“Long Cheap Talk,” R. Aumann & S. Hart (2003)

I wonder if Crawford and Sobel knew just what they were starting when they wrote their canonical cheap talk paper – it is incredible how much more we know about the value of cheap communication even when agents are biased. Most importantly, it is not true that bias or self-interest means we must always require people to “put skin in the game” or perform some costly action in order to prove the true state of their private information. A colleague passed along this paper by Aumann and Hart which addresses a question that has long bedeviled students of repeated games: why don’t they end right away? (And fair notice: we once had a full office shrine, complete with votive candles, to Aumann, he of the antediluvian beard and two volume tome, so you could say we’re fans!)

Take a really simple cheap talk game, where only one agent has any useful information. Row knows what game we are playing, and Column only knows the probability distribution of such games. In the absence of conflict (say, where there are two symmetric games, each of which has one Pareto optimal equilibrium), Row first tells Column that which game is the true one, this is credible, and so Column plays the Pareto optimal action. In other cases, we know from Crawford-Sobel logic that partial revelation may be useful even when there are conflicts of interest: Row tells Column with some probability what the true game is. We can also create new equilibria by using talk to reach “compromise”. Take a Battle of the Sexes, with LL payoff (6,2), RR (2,6) and LR=RL=(0,0). The equilibria of the simultaneous game without cheap talk are LL,RR, or randomize 3/4 on your preferred location and 1/4 of the opponent’s preferred location. But a new equilibria is possible if we can use talk to create a public randomization device. We both write down 1 or 2 on a piece of paper, then show the papers to each other. If the sum is even, we both go LL. If the sum is odd, we both then go RR. This gives ex-ante payoff (4,4), which is not an equilibrium payoff without the cheap talk.

So how do multiple rounds help us? They allow us to combine these motives for cheap talk. Take an extended Battle of the Sexes, with a third action A available to Column. LL still pays off (6,2), RR still (2,6) and LR=RL=(0,0). RA or LA pays off (3,0). Before we begin play, we may be playing extended Battle of the Sexes, or we may be playing a game Last Option that pays off 0 to both players unless Column plays A, in which case both players get 4; both games are equally probable ex-ante, and only Row learns which game we actually in. Here, we can enforce a payoff of (4,4) if, when the game is actually extended Battle of the Sexes, we randomize between L and R as in the previous paragraph, but if the game is Last Option, Column always plays A. But the order in which we publicly randomize and reveal information matters! If we first randomize, then reveal which game we are playing, then whenever the public randomization causes us to play RR (giving row player a payoff of 2 in Battle of the Sexes), Row will afterwards have the incentive to claim we are actually playing Last Resort. But if Row first reveals which game we are playing, and then we randomize if we are playing extended Battle of the Sexes, we indeed enforce ex-ante expected payoff (4,4).

Aumann and Hart show precisely what can be achieved with arbitrarily long strings of cheap talk, using a clever geometric proof which is far too complex to even summarize here. But a nice example of how really long cheap talk of this fashion can be used is in a paper by Krishna and Morgan called The Art of Conversation. Take a standard Crawford-Sobel model. The true state of the world is drawn uniformly from [0,1]. I know the true state, and get utility which is maximized when you take action on [0,1] as close as possible to the true state of the world plus .1. Your utility is maximized when you take action as close as possible to the true state of the world. With this “bias”, there is a partially informative one-shot cheap talk equilibrium: I tell you whether we are in [0,.3] or [.3,1] and you in turn take action either .15 or .65. How might we do better with a string of cheap talk? Try the following: first I tell you whether we are in [0,.2] or [.2,1]. If I say we are in the low interval, you take action .1. If I say we are in the high interval, we perform a public randomization which ends the game (with you taking action .6) with probability 4/9 and continues the game with probability 5/9; for example, to publicly randomize we might both shout out numbers between 1 and 9, and if the difference is 4 or less, we continue. If we continue, I tell you whether we are in [.2,.4] or [.4,1]. If I say [.2,.4], you take action .3, else you take action .7. It is easy to calculate that both players are better off ex-ante that in the one-shot cheap talk game. The probabilities 4/9 and 5/9 were chosen so as to make each player indifferent from following the proposed equilibrium after the randomization or not.

The usefulness of the lotteries interspersed with the partial revelation are to let the sender credibly reveal more information. If there were no lottery, but instead we always continued with probability 1, look at what happens when the true state of nature is .19. The sender knows he can say in the first revelation that, actually, we are on [.2,1], then in the second revelation that, actually, we are on [.2,4], in which case the receiver plays .3 (which is almost exactly sender’s ideal point .29). Hence without the lotteries, the sender has an incentive to lie at the first revelation stage. That is, cheap talk can serve to give us jointly controlled lotteries in between successive revelation of information, and in so doing, improve our payoffs.

