knowing for certain when there is no email waiting for you.
It’s a very interesting tradeoff and one we can precisely characterize with a little mathematics. But before we go into it, I want you to ask yourself a question and note the answer before reading on. On a typical day if you are deciding whether to check your inbox, suppose that the probability is p that you have new mail. What p is going to get you to get up and check? We know that you’re going to check if p=1 (indeed that’s what your mailbeep does, it puts you at p=1.) And we know that you are not going to check when p=0. What I want to know is what is the threshold above which its sufficiently likely that you will check and below which is sufficiently unlikely so you’ll keep on reading? Important: I am not asking you what policy you would ideally stick to if you could control your temptation, I am asking you to be honest about your willpower.
Ok, now that you’ve got your answer let’s figure out whether you should use your mailbeep or not. The first thing to note is that the mail arrival process is a Poisson process: the probability that an email arrives in a given time interval is a function only of the length of time, and it is determined by the arrival rate parameter r. If you receive a lot of email you have a large r, if the average time spent between arrivals is longer you have a small r. In a Poisson process, the elapsed time before the next email arrives is a random variable and it is governed by the exponential distribution.
Let’s think about what will happen if you turn on your mail notifier. Then whenever there is silence you know for sure there is no email, p=0 and you can comfortably go on working temptation free. This state of affairs is going to continue until the first beep at which point you know for sure you have mail (p=1) and you will check it. This is a random amount of time, but one way to measure how much time you waste with the notifier on is to ask how much time on average will you be able to remain working before the next time you check. And the answer to that is the expected duration of the exponential waiting time of the Poisson process. It has a simple expression:
Expected time between checks with notifier on =
Now let’s analyze your behavior when the notifier is turned off. Things are very different now. You are never going to know for sure whether you have mail but as more and more time passes you are going to become increasingly confident that some mail is waiting, and therefore increasingly tempted to check. So, instead of p lingering at 0 for a spell before jumping up to 1 now it’s going to begin at 0 starting from the very last moment you previously checked but then steadily and continuously rise over time converging to, but never actually equaling 1. The exponential distribution gives the following formula for the probability at time T that a new email has arrived.
Probability that email arrives at or before a given time T =
Now I asked you what is the p* above which you cannot resist the temptation to check email. When you have your notifier turned off and you are sitting there reading, p will be gradually rising up to the point where it exceeds p* and right at that instant you will check. Unlike with the notification system this is a deterministic length of time, and we can use the above formula to solve for the deterministic time T at which you succumb to temptation. It’s given by
Time between checks when the notifier is off =
And when we compare the two waiting times we see that, perhaps surprisingly, the comparison does not depend on your arrival rate r (it appears in the numerator of both expressions so it will cancel out when we compare them.) That’s why I didn’t ask you that, it won’t affect my prescription (although if you receive as much email as I do, you have to factor in that the mail beep turns into a Geiger counter and that may or may not be desirable for other reasons.) All that matters is your p* and by equating the two waiting times we can solve for the crucial cutoff value that determines whether you should use the beeper or not.
The beep increases your productivity iff your p* is smaller than
This is about .63 so if your p* is less than .63 meaning that your temptation is so strong that you cannot resist checking any time you think that there is at least a 63% chance there is new mail waiting for you then you should turn on your new mail alert. If you are less prone to temptation then yes you should silence it. This is life-changing advice and you are welcome.
Now, for the vapor mill and feeling free to profit, we do not content ourselves with these two extreme mechanisms. We can theorize what the optimal notification system would be. It’s very counterintuitive to think that you could somehow “trick” yourself into waiting longer for email but in fact even though you are the perfectly-rational-despite-being-highly-prone-to-temptation person that you are, you can. I give one simple mechanism, and some open questions below the fold.
Here’s a card game: You lay out the A,2,3 of Spades, Diamonds, Clubs in random order on the table face up. So that’s 9 cards in total. There are two players and they take turns picking up cards from the table, one at a time. The winner is the first to collect a triplet where a triplet is any one of the following sets of three:
- Three cards of the same suit
- Three cards of the same value
- Ace of Spaces, 2 of Diamonds, 3 of Clubs
- Ace of Clubs, 2 of Diamonds, 3 of Spades
Got it? Ok, this game can be solved and the solution is that with best play the result is a draw, neither player can collect a triplet. See if you can figure out why. (Drew Fudenberg got it almost immediately [spoiler.]) Answer and more discussion are after the jump.
Models of costly voting give rise to strategic turnout: in a district in which party A has a big advantage, supporters of party A will have low turnout in equilibrium in order to make the election close. That’s because only when the election is close will voters have an incentive to turnout and vote, which is costly.
Looking at elections data it is hard to identify strategic turnout. Low turnout is perfectly consistent with non-strategic voters who just have high costs of voting.
Redistricting offers an interesting source of variation that could help. Suppose that a state has just undergone redistricting and a town has been moved from a district with a large majority for one party into a more competitive district. Non-strategic voters in that town will not change their behavior.
But strategic voters will have different incentives in the new district. In particular we should see an increase in turnout among voters in the town that is new to the district. And this increase in turnout should be larger than any change in turnout observed for voters who remained in the district before and after redistricting.
There are probably a slew of testable implications that could be derived from models of strategic turnout based on whether the new district is more or less competitive than the old one, whether the stronger party is the same or different from the stronger party in the old district, and whether the town leans toward or against the stronger party in the new district.
