“The Nash Bargaining Solution in Economic Modeling,” K. Binmore, A. Rubinsten & A. Wolinsky (1986)

If we form a joint venture, our two firms will jointly earn a profit of N dollars. If our two countries agree to this costly treaty, total world welfare will increase by the equivalent of N dollars. How should we split the profit in the joint venture case, or the costs in the case of the treaty? There are two main ways of thinking about this problem: the static bargaining approach developed first by John Nash, and bargaining outcomes that form the perfect outcome of a strategic game, for which Rubinstein (1982) really opened the field.

The Nash solution says the following. Let us have some pie of size 1 to divide. Let each of us have a threat point, S1 and S2. Then if certain axioms are followed (symmetry, invariance to unimportant transformations of the utility function, Pareto optimality and something called the IIA condition), the bargain is the one that maximizes (u1(p)-u1(S1))*(u2(1-p)-u2(S2)), where p is the share of the pie of size 1 that accrues to player 1. So if we both have linear utility, player 1 can leave and collect .3, and player 2 can leave and collect 0, but a total of 1 is earned by our joint venture, the Nash bargaining solution is the p that maximizes (p-.3)*(1-p-0); that is, p=.65. This is pretty intuitive: 1-.3-0=.7 of surplus is generated by the joint venture, and we each get our outside option plus half of that surplus.

The static outcome is not very compelling, however, as Tom Schelling long ago pointed out. In particular, the outside option looks like a noncredible threat: If player 2 refused to offer player 1 more than .31, then Player 1 would accept given his outside option is only .3. That is, in a one-shot bargaining game, any p between .3 and 1 looks like an equilibrium. It is also not totally clear how we should interpret the utility functions u1 and u2, and the threat points S1 and S2.

Rubinstein bargaining began to fix this. Let players make offers back and forth, and let there be a time period D between each offer. If no agreement is reached after T periods, we both get our outside options. Under some pretty compelling axioms, there is a unique perfect equilibrium whereby player 1 gets p* if he makes the first offer, and p** if player 2 makes the first offer. Roughly, if the time between offers is D, player 1 must offer player 2 a high enough share that player 2 is indifferent between that share today and the amount he could earn when he makes an offer in the next period. Note that the outside options do not come into play unless, say, player 1′s outside option is higher than min{p*,p**}. Note also that as D goes to 0, all of the difference in bargaining power has to do with who is more patient. Binmore et al modify this game so that, instead of discounting the future, rather there is a small chance that the gains from negotiation will disappear (“breakdown”) in between every period; for instance, we may want to form a joint venture to invent some product, but while we negotiate, another firm may swoop in and invent it. It turns out that this model, with von Neumann-Morganstern utility functions for each player (though perhaps differing levels of risk aversion) is a special case of Rubinstein bargaining.

Binmore et al prove that as D goes to zero, both strategic cases above have unique perfect equilibria equal to a Nash bargaining solution. But a Nash solution for what utility functions and threat points? The Rubinstein game limits to Nash bargaining where the difference in utilities has to do with time preference, and the threat points S1 and S2 are equal to zero. The breakdown game limits to Nash bargaining where the difference in utilities has to do with risk aversion, and the threat points S1 and S2 are equal to whatever utility we would get from the world after breakdown.

Two important points: first, it was well known that a concave transformation of a utility function leads to a worse outcome in Nash bargaining for that player. But we know from the previous paragraph that this concave transformation is equivalent to a more impatient Rubinstein bargainer: a concave transformation of the utilities in the Nash outcome has to do with changing the patience, not the risk aversion, of players. Second, Schelling was right when he argued that the Nash threat points involve noncredible threats. As long as players prefer their Rubinstein equilibrium outcome to their outside option, the outside option does not matter for the bargaining outcome. Take the example above where one player could leave the joint venture and still earn .3. The limit of Rubinstein bargaining is for each player to earn .5 from the joint venture, not .65 and .35. The fact that one player could leave the joint venture and still earn .3 is totally inconsequential to the negotiation, since the other player knows that this threat is not credible whenever the first player could earn at least .31 by staying. This point is often wildly misunderstood when people apply Nash bargaining solutions: properly defining the threat point matters!

