Pollsters

Several new worries:

1. I think we now know that the "best" equilibrium changes discontinuously in changes in bias. We proved this for the case where b is in the neighborhood of 1/6. The upshot of this is that there's no general rnaking possible between centralized and decentralized information. In the neighborhood of b=1/6, decentralized is strictly better for b<=1/6 (in a neighborhood) and strictly worse (owing to the discontinuity) for b>1/6

2. What is the contribution of our model relative to Battaglini. Here's a link to the article. According to the abstract there:

We study policy advice by several experts with noisy private information and biased preferences. We highlight a trade-off between the truthfulness of the information revealed by each expert and the number of signals from different experts that can be aggregated to reduce noise. Contrary to models with perfectly informed experts, because of this trade-off, full revelation of information is never possible. However, almost fully efficient information extraction can be obtained in two cases. First, there is an equilibrium in which the outcome converges to the first best benchmark with no asymmetric information as we increase the precision the experts' signals. Second, the inefficiency in communication also converges to zero as the number of experts increases, even when the residual noise in the experts' private signals is large and all the experts have significant and similar (but not necessarily identical) biases

That's pretty close to what we're interested in.

Key differences:

1. The signals received by the experts are on some Q dimensional version of R having full support and consisting of the true state + normally distributed white noise.

2. The receiver has an improper uniform prior over R^Q.

This has a big impact on the results compared to what we find. In particular B finds:

a. Truthful revelation is impossible (cf us: If N is sufficiently small, truth-telling is an equilibrium. )

b. In the limit, you can get almost all the information. (cf us: For N large, full information is impossible)

So now, the big question: how much of a contribution is there relative to B? Since B is published in BE Journals, how likely it is for us to get to a general interest place?

Now, we could go to politics journals with a good hope of success, but is this even worthwhile. Or we could pitch to business and be less formal than B with hopes that the applied-ness gets us some mileage.

Bubbles- More on Global Deviations

We seem to be having terrible problems with dealing with the possibility of global deviations. Today, I started with the observation that if d>0, then the hazard rate turned infinite for some realizations of t_0 and, as a consequence, the marginal cost calculation was trivial (since it was infinite). The problem is that this makes no economic sense. Taking a step back, it seems that our calculation of the unconditional hazard rate was wrong---we integrated the ratio of f/(1-F) over all t_0. Really, You have to integrate each piece separately and then put them together fo obtain the right formulation for the hazard rate. There are some other little details to do with endpoints as well to deal with. Bottom line: We're not close to solving this issue.

Branding v14 - A Way Out

The key for us is to come up with a set of conditions whereby the equilibrium we derived under the "relaxed" framework is still an equilibrium with constraints. For finite N cases, this creates a big mess of complicated inequality constraints. The way out seems to be to study the limit case. Here, the required inequalities take on very simple forms. Making these assumptions as strict inequalities and then relying on the fact that everything in sight is continuous in N enables us to make the following statement:

 

Suppose phi<something and tau*(1 - delta) > v(sigma - m), then for N sufficiently large, the following is the set of symmetric Nash equilibria:

 

Firms all advertise at a level a element of (a_0, a_0 + delta)

 

etc.

 

The proof of the proposition is to show that, under the conjectured strategies, no firm has a profitable deviation. The appendix shows the feasibility of the proposed strategies in the limit case.

Revelation Principle--Relation to Other Literatures

Vijay shown a remarkably good ability to make connections between our rev principle stuff and other literatures. Specifically, we now know---from a theorem in Hildenbrand---that our model applies when the type space is the set of all preference relations. The reason is that the space of all compact (I think) preference relations is metrizable. That means that it is a complete, separable metric space. Hence, it is Borel equivalent to any other complete, separable metric space and that's good news for us.

In other news, we investigated the relation between our work and Segal, Jordan, Hurwicz on minimum "cost" information transmission. The older version of that literature, which ignores incentive contraints, is concerned with the "cheapest" way to transmit information on preferences to a mechanism deisgner with under full commitment. They're interested in smooth mappings from the message space to allocations. This smoothness restriction is a big deal since it means dimensionality matters. Our concerns are different---we care mainly about incentives and not at all about smoothness (if we did care about smoothness, then there would be no version of the revelation principle possible with our setup (I think?)

Branding v14

Spent a lot of time trying to tie up loose ends on the branding paper. The model that we have is clunky in the sense that there are a lot of unpleasant things to check to ensure that the number of loyal customers does not exceed all the customers. In the limit, of course, it must be the case that a symmetric equilibrium involves zero advertising since otherwise advertising would be unbounded. The model delivers that, but in an awkward way---we need to make assumptions on the parameter space that essentially reduce the advertising strategy space down to a singleton.

Mike's idea is to go with a strategy space retriction to ensure that things are well-behaved. I'm not averse to that, but don't like a strategy space restriction that changes with N. If we could fix an upper bound on advertising to ensure that we wouldn't have to worry about the problem of too many loyals, that would be fine. I'm not sure how to do this since, as the number of firms grows large, the total number of loyals froma deviation grows as well. Perhaps if one could make the right strategy space restriction on the limit game, it would work in all of the finite ones as well.