In many situations, such reinforcement learning is an essential strategy, allowing people to optimize behavior to fit a constantly changing situation. However, the Israeli scientists discovered that it was a terrible approach in basketball, as learning and performance are “anticorrelated.” In other words, players who have just made a three-point shot are much more likely to take another one, but much less likely to make it:
What is the effect of the change in behaviour on players’ performance? Intuitively, increasing the frequency of attempting a 3pt after made 3pts and decreasing it after missed 3pts makes sense if a made/missed 3pts predicted a higher/lower 3pt percentage on the next 3pt attempt. Surprizingly [sic], our data show that the opposite is true. The 3pt percentage immediately after a made 3pt was 6% lower than after a missed 3pt. Moreover, the difference between 3pt percentages following a streak of made 3pts and a streak of missed 3pts increased with the length of the streak. These results indicate that the outcomes of consecutive 3pts are anticorrelated.
This anticorrelation works in both directions. as players who missed a previous three-pointer were more likely to score on their next attempt. A brick was a blessing in disguise.
The underlying study, showing a “failure of reinforcement learning” is here.
Suppose you just hit a 3-pointer and now you are holding the ball on the next possession. You are an experienced player (they used NBA data), so you know if you are truly on a hot streak or if that last make was just a fluke. The defense doesn’t. What the defense does know is that you just made that last 3-pointer and therefore you are more likely to be on a hot streak and hence more likely than average to make the next 3-pointer if you take it. Likewise, if you had just missed the last one, you are less likely to be on a hot streak, but again only you would know for sure. Even when you are feeling it you might still miss a few.
That means that the defense guards against the three-pointer more when you just made one than when you didn’t. Now, back to you. You are only going to shoot the three pointer again if you are really feeling it. That’s correlated with the success of your last shot, but not perfectly. Thus, the data will show the autocorrelation in your 3-point shooting.
Furthermore, when the defense is defending the three-pointer you are less likely to make it, other things equal. Since the defense is correlated with your last shot, your likelihood of making the 3-pointer is also correlated with your last shot. But inversely this time: if you made the last shot the defense is more aggressive so conditional on truly being on a hot streak and therefore taking the next shot, you are less likely to make it.
(Let me make the comparison perfectly clear: you take the next shot if you know you are hot, but the defense defends it only if you made the last shot. So conditional on taking the next shot you are more likely to make it when the defense is not guarding against it, i.e. when you missed the last one.)
You shoot more often and miss more often conditional on a previous make. Your private information about your make probability coupled with the strategic behavior of the defense removes the paradox. It’s not possible to “arbitrage” away this wedge because whether or not you are “feeling it” is exogenous.
Faced with a morally ambiguous choice, you are sometimes torn between conflicting motivations. And it can get to the point where you can’t really figure out which one is really driving you. Are you calling your old girlfriend because only she can give you the right advice about your sick cat, or because you just want to hear her voice? Are you recommending your colleague for the committee because he’s the right guy for the job or because you don’t want to do it yourself? Do you write a daily blog because it’s a great way to hash out new ideas or because you just love the attention?
From a conventional point of view its hard to understand how we could doubt our own motivations. At the moment of decision we can articulate at a conscious level what the right objective is. (If not, then on what basis would we have to be suspicious of ourselves?) And we should evaluate all the possible consequences of the action that tempts us in light of that objective and make the best choice.
So self-doubt is a smoking gun showing that this conventional framework omits an important friction. Here’s my theory what that friction is.
Information comes in millions of tiny pieces over time. It is beyond our memory and our conscious capacity to recall and assemble all of those data when called upon to make a decision that relies on it. Instead we discard the details and just store summary statistics. When it comes time to make a decision, the memory division of our decision-making apparatus steps up and presents the relevant summary statistics.
The instinctive feeling that “I should do X” is what it feels like when the reported summary statistics point in favor of X. It has an instinctive quality because it is entirely pre-conscious. Conscious deliberation begins only after that initial inclination is formed.
At that stage your task is to verify whether the proposed course of action is consistent with your current motivation and the specific details of the situation you find yourself in. But that decision is necessarily made with limited information because you only have the summary statistics to go on.
Any divergence between your present frame of mind and the frame of mind that you were in when you recorded and stored those summary statistics can give you cause for doubting your instincts.
That suggests an interesting behavioral framework. The decision maker is composed of two agents, an Advisor and a Decider. The Advisor has all of the information about the payoffs to different actions and he makes recommendations to the Decider who then takes an action. The friction is that the Advisor and Decider’s preferences are different and the difference fluctuates over time. Thus, at any point in time the Decider must resolve a conflict between his own objective and the unknown objective of the Advisor.
I write all the time about strategic behavior in athletic competitions. A racer who is behind can be expected to ease off and conserve on effort since effort is less likely to pay off at the margin. Hence so will the racer who is ahead, etc. There is evidence that professional golfers exhibit such strategic behavior, this is the Tiger Woods effect.
We may wonder whether other animals are as strategically sophisticated as we are. There have been experiments in which monkeys play simple games of strategy against one another, but since we are not even sure humans can figure those out, that doesn’t seem to be the best place to start looking.
I would like to compare how humans and other animals behave in a pure physical contest like a race. Suppose the animals are conditioned to believe that they will get a reward if and only if they win a race. Will they run at maximum speed throughout regardless of their position along the way? Of course “maximum speed” is hard to define, but a simple test is whether the animal’s speed at a given point in the race is independent of whether they are ahead or behind and by how much.
