Variants of “Large Type Structures”

Its been brought to my attention that I engage in bad blogging behavior. In my last post, I made implicit reference to a previous post without properly connecting the dots.  This was in my discussion of what—in my last post—I referred to as “belief complete” vs. “hierarchy complete.” So, let me elaborate on what I meant.

Both ideas are meant to reflect properties of a “large type structure.”

A type structure is belief complete if it induces all possible beliefs about types.  (Formally, if the players’ belief maps are onto. The definition is originally from Brandenburger, 2003. But, due to publication lags, it was actually published earlier in Battigalli-Siniscalchi, 2002.) A type structure is hierarchy complete if it induces all possible hierarchies of beliefs.  (One way to formalize this condition is: A structure is hierarchy complete if, for each type that arises in some type structure, there is a type in the hierarchy complete type structure that induces the same hierarchy of beliefs.  For better or worse, I referred to this property as “terminal” in a publication. See my original post on the history of this concept.)

The canonical constructions in Mertens-Zamir, Brandenburger-Dekel, and Hiefetz-Samet are each both “belief complete” and “hierarchy complete,” at least when the underlying set of uncertainty is Polish. But, a priori, these are distinct concepts.  Even when the underlying set of uncertainty is finite: A hierarchy complete structure need not be belief complete.  (See this paper for an example.) A belief complete structure need not be hierarchy complete. (See this paper for a theorem.)

After writing my last post, I started wondering if maybe the terminology should be type complete vs. hierarchy complete.

PS: Sorry for the self-cites.  They were the easiest.  If I should be citing you instead, please let me know!

What’s New…

Its been a long time since I blogged—much to report book-wise. Some quick updates:

1. I was very proud to see that my portion of the book alone is now around 75 pages. But that seemed less impressive once I remembered how long my average paper is.
2. Re terms for large type structures: We are now differentiating between “belief complete” (e.g., all beliefs about types are present, as in Brandenburger 2003) vs. “hierarchy complete” (e.g., all beliefs about hierarchies are present).  This helps with some of the issues I raised in a previous post.
3. We’ve gotten some requests re advertising papers on this blog.  Sorry we have been slow to implement!  We are trying to figure out how to set up a secondary page that only has such announcements.  (If you know how—please do let us know!)

More on content soon! 

What is an Epistemic Condition? … Again

In part, motivated by a new paper with Jerry Keisler (Iterated Dominance Revisited) and in part motivated by book writing, I’ve been bugging my coauthors (Paolo, Marciano, and Jerry) and non-coauthors (Byung Soo) to have long conversations about (a) what do we mean by an epistemic condition? and (b) what makes such a requirement “good”? I’d like to share some excerpts from what I’ve written on (a) for the book—with no implication to coauthors and non-coauthors.  (I’ll blog in a couple of days about (b)—need to make sure I don’t write something mean.)

— Excerpt —

The starting point is the description of the strategic situation. This is given by—what we referred to as—an epistemic game.  …

It is worth pointing out how this step compares to a more traditional game-theoretic analysis. Traditionally, the starting point is a description of the rules of the game and the payoff functions.  … Such substantive assumptions are imposed by specifying the game G. But now there is a second set of assumptions to bring to the table—assumptions about players’ beliefs.  … Such substantive assumptions are imposed by specifying the type structure .

For a given a game G, there are many type structures compatible with G.  When we choose one compatible type structure and focus on the epistemic game , we explicitly impose substantive assumptions on the players’ beliefs. Such substantive assumptions about beliefs may reflect “the context” within which the game is played, e.g., cultural or societal norms, a history prior to the game as specified, etc.  Thus, when we restrict the analysis of a game to a particular type structure (or to a particular class of type structures), we impose—what we’ll call— a contextual assumption.

Whether a particular contextual assumption is vs. is not of interest depends on the application at hand.  For instance, consider a game between borrowers and lenders.  It may be of interest to impose the contextual assumption that “it is transparent to lenders and borrowers that `borrowers do repay lenders,”’ e.g., given the history of interactions between the particular borrowers and lenders.  Or, the particular borrowers and lenders may be new to the market and, so, cannot come to such a conclusion.  In that case, it may be of interest to impose the contextual assumption that “borrowers and lenders consider all possible beliefs,” i.e., the type structure is complete.  The latter form of a contextual assumption is, in a sense, the “neutral” case, i.e., where the context in which G is played does not limit the players’ beliefs.

