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“Limited Commitment: Mechanism Design Meets Information Design” Professor Vasiliki Skreta #1 — Transcript

by UTMD 東京大学マーケットデザインセンター UTokyo Market Design center · 16,436 words · 2,524 segments · language en · Watch on YouTube

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0:00First of all, thank you very much for
0:01trusting me to come um to give you these
0:04lectures. Um I will talk about a a body
0:07of literature um on mechanism design and
0:11I know a lot of you know a lot about
0:12mechanism design in economic theory. Um
0:16so this is um a a special um part of
0:19mechanism design where the designer is
0:22also a strategic player and there are a
0:24number of subtleties that arise in this
0:26type of environment that are also
0:28transferable in other situations where
0:31we are thinking about the designer as
0:32being a player in a strategic setting.
0:36So uh the title of this lecture is
0:38limited commitment mechanism design with
0:41information design and I will talk about
0:44a pure theory but also I will apply uh
0:47the tools and the theory we're going to
0:49develop in the first lecture to a to a
0:52problem of designing histories or data
0:55sets if you like from past purchases
0:58which are what the designer is going to
1:01design in the first interaction that
1:03then the designer which can be the same
1:05designer another designer can exploit at
1:08a subsequent interaction. So if you
1:10think about a vertical relationship, an
1:12upstream firm collects data and sell it
1:14to a downstream firm. That's going to be
1:16our second lecture. This um lecture is
1:19drawn on work I've been doing for since
1:22my thesis but primarily on work I've
1:25done recently with a a extremely
1:29talented and gifted theorist Lara Deval
1:32who is also a very dear friend of mine.
1:35So um most of the work will most of the
1:38lecture will be based on work joint work
1:41with um with Lara. So, as I was alluding
1:44to a minute ago, the first lecture is
1:46going to talk about um uh the background
1:49what is the problem we're looking at and
1:51it's going to give you some perspective
1:53on some the first seminal papers by Ben
1:56Strauss on the topic and then we're
1:59going to talk about our revelation
2:01principle under limited commitment and
2:04introduce to you the class of mechanisms
2:07that are without loss of generality in
2:09those settings which we call direct
2:11blackwell mechanisms.
2:13direct black mechanisms are a class of
2:15mechanisms that I I will show you in a
2:17minute and um includes the standard
2:21direct revelation mechanisms that we use
2:24that we are more familiar with. So it's
2:26a class that is encompassing the
2:29standard canonical class if I'm making
2:31any sense. And then I'm going to
2:33illustrate the theory with a very simple
2:36two type two period model and show you
2:39how we can use this new revelation
2:42principle to express um the designer's
2:46problem again as a constraint
2:47optimization problem which is much
2:49simpler to analyze than a complicated
2:51game with a player who is choosing
2:53mechanisms.
2:55And the second lecture is going to think
2:57about the application of thinking about
3:00mechanism design as data set design in a
3:02way. So we're going to think about some
3:05um the the application will be about how
3:08um purchase history and product
3:11personalization
3:12um are shaped in a in an upstream
3:15downstream interaction where the firm
3:18can uses past purchase history with
3:21price discriminate in the second bill.
3:23And we're going to see some type of how
3:26this language we have we're going to
3:28develop in the first lecture is going to
3:30help us find what is the optimum um
3:34product line for the upstream seller and
3:36how this design of um product affects
3:42pricing and welfare and so forth
3:45which are the things we care about econ
3:47in economic settings. Right now we want
3:49to think about after we design the
3:51institution what are the implications
3:53what are the new frictions we get and
3:54what are the the welfare implications or
3:57any regulatory um uh issues that we
4:01could address with the framework.
4:04Sorry that was the wrong direction. Okay
4:07so let's start uh with a um with a kind
4:10of broad picture and um where we are. So
4:14as we all know I guess most of a lot of
4:17you know a lot about mechanism design
4:19but if something is unclear please raise
4:23your hand and don't be this is a class
4:26this is not a seminar this lecture is
4:29primarily for you not for me so I would
4:32really like to feel comfortable to raise
4:33your hand and ask whatever you like it
4:36doesn't have to be as it has to be a any
4:40any question is is a good question so
4:42please ask anything you want it's the
4:45lecture is for you. So um
4:49so we kind of know all of us more or
4:52less mechanism design is a part of
4:55economics that thinks about designing
4:57markets or institutions and it's sort of
5:00a I read this quote once in the
5:02economics that I thought is wonderful.
5:04It's mechanism design is the helping
5:06hand to the invisible hand. So how we
5:10help kind of markets that perform better
5:12and it has been successfully applied to
5:14many settings from auctions,
5:16regulations, social insurance, taxations
5:18and so forth. So um when we're thinking
5:23about designing what is the best
5:24institution of course that is a very
5:26complicated problem because we're
5:28thinking about what is the set of rules
5:30among all set of rules that um is the
5:34best according to some objective. And
5:36this question is even even writing down
5:39all set of rules is a very complicated
5:43thing to do and but we have seen
5:46tremendous progress in mechanism design
5:49thanks to a fundamental result which is
5:52is called the revelation principle. This
5:54result underpins the broad applicability
5:57of mechanism design and in particular in
5:59a dynamic setting. This result says well
6:02take an arbitrary sequence of mechanisms
6:05together with the agent strategy and
6:08think about uh now the um what is the
6:12the outcomes we are going to get when we
6:14think about the agent best responding to
6:16the sequence of mechanisms. So the se
6:19the mechanisms can be arbitrary and
6:21complicated. The agent strategy can be
6:23arbitrary and complicated but it has to
6:25be some optimal response in a strategic
6:27setting. The revelation principle says
6:29in a dynamic setting that we can rep
6:32replicate the same distribution over
6:34outcomes which is joint distributions
6:36over types and ultimate allocations who
6:39gets what with a sequence of direct
6:42revelation mechanisms
6:44where the strategy space of the agent is
6:47simply to make a report about their
6:49diet. And it and it is not just that we
6:51get a simple class of mechanisms. We
6:54also get simple behavior in particular
6:57telling the truth and participating
7:00in with a using that as a as the best
7:03response with our sequence of direct
7:05mechanism is enough to replicate um the
7:09thing that what arises from an arbitrary
7:11sequence of mechanisms. And this makes
7:14the problem very well tractable because
7:17essentially that what we get sorry is
7:20that um because we can basically replace
7:25the agent here which is strategic layer
7:28is essentially captured with the two
7:31constraints telling the truth and
7:33participating and effectively transforms
7:36the equilibrium problem into a
7:38constraint optimization problem. So
7:41instead of solving for equilibrium we
7:43are solving a decision problem much
7:46simpler.
7:48Um
7:51so now the problem is that the
7:53applicability of of the revelation
7:55principle relies on the designer's
7:58availab ability to commit to the entire
8:01sequence of mechanisms
8:03and this commitment assumption is often
8:05unrealistic. So if you think about firms
8:09they accumulate data about consumers
8:11over time and they're tempted to use
8:13this data to change the pricing the
8:15terms of contracts the same in a similar
8:19way governments benevolent governments
8:21are long lived but they cannot bind the
8:24successors so if I'm a government I can
8:26design a policy for today but no one
8:29promises that this policy is going to be
8:30adopted by uh the future incumbents
8:35so in practice practice um that's a
8:38quote from the Nobel Prize committee
8:39when they're commenting on Je roll's
8:41work in pract on regulation in practice
8:44regulation is seldom once and for all
8:47but rather an activity that take place
8:49over an extended period the regulator
8:52may be unable to commit to a regulatory
8:54policy over the relevant time span so
8:59it's not just that you know the
9:01assumption is unrealistic the optimal
9:04mechanisms we get are hard to swallow.