Final RAND version (IDEAS). There has been substantial work since the 80s on the problem of bargaining, particularly in trying to construct models where delay is generated, since Rubinstein guarantees agreement immediately and real-world bargaining rarely ends in one step; unsurprisingly, these newer papers tend to rely on difficult manipulation of theorems using asymmetric information.

“Being Realistic about Common Knowledge: A Lewisian Approach,” C. Paternotte (2011)

(Site note: apologies for the recent slow rate of posting. In my defense, this is surely the first post in the economics blogosphere to be sent from Somalia, where I am running through a bunch of ministerial and businessman meetings before returning to the US for AEA. The main AEA site is right down the street from my apartment, so if you can’t make it next week, I will be providing daily updates on any interesting presentations I happen across. Of course, I will post some brief thoughts on the Somali economy as well.)

We economists know common knowledge via the mathematical rigor of Aumann, but priority for the idea goes to a series of linguists in the 1960s and to the superfamous philosopher David Lewis and his 1969 book “Conventions.” Even within philosophy, the formal presentation of Aumann has proven more influential. But the economic conception of common knowledge is subject to some serious critiques as a standard model of how we should think about knowledge. One, it is equivalent to an infinite series of epistemic iterations: I know X, know you know that I know X, and so on. Second, and you may know this argument via Monderer and Samet, the standard “common knowledge is created when something is announced publicly” is surely spurious: how do I know that you heard correctly? Perhaps you were daydreaming. Third, Aumann-style common knowledge is totally predicated on deductive reasoning: every agent correctly deduces the effect of every new piece of information on their own knowledge partition. This is asking quite a bit, to say the least. The first objection is not too worrying: any student of game theory knows the self-evident event definition of common knowledge, which implies that epistemic iteration definition. Indeed, you can think of the “I know, know that you know, know you know that I know, etc.” iterations as the consequence of knowing some public event. Paternotte gives the great example of any inductive proof in mathematics: knowing X holds for the first element and X holding for element i implies it holds for i+1 is not terribly cognitively demanding, but knowing those two facts implies knowledge of an infinite string of implications. The second objection, fallibility, has been treated with economists using p-belief: assign a probability distribution to the state space, and talk about having .99-common belief rather than common knowledge. The third, it seems, is less readily handled.

But how did Lewis think of common knowledge? And can we formalize his ideas? What is then represented? This paper is similar to Cubitt and Sugden (2003, Economics and Philosophy), though it strikes me as the more interesting take. Lewis said the following:

It is common knowledge among a population that X iff some state of affairs holds such that
1: Everyone has reason to believe that A holds
2: A indicates to everyone that everyone has reason to believe that A holds, and
3: A indicates to everyone that X.

Note that the Lewisian definition is not susceptible to the three arguments noted above. Agents don’t necessarily believe something, but rather just have reason to do so. They know how each other reason, but the method of reasoning is not necessarily deductive. Let’s try to formalize those conditions in a standard state space world. Let B(p,i)E be the belief operator of agent i: B(.7,John):”It rains today” means John believes with probability .7 that it will rain today. Condition 1 in Lewis looks like claiming that all agents believe with p>.5 that A holds (have a “reason to believe A”). The word “A indicate X” should mean that there is a reasoning function of agent i, f(i), such that if A is believed with p>.5, then so is X (we will need some technical conditions here to ensure the function f(i) is defined uniquely for a given reasoning standard).