And if the animals learn that one of them is especially fast, do they ease off when racing against her? Do the animals exhibit a tiger Woods effect?
There are of course horse-racing data. That’s not ideal because the jockey is human. Still there’s something we can learn from horse racing. The jockey does not internalize 100% of the cost of the horse’s effort. Thus there should be less strategic behavior in horse racing than in races between humans or between jockey-less animals. Dog racing? Does that actually exist?
And what if a dog races against a human, what happens then?
Doctors sometimes resist prescribing costly diagnostic procedures, saying that the result of the test would be unlikely to affect the course of treatment. But what we know about placebo effects for medicine should have implications also for the value of information, even when it leads to no objective health benefits.
I have a theory of how placebos work. The idea is that our bodies, either through conscious choices that we make or simply through physiological changes, must make an investment in order to get healthy. Being sick is like being, perhaps temporarily, below the threshold where the body senses that the necessary investment is worth it. A placebo tricks the body into thinking that we are going to get at least marginally more healthy and that pushes above the threshold triggering the investment which makes us healthy.
The same idea can justify providing information that has no instrumental value. Suppose you have an injury and are considering having an MRI to determine how serious it is. Your doctor says that surgery is rarely worthwhile and so even if the MRI shows a serious injury it won’t affect how you are treated.
But you want to know. For one thing the information can affect how you personally manage the injury. That’s instrumental value that your doctor doesn’t take into account.
But even if there were nothing you could consciously do based on the test result, there may be a valuable placebo reason to have the MRI. If you find out that the injury is mild, the psychological effect of knowing that you are healthy (or at least healthier than you previously thought) can be self-reinforcing.
The downside of course is that when you find out that the injury is serious you get an anti-placebo effect. So the question is whether you are better off on average when you become better informed about your true health status.
If the placebo effect works because the belief triggers a biological response then this is formally equivalent to a standard model of decision-making under uncertainty. Whenever a decision-maker will optimally condition his decision on the realization of information, then the expected value of learning that information is positive.
In the past few weeks Romney has dropped from 70% to under 50% and Gingrich has rocketed to 40% on the prediction markets. And in this time Obama for President has barely budged from its 50% perch. As someone pointed out on Twitter (I forget who, sorry) this is hard to understand.
For example if you think that in this time there has been no change in the conditional probabilities that either Gingrich or Romney beats Obama in the general election, then these numbers imply that the market thinks that those conditional probabilities are the same. Conversely, If you think that Gingrich has risen because his perceived odds of beating Obama have risen over the same period, then it must be that Romney’s have dropped in precisely the proportion to keep the total probability of a GOP president constant.
It’s hard to think of any public information that could have these perfectly offsetting effects. Here’s the only theory I could come up with that is consistent with the data. No matter who the Republican candidate is, he has a 50% chance of beating Obama. This is just a Downsian prediction. The GOP machine will move whoever it is to a median point in the policy space. But, and here’s the model, this doesn’t imply that the GOP is indifferent between Gingrich and Romney.
While any candidate, no matter what his baggage, can be repositioned to the Downsian sweet spot, the cost of that repositioning depends on the candidate, the opposition, and the political climate. The swing from Romney to Gingrich reflects new information about these that alters the relative cost of marketing the two candidates. Gingrich has for some reason gotten relatively cheaper.
I didn’t say it was a good theory.
Update: Rajiv Sethi reminded me that the tweet was from Richard Thaler. (And see Rajiv’s comment below.)
You and your partner have to decide on a new venture. Maybe you and your sweetie are deciding on a movie, you and your co-author are deciding on which new idea to develop, or you and your colleague are deciding which new Assistant Professor to hire.
Deliberation consists of proposals and reactions. When you pitch your idea you naturally become attached to it. Its your idea, your creation. Your feelings are going to be hurt if your partner doesn’t like it.
Maybe you really are a dispassionate common interest maximizer, but there’s no way for your partner to know that for sure. You try to say “give me your honest opinion, I promise I have thick skin, you won’t hurt my feelings.” But you would say that even if it’s a little white lie.
The important thing is that no matter how sensitive you actually are, your partner believes that there is a chance your feelings will be hurt if she shoots down your idea. And she might even worry that you would respond by feeling resentful towards her. All of this makes her reluctant to give her honest opinion about your idea. The net result is that some inferior projects might get adopted because concern for hurt feelings gets in the way of honest information exchange.
Unless you design the mechanism to work around that friction. The basic problem is that when you pitch your idea it becomes common knowledge that you are attached to it. From that moment forward it is common knowledge that any opinion expressed about the idea has the chance of causing hurt feelings.
So a better mechanism would change the timing to remove that feature. You and your partner first announce to one another which options are unacceptable to you. Now all of the rejections have been made before knowing which ones you are attached to. Only then do you choose your proposal from the acceptable set.
If your favorite idea has been rejected then for sure you are disappointed. But your feelings are not hurt because it is common knowledge that her rejection is completely independent of your attachment. And for exactly that reason she is perfectly comfortable being honest about which options are unacceptable.
This is going to work better for movies, and new Assistant Professors than it is for research ideas. Because we know in advance the universe of all movies and job market candidates.