9:07So they are time inconsistent. For
9:09example, if we're thinking about um the
9:11optimal mechanism of a sell for a seller
9:14of a durable good. So you are a seller,
9:16you have a unit of a durable good. Let's
9:18say it's I don't know a fringe,
9:21something durable, you set the optimal
9:24mechanism is to charge a constant price.
9:26So you choose a price and then you never
9:29change it. But if you choose a price and
9:31you don't sell the good then you're
9:33committed to remove the object from the
9:35market. Uh the same thing in in context
9:40of taxation the same contract has to be
9:42repeated every period but that's not um
9:45sequentially rational because once the
9:47regulator realizes oh this firm is
9:49actually an efficient firm and can
9:51produce at a lower cost then the
9:53regulator wants to demand you know more
9:56production and less subsidy for the
9:58firm.
10:00Um
10:01so uh that's where we are and that's
10:05sort of the we're going to be thinking
10:07about now me a mechanism designer who
10:10has
10:12imperfect commitment.
10:14So under limited commitment the designer
10:17is now a player. The designer
10:19strategically exploits revealed
10:21information by the agent. Um and because
10:24the the design the agent expects well
10:27today there's a mechanism we know the
10:29rules that's great but whatever the
10:31designer learns
10:33about me uh from this interaction that
10:36could be exploded the agent is reluctant
10:39to reveal private information and the
10:42intuition is that revealed either
10:44through reporting but ultimately you
10:47know through behavior they lose
10:48information rents and in that sense the
10:52classic revelation principle no longer
10:54applies in in particular. It is not just
10:57the case that we don't have, you know,
10:59truthtelling as a canonical best
11:02response. Truth telling may not even be
11:04a best response. So, Lavon and Zola have
11:08a seminal paper in econometric 1988.
11:11Actually, this is a paper in econometric
11:13that has a problem but not a full
11:15solution. They show that in a two period
11:19regulation problem under limited
11:20commitment a standard sort of two period
11:23take you know the standard regulation
11:24problem bar on you know bas or baron
11:28meers type of problem repeat it twice
11:31you cannot find the optimum the and
11:33actually they're showing that in the
11:34first period it might be impossible to
11:37separate a continuum of types
11:40so the core conceptual difficulty is
11:44following the designers 's current
11:46payoff depends on future choices, future
11:50mechanism choices which depend on
11:53posterior beliefs about agent style. So
11:57I'm in period two. I'm going to choose
11:59the mechanism now that I I have I have
12:03some I have observed the behavior of the
12:05agent. I've updated my belief. So now
12:07I'm going to choose a second period
12:08mechanism. This mechanism depends on my
12:11beliefs. But these me these beliefs
12:14about the agent's type are themselves
12:16mechanism dependent because the agent
12:19the agent in period one thinks about
12:22well what will my behavior in period one
12:25due to the principal's belief in period
12:28two because this will affect the
12:29mechanism I will see tomorrow. So this
12:32creates this conceptual difficulty that
12:35the mecha beliefs are past and future
12:40mechanism dependent
12:43and the lack of commit and then I'm
12:46quoting from the book laon and have a
12:49big book on regulation. This was sort of
12:51a back in the day when I was young. This
12:54was sort of a a nice book to read when
12:57you were interested in regulation, which
12:59was sort of the older term for market
13:01design, sort of the more the loose term
13:04uh of thinking about, you know, how to
13:06fix markets. And this is sort of was
13:08considered one of the kind of bibles of
13:10regulation that was written by Lafon and
13:13Chiro. And they they have several
13:16chapters on imperfect commitment and
13:18they write in the book the lack of
13:20commitment in repeated adverse election
13:22situations leads to substantial
13:24difficulties for contract theory. So in
13:28a sense
13:30the lit the literature a little bit
13:32paused at the beginning of the 90s. Yes
13:34please.
13:34>> Just a clipation question. Yes.
13:37>> Uh so you are calling it a limited
13:39committed case not the no committing
13:41case. Um so is that because you assuming
13:45that the mechanism designer can at least
13:47commit to the current mechanism
13:49>> to the current? Yes. And actually this
13:51is an excellent question because in a in
13:53one slide or two slides I will tell you
13:55that although the commitment benchmark
13:58is very well defined lack of commitment
14:01has many can take many shapes because we
14:04can think about designing a contract
14:07that is a 100 period long but we're
14:09going to we allow for the possibility of
14:12renegotiation. We can so let's think
14:15about we have a 100 period relationship.
14:17Commitment says we're going to write the
14:19contract that fully specifies everything
14:21that's going to happen in the next
14:22hundred years. That's commitment.
14:25Limited commitment could mean I'm going
14:27to write 10 period contracts or I'm
14:29going to write a long contract that I'm
14:31going to allow to renegotiate or I'm
14:33going to write period by period
14:36contracts or I'm going to write a and
14:39even in the long-term contract that
14:41could allow for for certain levels of
14:44memory of the contract. Maybe the cond
14:47cannot remember you know 80 messages. So
14:51all of those things are part of the
14:53design some lit. So what we're going to
14:55do today this is actually not um I'm not
14:58opening a tangent. I'm I'm answering
15:00something that I was going to say later,
15:02but I think it's important for us to
15:04kind of think about this. What we're
15:06going to be thinking today as I'm going
15:07to talk about more specifically in a
15:09minute is com I'm I'm writing a
15:12mechanism for today, but I'm not com I'm
15:15not specifying anything contractually
15:18from tomorrow onwards. And this is what
15:21some people call no commitment, some
15:24people call limited commitment. I'm
15:26gonna call it in the chapter you know
15:29that I was kindly asked to I call it
15:32kind of spot commitment. Um so in that
15:36that's that's the thing but in all of
15:38those situations that I'm describing. So
15:41for example this is the notion of
15:43commitment using laon and ti and this is
15:46the notion of commitment of limited
15:48commitment using the classic papers.
15:50That's what I mean with commitment. And
15:52this is the notion of limited
15:53commitment. I'm sorry. Um,
15:57>> all right. Is are we good?
15:58>> Yeah. Thank you.
15:58>> Okay.
16:01So,
16:03uh, this was actually a good question
16:05because I'm I h I'm going to comment on
16:07this. Um,
16:09but I haven't I should have done it a
16:12bit earlier. So, thank you for asking
16:14this question.
16:16All right. So with that in mind, let me
16:20just go back one second. So with that in
16:23mind, um think about what what I will be
16:27saying in the next couple of slides as
16:30we commit to the mechanism today but not
16:32to anything tomorrow.