What is interesting is that this definition is tightly linked to standard Monderer-Samet common p-belief. For every common p-belief, p>.5, there are a set of parameters for which Lewisian common knowledge exists. For every set of parameters where Lewisian common knowledge exists, there is at least .5-common belief. Thus, though Lewisian common knowledge appears to be not that strict, it in fact is in a strong sense equivalent to common p-belief, and thus implies any of the myriad results published using that simpler concept. What an interesting result! I take this to mean that many common complaints about common knowledge are not that serious at all, and that p-belief, quite standard these days in economics, is much more broadly applicable than I previously believed.

http://www.springerlink.com/content/n81219v23334n610/ (GATED. Philosophy community: you have to do something about the lack of working papers freely accessible! Final version in Synthese 183.2 – if you are a micro theorist, you should definitely be reading this journal, as it is definitely the top journal in philosophy publishing analytic, formal results in theory of knowledge.)

“Centralizing Information in Networks,” J. Hagenbach (2011)

Ah…strategic action on network topologies. There is a wily problem. Tons of work has gone into the problem of strategic action on networks in the past 15 years, and I think it’s safe to say that the vast majority is either trivial or has proved too hard of a problem to say anything useful at all. This recent paper by Jeanne Hagenbach is a nice exception: it’s not all obvious, and it addresses an important question.

There is a fairly well-known experimental paper by Bonacich in the American Sociological Review from 1990 in which he examines how communications structure affects the centralizing of information. A group of N players attempt to gather N pieces of information (for example, a 10-digit string of numbers). They each start with one piece. A communication network is endowed on the group. Every period, each player can either share each piece of information they know with everyone they are connected to, or hide their information. When some person collects all the information, a prize is awarded to everybody, and the size of the prize decreases in the amount of time it took to gather the info. The person (or persons) who have all of the information in this last period are awarded a bonus, and if there are multiple solvers in the final period, the bonus is split among them. Assume throughout that the communications graph is undirected and connected.

Hagenbach formalizes this paper as a game, using SPNE instead of Nash as a solution concept in order to avoid the oft-seen problem of networks where “everybody do nothing” is an equilibrium. She proves the following. First, if the maximum game length is at least N-1 periods, then every SPNE involves information being aggregated. Second, in any game where a player i could potentially solve the puzzle first (i.e., the maximum length of shortest paths of player i to other players is less than the maximum time T the game lasts), there is an SPNE where she does win, and further she wins in the shortest possible amount of time. Third, for a group of communication networks that includes graphs like the tree and the complete graph, then every SPNE is solved by some player is no more than N-1 periods. Fourth, for other simple graph structures, there are SPNEs for which an arbitrary amount of time passes before some player solves the game.

The intuition for all of these results boils down to the following. Every complete graph involves at least two agents connected to each other who will potentially each hold every piece of information the opponent lacks. When this happens, we are in the normal Game of Chicken. Since the problem has a final period T and we are looking for SPNE, in the final period T the two players just play chicken with each other, and chicken has two pure strategy Nash equilibria: I go straight, you swerve, or you go straight and I swerve. Either way, one of us “swerves”/shares information, and the other player solves the puzzle. The second theorem just relies on the strategy where whichever player we want to solve the puzzle refuses to share ever; every other player can only win nonzero payoff by getting their information to her, and they want to do so as quickly as possible. The fourth result is pretty interesting as well. Consider a 1000 period game, with four players arranged in a square: A talks to B and D, B talks to A and C, C to B and D, and D to A and C. We can be in a situation where B needs what A has, and A needs what B has, but not be in a duel. Why? Because A may be able to get the information from C, and B the information he needs from D. Consider the following hypothesized SPNE, though: everyone hides until period 999, then everyone passes information on in 999 and 1000. In this SPNE, everyone solves the puzzle simultaneously in period 1000 and gets one-fourth of the bonus reward. If any player deviates and, say, shares information before period 999, then the other players all play an easily constructed strategy whereby the three of them solve the following period but the deviator does not. If the reward is big enough, then all the discounting we need to get to period 1000 will not be enough to make anyone want to deviate.