Final published Econometrica 2003 copy (IDEAS). Sequential cheap talk has had many interesting uses. I particularly enjoyed this 2008 AER by Alonso, Dessein and Matouschek. The gist is the following: it is often thought that the tradeoff between decentralized firms and centralized firms is more local control in exchange for more difficult coordination. But think hard about what information will be transmitted by regional managers who only care about their own division’s profits. As coordination becomes more important, the optimal strategy in my division is more closely linked to the optimal decision in other divisions. Hence I, the regional manager, have a greater incentive to freely share information with other regional managers than in the situation where coordination is less important. You may prefer centralized decision-making when cooperation is least important because this is when individual managers are least likely to freely share useful information with each other.

“Welfare Gains from Optimal Pollution Regulation,” J. M. Abito (2012)

Mechanism design isn’t just some theoretical curiosity or a trick for examining auctions. It has, in the hands of skilled practitioners like David Baron, Jean-Jacques Laffont, Jean Tirole and David Besanko (an advisor of mine!), had a huge impact on economic regulation. Consider regulating a natural monopoly that has private information about its costs. In the standard sorting problem, I am going to have to pay information rents to firms that have low costs, since otherwise they will claim to have high costs and thus get to charge higher prices. If funds are costly – and the standard estimate in US public finance is that the marginal dollar of taxation imposes 30 cents of deadweight loss on society – then those information rents are a welfare loss and not just a transfer. Hence I may be willing to sacrifice some allocative efficiency by, for example, randomizing over all firms who claim to be at least somewhat efficient rather than paying a large information rent to learn exactly who the efficient firm is. Laffont’s 1994 Econometric Society address covers this basic point quite nicely.

Mike Abito, on the job market here at Northwestern, notes that few real-world policies actually take account of this tradeoff. Consider a regulator who wants polluting firms to abate their pollution when economically feasible. If the distribution of abatement costs is widely dispersed, then low cost firms have a large incentive to claim high costs and therefore avoid paying for abatement. Especially in this case, it may be worthwhile to sacrifice some allocative efficiency in an optimal pollution abatement scheme, having low cost firms not abate as much as they would if the regulator wanted all information about each firm’s costs. In order to design the optimal pollution regulation scheme, then, we need to know the distribution of marginal abatement costs, which is not something we know immediately from data. In particular, consider regulating SO2 among power plants. Hence, to the world of theory, my friends! (And, briefly, why not just sell pollution permits? If you give away the permits to each plant, then the same informational issue arises, and you do not earn any tax revenue that could offset distortionary taxes elsewhere in the economy.)

Let a power plant, at some cost and effort level, produce some bundle of electricity and SO2. Observed costs alone are not enough if firms have inherently high costs, since firms may appear to have high costs when in fact they are simply exerting low effort. Abito notices that power plants are both rate regulated – meaning that they are natural monopolies whose rates are set by a government agency that estimates their costs – and regulated for pollution reasons. By writing down an auditing game, you see that in the periods the firm is being watched for rate-setting purposes, they exert low effort. They do exert effort in future periods, since any cost reduction comes to them as profits. Indeed, if you look at, for instance, heat generation during years where the plants are being watched, the amount of heat generated declines by roughly the same amount as effort is estimated to decline in the model, so the hypothesized equilibrium of the auditing game is not totally out of line with the data.

What this wedge between cost efficiency in years when the plant is being watched and in other years gives us is an estimate of the cost function, including disutility of effort, which generates some bundle of SO2 and electricity. In fact, it gives us just enough of an exclusion restriction to estimate the distribution of marginal abatement costs of SO2 using techniques from dynamic structural IO. Once we have estimated that distribution, we can solve for numerical estimates of the welfare gain from various abatement policies. Laffont long ago showed that the optimal pollution regulation under this private information, assuming we know the distribution of marginal abatement costs, involves a bundle of type-dependent emission taxes and type-dependent transfers which give the least efficient firm zero profits, but which also lead to less effort and less pollution abatement for more efficient firms that you would get with full information; again, this is just the tradeoff between information rents and allocative efficiency. Such a heterogeneous policy might be tough to implement in practice, however. Welfare gains from the optimal policy instead of a uniform emissions standard, given the estimated distribution of marginal abatement costs, are equal to about 10% of the entire variable cost of the average plant. A uniform emissions tax (rather than a standard which imposes a maximum amount of emissions) captures something like 60-70% of this improvement, and is easier to implement.

More generally, the gain to society of using regulatory regimes that condition on the underlying properties of each firm really depends on properties like the distribution of marginal abatement costs which atheoretically can never be known, but which with the use of proper structure can actually be estimated. What is particularly cool here is that, unlike most earlier work, the underlying firm properties are estimated without assuming that the regulator is already optimizing, an assumption that is simply false in the case of pollution regulation. Good stuff.