16:35Okay. So we have two approaches that
16:38have emerged. the one approach. H so the
16:41literature basically kind of stopped
16:43because it you know la kind of tried for
16:46about a decade and then it was they
16:50stopped and then then there is a
16:53you know some work that starts around
16:54the late 90s
16:57um beginning of 2000. So my paper was
16:59published in 2006. I started working on
17:01it probably 1999. You know how it goes
17:04with your first chapter for your thesis
17:06how long it takes to get published. at
17:08least in my case. Um so in in B and
17:12sprrow they thought about design proving
17:16a relation principle
17:18and I in they prove some results that
17:22showed well if you have finitely many
17:24types you're going to get the truth
17:25telling with positive probability. I
17:27wanted to solve a problem with a
17:29continuum of types where we know trutht
17:31telling does not hold. their results
17:34were not actually I was not aware of
17:36their results when I started on my um h
17:39chapter. So when I started working on
17:41the problem I started thinking about it
17:44without thinking about revelation risk
17:46the relation risk at all. And what I did
17:49I thought okay let's think about this
17:51game and think about the implications of
17:54the designer's limited commitment and
17:55agents behavior on the set of
17:57implementable outcomes basically on what
18:00are the joint distributions of our
18:02locations and types we can get at the
18:03end of the game and so let me just give
18:08you I will briefly give you this slide
18:10and then I will stop with the
18:11generalities I know it's frustrating you
18:13are all like highbrow theories you want
18:15to see equations there's going to be
18:16lots of them coming, I promise. But this
18:20this slide is going to give you sort of
18:22the recipe I used. And why am I
18:25insisting on some old you know approach?
18:28Because even though we have a revelation
18:31principle today, these problems are
18:33inherently hard. So sometimes when you
18:35want to solve a problem, relying on
18:38different ideas might be useful. And so
18:41what I did in my paper I thought well a
18:45choice of mechanisms induces a a game
18:47for the agent and the as I said the
18:51mechanisms coupled with the agent
18:52strategy induced and outcome
18:53distribution and the information the
18:56designer learns about the agent. So then
18:58I asked if we think about this long game
19:01where you know the the designer is
19:03choosing the selling mechanisms it was a
19:05durable good problem I was looking at
19:07what are the implications of the agent
19:09strategy being a best response to the
19:11mechanisms and actually if we thought if
19:14we think about the implementable
19:15outcomes in a buyer seller relationships
19:17that's basically an expected discounted
19:19probability of trade and an expected
19:21discounted transfer. So then if you
19:23think about these composite objects, you
19:25can think of them as every type of the
19:28of the buyer is going to have one of
19:29those probabilities and one of those
19:32transfers and an implication of best
19:34response is actually truthful. This kind
19:36of it has to satisfy the self-
19:38selectivity constraints. So this self-
19:40selectivity that we call truthtelling is
19:44essentially an indication of best
19:45response. But then you could use these
19:48composite objects to think about what
19:50the sequential rationality tells me
19:51about these objects. And that's
19:53basically what I did. And with with
19:56using kind of this sort of like very um
20:01approach that is not grounded in any
20:02canonical language. I was able to give
20:05enough structure on what are the PP PB
20:09implementable outcome distributions that
20:11I was able to optimize over that and
20:13show that the optimal PB implementable
20:16um outcome arises at simply posted price
20:20in every period. Of course that prove
20:22was complicated because I had to show um
20:26some steps but in any case I'm not going
20:27to go over that paper. that this le this
20:31at this level of generality you can use
20:33this approach without thinking about
20:36what is the cardinality of the D space
20:38or the nature of the interaction because
20:40it just thinks about the strategic
20:42setting and the implications of
20:44strategic behavior on outcomes which are
20:48very natural for a game theorist I
20:51think. Um so the difficulty in the
20:54analysis of this problem is something
20:56that is a little bit subtle and it's
20:59what I one can call it a failure of
21:02taxation principle. So in many sort of
21:04buyer seller relationships or in many um
21:09firm consumer relationship we could
21:11think about the mechanism as a direct
21:13mechanism. So I import my type theta and
21:16I get a quality Q and a transfer. But we
21:19could also think about the the buyer
21:21choosing a out of a menu of qualities
21:24and transfers. So I'm just going to
21:26offer quality of Q1, Q2, Q3 and I'm
21:30going to ask the buyer choose one. And
21:32of course the taxation principle says
21:35well it's with a generality asking the
21:37buyer to report or asking them to make a
21:40choice out of the menu. Are you with me?
21:42Now when you have a dynamic
21:44relationship, you could have two
21:46different messages or two different
21:48strategies leading to the same element
21:51in the menu. So it says an orange for
21:54like 100 yen. So this item in the menu
21:58can be encoding different messages and
22:02the messages could be used by different
22:04types. And when I see messages, I not
22:07only see the same menu choice because I
22:09I get different information that I use
22:11differently in the future. And that
22:14creates a subtlety. So you could have 10
22:17messages lead to the same allocation,
22:19but because these 10 messages are used
22:21by different types differentially, they
22:23have different posteriors attached to
22:25them and different continuations.
22:28Um, so this is why this type of approach
22:33although it's useful can get quite messy
22:37quickly. Oh, so now I'm going the wrong
22:39way. Sorry. Okay. So instead what we did
22:45with our paper is we developed a
22:48revelation principle for mechanism
22:50design with limited commitment. And
22:54what we what this result helps us do is
22:57to characterize a class of mechanisms
22:59and strategies that are enough to
23:01implement any outcome distribution that
23:04can be implemented under a limited
23:06commitment
23:08and
23:09um so now I'm going to
23:14question so we're going to study we
23:17study an uniform designer who interacts
23:20over time with a privately and
23:21persistently informed agent and the
23:24designer offers spot mechanisms
23:27can commit to today's but not to future
23:29mechanisms. So I don't know
23:33can I use this? I don't know how to use
23:35the how to press the button. So there
23:38I'm going to give you those slides. So
23:40and in the I don't want to spend a lot
23:42of time talking about there is a whole
23:44taxonomy
23:46of you know I put some taxonomy here.
23:48For example there is even no commitment
23:50to the current mechanism. There is even
23:52now some things of committing to smart
23:55contracts. We are in this spot. a new a
23:58new one period contract is offered in
24:00each period. No interal linkages and
24:05um if you don't mind I'll just go back
24:07and leave the taxonomy for for you to
24:10kind of look at later.
24:14So what we're going to do in the next
24:15steps we're going to overview the result
24:18the relation principle for for this
24:20particular class of limited commitment
24:23spot commitment. We didn't write a
24:26revelation principle that says what
24:27happens when you can come into 10
24:28periods 100 periods between negotiation
24:32or where we can think about mechanisms
24:34that have memory. We did not do that.
24:37And I think
24:40one of one of prestigious colleagues of
24:43yours like Takuro Yamashita has a paper
24:46working with
24:48um Nicolola Lomis on they thinking about
24:52a different form of commitment where
24:53there's a long-term mediator and can
24:56actually h store information and make
24:59recommendations to the designer. Um, so
25:02this is not what we're going to be
25:03thinking about here. Here we're going to
25:05be thinking about a current mechanism
25:07who who doesn't have memory and takes
25:11reports today determines interactions
25:14today and nothing for tomorrow.
25:18Um, and within this framework, we're
25:20going to introduce a new class, a class
25:23that is without loss of generality,
25:24which we're going to call direct black
25:26mechanisms. And then we're going to show
25:28how to apply these results.
25:33So limited commitment if you like is a
25:35form of contractual incompleteness
25:38because we are contractually specifying
25:40what is going to happen today only but
25:43nothing tomorrow. So there are other
25:45forms of contractual incompleteness
25:48because for example we might not have
25:49enough language to describe all
25:51contingencies. This is not the same
25:54here. We're going to have enough
25:55language to talk about contractual
25:58outcomes today, but we're not going to
26:00specify anything about tomorrow.
26:05So, this type of limited commitment that
26:07we're going to be looking at is relevant
26:09for the classic settings of regulation,
26:11forc auctions, and so forth where we're
26:14thinking about one period commitment,
26:16political economy settings, because this
26:18one period could be your electoral
26:21cycle. It could be four years. It
26:23doesn't have to be one you know one
26:24year. So we are thinking about our
26:27period as four years in in situations
26:30where we have four years elections.