Bubbles v13

We're still stuck trying to prove the absence of global deviations to the equilibrium (supposed) we constructed. Today's approach was to show the monotonicity of the hazard rate under various conditions to changes in the "s" plan. The problem is that, even if one can show that thingd are okay with respect to the marginal distribution, one still needs to show monotonicity with respect to the hazard rate of the convolution of t_0 and the third lowest of 5 draws of the delay time conditional on t_i. This is not likely to be a friendly object.

Now, is there some other way to prove this? A necessary condition for an equilibrium is that MB=MC for a given s. If we could show that there is a unique zero (we already know of one) and that things are locally well-behaved, then we'd be in business. The problem is that the mathematical object over which we are finding zeros is not a very friendly guy in "s". Is it possible to do a series expansion (but around what point? it would still only be local). This would seem to be a dead end.

My thought is that it helps to play with the particular parameter values we are using in the experiment. At the very least, we'd know that the objects that we're studying in the lab are indeed equilibria. It seems like knowing at least this is necessary before proceeding any further in bashing our heads against the wall trying to solve this puzzle.

Yeesh.

Too Much Self-Control

In my latest conversation with Paul Gertler, we made the observation that, while the literature has mostly focused on the problem of too little self-control leading to a bias toward present consumption, we both know a lot of people with the opposite bias--too much self-control. These people put off spending into the indefinite future and ultimately die with large estates.

In terms of the distribution of wealth at retirement, this hypothesis would lead to a bimodal distribution: those with the usual self-control problem would end up with little in the way of assets while those with the opposite problem would end up with a lot in the way of assets. This would look different from the expected bell curve if people had perfect self-control and were following the life-cycle hypothesis. Of course, we don't actually know whether the curve looks like this.

One can think of it a different way: what would the savings patterns look like over time? Under the life-cycle theory, one should see high savings rates over working years which declline and ultimately turn negative in retirement. Certainly, as a baseline, one should see a strong tendency toward dissaving in retirement if there were few individuals with too much self-control. Thus, a different table one might be interested in are the savings rates, controlling for income stream, over time. Under the alternative hypothesis, these savings rates would be negative for those with too little self-control and positive (even in retirement) for those with too much.

Whay is all this interesting? Suppose that one is considering a privatization scheme for social security. Now those with too little self-control will end up spending their way into oblivion. They need the commitment device of forced savings. OTOH, those with too much self-control will use this new and improved savings instrument to save more. The upshot is that this increased savings could actually lead to a mini-recession in the short-term and even more dysfunctional savings behavior in the long-term.

The idea of the paper is to combine some stylized facts (which ones?) about savings with a toy model of the too much self-control guys along with the policy implications and aim this at AER.

Directing Traffic

What is the likely impact of the traffic paper? Right now, it simply shows that changes in the economics of different road networks lead to changes in behavior as predicted by economic theory. One of the changes we studied is something called the Pigou-Knight-Downs paradox. However, this paradox seems to go by many different names. It would be good to figure out what exactly is the right name for it. Richard Arnott, a prominent researcher in this area, calls this:

The single most important result in road planning.

Or something close to that.

Arnott's paper Alleviating Traffic Congestion, which is unpublished, is apparently a big deal in the field of traffic design. I should read this more carefully.

We're trying to figure out where to go with Traffic. Our current plan is to write it in a slightly better fashion and circulate it for comments. Our current thinking is to go for a top field journal like Journal of Public Economics or Journal of Urban Economics. The research agenda going forward would then be to try another traffic experiment with a view toward placing it in a field journal followed by an attempt at a magnum opus.

Ebay v Yahoo

Met with my RA on this topic. She's now run a ton of regressions and other tests comparing the results of a series of field experiments on coin auctions. Here findings thus far:

1. The ending rule on Yahoo simply makes no difference to the number of bidders or the revenues made in the auction. She notes that the ending rule is nt displayed prominently on the Yahoo page and this might make it not salient to bidders. Another possibility is that practically all Yahoo auctions are run under the eBay ending rule (check this), so bidders may not even think to look at the ending rule in the first place.

Indeed, a simple test of this idea is simply to count the number of matched pairs of auctions where the eBay rule made more money than the ggg rule. Out of 24 matched pairs, in 12 cases a hard close did better and in 12 the ggg rule did better. This is certainly consistent with a fair coin toss.

2. Revenues and numbers of bidders are higher on eBay than on Yahoo all else equal. This is not surprising. What is sort of surprising is that the 70% reserve price does not affect revenues on Yahoo but does affect revenues on eBay. One would think that, given the smaller number of bidders, the reverse would be true. Conditional on making a sale, the reserve price is more likely to be binding when there are more bidders. Even controlling for the number of bidders, the site location makes a difference.

This led me to think of the following hypothesis: Roth and Ockenfels compare eBay and Amazon directly and conclude that differences in the ending rule drive differences in revenues. They then show that this holds in controlled laboratory experiments. Presumably, the ending rule is more salient in the lab experiments than on the Yahoo site. On the other hand, we're seeing differences in the strategy of bidders on Yahoo versus eBay. Suppose that a part of what O & R were seeing is attribuatable to heterogeneities among bidder types. Could this lead them to conclude that the ending rule was driving the outcome when, in fact, other selection characteristics were responsible? I don't really know, but this is the line I'm pursuing.

Not sure how interesting all this is.