November 2012 Working Paper (Not available on IDEAS). There are a number of interesting papers in environmental economics on the job market this year. Lint Barrage at Yale discusses how carbon taxes and other taxes should interact in optimal fiscal policy. In particular, since carbon in the atmosphere lowers the productive capacity of assets (like agricultural land) in the future, not taxing carbon is identical to taxing capital, producing the same distortion. When the economy already has distortionary taxation, the optimal rate of carbon taxation will need to be adjusted. Joseph Shapiro from MIT estimates the environmental damage from CO2 produced in international trade. It is two orders of magnitude smaller than the gains from that trade, and a small carbon tax on international shipping is optimal. In a separate paper, Shapiro and coauthors find that US mortality during heat waves declined massively over the twentieth century, that all of the decline appears to be linked to adoption of air conditioning, and hence that mitigation of some negative health impacts of climate change in poor countries will likely be handled by A/C. Since A/C uses electricity, non-carbon methods of generating that electricity are critical if we want to avoid making climate change worse while we mitigate these impacts.

“How Better Information Can Garble Experts’ Advice,” M. Elliott, B. Golub & A. Kirilenko (2012)

Ben Golub from Stanford GSB is on the market this year following a postdoc. This paper, which I hear is currently under submission, is a simple and straightforward theoretical point, but it does have some worrying implications for public policy. Consider a set of experts which society queries about the chance of some probabalistic event; the authors mention the severity of a flu, or the risk of a financial crisis, as examples. These experts all have different private information. Given their private information, and the (unmodeled) payoff they receive from a prediction, they weigh the risk of type I and type II errors.

Now imagine that information improves for each expert (restricting to two experts as in the paper). With the new information, any possible set of type I and type II errors is still possible, and there is now the possibility of making predictions with strictly fewer type I and type II errors. This means that the “error frontier” expands outward for each expert. To be precise, if each agent gets a signal in [0,1] whose cdf is G(i) for expert i if the event will actually occur. A new signal that generates a second cdf G2(i) which first order stochastically dominates G(i) is an information improvements. Imagine both experts receive information improvements. Is this socially useful? It turns out that is it not necessarily a good thing.

How? Imagine that expert 1 is optimizing by making x1 type I errors and y1 type II errors given his signal, and expert 2 is optimizing by making x2 type I errors and y2 type II errors. Initially expert 1 is making very few type I errors, and expert 2 is making very few type II errors. Information improves for both, pushing out the “error frontier”. At the new optimum for expert 1, he makes more type I errors, but many fewer type II errors. Likewise, at the new optimum, expert 2 makes more type II errors and fewer type I errors. Indeed, it can be the case that expert I after the information improvement is making more type I and type II errors than expert 2 did in her original prediction, and that expert II is now making more type I and type II errors than expert 1 did in his original prediction. That is, the new set of predictions are a Blackwell garbling of the original set of predictions, and hence less useful to society no matter what decision rule society uses when applying the information to some problem. Note that this result does not depend on experts trying to out-guess each other or anything similar.

Is such a perverse outcome unusual? Not necessarily. Let both experts be “envious” before new information arrives, meaning the both experts prefer the other’s bundle of type I and type II errors to any such bundle the expert can choose himself. Let the agents payoffs not depend on the prediction of the other agents. Finally, Let the new information be a “technology transfer”, meaning a sharing of some knowledge already known to one or both agents. That is, after the new information arrives, the error frontier of both agents lies within the convex hull of their original combined error frontiers. With envious agents, there is always a technology transfer that makes society worse off. All of the above holds even when experts are not required to make discrete {0,1} predictions.

This is all to say that, as the authors note, “better diagnostic technology need not lead to better diagnoses”. But let’s not go too far: there is no principal-agent game here. You may wonder if society can design payment rules to experts to avoid such perversity. We have a literature, now large, on expert testing, where you want to avoid paying “fake experts” for information. Though you can’t generally tell experts and charlatans apart, Shmaya and Echenique have a paper showing that there do exist mechanisms to ensure that, at least, I am not harmed “too much” by the charlatans’ advice. It is not clear whether a mechanism exists for paying experts which ensures that information improvements are strictly better for society. By Blackwell’s theorem, more information is strictly better for the principal, so incentivizing the experts to express their entire type I-type II error frontier (which is equivalent to expressing their prior and their signal) would work. How to do that is a job for another paper.

July 2012 working paper (unavailable on Repec IDEAS).

“Paternalism, Libertarianism and the Nature of Disagreement,” U. Loginova & P. Persson (2012)

Petra Persson is on the job market this year from Columbia. Her CV is pretty incredible – there’s pure theory, cutting edge empirical techniques, policy work, networks, behavioral and more. Her job market paper is about the impact of social insurance policy on seemingly unrelated markets like marriage, and I’ll discuss it briefly at the end of the post, but I want to focus on another paper on hers which struck me as quite interesting.

Imagine a benevolent ruler who has private information about some policy, such as the relative safety of wearing seatbelts. This ruler can either tell citizens the information, or lie, or coerce them to take such action. Naive libertarianism suggests that we should always be truthful is altruistic; consumers can then weigh the information according to their preferences and then choose the policy optimal for them.