26:33And we could also think about
26:36this type of limited commitment as a
26:38setting where we're thinking about an
26:40initial interaction that leaks
26:42information. So there areformational
26:44externalities we and this is relevant to
26:47vertical contracting or after markets as
26:50we're going to explore in the second
26:52lecture or and also in privacy design
26:56situations where the designer the
26:58upstream designer who is who is a firm
27:01let's say Amazon is designing its
27:04platform or its pricing policy is also
27:07um h functioning as a data broker or
27:11selling the data set to another firm.
27:13because this kind of information
27:15friction of what is the abstract from
27:18learning from me and how this is going
27:20to exploit me is at the heart of this
27:23commitment problem here and also in
27:25those types of settings.
27:28Okay, so this is sort of a big slide
27:30with what we're going to be doing but we
27:32have a lot of time so I hope maybe it's
27:34too many just too many tles or rather
27:37than anything else. So let's think about
27:40our game. So we're going to what is
27:44actually um different from my work and
27:48Laura with the works on of best services
27:51where we're fully describing a strategic
27:53game an excessive form game and then
27:56we're going to be thinking about the
27:59the equilibrium outcomes of that game.
28:02So this game has two players and
28:05principal and an agent. So notice an
28:07important thing we have not done is we
28:10don't have we have not tackled the case
28:12of multiple agents and that's what I was
28:14asked in the world congress you know we
28:16worked on it a bit throughout this year
28:18but this is not this the having multiple
28:21agents has other issues that come up
28:26that you know there are so many new
28:29issues that come up with these problems
28:30that one can write many papers even to
28:34um investigate some of The early
28:36concerns that arise when you have
28:38multiple agents in particular the
28:40principal will might have might be
28:43learning things about players that are
28:45that are not common knowledge. So you
28:47have an informed principal on top of
28:49your meeting committee because I can be
28:51learning things. I'm a principal. I see
28:53messages of different of all of you but
28:56you don't see the messages I'm seeing.
28:58So I'm start to get more information
29:00than the agents themselves.
29:03Okay. So let's close this parenthesis
29:05and let's go back to this. In this game
29:08now there is one principle and one
29:11agent.
29:12>> Okay they are going to interact over
29:16some number of periods that could be
29:17infinite. The principal holds the
29:20bargaining power. So they will the
29:21Prisma will be designing mechanisms and
29:24the agent has private information
29:26persistent for the for most of the talk
29:29which we're going to index by data
29:31distributed according to some prior each
29:34period and allocation a is determined
29:37this allocation can be equality in a
29:40transfer can be a match for those of you
29:43who like matching can be uh which I h
29:47which candidate won if you're thinking
29:48about an election or which item in the
29:51agenda prevailed in current in current
29:53discussions. There is also an allocation
29:56a star that represents the agent's
29:58outside option.
30:00We are going to allow the set of visible
30:03allocations in period t to depend on
30:06past allocations in per in the previous
30:09period.
30:09>> That's a clarifying question about a
30:12star. Once a get a star, he exits or
30:16>> no he come back of course. Yes. Okay.
30:19>> So the getting an outside option does
30:22not terminate the relationship.
30:24>> Okay.
30:25>> Um
30:27so in particular if you reject the
30:30mechanism and result to a star then the
30:33principal can come back and offer
30:34another mechanism. So in a sense this is
30:37also a first step of thinking about
30:39negotiating over mechanisms but the only
30:43one part is making the proposal. So the
30:45same thing we talk about bargaining
30:47where you have one proposer making
30:49proposing the split. So here you have
30:51one proposer proposing the the the peace
30:55treaty mechanism for example. So I I am
30:58I have the power to you know propose the
31:01uh the trade rules and the other country
31:05says yes or no but you know although
31:08there's the bargaining power is on one
31:10party the other party has some power
31:12because it can say no no no no this is
31:15part of this process
31:18um
31:19so h let me come back to what I wanted
31:22to say here so the set there is a
31:25sequence of allocations
31:27Um and this sequence of allocations can
31:30determine what is the set of feasible
31:35allocations we can be drawing today. For
31:38example, if the past allocations are
31:40investment decisions or tasks you did
31:44and there's some learning by doing. What
31:46are the feasible things we can get today
31:48can depend on all these bad choices or
31:50if we're thinking about addictive
31:52behavior. So if we're thinking about the
31:54seller selling an addictive good, the
31:56level of consumption in the past can
31:58affect what you know they can be
32:01offering you today. Of course, this is a
32:03terrible example, you know, but you can
32:05I'm just it could be learning by doing,
32:08but the the framework is very general
32:11>> and also the framework is so general
32:13that are the payoffs of the designer and
32:15the agent depend on the type of the
32:18design of the agent which is persistent
32:21and we have an a paper and we generalize
32:24this uh result to settings where the
32:27designer where sorry where the agents
32:29type evolves over time. the evolving
32:32type is actually simpler in a way
32:34because there's when I'm learning about
32:37theta one if the persistence is not very
32:40high the agent doesn't worry about me
32:42learning theta 1 because theta 2 is
32:44going to be drawn and it's not going to
32:47be exactly the same but if if I if I
32:50learn that your type is a 100 and I know
32:52it forever that's great because your
32:55type is stuck so then you lose all your
32:57information r so that's why in the
33:00benchmark Mark paper, we have full
33:02persistence. Does it make sense? All
33:05right, good. As I said, I'm I'm dumping
33:09a lot of information on you, but these
33:11lectures are for you and understanding
33:13this, I think, is important. So, stop
33:16me. So the payoffs depend at the end of
33:20the game we're going to have a sequence
33:21of allocations
33:24um realized obviously during the game or
33:28for the exander perspective all of those
33:29things are going to be stoastic but at
33:31the end at the end of the game there's
33:34going to be some a of t from the exander
33:37perspective there's going to be
33:38distribution over those things but at
33:41the once we have a realized vector of
33:43allocations a t this is the designer pay
33:45of W and this is the agent's payoff.
33:48Notice that these expressions are not
33:51times separable necessarily and we don't
33:54necessarily allow for discounting. So
33:57time separability and discounting are
33:59special cases. We also don't impose
34:03transfers.
34:04A can be a a pair of qualities and
34:09transfers but it doesn't have to. A
34:12could be a match. A could be a
34:14candidate. So the setting is not
34:17restricted to settings with transferable
34:19utility or to settings where we have
34:21time severable payoffs or discounting
34:24any payoff structure works. Any um set
34:28of allocation works in the math got
34:32really messy in the paper because this
34:34space we take it to be the set of
34:36current allocations is is taken to be a
34:38polish space. So you really you can take
34:42this to be finite a continum whatever
34:44you want as long as it's a polish space.
34:47Are you with me? All right. Okay. So now
34:51what is the set of mechanisms that um
34:55determine a period by period
34:57interactions?
34:58So a mechanis so we're going to write
35:00down an extensive form and we're going
35:02to assume that designers choosing a
35:05mechanism that has this form. It
35:08consists of a set of input messages M
35:12a set of output messages S and a mapping
35:15that will map an an input message to a
35:18distribution over the distributions over
35:22output messages and allocations.