Research ideas and other creative ventures are different because there is no way to enumerate all of the possibilities beforehand and reject the unacceptable ones. Indeed the real value of a collaborative relationship is that the partners are bringing to the table brand new previously unconceived-of ideas. This makes for a far more delicate relationship.
We can thus classify relationships according to whether they are movie-like or idea-like, and we would expect that the first category are easier to sustain with second-best mechanisms whereas the second require real trust and honesty.
(inspired by a conversation with +Emil Temnyalov and Jorge Lemus)
I was talking to someone about matching mechanisms and the fact that strategy-proof incentives are often incompatible with efficiency. The question came up as to why we insist upon strategy-proofness, i.e. dominant strategy incentives as a constraint. If there is a trade-off between incentives and efficiency shouldn’t that tradeoff be in the objective function? We could then talk about how much we are willing to compromise on incentives in order to get some marginal improvement in efficiency.
For example, we might think that agents are willing to tell the truth about their preferences as long as manipulating the mechanism doesn’t improve their utility by a large amount. Then we should formalize a tradeoff between the epsilon slack in incentives and the welfare of the mechanism. The usual method of maximizing welfare subject to an incentive constraint is flawed because it prevents us from thinking about the problem in this way.
That sounded sensible until I thought about it just a little bit longer. If you are a social planner you have some welfare function, let’s say V. You want to choose a mechanism so that the resulting outcome maximizes V. And you have a theory about how agents will play any mechanism you choose. Let’s say that for any mechanism M, O(M) describes the outcome or possible outcomes according to your theory. This can be very general: O(M) could be the set of outcomes that will occur when agents are epsilon-truth-tellers, it could be some probability distribution over outcomes reflecting that you acknowledge that your theory is not very precise. And if you have the idea that incentives are flexible, O can capture that: for mechanisms M that have very strong incentive properties, O(M) will be a small set, or a degenerate probability distribution, whereas for mechanisms M that compromise a bit on incentives O(M) will be a larger set or a more diffuse probability distribution. And if you believe in a tradeoff between welfare and incentives, your V applied to O(M) can encode that by quantifying the loss associated with larger sets O(M) compared to smaller sets O(M).
But whatever your theory is you can represent it by some O(.) function. Then the simplest formulation of your problem is: choose M to maximize V(O(M)). And then we can equivalently express that problem in our standard way: choose an outcome (or set of outcomes, or probability distribution over outcomes ) O to maximize V(O) subject to the constraint that there exists some mechanism M for which O = O(M). That constraint is called the incentive constraint.
Incentives appear as a constraint, not in the objective. Once you have decided on your theory O, it makes no sense to talk about compromising on incentives and there is no meaningful tradeoff between incentives and welfare. While we might, as a purely theoretical exercise, comment on the necessity of such a tradeoff, no social planner would ever care to plot a “frontier” of mechanisms whose slope quantifies a rate of substitution between incentives and welfare.
I stopped following Justin Wolfers on Twitter. Not because I don’t want his tweets, they are great, but because everyone I follow also follows Justin. They all retweet his best tweets and I see those so I am not losing anything.
Which made me wonder how increasing density of the social network affects how informed people are. Suppose you are on a desert island but a special desert island which receives postal deliveries. You can get informed by subscribing to newspapers but you can’t talk to anybody. As long as the value v of being informed exceeds the cost c you will subscribe.
Compare that to an individual in a dense social network who can either pay for a subscription or wait around for his friends to get informed and find out from them. It won’t be an equilibrium for everybody to subscribe. You would do better by saving the cost and learning from your friends. Likewise it can’t be that nobody subscribes.
Instead in equilibrium everybody will subscribe with some probability between 0 and 1. And there is a simple way to compute that probability. In such an equilibrium you must be indifferent between subscribing and not subscribing. So the total probability that at least one of your friends subscribes must be the q that satisfies vq = v – c. The probability of any one individual subscribing must of course be lower than q since q is the total probability that at least one subscribes. So if you have n friends, then they each subscribe with the probability p(n) satisfiying 1 – [1 - p(n)]^n = q.
(Let’s pause while the network theorists all rush out of the room to their whiteboards to solve the combinatorial problem of making these balance out when you have an arbitrary network with different nodes having a different number of neighbors.)
This has some interesting implications. Suppose that the network is very dense so that everybody has many friends. Then everyone is less likely to subscribe. We only need a few people to be Justin Wolfers’ followers and retweet all of his best tweets. Formally, p(n) is decreasing in n.
That by itself is not such a bad thing. Even though each of your friends subscribes with a lower probability, on the positive side you have more friends from whom you can indirectly get informed. The net effect could be that you are more likely to be informed.
But in fact the net effect is that a denser network means that people are on average less informed, not more. Because if the network density is such that everyone has (on average) n friends, then everybody subscribes with probability p(n) and then the probability that you learn the information is q + (1-q)p(n). (With probability q one of your friends subscribes and you learn from them, and if you don’t learn from a friend then you become informed only if you have subscribed yourself which you do with probability p(n).) Since p(n) gets smaller with n, so does the total probability that you are informed.
Another way of saying this is that, contrary to intuition, if you compare two otherwise similar people, those who are well connected within the network have a tendency to be less informed than those who are in a relatively isolated part of the network.