But note something interesting. On some issues, one subset of politicians has libertarian leanings, while on others, a different subset has those leanings. For instance, a politician may favor legal assisted suicide but insist on mandatory seatbelt rules, while another politician may be against the mandatory belt and also against legal assisted suicide. Politicians can even vary in how libertarian they wish to be depending on who the policy affects. Witness that many politicians favor legalizing marijuana but very few favor legalizing it for 16 year olds. What explains this behavior?

Loginova and Persson examine this theoretically. Take a population of citizens. There are two possible states, 0 and 1. They can either think each state equally likely yet have different heterogeneous preferences from the politician (measured with a Crawford-Sobel style quadratic loss, though this isn’t a critical model) or they can have identical preferences as the politician yet have different heterogenous (prior) beliefs about the probability of each state. The politician can be altruistic to varying degrees – more altruism means he, according to his own prior, puts more and more weight on the utility of the agent. The politician gets a noisy signal about the true state. To limit the extent of opposing beliefs, the politician is restricted to having the same prior as the median citizen.

If the politician can only advise or not advise, when does he make a truthful public announcement? If he disagrees on preferences with the citizens, then the more altruistic, the more likely he is to announce truthfully, for the standard libertarian reason: the citizens know their own preferences, and the better informed they are, the better they can maximize their own welfare. If, however, he disagrees on priors with the citizens, then the more altruistic, the less likely he is to announce truthfully: altruism means I care about the citizen’s welfare, but since they have priors that are in my eyes wrong, the citizens know that even when I am altruistic I have incentive to lie so that citizens take actions that are optimal according to my prior, therefore truthful communication cannot be sustained.

Now what if the politician could (at a cost to him) force all individuals to take an individual action? With preference disagreement, an altruistic politician would never do this, both because he can send all the information to citizens with a free message and also because a mandate does not respect heterogeneity of preferences. Even if action 0 is better than action 1 for 90% of the population, an altruistic principle also cares about the other 10%. With disagreement about priors, however, an altruistic politician is more likely to impose a mandate the more altruistic he is. Even though citizens have heterogeneous priors, the principle thinks all of them are wrong, and hence is not worried about heterogeneity when imposing a mandate. Since we noted in the last paragraph that altruistic politicians who have different priors from citizens will not be able to credibly send their information, the mandate allows the politician’s private information to be used in the citizen’s actions.

Finally, what if the politician can send individual-level messages or enforce individual mandates? A politician with preference disagreement need to be fairly altruistic before his public message is credible; in fact, he needs to be able to credibly persuade the individual with the average disagreement in order for his public signal to be credible. If he is not altruistic enough, he can still credibly persuade those agents who have only a limited amount of preference disagreement with him. If mandates are possible, the politician with limited altruism will force individuals whose preferences are very different from the politician to take the politician’s desired action, but since preferences of the politician and the agents are more aligned when altruism is higher, the share of citizens who face a mandate declines as the politician’s altruism increases. Likewise, a politician with disagreement about priors can only truthfully send information when his altruism is low. If the politician is very altruistic, even though the public signal will not be believed, a politician can still credibly send information to those whose priors are similar to the politician. The politician with low levels of altruism will only mandate the action of agents with extreme beliefs, but as altruism increases, more and more citizens will face a mandate.

Very good – the use of paternalistic policies, and the extent to which they are targeted at individuals, depends qualitatively on whether the politician disagrees with the agents about their preferences or about their knowledge, and the extent to which mandates are applied on certain groups depends on how extreme their preferences or beliefs are. There is nothing inherently contradictory in an altruistic politician taking the libertarian side on one issue and the paternalistic side on another.

July 2012 working paper (No IDEAS version). Petra has many other interesting papers. In her job market paper, presented here last week, she shows that social insurance, in this case a widow’s benefit in Sweden, can have major affects in other markets. In particular, a really nice regression discontinuity shows that the benefit was leading to a huge number of extra marriages, that these were more likely to end in divorce, that intrahousehold bargaining was affected, and much more (Jeff at Cheap Talk has a longer description). Her paper Circles of Trust notes a reason for cliquish behavior in some labor markets. If I have information whose value declines with use (such as a stock tip) and I am altruistic, I may wish to tell my friends the info. But I worry that they will tell their friends, who I don’t know and hence don’t really care about. If my friend could commit not to tell his friends, I would give him the info. How can we commit ex-ante? Make our social networks a clique. I would bet that this phenomenon explains hiring in, say, small hedge funds to a great extent.

“Bayesian Persuasion,” E. Kamenica & M. Gentzkow (2011)

Kamenica and Gentzkow recently published this gloriously-titled extension of the cheap talk literature in AER. Recall that in Crawford-Sobel cheap talk, a sender and receiver differ in their preferred action, and the sender holds better information about the true state. The receiver cannot commit to an action conditional on the message, so this is not a principal-agent problem. Even though both people know the sender is biased, there are equilibria that are partially informative, where the sender credibly just tells you what set of states the true state is in, and the receiver updates based on that knowledge. These equilibria are not always good for the sender, nor for the receiver – both may be better off in the “babbling equilibrium” where the signal is totally ignored.