35:25So M is a set of input messages and
35:28we're going to assume that its cardality
35:31it's greater or equal to the cardality
35:33of the type space. So if the type space
35:35a continum m is has you know it's a
35:37continum if a type space has a million
35:40types we we are going to allow the the m
35:44to have a lot at least a million
35:46messages and then s is a set of output
35:48messages
35:51>> so
35:52>> just to make sure so the output message
35:54is the public information
35:56>> it's going to I'm going to write down
35:57the the timing and what is observable
36:00>> okay
36:01>> yes but that but because we in a lecture
36:04and and we're going yes it's public
36:07>> and it's important that it's public
36:10>> okay thank you
36:11>> because you could also think about the
36:12output message being a message for the
36:15agent and a message for the principal
36:17>> that will create the issues that I was
36:20alluding to earlier
36:21>> that we don't know what was the output
36:23right
36:24>> but now think about so an intuitive way
36:27to think about actually let me proceed
36:30the way I planned it now
36:32>> so this is a this is a short-term
36:34mechanism. So in in contrast to some
36:38other works that talk about smart
36:40contracts and so forth that you might
36:41have seen in the literature that were
36:43published subsequently, this mechanism
36:46does not depend on past input messages
36:50since the principle can offer only spot
36:53mechanisms. So the the the principle
36:57tomorrow
36:58knows only the past output messages and
37:01allocations but not the messages that
37:03were inputed in the in the previous
37:05period and the mechanism cannot be
37:08carrying on past input messages. Every
37:11period has to depend on current input
37:14messages. Make sense?
37:18>> Yes. Why are you achieving this descent?
37:21Practically it's possible for me to
37:24depend my second period mechanism on the
37:27first period.
37:29>> Yes,
37:29>> that's what you exclude it. See, is that
37:32what you said?
37:34>> Yes, we we are excluding the current
37:36mechanism to be dependent on past input
37:39messages because if we start introd Who
37:42saw the first period input message? is
37:46the mediator or
37:47>> there is no mediator here. So the this
37:49is a relationship. So here we're trying
37:52to we you could of course think about a
37:56more rich contractual environment where
37:59you have a a mediator collecting
38:02messages from the agent and then maybe
38:06suggesting mechanisms to the designer
38:08today. But this is a different time.
38:11This is a different set. This is a
38:12different natural expensive farm game
38:15that's not the one we studied.
38:18>> So in the first period there's an input
38:20message right? Yes.
38:22>> So who saw this uh input and implement
38:25first period mechanism? Is it mediator
38:28or
38:29>> so the so the principal chooses a
38:32computer.
38:33>> Yeah.
38:33>> A you put your type in a computer and
38:38the computer speeds an output.
38:39>> I see. So imagine the computer as being
38:41a garbling device.
38:43>> Okay.
38:43>> So I'm putting my message in. I am
38:45seeing the the design the I can choose a
38:49computer that actually tells me what is
38:51the end
38:52>> and then I will select.
38:55>> Okay.
38:56>> Yeah. And yeah.
39:00>> Yes. But of course there are many other
39:03ways to think about mechanisms and
39:06imperfect commitment. This is the thing
39:08we thought is the most natural given you
39:13know um the the work on spot commitment
39:16of laon zero. Okay. So this is exactly
39:19the timing that we're thinking.
39:22Okay. So now because this is what the
39:25designer is choosing every so this is a
39:26function this is a complicated space and
39:28to make the designer's problem
39:31um well defined we are going to assume
39:34that there is some exogenous collection
39:36of messages so you know now you're
39:39writing a game usually when we write a
39:41game we write down this is the action
39:42set for a player here the action set of
39:45a player consists of these functions are
39:47you with me to make this the selection
39:49the choice sort of
39:52um well defined we are going to assume
39:54that there's some arbitrary collection
39:56of messages the gazillion you know
39:59collections this has many many many
40:01elements but you can choose a me an
40:04input messages and an output message
40:06space and any mapping five measurable
40:10from m to distributions over s um sj
40:14comma the allocations are you with me
40:18the collection doesn't mean anything it
40:20can be an arbitrary collection we're
40:21just making this assumption. This is not
40:23really an assumption. We're just writing
40:25this this way to make the choice well
40:27defined.
40:29>> Yes.
40:30>> So you're going to focus on this type of
40:33mechanism and give us the reveal
40:35revelation principle. Yes. Okay. So
40:36there could be other mechanisms that
40:38fall outside of this class where the
40:41revelation principle may or may not this
40:43so that I understand where you're going.
40:46>> Yes. So this is actually um you're
40:50correct but not let me change your words
40:52a bit. So the revelation principle
40:56we prove says if your if you are
41:01studying this any type of dynamic
41:03relationship of the form I just wrote
41:05down
41:06which this with these mechanisms our
41:09mechanisms replicate all outcomes.
41:14If we have a different extensive form in
41:17particular one where the designer is
41:20offering mechanisms that can depend on
41:22past messages we could have outcomes
41:26that we don't replicate with our
41:29mechanisms.
41:30>> So there could be different outcomes. We
41:32don't know if it's going to be and in in
41:36some games we know for example from from
41:38the work of um Gapuru that in like buyer
41:43seller or the work of Georgiadis Bowski
41:46and and Zenesh that if you allow for
41:49smart contracts
41:51like contracts that have some memory and
41:53there's a mediator uh operating those
41:56contracts the mediator introduces some
41:58interular commitment because the
42:01mediator can sort from store information
42:03for you and keep you from accessing it
42:06if you're an EVA.
42:08So when you have those type of um h
42:12richer contractual spaces, you can do
42:16more usually although it's not 100%
42:19clear and actually those papers they've
42:22been trying to prove revelation
42:23principles along those lines. we have
42:25but to the best of our knowledge we
42:27don't know of any analogous results at
42:31the moment that doesn't mean they are
42:32not possible it means that they have not
42:35been formulated
42:39but you know so if you buy a notion of
42:43limited commitment which corresponds to
42:45the notions of laonti this class of
42:48mechanisms that I'm going to give you
42:51are replicating all outcomes
42:54of that particular game.
42:56>> So just in case so so I'm a bit confused
42:59about what you mean by the short-term
43:00mechanisms. So yes, so so you say like
43:03you know the so in period 2 I have to
43:06choose a mechanism
43:07>> again
43:07>> right you know then so the choice of the
43:11mechanism I'm giving to your agent
43:14>> yes
43:14>> does not depend on your
43:17first period input message does it mean
43:20>> yes
43:21>> okay I see so that means like so so you
43:24said like but still what happens in the
43:27first period actually gives me some
43:28extra information about so so that does
43:30that come from the like an out messages.
43:33>> Yes. And the allocations
43:35>> and ah okay. So the allocation is also
43:37information.
43:38>> Yes. But let me look let's discuss them
43:41the timing.
43:43>> So in first of all the agent has learned
43:45their type. I did not put that in the
43:47timing. Okay. The agent knows data and
43:50then we are in BT. Now the principal and
43:53um the agent observed the realization of
43:55a public randomization device. And I
43:58will explain why we need this logger.
44:00Okay.
44:01Um the principal offers the agent a
44:04mechanism of the what I just told you.
44:08All right. Then uh observing the
44:10mechanism, the the agent decides whether
44:13to accept the mechanism or reject.
44:16If the agent accepts the mechanism, the
44:18agent privately reports a message M. And
44:21then S and A are drawn from five. And
44:24this is publicly observed. All right,
44:26these are the questions you were already
44:27asking. If the agent rejects
44:31and then we move to per t plus one. If
44:33the agent rejects then the allocation is
44:36is determined and we move to t plus one
44:39as you asked. So now so this is the
44:42timing.