All of this is based on a symmetric equilibrium. So one way to think about this is as a theory for why we see hierachies in information transmission, as represented by an asymmetric equilibrium in which some people subscribe for sure and others are certain not to. At the top of the hierarchy there is Justin Wolfers. Just below him we have a few people who follow him. They have a strict incentive to follow him because so few others follow him that the only way to be sure to get his tweets is to follow him directly. Below them is a mass of people who follow these “retailers.”
I was working on a paper, writing the introduction to a new section that deals with an extension of the basic model. It’s a relevant extension because it fits many real-world applications. So naturally I started to list the many real-world applications.
“This applies to X, Y, and….” hmmm… what’s the Z? Nothing coming to mind.
But I can’t just stop with X and Y. Two examples are not enough. If I only list two examples then the reader will know that I could only think of two examples and my pretense that this extension applies to many real-world applications will be dead on arrival.
I really only need one more. Because if I write “This applies to X, Y, Z, etc.” then the Z plus the “etc.” proves that there is in fact a whole blimpload of examples that I could have listed and I just gave the first three that came to mind, then threw in the etc. to save space.
If you have ever written anything at all you know this feeling. Three equals infinity but two is just barely two.
This is largely an equilbrium phenomenon. A convention emerged according to which those who have an abundance of examples are required to prove it simply by listing three. Therefore those who have listed only two examples truly must have only two.
Three isn’t the only threshold that would work as an equilibrium. There are many possibilities such as two, four, five etc. (ha!) Whatever threshold N we settle on, authors will spend the effort to find N examples (if they can) and anything short of that will show that they cannot.
But despite the multiplicity I bet that the threshold of three did not emerge arbitrarily. Here is an experiment that illustrates what I am thinking.
Subjects are given a category and 1 minute, say. You ask them to come up with as many examples from that category they can think of in 1 minute. After the 1 minute is up and you count how many examples they came up with you then give them another 15 minutes to come up with as many as they can.
With these data we would do the following. Plot on the horizontal axis the number x of items they listed in the first minute and on the vertical axis the number E(y|x) equal to the empirical average number y of items they came up with in total conditional on having come up with x items in the first minute.
I predict that you will see an anomalous jump upwards between E(y|2) and E(y|3).
This experiment does not take into account the incentive effects that come from the threshold. The incentives are simply to come up with as many examples as possible. That is intentional. The point is that this raw statistical relation (if it holds up) is the seed for the equilibrium selection. That is, when authors are not being strategic, then three-or-more equals many more than two. Given that, the strategic response is to shoot for exactly three. The equilibrium result is that three equals infinity.
Usain Bolt was disqualified in the final of the 100 meters at the World Championships due to a false start. Under current rules, in place since January 2010, a single false start results in disqualification. By contrast, prior to 2003 each racer who jumped the gun would be given a warning and then disqualified after a second false start. In 2003 the rules were changed so that the entire field would receive a warning after a false start by any racer and all subsequent false starts would lead to disqualification.
Let’s start with the premise that an indispensible requirement of sprint competition is that all racers must start simultaneously. That is, a sprint is not a time trial but a head-to-head competition in which each competitor can assess his standing at any instant by comparing his and his competitors’ distance to a fixed finished line.
Then there must be penalty for a false start. The question is how to design that penalty. Our presumed edict rules out marginally penalizing the pre-empter by adding to his time, so there’s not much else to consider other than disqualification. An implicit presumption in the pre-2010 rules was that accidental false starts are inevitable and there is a trade-off between the incentive effects of disqualification and the social loss of disqualifying a racer who made an error despite competing in good faith.
(Indeed this trade-off is especially acute in high-level competitions where the definition of a false start is any racer who leaves less than 0.10 seconds after the report of the gun. It is assumed to be impossible to react that fast. But now we have a continuous variable to play with. How much more impossible is it to react within .10 seconds than to react within .11 seconds? When you admit that there is a probability p>0, increasing in the threshold, that a racer is gifted enough to reach within that threshold, the optimal incentive mechanisn picks the threshold that balances type I and type II errors. The maximum penalty is exacted when the threshold is violated.)
Any system involving warnings invites racers to try and anticipate the gun, increasing the number of false starts. But the pre- and post-2003 rules play out differently when you think strategically. Think of the costs and benefits of trying to get a slightly faster start. The warning means that the costs of a potential false start are reduced. Instead of being disqualified you are given a second chance but are placed in the dangerous position of being disqualified if you false start again. In that sense, your private incentives to time the gun are identical whether the warning applies only to you or to the entire field. But the difference lies in your treatment relative to the rest of the field. In the post-2003 system that penalty will be applied to all racers so your false start does not place you at a disadvantage.
Thus, both systems encourage quick starts but the post 2003 system encouraged them even more. Indeed there is an equilibrium in which false starts occur with probability close to 1, and after that all racers are warned. (Everyone expects everyone else to be going early, so there’s little loss from going early yourself. You’ll be subject to the warning either way.) After that ceremonial false start the race becomes identical to the current, post 2010, rule in which a single false start leads to disqualification. My reading is that equilibrium did indeed obtain and this was the reason for the rule change. You could argue that the pre 2003 system was even worse because it led to a random number of false starts and so racers had to train for two types of competition: one in which quick starts were a relevant strategy and one in which they were not.