Contextual assumptions impose restrictions on the beliefs players “consider possible.” The choice of the contextual assumption is based on a story external to the game G, e.g., some prior unmodeled history that leads players to “rule out” certain beliefs. (For instance, it may be “transparent” to drivers that UK drivers all drive on the left-hand side of the road.) Given the contextual assumption, a natural next-step is to ask whether these beliefs are, in a sense, “reasonable.” That is, might “strategically sophisticated” players hold those beliefs?

To answer this question, we must formalize the idea of “strategic sophistication.” This is what the, so-called, epistemic conditions are meant to capture.  In particular, the epistemic conditions will map an epistemic game to a subset of the states within .  The idea is that this mapping is chosen so that the output is the “strategically sophisticated states” within the epistemic game .  So, for instance, RmBR and RCBR are examples of epistemic conditions—each reflecting different levels of strategic sophistication. Note, the mapping can restrict both the players’ behavior and the players’ beliefs. …

— Back to Reality —

When we justify a solution concept, we often times impose both contextual assumptions and epistemic conditions.  But, we still use the phrase “epistemic conditions for Solution Concept X.” Maybe the phrase is not appropriate?

Hello world, again!

Hi all,

today we have decided to open our blog to the wide world. Please, wide world: be gentle! Check the About page for details and disclaimers.

Get Rid of the Universal Type Structure

This is a plea for the elimination of the phrase universal type structure.  Why?  Primarily because its often not clear what formal object the authors have in mind.

First a bit of a history lesson: The first two papers in the literature are Armbruster-Boge (AB) and Boge-Eisele (BE).  (Sorry, Mertens-Zamir was not the first paper.)  They each sought to answer different questions.

• AB sought to answer, what I’ll call, the ordinal question.  We have that each player faces uncertainty about a parameter, each player faces the uncertainty about the other players’ beliefs about the parameter, each player… The question is: Does this reasoning ever stop?  That is, does there exist an ordinal so that the players’ hierarchies of beliefs up to that ordinal uniquely determine all subsequent beliefs?
• BE sought to answer: Does there exist a large type structure so that, for each “small” type structure, there is a hierarchy preserving map from the “small” structure to the large type structure?

I believe AB used something like the phrase multi-orbit system to describe their type structure.  (But, I can’t find the paper right now.)  BE said their type structure was terminal.

Then comes along Mertens-Zamir (MZ).  They had everything but the kitchen sink and they called their type structure universal.  Its hard to know—ex post—what question they were vs. were not interested in.

Next comes along Brandenburger-Dekel (BD).  From their writing, it is hard to argue that they were interested in anything but the ordinal question.  This is clear from their math.  But, it is also the sense that I got from having banged Adam over his head fifty times saying “Are you sure this is what you meant?”  My impression from talking to Adam then is that–not only were they only concerned with the ordinal question—they (at the time) saw MZ as being (only) interested in the ordinal question.

I cannot find the phrase universal in BD.  But, my impression is that, post BD and prior to other work, people used the phrase universal when referring both to MZ and BD.

Come along Heifetz-Samet (HS).  They are explicitly interested in a version of the question raised by BE: Does there exist a large type structure, so that for each type structure there is a unique type morphism from the small type structure to the large type structure?  If the answer to the question is yes, then so is the answer to BE’s question.  But the converse need not hold for two reasons.  First, even if each hierarchy morphism is a type morphism, the hierarchy morphisms need not be unique (i.e., if there are redundancies).  Second, there may be a hierarchy morphism from a small structure to a large structure, even though it is not a type morphism. (This does not arise in HS’s construction.)

HS used the phrase universal to refer to their property.  They were the first paper to define the term and so, its not surprising that there were lots of follow-up papers that used the phrase universal in exactly this way.  I will refer to such papers as the “HS-camp.”

On the one hand, HS clarified what they meant, in a way that was missing from the previous literature.  But, by picking the phrase universal, it also caused some confusion.  Many people did—and still do—think of the phrase  universal as the particular type structures constructed by MZ or BD.  But HS meant it as a property of an arbitrary type structure, one based on embedding type structures.  Still others think of it as “the answer to the ordinality question.”

This has led to a situation where (i) people often don’t formally say what they mean by universal in their papers (thinking it is obvious?) and (ii) it is often unclear what they do mean (precisely because there is no consensus). So, with this, I propose that we need to coordinate on new terminology.