Kamenica and Gentzkow consider a problem where the agent picks the signal structure, and then sends a verifiable signal; this avoids worries about the babbling equilibrium. Their example is a prosecutor, who legally cannot lie, but can choose how to collect information. Both judge and prosecutor have a prior of .3 that the defendant is guilty. The prosecutor earns payoff 1 from a guilty conviction, and 0 from innocence. The judge earns payoff 1 for a correct conviction/acquittal, and 0 from an incorrect one. The prosecutor bias is known. Nonetheless, the prosecutor can get the defendant convicted 60% of the time! How? Collect two types of evidence. We want to use signals to generate a posterior that the defendant is guilty with probability of exactly .5 (in which case the expected-utility maximizing judge convicts), and a posterior that the defendant is innocent with probability 1. That is, half of the time that the prosecutor says “guilty”, the defendant is innocent, and always if the prosecutor says “innocent”, the defendant is innocent. This means the prosecutor says guilty 60% of the time, and innocent 40% of the time. These are Bayesian rational given the prior of guilt with p=.5.

Note what’s going on. From the prosecutor’s perspective, it doesn’t matter when the judge thinks the defendant is guilty with probability 1 or p=.8 or p=.55. As long as p(guilty)>=.5, the judge will convict, and the prosecutor will get payoff 1. So collecting evidence that is really strong is worthless. Better to collect evidence that is just strong enough to get a conviction; by doing so, the prosecutor can give the same evidence for lots of innocent people as well as the truly guilty! This principle applies broadly; the authors show that a car dealership will want a buyer to just barely believe the car is a good match for her if she should buy, and tell all others that the car is a terrible match.

The math here is interesting. Subgame perfection, plus a short proposition, means that the agent’s action will be a deterministic function of his posterior. The sender’s payoff is a function of the agent’s action. Hence the sender’s payoff is a function of the induced posterior. Whenever there are convexities in the sender payoff as a function of agent’s action, the sender should choose signals (or randomize) to “smooth out” that convexity. In the example, the prosecutor payoff is 0 if the judge’s posterior of guilt is below .5, and 1 if equal to or above .5. That function has a convexity between 0 and .5. We can choose signals in a verifiable way such that if the prior is some p in (0, .5), the overall probability of conviction rises to 2p; just choose evidence such that the p percent of guilty defendants, plus another p percent of innocent defendants, all have the same just-strong-enough evidence against them.

Final AER version (IDEAS page)

“Dynamic Lemons Problem,” I-K Cho & A. Matsui (2012)

Here’s a result so fresh that the paper doesn’t even exist yet; Cho (yes, the Cho from Cho-Kreps) presented a version of the following at a seminar here recently and I wanted to jot some notes down while it’s fresh in my mind.

Take the standard Akerlof problem. There is a unit mass of buyers, a unit mass of high quality sellers, and a unit mass of low quality sellers. Quality is known only to the individual seller. All buyers value high quality goods at H and low quality goods at L. Low quality sellers have a reservation value of 0 and high quality sellers a reservation value of C, where H-C>L. Thus, efficiency is maximized by each high quality seller selling at some price to a buyer. Car quality is unobservable, so assume there is a pooling equilibrium where high and low quality cars are sold at the same price. Then consumers value these cars at (H+L)/2. So if (H+L)/2 is less than C, high quality sellers will be unwilling to sell, only low quality sellers will remain in the market, and we say the market has collapsed. It is a pretty robust result in these types of asymmetric information models that the lowest quality types always remain in the market, and the high quality types are often pushed completely out.

What if this market, however, were dynamic? What we mean here is that, in period one, buyers and sellers match randomly with each other; there are twice as many sellers as buyers, so only half the sellers match. In any given match, a price is offered for the transaction. If either party rejects the price, no match is made, payoffs are zero this period, and both buyer and seller rejoin the unmatched pool in the next period. If it is accepted, the good is sold, its value is realized, and the seller receives payoff of the price minus his cost, while the buyer receives either H or L minus the price, depending on what quality the good ended up being. This relationship is maintained, with precisely the same payoffs, every period into the future, except that with probability (1-delta) the relationship ends and both buyer and seller rejoin the unmatched pool. We then move to the next period and everyone who is currently unmatched is randomly matched again. We are interested in the existence of price(s) that form an undominated stationary equilibrium as the period length (and hence discounting between periods) goes to zero.

Will the static logic, that all buyers match with low quality types at a price somewhere between 0 and L, maintain? It will not. If that were an equilibrium, then every seller unmatched will be a high quality seller. So an individual buyer who rejoins the unmatched pool will match almost certainly with a high quality seller, and if the price is C+epsilon, the seller will accept the offer and the buyer will improve his payoff. In any equilibrium, then, there must be at least two prices being offered, and at least some high quality sellers must match.