Is there any better system? Here’s a suggestion. Go back to the 2003-2009 system with a single warning for the entire field. The problem with that system was that the penalty for being the first to false start was so low that when you expected everyone else to be timing the gun your best response was to time the gun as well. So my proposal is to modify that system slightly to mitigate this problem. Now, if racer B is the first to false start then in the restart if there is a second false start by, say racer C, then racer C and racer B are disqualified. (In subsequent restarts you can either clear the warning and start from scratch or keep the warning in place for all racers.)
Here’s a second suggestion. The racers start by pushing off the blocks. Engineer the blocks so that they slide freely along their tracks and only become fixed in place at the precise moment that the gun is fired.
(For the vapor mill, here are empirical predictions about the effect of previous rule-regimes on race outcomes:
- Comparing pre-2003, under the 2003-2009 you should see more races with at least one false start but far fewer total false starts per race. The current rules should have the least false starts.
- Controlling for trend (people get faster over time) if you consider races where there was no false start, race times should be faster 2003-2009 than pre-2003. That ranking reverses when you consider races in which there was at least one false start. Controlling for Usain Bolt, times should be unambiguously slower under current rules.)
Experiments concerning the effect of publishing calorie counts on restaurant menus tend to show little effect on choices. In the experiments that I know of, choices before and after publishing calorie counts are compared. But this form of test cannot be considered conclusive. Some people were overestimating the calories and they might cut back, some were underestimating and they might eat more. There is no reason to expect that the aggregate change should be positive or negative.
A better experiment would be to use a restaurant where calorie counts are already published and manipulate them. Will people change their choices when you add 5% to the reported calories? 10%? What is the elasticity? It’s a safe guess that there would be little response for small changes and a large response for very large changes. Any response at all would prove that their is value is publishing calorie counts because it would prove that this information is useful for choices.
The only question that would remain is how those welfare gains measure up against the cost of collecting and publicizing the information.
Bandwagon effects are hard to prove. If an artist is popular, does that popularity by itself draw others in? Are you more likely to enjoy a movie, restaurant, blog just because you know that lots of other people like it to? It’s usually impossible to distinguish that theory from the simpler hypothesis: the reason it was popular in the first place was that it was good and that’s why you are going to like it too.
Here’s an experiment that would isolate bandwagon effects. Look at the Facebook like button below this post. I could secretly randomly manipulate the number that appears on your screen and then correlate your propensity to “like” with the number that you have seen. The bandwagon hypothesis would be that the larger number of likes you see increases your likeitude.
Clearly the reason that sex is so pleasurable is because that motivates us to have a lot of it. It is evolutionarily advantageous to desire the things that make us more fit. Sex feels good, we seek that feeling, we have a lot of sex, we reproduce more.
But that is not the only way to get motivated. It is also advantageous to derive pleasure directly from having children. We see children, we sense the joy we would derive from our own children and we are motivated to do what’s necessary to produce them, even if we had no particular desire for the intermediate act of sex.
And certainly both sources of motivation operate on us, but in different proportions. So it is interesting to ask what determines the optimal mix of these incentives. One alternative is to reward an intermediate act which has no direct effect on fitness but can, subject to idiosyncratic conditions together with randomness, produce a successful outcome which directly increases fitness. Sex is such an act. The other alternative is to confer rewards upon a successful outcome (or penalties for a failure.) That would mean programming us with a desire and love for children.
The tradeoff can be understood using standard intuitions from incentive theory. The rewards are designed to motivate us to take the right action at the right time. The drawback of rewarding only the final outcome is that it may be too noisy a signal of whether he acted. For example, not every encounter results in offspring. If so, then a more efficient use of rewards to motivate an act of sex is to make sex directly pleasurable. But the drawback of rewarding sex directly is that whether it is desirable to have sex right now depends on how likely it is to produce valuable offspring. If we are made to care only about the (value of) offspring we are more likely to make the right decision under the right circumstances.
Now these balance out differently for males than for females. Because when the female becomes pregnant and gives birth that is a very strong signal that she had sex at an opportune time but conveys noisier information about him.That is because, of course, this child could belong to any one of her (potentially numerous) mates. Instilling a love for children is therefore a relatively more effective incentive instrument for her than for him.
As for love of sex, note that the evolutionary value of offspring is different for males than for females because females have a significant opportunity cost given that they get pregnant with one mate at a time. This means that the circumstances are nearly always right for males to have sex, but much more rarely so for females. It is therefore efficient for males to derive greater pleasure from sex.
(It is a testament to my steadfastness as a theorist that I stand firmly by the logic of this argument despite the fact that, at least in my personal experience, females derive immense pleasure from sex.)
Drawing: Misread Trajectory from www.f1me.net
How does the additional length of a 5 set match help the stronger player? Commenters to my previous post point out the direct way: it lowers the chance of a fluke in which the weaker player wins with a streak of luck. But there’s another way and it can in principle be identified in data.
To illustrate the idea, take an extreme example. Suppose that the stronger player, in addition to having a greater baseline probability of winning each set, also has the ability to raise his game to a higher level. Suppose that he can do this once in the match and (here’s the extreme part) it guarantees that he will win that set. Finally, suppose that the additional effort is costly so other things equal he would like to avoid it. When will he use his freebie?
Somewhat surprisingly, he will always wait until the last set to use it. For example, in a three set match, suppose he loses the first set. He can spend his freebie in the second set but then he has to win the third set. If he waits until the third set, his odds of winning the match are exactly the same. Either way he needs to win one set at the baseline odds.