44:43>> So maybe this is obvious but when you
44:45say the agent privately report it does
44:47it mean that the mechanism designer
44:49doesn't observe n
44:51>> exactly.
44:52>> So can you actually specify the history
44:53of the the information for the
44:55principle? Yes. Okay. Good. Good.
44:59>> Yes. Yes.
44:59>> No, it's good.
45:00>> So, yes. So, bear with me one second.
45:03Yes, you are right on this. You are
45:05really very pay attentive and I'm really
45:08grateful. So, the above defines a
45:10mechanism selection game
45:13G where we can index it by the actually
45:16I should have put the collection of
45:18messages here. I was changing a little
45:20bit the way we had presented this in the
45:22past.
45:24So this h this collection here should be
45:26indexed by the collection of messages.
45:28All right. So the public histories are
45:31as follows. This is what everyone sees.
45:33We see the lottery at the beginning.
45:37Then there is a choice of mechanism and
45:38one. No these are not the messages. This
45:40is the mechanism. It's be it's supposed
45:42to be bold. It doesn't look like bold
45:44but it's different from the okay this is
45:47the mechanism. Then pi1 is the agent's
45:50participation decision
45:53and then we see x the output message and
45:56a1 the participation. So if you don't
45:59participate this output message is the
46:02empty set and this is a star but not to
46:04make the notation a mess we just say we
46:08see whether you participate or not. So
46:10that's the public history. All right.
46:13The private history of the agent indexed
46:16by HDA
46:18is the same as the public history. But
46:20in every period, the agent knows what
46:24they report in the mechanism.
46:27Which means that the agent here
46:30accumulates private information because
46:33in PR 100 I know theta and I know the 99
46:37messages I said in the past and this is
46:40not known by the design.
46:43So now now
46:45the designer's belief at history t
46:49given some public some public ht
46:54are beliefs about data and about the
46:58private histories of the agent which are
47:00consistent with this public history.
47:04And notice that the public history
47:06itself contains a lot of information
47:08because the theta which is inputed which
47:12which the theta determines
47:16the M's that go into the mechanism. The
47:19M's that go into the mechanism determine
47:21what S's and A's and C.
47:24So this contains information but not
47:27directly the M's at this point.
47:32Is this better now?
47:33>> Yeah. Okay. No, that's good.
47:34>> Okay. Yes. I could not start with this
47:36slide though. I had to.
47:38>> Yeah. Right. Right. Yeah.
47:39>> So now what are the strategies of these
47:41players? So the principal strategies and
47:44at every history describes the possibly
47:47random choice of mechanisms. So the
47:49principle could be mixing over
47:50mechanisms.
47:53Um and the agent strategy at when the
47:56type is played at this history and given
47:59this mechanism proposed
48:01a period t it specifies the probability
48:04of of accepting the the
48:08mechanism and the reporting strategy
48:12which specifies the probability
48:14distributions over the messages
48:17of the mechanism proposed.
48:21All right.
48:23So,
48:24so now we're going to be thinking about
48:28um so we are think so these are the
48:30strategies. We have our histories. This
48:33is a game a dynamic game and we're going
48:37to be thinking h we're going to be
48:39focusing on assessments of this game
48:41that are perfect bas and equilibrium and
48:43I'm going to write down in formally the
48:45definition I the definition fully is in
48:48the paper but the princip requires that
48:52the principal and the agent strategy are
48:54sequentially rational and that beliefs
48:57are determined by a base rule uh
49:00whenever possible. So what is an
49:02equilibrium outcome here? The prior
49:08mu1 and the assessment
49:11um sigma p sigma a mu induce a
49:14distribution over the terminal nodes of
49:16the game and then we project
49:21um those distributions
49:23on to the set of types and a of ts.
49:27Okay. So we get the distribution over
49:29terminal nodes. It's a g complicated
49:31game but we project those terminal nodes
49:34onto the thing that we care about which
49:36is types and allocations
49:39and
49:41um
49:43the problem is to replicate outcomes the
49:46mechanisms
49:48we are thinking are encoding not only
49:51the rules that determine the current
49:53allocation but also the information the
49:56designer obtains from the interaction
49:58and that's sort of the role of the
49:59output messages because the output if
50:02the if the mechanism is fully revealing
50:04the M is invertible. So I can learn the
50:08M from the S but I could be garbling the
50:11S. So the M. So for example I could be
50:14putting all M's to one output message
50:17and then one allocation then I learn
50:20nothing. It's like a full pooling
50:22mechanism if you like. But with the
50:24revelation brief with the standard
50:26mechanisms we have where we're thinking
50:29about each message being fully observed
50:32we are sort of ting our hands a little
50:34bit and that was what was initially done
50:36in the literature. So let me start with
50:39a little bit motivating where these do
50:42these mechanisms come from right because
50:44they look like
50:46>> of course
50:48>> what about of belief
50:50>> excuse me
50:51>> what about of belief
50:54>> so all beliefs are going to be an
50:56important part of this problem. So when
50:58we when I say we replicate all outcomes,
51:01a lot of the work we had to do in the
51:03proof is what happens in off path where
51:08we have
51:10our beliefs are not being done by base
51:12rule and we see messages that arise
51:14response that could probably be zero h
51:17in all all of those things have to take
51:19to be taken care of. So the way we the
51:21way the proof works you are going to see
51:24at some point that we guarantee
51:25participation only for types that have
51:27positive probability. And this is some
51:31of the
51:33detailed work that had to be done to to
51:36take to be able to replicate all
51:37outcomes with this very language of
51:40beliefs. So there are going to be but we
51:43have worked with in taking care of
51:45everything off all messages that have
51:47zero probability because that we are we
51:50when we replicate when we take this
51:52fully extensive on game with all these
51:55histories a lot of them are never
51:56arising in equilibrium we have to
51:58specify play everywhere
52:01everywhere and the specification has to
52:04happen with our our class of mechanisms
52:08I I won't have time to tell you about
52:10all the details but I Just want to tell
52:11you, reassure you that we have the help
52:15with all of this.
52:16>> By the way, that's probably that's going
52:18to coming up sometime soon, but you
52:20know, does the governing actually help
52:22the the mechanism designers? Yes.
52:23>> Oh, okay. I see.
52:24>> Yes.
52:25>> That's why you want to you want to
52:27>> we want we I mean it's not that we want
52:29it. I think I mean we want Let me
52:32explain where Let me
52:33>> Yeah.
52:34>> You're a couple of You're always two or
52:36three slides ahead. Okay.
52:37>> Okay.
52:38>> So, that's that's great for you. it it
52:40maybe it's frustrating because you're
52:42quick and you're like where where is she
52:44going to say this? I'm sorry but we're
52:45gonna we're gonna get there.
52:47>> Okay.
52:47>> Uh okay.
52:49So let's think about um the mechanisms
52:54we see in Myers's 1982 paper and related
52:58force by force in in the 80s.
53:02So I guess the this is not 100% a fair
53:06comparison because a model of Marrison
53:08is a model of with multiple agents and
53:11he allows contractable outcomes when
53:14which means that there are the age that
53:15we talk about here but he also allows
53:17for fully non-contractable decisions or
53:20actions that are the actions that the
53:22agent can be choosing. So all of those
53:25things we don't have actions we don't
53:26have many agents here. Are you with me?
53:29So we don't have moral hazard in
53:31particular in this model with Laura
53:34and we only have contractable decisions.