Note, now, the incentive for low quality sellers. By pretending to be a high quality seller (only accepting the price C+epsilon, and not the price at which only low types would accept), the low quality sellers improve their payoff. So, again, in equilibrium, low quality sellers will mix between accepting any offer and only accepting the high offer in which they pretend to be high quality types. Cho shows that this deception leads to a couple worrying results: first, this deception means that, in the limit, buyer’s expected payoffs go to zero in any equilibrium, and second, that some buyers do not match for arbitrarily long numbers of periods. The second result obtains because buyers are getting arbitrarily close to zero payoff, so the harm from waiting is very low, and if they accept a low quality seller masquerading as a high quality seller, they will receive a negative payoff.

Note that all of the intuition above appears (at least to me) robust to allowing a continuum of seller types or to allowing sellers and buyers to break existing relationships the period after they begin. What remains to be seen, though, is what economic problem looks like Mortensen-Pissarides plus Akerlof lemons. I’d be surprised, given the importance of relational contracts in the world of finance, if we couldn’t find something suitable in that venue.

http://www.kellogg.northwestern.edu/research/math/seminars/2011-12/In-Koo_Cho_Abstract.pdf (Extended abstract – I’m fairly certain no draft of this paper is out yet)

“Agreeing to Disagree: The Non-Probabilistic Case,” D. Samet (2010)

Aumann famously showed that two Bayesian agents with common priors cannot “agree to disagree” about a posterior that is common knowledge. One might wonder, does this generalize to decision functions other than Bayesianism? In the early 80s, Cave (1983) and Bacharach (1985) did precisely that, stating that likemindedness (we take the same decision if we have the same information) and a sure thing principle that only implicitly used the knowledge operator. This recent paper in GEB by Dov Samet shows that the sure thing principle they use is problematic, and rederives conditions necessary for agreement.

The problem essentially is this. Give me two agents with two information partitions. I want to say that A is more knowledgeable than B if A’s knowledge is given by a set E and B’s knowledge by a set E+F, where + here represents the union operator. The problem is that this is impossible with the standard partitional formulation of knowledge that philosophers and economists use. If two agents do not have exactly the same information, then each one knows something the other does not. This is true even if one has an information partition that is a strict refinement of the other. Why? Let A know event G at state w. Let B not know event G. The knowledge operator itself also defines events, and by a property of knowledge, A does not know that A does not know G at w, while B knows that he does not know G.

The intuition from Cave and Bacharach can still work, though. Let [j>=i] be all the states where no matter what event E occurs, j knows E whenever i knows it. Assume that if i knows j is at least as knowledgeable at he is in state w, then i takes the same decision as j. Finally assume that if we add a third agent who knows less than i or j at w, then all three agents make the same decision. When these assumptions hold, agents cannot agree to disagree.

Samet quotes a story from Aumann that sums up how the theory works. Alice and Bob are detectives. Bob collects data until 5 with his partner Alice. Alice then stays at work until late at night collecting more information. Both were trained in the same academy, and therefore make the same decisions if they have the same information. Intuitively, if Bob knows that Alice has every bit of information he has plus more, then he should just make the same decision in the end as Alice about the guilt of the suspect. The conditions in the prior paragraph capture this intuition.

http://www.tau.ac.il/~samet/papers/generalized-agreement-theorem.pdf (Final GEB version)

“Organizations as Information Processing Systems,” R. Daft & R. Lengel (1983)

I don’t believe this paper is well-known by economists, but it has been hugely influential for management and media studies. The theory in this paper is qualitative in the same way economic theory is, but is not mathematical. In this post, I’ll try to reinterpret the main ideas mathematically.

Firms face two primary types of uncertainty. First, the outside environment is uncertain. Second, the internal environment is uncertain. When speech is vague, a manager may misinterpret what the true state of the world is, or subordinates may misinterpret the goals of the organization. When speech is precise, it can be very costly to interpret. Indeed, precise speech about unclear goals is basically worthless: two subordinates may precisely state the answer to two different problems, both of which are different from what the manager wanted to know.

Choice of media, then, can vary. Sometimes speech within an organization is very formal: quantitative models, memos, etc. Sometimes it is informal: face-to-face meetings, informal legends, company lore. The informal speech is able to discuss a broader set of ideas, but with greater ambiguity. The formal speech can present specific ideas exactly, but nothing more. This tradeoff roughly implies the following: when the purpose of a discussion is equivocal or unclear, informal speech should be used to “get us on the same page”. When a discussion involves something routine, precise speech can be used. This has a number of implications: for example, informal communication will be most common at the goal setting stage, or when two different departments are beginning to work together on a task, but formal communication will be most common within a division or after goals have been agreed upon by all parties or when the external environment has less uncertainty.