The advantage of waiting until the third set is that this allows him to avoid spending the effort in a losing cause. If he uses his freebie in the second set, he will have wasted the effort if he loses the third set. Since the odds of winning are independent of when he spends his effort, it is unambiguously better to wait as long as possible.
This strategy has the following implications which would show up in data.
- In a five set match, the score after three sets will not be the same (statistically) as the score in a three set match.
- In particular, in a five-set match the stronger player has a lower chance of winning a third set when the match is tied 1-1 than he would in a three set match.
- The odds that a higher seeded player wins a fifth set is higher than the odds that he wins, say, the second set. (This may be hard to identify because, conditional on the match going to 5 sets, it may reveal that the stronger player is not having a good day.)
- If the baseline probability is close to 50-50, then a 5 set match can actually lower the probability that the stronger player wins, compared to a 3 set match.
This “freebie” example is extreme but the general theme would always be in effect if stronger players have a greater ability to raise their level of play. That ability is an option which can be more flexibly exercised in a longer match.
Apparently it’s biology and economics week for me because after Andrew Caplin finishes his fantastic series of lectures here at NU tomorrow, I am off to LA for this conference at USC on Biology, Neuroscience, and Economic Modeling.
Today Andrew was talking about the empirical foundations of dopamine as a reward system. Along the way he reminded us of an important finding about how dopamine actually works in the brain. It’s not what you would have guessed. If you take a monkey and do a Pavlovian experiment where you ring a bell and then later give him some goodies, the dopamine neurons fire not when the actual payoff comes, but instead when the bell rings. Interestingly, when you ring the bell and then don’t come through with the goods there is a dip in dopamine activity that seems to be associated with the letdown.
The theory is that dopamine responds to changes in expectations about payoffs, and not directly to the realization of those payoffs. This raises a very interesting theoretical question: why would that be Nature’s most convenient way to incentivize us? Think of Nature as the principal, you are the agent. You have decision-making authority because you know what choices are available and Nature gives you dopamine bonuses to guide you to good decisions. Can you come up with the right set of constraints on this moral hazard problem under which the optimal contract uses immediate rewards for the expectation of a good outcome rather than rewards that come later when the outcome actually obtains?
Here’s my lame first try, based on discount factors. Depending on your idiosyncratic circumstances your survival probability fluctuates, and this changes how much you discount the expectation of future rewards. Evolution can’t react to these changes. But if Nature is going to use future rewards to motivate your behavior today she is going to have to calibrate the magnitude of those incentive payments to your discount factor. The fluctuations in your discount factor make this prone to error. Immediate payments are better because they don’t require Nature to make any guesses about discounting.
Here is a problem at has been in the back of my mind for a long time. What is the second best dominant-strategy mechanism (DSIC) in a market setting?
For some background, start with the bilateral trade problem of Myerson-Satterthwaite. We know that among all DSIC, budget-balanced mechanisms the most efficient is a fixed-price mechanism. That is, a price is fixed ex ante and the buyer and seller simply announce whether they are willing to trade at that price. Trade occurs if and only if both are willing and if so the buyer pays the fixed price to the seller. This is Hagerty and Rogerson.
Now suppose there are two buyers and two sellers. How would a fixed-price mechanism work? We fix a price p. Buyers announce their values and sellers announce their costs. We first see if there are any trades that can be made at the fixed price p. If both buyers have values above p and both sellers have values below then both units trade at price p. If two buyers have values above p and only one seller has value below p then one unit will be sold: the buyers will compete in a second-price auction and the seller will receive p (there will be a budget surplus here.) Similarly if the sellers are on the long side they will compete to sell with the buyer paying p and again a surplus.
A fixed-price mechanism is no longer optimal. The reason is that we can now use competition among buyers and sellers and “price discovery.” A simple mechanism (but not the optimal one) is a double auction. The buyers play a second-price auction between themselves, the sellers play a second-price reverse auction between themselves. The winner of the two auctions have won the right to trade. They will trade if and only if the second highest buyer value (which is what the winning buyer will pay) exceeds the second-lowest seller value (which is what the winning seller will receive.) This ensures that there will be no deficit. There might be a surplus, which would have to be burned.
This mechanism is DSIC and never runs a deficit. It is not optimal however because it only sells one unit. But it has the viture of allowing the “price” to adjust based on “supply and demand.” Still, there is no welfare ranking between this mechanism and a fixed-price mechanism because a fixed price mechanism will sometimes trade two units (if the price was chosen fortuitously) and sometimes trade no units (if the price turned out too high or low) even though the price discovery mechanism would have traded one.
But here is a mechanism that dominates both. It’s a hybrid of the two. We fix a price p and we interleave the rules of the fixed-price mechanism and the double auction in the following order
- First check if we can clear two trades at price p. If so, do it and we are done.
- If not, then check if we can sell one unit by the double auction rules. If so, do it and we are done.
- Finally, if no trades were executed using the previous two steps then return to the fixed-price and see if we can execute a single trade using it.
I believe this mechanism is DSIC (exercise for the reader, the order of execution is crucial!). It never runs a deficit and it generates more trade than either standalone mechanism: fixed-price or double auction.
Very interesting research question: is this a second-best mechanism? If not, what is? If so, how do you generalize it to markets with an arbitrary number of buyers and sellers?
A buyer and a seller negotiating a sale price. The buyer has some privately known value and the seller has some privately known cost and with positive probability there are gains from trade but with positive probability the seller’s cost exceeds the buyers value. (So this is the Myerson-Satterthwaite setup.)