What does this all mean for social science? Essentially, if I want information to be shared and I have both team and individual bonuses, then no matter what individual and team bonuses I give, the information will be properly aggregated by strategic agents quite quickly if I make communication follow something like a hierarchy. Every (subgame perfect) equilibrium involves quick coordination. On the other hand, if the individual and team bonuses are not properly calibrated and communication involves cycles, it may take arbitrarily long to coordinate. I think a lot more could be done with these ideas applied to traditional team theory/multitasking.

One caveat: I am not a fan at all of modeling this game as having a terminal period. The assumption that the game ends after T periods is clearly driving the result, and I have some hunch that simply using a different equilibrium concept than SPNE and allowing an infinite horizon, you could solve for very similar results. If so, that would be much more satisfying. I always find it strange when hold-up problems or bargaining problems are modeled as having necessarily a firm “explosion date”. This avoids much of the great complexity of negotiation problems!

http://hal.archives-ouvertes.fr/docs/00/36/78/94/PDF/09011.pdf (2009 WP – final version with nice graphs in GEB 72 (2011). Hagenbach and a coauthor also have an interesting recent ReStud where they model something like Keynes’ beauty contest allowing cheap talk communication about the state among agents who have slight heterogeneity in preferences.

“The Temporal Structure of Scientific Consensus Formation,” U. Shwed & P. Bearman (2010)

This great little paper about the mechanics of scientific consensus appeared in the last copy of the American Sociological Review. The problem is the following: how can we identify when the scientific community – seen here as an actor, following the underrated-among-economists-yet-possibly-crazy philosopher Bruno Latour – achieves consensus on a topic? When do scientists agree that cancer is caused by sun exposure, or that smoking is carcinogenic, or that global warming is caused by man? The perspective here is a school in sociology known as STS that is quite postmodern, so there is certainly no claim that this scientific consensus means we have learned the “truth”, somehow defined. Rather, we just want to know when scientists have stopped arguing about the basics and have moved on to extensions and minor issues. Kuhn would call this consensus “normal science”, but Latour and the STS guys often refer to it as “black boxing,” in which scientific consensus allows scientists to state something like “smoking causes cancer” without having to defend it. Economists contains many such black boxed facts in its current paradigm: agents are expected utility maximizers, for example. (Note that “the black box”, in economics, can also refer to growth in multifactor productivity, as in the title of Nate Rosenberg’s book on innovation; this is somewhat, but not entirely, the same concept).

But how do we identify which facts have been black boxed? Traditionally, sociologists of science have used expert conclusions. For instance, IPCC reports survey experts on climate change. The first IPCC report in the 1980s did not identify climate change as anthropogenic, but all future reports did. The problem is that such expert reports are not available for all problems where we wish to investigate consensus, and second that it is in some sense “undemocratic” to rely on expert judgments alone. It would be better to have a method whereby an ignorant observer can look down from on high at the world of science and pronounce that “topic A is in a state of consensus and topic B is not.”

This is precisely what Bearman and Shwed do. They construct citation networks, using keyword searches, on a number of potentially contested ideas over time, ranging from those which are traditionally considered to have little epistemic rivalry (coffee causes cancer) to those with well-known scientific debates (the nature of gravitational waves). They use a dynamic method where the sample at time t includes all papers which were cited within the past X years (where X is the median age of papers cited by new research at time t), as well as any older articles in the same field which were cited by those papers. They then examine the modularity of the citation network. A high level of modularity, in network terms, essentially means that the network is made up of relatively distinct communities vis-a-vis a random graph. Since citations to papers viewed favorably are known to be more common, high modularity means there are multiple cliques who cite each other, but do not cite their epistemic rivals.

With this in hand, the authors show that areas considered by expert studies to have little rivalry do indeed have flat and low levels of modularity. Those traditionally considered to be contentious do indeed show a lot of variance in their modularity, and a high absolute level thereof. The “calibrating” examples show evidence, in the citation network, of consensus being reached before any expert study proclaimed such consensus. In some sense, then, network evaluation can pinpoint scientific consensus faster, and with less specialized knowledge, than expert studies. Applying the methodology to current debates, the authors see little contention over the non-carcinogenicity of cell phones or the lack of a causal relation between MMR vaccines and autism. The methodology could obviously be applied to other fields – literature and philosophy would both be interesting cases to examine.