53:37So in Marrison's paper there is a set of
53:40input messages and a set of output
53:42messages and pi assigns to each input
53:46message a joint distribution over
53:48messages and allocations.
53:52>> That's what's the assumption on ML and
53:55are they just the polish spaces?
53:57>> Yes.
53:58>> Okay. and they have some constraints on
54:00the cardality. So we assume that the
54:03type that um if the if the type space is
54:06finite for example and they said is
54:09finite then it has to have more types
54:12more messages than types and also we
54:14assume that the colle that the
54:15collection in in extensive form connect
54:19the output messages also contain the set
54:22of beliefs about the types.
54:24Yes. Okay. So uh so what the the way
54:29this mechanism work the agent as a
54:31function of the type sends a message the
54:33message goes into five and then with the
54:36principle CS sa.
54:40So
54:42the revelation principle under
54:43commitment says without laws of
54:46generality
54:47communication is direct so messages
54:49equal types. Communication is
54:52observable.
54:53M and S have the same cardality. So PH
54:56is invertible and the output messages
55:00are redundant due to full
55:01contractability. So when we have full
55:04contractability,
55:05we don't have basically any actions like
55:07effort. We don't need these output
55:09messages. We we just think as M and S
55:14being the same thing. So that that means
55:15the file is invertible. So I see the S
55:17and the M and communication is truthful.
55:20So when we have full commitment we just
55:22say the mechanism is a mapping from fade
55:25off to distributions over ultimate
55:27allocations.
55:29Yes.
55:32Now um best and stra is the first paper
55:36that I was alluding to that uh wrote the
55:39revelation principle in these settings.
55:41So they look again, they look at a at a
55:43setting with one agent and and finite
55:46has finally many types and they
55:51assume
55:52from the get-go they look at the
55:55mechanisms along the lines we wrote but
55:57they have some even further
55:59restrictions.
55:59>> I think I think it's sort of lost in the
56:02last slide. What was the revelation
56:04principle?
56:05>> This is under commitment. commitment
56:07>> under commitment we will get X=
56:12>> and communication is observable so
56:16>> S is
56:16>> we are going to have an invertible five
56:18so we the S is and because we have full
56:21contractability the S is redundant we
56:24don't have any S we just have
56:27the mapping from theta to distribution
56:31over age that's the revelation and the
56:33commit
56:33>> and the commitment
56:36Now we move to the first paper with
56:38limited commitment best structure here
56:42those authors
56:45they start with a class of mechanisms
56:47that is even more restricting than ours
56:51that's how science progresses though
56:52this was a big step I mean times is step
56:55by step they assume they start by
56:58assuming communication is observable so
57:01M and S have the same cardinal f
57:04invertible
57:05they don't allow for randomization in
57:07the allocation. So that's actually a big
57:10restriction because we need the
57:12randomizations. So each output message
57:16or input message in in their case is
57:18attached to one allocation. So a of n
57:22and then we also have a reduced form way
57:25to capture limited commitment
57:28which means that the principal observes
57:30m because it's observable update the
57:34beliefs and then chooses ym that
57:36maximizes his payoff in the second
57:39period. So there is only like a two
57:42period interaction and the second period
57:44is a reduced form way to capture limited
57:47commitment and in that in that sense
57:50they don't even write down an extensive
57:51form so they don't have to do to deal
57:54with histories opa or any of that and
57:57that is important because in in some of
58:00the games a lot of things can happen
58:03when you allow to to not optimize you
58:08can like in repeated games that you know
58:09here are well expert on repeating games.
58:12A lot of the good outcomes are sustained
58:14by precisely by having suboptimal play
58:18specified in some nodes of the game.
58:22>> All right.
58:24>> So, so
58:24>> oh no back
58:26>> please. Yes.
58:28>> So here are you assuming that there are
58:30two periods?
58:31>> They are assuming that there are two
58:33periods.
58:33>> Okay. So
58:34>> we are assuming
58:35>> no no this two period. Okay. That was my
58:39question.
58:39>> Yes. Yes. Yes. That's what they they
58:41have the main so the the body of the
58:43paper is written by these two period and
58:45then they have an extension to multiple
58:47periods. But that extension in our view
58:49is kind of problematic because it
58:51imposes some marovian structure which is
58:54with loss of generality. Okay. So under
58:57those assumption
59:00they so first let me say some things. So
59:03the the part communication is observable
59:05is no longer without loss of generality
59:08due to limited contractability.
59:11So although in the revelation principal
59:13commitment we got that full
59:15observability is fine.
59:18Let me go back because I
59:21with with the relation princip
59:24commitment
59:26the and the we can take we can assume
59:30that communication is observable because
59:33you commit to it. So you'll see I see
59:36your type but I have already committed
59:38what I'm going to be doing to your type
59:41and and then that's it.
59:44When we have limited commitment,
59:46communication is no longer um observable
59:50communication no longer with loss of
59:52generality due to limited
59:53contractability
59:54because the principle only commits to
59:56data location.
1:00:00So what the these authors show is that
1:00:05under those conditions so communicate
1:00:08observable communication no
1:00:09randomization and reduced form of
1:00:11limited commitment if the principal
1:00:13earns his highest payoff consistent with
1:00:16the agents payoff then without loss of
1:00:18generality communication is direct. So
1:00:22they are thinking about the bar frontier
1:00:24of payoffs,
1:00:26not all payoffs, not all outcomes. And
1:00:28they are showing that we can replicate
1:00:31the frontier with direct communication.
1:00:34However, we're going to lose
1:00:35truthtellingness.
1:00:37That's the revelation principle of best
1:00:39straps. So that's still a simplification
1:00:42because because before researchers they
1:00:46were trying to do two type models and
1:00:48they thought how many messages do we
1:00:50need? Do we need 10? Do we need 20? This
1:00:53result said, "Look, you need two." That
1:00:56is a big result. But if you have three
1:00:59types, you need three. And if you allow
1:01:01for mixing and you have more than two
1:01:03types, good luck because each type in
1:01:07those settings, you can be mixing
1:01:08upwards and downwards actually. So how a
1:01:11player is reporting is not being done by
1:01:13this report, this result. And of course
1:01:16these authors are very smart and they
1:01:18realize oh why did we assume this
1:01:22but you know let me kind of also tell
1:01:25you some background at this point the
1:01:28sort of the intellectual background is
1:01:29sort of the is more the to lose approach
1:01:32of mechanism design where there's a lot
1:01:33of you know taxation principle type of
1:01:36arguments and not really thinking about
1:01:37the communication but they start reading
1:01:40my 82 and they said hm maybe we should
1:01:43start using the mechanisms in 1982 where
1:01:46the me the information
1:01:49you know there is some communication in
1:01:51the mechanism. So then in 2007 they
1:01:54published another paper. Sorry it
1:01:56doesn't come across very well. Something
1:01:58happened but this is best and stra 2007
1:02:01barely visible but hopefully given that
1:02:04I'm saying it somewhat.
1:02:08Um so they this is sort of the second
1:02:12version. So that now they're adding
1:02:14noise to the communication. That's a big
1:02:16innovation. So now they remove this
1:02:19assumption of full observability and
1:02:22they maintain the assumption of um
1:02:25reduced form commitment. But now so now
1:02:27the principal just observes the output
1:02:29message updates and chooses a y of s
1:02:33that maximizes payoff and they maintain
1:02:36the assumption of no randomization in
1:02:38the allocation. So each method is
1:02:40attached to one allocation and they show
1:02:43under those assumptions that communi
1:02:45without also generality communication is
1:02:48going to be direct and truthful. So
1:02:51that's a big now a big second step of
1:02:54progress in science. We started with two
1:02:57types. No idea how many messages we need
1:03:00to have. First big result 2001. Well you
1:03:04don't need more messages than types if
1:03:06you have finally many types.