What terminology should be used? Here’s one proposal:

1. Stop using the phrase universal all together.  Its too confusing.
2. Reference the ordinality question as the ordinality question.  (Note, a type structure can be an answer to the ordinality question.  There is no “property of a type structure involved.)
3. MZ, BD, and HS each provide (different) constructions of a large type structures.  Reference these as the canonical construction of a large type structure and point out which construction you are referencing.
4. Find new terminology for the embedding property, i.e., not the phrase universal.

On the embedding property:  People in the HS-camp typically argue that their large structure is terminal (in the sense of category theory), but the large structure in BE may not be terminal (since the hierarchy morphism need not be unique).  My gut reactions to this:

• If one takes as given that two types that induce the same hierarchies beliefs are equivalent, then hierarchy morphisms are unique upto equivalence classes.
• The phrase terminal is certainly meant to hint to category theory.  But, I am not convinced it **must** be used **exactly** as in category theory—i.e., that equivalence classes must be ruled out. (Question: Would a pure category theorists insist on ruling out equivalence classes?)
• So, in my view, if you want to argue that terminal is bad terminology for the BE question (i.e., mapping via hierarchy morphisms) but good terminology for the HS question (i.e., embeddings via type morphisms), you must argue that hierarchy morphisms are not unique up to an equivalence class.  That is, you must then argue that types convey more information than hierarchies morphisms and so there is “information lost” by looking at hierarchy morphisms instead of a unique type morphism.

For many of the questions we are interested in, we would not distinguish two types that induce the same hierarchies of beliefs.  That is, those types are “behaviorally equivalent” for much of the analyses we are interested in.  (The one exception comes in when the analyst does not fully specify the full set of parameters that the player is uncertain about.  See, Ely-Peski and Liu on this point.)  So, to me, it doesn’t seem horrific to use the phrase terminal to capture the BE question, with the idea being that the HS question is a special case of the BE question—one that is interesting mathematically.

But I can also see an argument that, perhaps in the future, we may not see these as being equivalence classes and, so, we should not adopt two phrases: weak terminality and strong terminality?

What is an Epistemic Condition?

If we’re going to talk epistemic game theory, certainly we should have an answer!

Let’s start with the dictionary definition.  Merriam Webster tells me that an epistemic condition is:  a condition relating to knowledge or knowing.  But there are several ways this definition can be applied.

1. The definition I have been implicitly adopting in practice—admittedly without much thought—is that an epistemic condition is a restriction on on which states we consider possible.  This restriction is based on strategic reasoning.  For instance, we only look at states that are consistent with “rationality, belief of “rationality,” belief of “rationality and belief of ‘rationality,’” etc.  Certainly, belief of an event E should be part of an epistemic condition.  But, I guess I am sneaking in a no-no when I add in rationality—this is about maximizing and not a condition about knowledge per se. (Though, see the definition of “rationality” in BFK’s “Admissibility in Games.”)  Also, note, I am taking a broad reading of MW’s word “knowledge” … really changing it to “thinking.”

2. In 1, I implicitly fixed both the game and the type structure.  When I fix the type structure, I am implicitly imposing a “restriction” on what beliefs players consider possible.  Note, the “restriction” can be “players consider all beliefs possible,” i.e., as in the canonical construction of a large type structure in Mertens Zamir, Brandenburger Dekel, Heifetz Samet, etc.  Such a restriction is also a condition about knowledge—when knowledge is broadly defined to include (an informal notion of which beliefs) players consider possible.  So shouldn’t this be part of an epistemic condition? Indeed, the earlier epistemic game theory literature (which focused on “providing epistemic conditions for a solution concept”) included such a restriction as part of the phrase “epistemic conditions.”

But, to me, there s a different flavor to 1 vs. 2. Under 1, I fix the description of the strategic situation—i.e., the rules of the game, payoffs, beliefs—and then I impose conditions on players’ knowledge.  Under 2, I fix part of the description of the strategic situation—i.e., the rules of the game and payoffs—and then I impose a restriction both on the type structure (i.e., which epistemic game I study) and also on players’ knowledge within that type structure.  While both are clearly important, maybe we need two different terms for these conditions? (to clarify which exercise we are up to!)

Now, of course, beyond the question of “What is an Epistemic Condition?” there is also the question of “What is a Good Epistemic Condition?.”  I’ll have more to say on that at a later time.

Hello World, 2

Thanks Marciano for setting this up!   Perhaps we should change the picture on the main blog page to be a book picture.  (The picture?)  We probably should have taken Phoenix book pictures… oh well….