One final note: this article is published in a sociology journal. I would greatly encourage economists to sign up for eTOC emails from our sister fields, which often publish content on econ-style topics, though often with data, tools and methodologies an economist wouldn’t think of using. In sociology, I get the ASR and AJS, though if you like network theory, you may want to also look at a few of the more quantitative field journals. In political science, APSR is the journal to get. In philosophy, Journal of Philosophy and Philosophical Review, as well as Synthese, are top-notch, and all fairly regularly publish articles on knowledge which would not be out of place in a micro theory journal. I read Philosophy of Science as well, which you might want to take a look at if you like methodological questions. The hardcore econometricians and math theory guys surely would want to look at journals in stats and mathematics; I don’t read these per se, but I often follow citations to interesting (for economics) papers in JASA, Annals of Statistics and Biometrika. I’m sure experimentalists should be reading the psych and anthropology literature as well, but I have both little knowledge and little interest in that area, so I’m afraid I have no suggestions; perhaps a commenter can add some.

http://asr.sagepub.com/content/75/6/817.full.pdf+html (Final version, ASR December 2010. GATED. Not only could I not find an ungated working paper version, I can’t even find a personal webpage at all for either of the authors; there’s nothing for Shwed and only a group page including a subset of Bearman’s work. It’s 2011! And worse, these guys are literally writing about epistemic closure and scientific communities. If anyone should understand the importance of open access for new research, it’s them, right?)

“On Consensus through Communication with a Commonly Known Protocol,” E. Tsakas & M. Voorneveld (2010)

(Site note: I will be down in Cuba until Dec. 24, so posting will be light until then, though I do have a few new papers to discuss. I’m going to meet with some folks there about the recent economic reforms and their effects, so perhaps I’ll have something interesting to pass along on that front.)

A couple weeks ago, I posted about the nice result of Parikh and Krasucki (1990), who show that when communication is pairwise, beliefs can fail to converge under many types of pre-specified orders of communication. In their paper, and in every paper following it that I know of, common knowledge of the order of communication is always assumed. For instance, if Amanda talks with Bob and then Bob talks with Carol, since only common knowledge of the original information partitions is assumed, for Carol to update “properly” she needs to know whether has Bob has talked to Amanda previously.

In a paper pointed out by a commenter, Tsakas and Voorneveld point out through counterexample just how strict this requirement is. They expand the state space to include knowledge of the order of communication (using knowledge in the standard Aumann way). It turns out that with all of the necessary conditions of Parikh and Krasucki holding, and uncertainty about whether a single act of communication occurred, consensus can fail to be reached. What’s worrying here from a modeling perspective is that it is really convenient to model communication as a directed graph, where A links to B if A talks to B infinite times. I see the Tsakas and Voorneveld result as giving some pause to that assumption. In particular, in the example, all agents have common knowledge of the communications graph, since the only uncertainty is in one period and therefore no uncertainty about the structure of the graph.

There is no positive result here: we don’t have useful conditions guaranteeing belief convergence under uncertainty about the protocol. In the paper I’m working on, I restrict all results to “regular” communication, meaning the only communication is through formal channels that occur infinite times, and because of this I only need to assume knowledge of the graph.

http://edocs.ub.unimaas.nl/loader/file.asp?id=1490 (Working Paper. Tsakas and Voorneveld also have a 2007 paper on this topic that corrects some erroneous earlier work: https://gupea.ub.gu.se/dspace/bitstream/2077/4576/1/gunwpe0255.pdf. In particular, even if consensus is reached, information only becomes common knowledge among under really restrictive assumptions. This is important if, for instance, you are studying mechanisms on a network, since many results in game theory require common knowledge about what opponents will do: see Dekel and Brandenburger (1987) and Aumann and Brandenburger (1995), for instance. I’ll have more to say about this about this once I get a few more results proved.)