Do three treatments.
- The experimenter fixes a price in advance and the buyer and seller can only accept or reject that price. Trade occurs if and only if they both accept.
- The seller makes a take it or leave it offer.
- The parties can freely negotiate and they trade if and only if they agree on a price.
Theoretically there is no clear ranking of these three mechanisms in terms of their efficiency (the total gains from trade realized.) In practice the first mechanism clearly sacrifices some efficiency in return for simplicity and transparency. If the price is set right the first mechanism would outperform the second in terms of efficiency due to a basic market power effect. In principle the third treatment could allow the parties to find the most efficient mechanism, but it would also allow them to negotiate their way to something highly inefficient.
A conjecture would be that with a well-chosen price the first mechanism would be the most efficient in practice. That would be an interesting finding.
A variation would be to do something similar but in a public goods setting. We would again compare simple but rigid mechanisms with mechanisms that allow for more strategic behavior. For example, a version of mechanism #1 would be one in which each individual was asked to contribute an equal share of the cost and the project succeeds if and only if all agree to their contributions. Mechanism #3 would allow arbitrary negotation with the only requirement be that the total contribution exceeds the cost of the project.
In the public goods setting I would conjecture that the opposite force is at work. The scope for additional strategizing (seeding, cajoling, guilt-tripping, etc) would improve efficiency.
Anybody know if anything like these experiments have been done?
This is the third and final post on ticket pricing motivated by the new restaurant Next in Chicago and proprietors Grant Achatz and Nick Kokonas new ticket policy. In the previous two installments I tried to use standard mechanism design theory to see what comes out when you feed in some non-standard pricing motives having to do with enhancing “consumer value.” The two attempts that most naturally come to mind yielded insights but not a useful pricing system. Today the third time is the charm.
Things start to come in to place when we pay close attention to this part of Nick’s comment to us:
we never want to invert the value proposition so that customers are paying a premium that is disproportionate to the amount of food / quality of service they receive.
I propose to formalize this as follows. From the restaurant’s point of view, consumer surplus is valuable but some consumers are prepared to bid even more than the true value of the service they will get. The restaurant doesn’t count these skyscraping bids as actually reflecting consumer surplus and they don’t want to tailor their mechanism to cater to them. In particular, the restaurant distinguishes willingness to pay from “value.”
I can think of a number of sensible reasons they would take this view. They might know that many patrons overestimate the value of a seating at Next. Indeed the restaurant might worry that high prices by themselves artificially inflate willingness to pay. They don’t want a bubble. And they worry about their reputation if someone pays $1700 for a ticket, gets only $1000 worth of value and publicly gripes. Finally they might just honestly believe that willingness to pay is a poor measure of welfare especially when comparing high versus low.
Whatever the reason, let’s run with it. Let’s define 
where 
Now let’s consider the optimal pricing mechanism for a restaurant that maximizes a weighted sum of profit and consumer’s surplus, where now consumer’s surplus is measured as the difference between 
![(1 - \alpha) W(v) + (2 \alpha - 1) [v - \frac{1-F(v)}{f(v)} ]](http://s0.wp.com/latex.php?latex=%281+-+%5Calpha%29+W%28v%29+%2B+%282+%5Calpha+-+1%29+%5Bv+-+%5Cfrac%7B1-F%28v%29%7D%7Bf%28v%29%7D+%5D&bg=ffffff&fg=545454&s=0 "(1 - \alpha) W(v) + (2 \alpha - 1) [v - \frac{1-F(v)}{f(v)} ]")
And now we have something! Because if 
The optimal mechanism is a hybrid between an auction and a lottery. It has no reserve price (over and above the cost of service) so there are never empty seats. It earns profits but eschews exorbitant prices.
It has clear advantages over a fixed price. A fixed price is a blunt instrument that has to serve two conflicting purposes. It has to be high enough to earn sufficient revenue on dates when demand is high enough to support it, but it can’t be too high that it leads to empty seats on dates when demand is lower. An auction with rationing at the top is flexible enough to deal with both tasks independently. When demand is high the fixed price (and rationing) is in effect. When demand is low the auction takes care of adjusting the price downward to keep the restaurant full. The revenue-enhancing effects of low prices is an under-appreciated benefit of an auction. Finally, it’s an efficient allocation system for the middle range of prices so scalping motivations are reduced compared to a fixed price.
Incentives for scalping are not eliminated altogether because of the rationing at the top. This can be dealt with by controlling the resale market. Indeed here is one clear message that comes out of all of this. Whatever motivation the restaurant has for rationing sales, it is never optimal to allow unfettered resale of tickets. That only undermines what you were trying to achieve. Now Grant Achatz and Nick Kokonas understand that but they are forced to condone the Craigslist market because by law non-refundable tickets must be freely transferrable.
But the cure is worse than the disease. In fact refundable tickets are your friend. The reason someone wants to return their ticket for a refund is that their willingness to pay has dropped below the price. But there is somebody else with a willingness to pay that is above the price. We know this for sure because tickets are being rationed at that price. Granting the refund allows the restaurant to immediately re-sell it to the next guy waiting in line. Indeed, a hosted resale market would enable the restaurant to ensure that such transactions take place instantaneously through an automated system according to the same terms under which tickets were originally sold.
Someone ought to try this.