Clearly, the intersection of language and economics is far more general. For example, equivocality is often introduced on purpose: people speak vaguely, for example, in order than common knowledge does not develop. An example, after a first date: “Would you like to come up to my apartment for some coffee?” Further, vague and precise speech are more than simply vague or precise, but rather are vague and precise in particular ways. Poetry is quoted rather than a meaningless stream of words, for example. Neither the authors or I have much to say on these extensions, but it is definitely an open field right now for some interested researcher.

How might you model the ideas of the present paper mathematically? (Of course, you might ask why these ideas should be modeled mathematically anyway, but I have discussed many times here why social science theory ought be formal, and to the extent that it’s formal, the tools of mathematical logic allow the cleanest possible transmission of ideas and derivation of unexpected consequences, so I won’t rehash those arguments here. Indeed, the whole “should we be formal” discussion seems a bit too meta in the context of this post…) Let the relevant true state be a number in [0,1]^n. Let transmission of the exact state be increasing in its dimension, perhaps linearly. Let transmission of imprecise information be increasing less than linearly, perhaps logarithmically. Imprecise states are interpreted by the receiver with error (something like the truncated exponential version of a normal distribution to ensure we stay in [0,1]^n). Loss functions of the final decision made by the receiver depend on distance from the true state. What should a manager do? Well, on simple decisions where the relevant state is only a point on the line segment [0,1], getting the exact state is cheap, so subordinates should send the manager fairly precise information like a statistical estimate in a memo. On complex decisions, where the relevant state is a point in the 100-dimension [0,1] hypercube, learning the true state will be very expensive (it may require the manager to read a 1000 page quantitative report, for instance), but learning an approximate state will be relatively cheap (it may involve some face-to-face conversations). Once the model is formalized like this, then we can answer questions like “Should management communicate via a hierarchy or not?” I have some plans for work along these lines, using some ideas about transmitting counterfactuals given a set of information partitions, and would definitely appreciate comments concerning how to model this type of media richness.

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA128980&Location=U2&doc=GetTRDoc.pdf (Working paper)

“Relational Incentive Contracts,” J. Levin (2003)

In the real world, there is a lot of information that simply can’t be contracted on, whether for legal reasons, for information verification reasons, or for cost of contracting reasons. Nonetheless, we still see attempts to maintain incentive structures even without contracts when a relationship in repeated: consider a worker in a long-term relationship with a firm who expects a bonus, given at the firm’s discretion, each year. Jon Levin – fair bet for this year’s Clark? – calls these “relational contracts,” where the incentive to not break the implicit equilibrium contract comes from a desire to not get minmax punished for the rest of the game. What might an optimal repeated relational contract look like if this is the only incentive agents have not to deviate ex-post, under various informational assumptions?

In general, this will be a very difficult problem; even today, fully specifying general optimal mechanisms has made little progress since Tirole and Laffont (1988). Levin has a clever trick, though, that shows some intuition from the auction theory literature. If every actor in the game has quasilinear utility and is risk-neutral, then there is no scope for risk-offloading in the optimal contract, and further simple money transfers in any period can be used, in one shot, to stand in for potentially complicated multi-period reward and punishment strategies. In particular, if any self-enforcing contract can achieve total average surplus per period s, then any outcome given each player at least her minmax with total surplus below s is achievable in equilibrium. This is not just a variation of Fudenberg-Levine-Maskin’s 1994 folk theorem for repeated games (since the discount rate is arbitrary here), but rather just comes from making one of the actors pay the other a lump sum in period one: incentives at all future times do not change and each actor still gets at least the minmax, so the equilibrium remains. But now note that the maximum social surplus that can be achieved is achievable with a stationary incentive structure, meaning incentives that depend only on current period variables. The reason is that if I’m going to maintain some incentive with a complicated string of rewards and punishments in the future, those rewards and punishments have an equilibrium expected value to the actor. By risk-neutrality and quasilinearity, I can just transfer the expected sum of that string to (or from) the actor in the current period. There is a brief argument ensuring that in equilibrium, since the principle’s action is perfectly observed by both agents, there is no reason the principle would destroy or create social surplus in the future, so the total social surplus is just a fixed value to be moved shifted among the two players.

With these nice properties in hand, it turns out that optimal relational contracts have a relatively simple form. With perfect information, the relevant constraints are that neither the principle nor the agent wants to walk away from the promised continuation utility, which is just the discounted difference of all future stage game payoffs higher that the outside option. The IC constraint inducing optimal effort for the agent is the normal one, but there is also a dynamic constraint which requires the largest total payment in any period to the agent minus the smallest total contingent payment to be bounded, since if not, one of the actors has an incentive to walk away and take their outside option forever at the end of the current contract instead of paying the specified bonus transfer. This limitation on incentives is essentially the cost of not being able to contract.

What about the limited information cases, moral hazard and adverse selection? Let the agent have a cost of production that is unobservable by the principle, and let that agent choose a level of effort which is observable. Make the standard assumptions on the cost function that allow full separation of types in the static hidden information problem. The lack of contracts in the dynamic problem give an highest-total-surplus equilibrium where equilibrium effort for all types is lower than the first-best. By self-selection arguments, getting more effort from a higher cost type means raising the slope of the bonus schedule in effort. But the total variance in incentives is bounded as described above. So sometimes, relatively high cost types are all pooled at a suboptimal level of effort.

If there is moral hazard rather than hidden information (agent’s cost is observed by everyone, but not agent’s effort), assuming normal Rogerson conditions so the first-order approach can be used to solve the program, risk-neutrality allows us to use a “one step” incentive system: if output is high enough, pay the maximal bonus, else pay the minimal one.

A couple final notes. In the case of subjective performance measures (only the principal observes final output which has some stochastic component), the optimal contract is a termination contract: if output is sufficiently low, terminate the job, else pay a bonus. The reason termination is necessary is that the principal must be punished for trying to cheat the agent by reporting low output, and terminating the job punishes the principal by giving him only his outside option forever. Second, there’s no worry here about unrealistically using an infinite game, since we discount: you can let some exogenous chance of the contract ending at any time enter the problem through the discount rate and by risk-neutrality this interpretation is not worrisome.

http://www.stanford.edu/~jdlevin/Papers/RIC.pdf (Final version, AER 2003)

“Nash Implementation with Partially Honest Individuals,” B. Dutta & A. Sen (2009)

Sometimes a lovely, unintuitive result seems a too simple once you see the proof. Here’s one, from Dutta and Sen, that I bet you didn’t expect.

Consider Nash implementability of social choice functions; that is, we want to know if, for some social choice correspondence (a mapping from preferences R to outcomes), there exists a game form whose only Nash equilibrium outcomes given preferences is in the social choice correspondence (scc). Maskin famously showed that, with three or more players, any social choice correspondence satisfying No Veto Power (if (n-1) of the players top-rank an outcome A, then the scc must select it) and Maskin Monotonicity (roughly, if the scc selects outcome A under preferences R, but not under R’, then at least one agent must reverse their ranking of a and some other outcome b). It turns out that Maskin Monotonicity is a super-strong assumption, actually: the scc must be dictatorial (Muller-Satterthwaite), and if it is a function, then the scc must be constant regardless of preferences (Saijo 1987, JET).

Dutta and Sen say, fine, but what if a single agent, whose identity is not known to the mechanism designer, has lexicographic preferences for honesty. That is, the agent maximizes her preferences while playing the designer’s chosen game form, but when two actions (here, the relevant action space is just revelation of the preference ordering) give the same outcome, and one of those actions is truthful, then the agent takes the truthful action. It turns out that this simple assumption allows any scc with three or more agents to be implemented!

The proof is simple if you know Maskin’s result. The game is the same as in Maskin: each agent’s strategy is to reveal the preference ordering of all agents, a recommended action, and an integer. As in Maskin, this is a game of complete information, so every agent but the designer knows other agent’s preference ordering; Matt Jackson has a paper on Bayesian implementability if you don’t like this assumption. If at least (n-1) reveals the same thing, and if the action A recommended is such that f(R)=A in the specified social choice correspondence, then that action is chosen. Otherwise, the action announced by the agent who chose the highest integer is implemented.

No veto power alone gets us everything except ensuring that there is no equilibrium where every agent deviates to some false preference orderings R’ and action a’. To show there is no such equilibrium, we can use Maskin Monotonicity. Alternatively, just note that with lexicographic honesty, such a deviation cannot be an equilibrium. By the (n-1) part of the game outcome above, the honest agent can deviate to (R,a) and not change the outcome. In that case, he prefers the honest revelation (R,a). So everyone revealing (R’,a’) is not an equilibrium. This is literally the whole proof. Basically, Maskin’s proof involved a snitch who is incentivized by monotonicity to deviate and entire the integer subgame when everyone reveals (R’,a’). Here, lexicographic honesty does the same job.

A few final notes: Dutta and Sen, of course, prove many more results, particularly for the more difficult problem of two-agent implementability; the general difficulty there is that one person deviates, since there are only two agents, you don’t know who deviated as the designer. They also show, with some reasonable restrictions on mechanism types, that if even there is an epsilon chance that a single agent may have lexicographic honesty, that is enough to allow existence of a mechanism implementing any scc. This paper does not get around a well-known objection to the Maskin mechanism, however: the strategy space is not compact. In particular, the integers are unbounded. Lombardi and Yoshihara (2011) show that some reductions in the strategy space far weaker than requiring compactness and requiring that agents know only their own preferences nonetheless drastically change what is Nash implementable with lexicographically honest players.

http://www2.warwick.ac.uk/fac/soc/economics/research/papers_2009/twerp_920.pdf (2009 Working Paper; hat tip to Dimitrios Diamantaras for the reference)