1:03:09second paper now but this paper had the
1:03:13problem the 2001 paper had the problem
1:03:15of well we know the messages we don't
1:03:18know the behavior so I want to kind of
1:03:20pause here because a lot of students in
1:03:22the cl in the lecture the revelation
1:03:25principle is not just about canonicity
1:03:27of the class of mechanisms it's not tell
1:03:29you play second prize auction it tells
1:03:32you how to play the auction
1:03:34it tells you canonical language rules
1:03:38and play two things
1:03:41and that's why it sort of puts all it
1:03:45you nails the problem because it tells
1:03:46you look you're going to look at second
1:03:48price auction I'm just making it like a
1:03:50specific class of games let's call it
1:03:53direct revelation games and it tells you
1:03:56exactly what behavior has to be can be
1:04:00expected here the first paper did not
1:04:04pin down the language but not the
1:04:05behavior the second paper did both. So
1:04:08why did we write a third paper? Well, we
1:04:13don't know still first of all there are
1:04:15two things here. There is a reduced form
1:04:17of capturing limited commitment two
1:04:19periods and we don't know what are the
1:04:21output messages but the output messages
1:04:24are what are what is needed for the
1:04:26current allocation and for the future
1:04:29allocation and this paper did not say
1:04:31what those are.
1:04:34Are you with me?
1:04:36So now
1:04:39this is where we come in. So our result
1:04:43says we I for the extensive form I wrote
1:04:46down. So now we are in a fully specified
1:04:49game. So we are going back to the
1:04:51original game with the histories we
1:04:53wrote. Are you all with me? And
1:04:55everything. And now
1:04:59we show that communication is going to
1:05:01be direct. And this argument, the first
1:05:06step of the argument is like the one in
1:05:08Bess 2007. But our argument is a bit
1:05:11harder because the agent accumulated
1:05:14information and communication is direct
1:05:16here. And notice
1:05:19my type it's I'm always communicating
1:05:22theta. I'm not going to communicate in
1:05:24period 20 m1 m2 m3 m18 n theta. I'm only
1:05:30communicating the pay of relevant type
1:05:33not the things I was saying before.
1:05:36So this is so the the canonicity of this
1:05:39language given the extensive form we
1:05:42wrote is not so obvious if you're
1:05:45thinking about you know down the line
1:05:48and then the output messages in every
1:05:50period are beliefs about types.
1:05:55So now and equilibrium is truthful.
1:06:00So equilibrium output messages coincide
1:06:02with the principal equilibrium belief.
1:06:05We have we have um
1:06:11the meaning of the me there is literally
1:06:15the messages that are communicated have
1:06:17literal meaning. So they have the truth
1:06:19and the output messages are literal. And
1:06:22the fact that we're insisting on having
1:06:25output messages being literal is what is
1:06:28very delicate to deal with or off
1:06:31because we cannot assign whatever
1:06:34because the beliefs are a big space. It
1:06:36has a lot of cardality. You know you can
1:06:39embed any space you want in this space
1:06:42by losing the literary meaning of it.
1:06:45But if you want the beliefs to have a a
1:06:48meaning that also ties your hand a bit.
1:06:52So um
1:06:55okay
1:06:56can can you sh so
1:06:59>> so from from these things uh in on the
1:07:03given path
1:07:04>> yes
1:07:04>> does this actually mean that the uh the
1:07:08uh the principal uh learns the type uh
1:07:12at the end of the period one? No, no,
1:07:15no. Because he doesn't see the
1:07:16>> T.
1:07:19Maybe I'm confused. So the communication
1:07:21being truthful
1:07:22>> but the let me pause here. The
1:07:26communication is not fully observable. M
1:07:28is not observable. What is observable is
1:07:31the output message.
1:07:33>> Right? So sorry.
1:07:35>> So I'm not going to learn theta.
1:07:38I'm going to learn the input. So this is
1:07:41M. I'm not going to see M.
1:07:44I am going to see the beliefs about data
1:07:47and the allocation.
1:07:48>> Yeah.
1:07:50>> And
1:07:51>> oh sorry sorry I thought I misspelled
1:07:53the slide now and I I think I
1:07:55understand.
1:07:55>> So and so this is what now what I've
1:07:58told you so far is that the mechanism
1:08:00can take input messages data and speed
1:08:03out beliefs and allocations. But we also
1:08:06prove another thing. We can decompose
1:08:09it. So this is a transition probability.
1:08:11It's theta. It's a it's a kernel from
1:08:14theta to beliefs and allocations. We can
1:08:17decouple this transition probability
1:08:20into two transitions probabilities. One
1:08:23from theta to beliefs that's and then
1:08:26one from beliefs to allocations. So
1:08:29basically
1:08:30let me pause here. Suppose you have a
1:08:32mechanism that is full pooling. Okay.
1:08:36the the theta is going to be mapped to
1:08:38the prior and then I'm going to map the
1:08:41prior the single thing to a quantity
1:08:44inequality or a match or something.
1:08:47If the if the community if we have a
1:08:49full if we had um a a fully separate if
1:08:54we if we have each data mapped to the
1:08:56direct measure that's like a direct
1:08:59mechanism because I'm mapping suppose I
1:09:01have two types which I can separate even
1:09:03under the limit commit and then I have
1:09:05one type mapped to the dra on it on it
1:09:07each side map on the dra on itself then
1:09:10the alpha the allocation is going to map
1:09:13the dra on allocations which is like the
1:09:16direct mechanism
1:09:17So the direct mechanisms the direct
1:09:20revelation mechanisms are a special case
1:09:23of this class of mechanisms where that
1:09:25each data is mapped to the direct
1:09:28measure on itself and that's it. Do you
1:09:32see the do you see that? Yeah. And do
1:09:34you think that the concepts of the like
1:09:36the false property? Because again the
1:09:38force property means like you do not
1:09:41learn from the A, right? You know if you
1:09:44have the
1:09:44>> No, you learn from the A. You learn you
1:09:47learn but the learning from the A is
1:09:49already encoded.
1:09:50>> Yeah. It must be already encoded. So
1:09:51that means like effectively you do not
1:09:53learn from.
1:09:54>> Effectively you don't learn from.
1:09:55>> Right.
1:09:57So basically that's what we do is
1:09:59basically
1:10:00>> the way that the proof works is you take
1:10:02all of those histories and then you
1:10:04orthogonalize everything. You you
1:10:06basically and that's also some kind of
1:10:08it talks a little bit how we we why we
1:10:11need the the
1:10:12>> the reason we need the public
1:10:14randomization device is to play with
1:10:17kind of mixing
1:10:18>> okay
1:10:19>> and to analyze the information. So we
1:10:21take the histories and then everything
1:10:24that is nonve relevant can been dumped
1:10:26into this omega everything that is more
1:10:30complicated than what I thought. Okay
1:10:33lecture now so I'm not saying all of
1:10:35those things in the lecture
1:10:36>> because this proof is you know
1:10:39>> yeah it's very delicate
1:10:42>> it's a very it took us like years to do
1:10:45this and maybe other smarter people can
1:10:47do it shorter but it took a long time to
1:10:50do this.
1:10:51Um
1:10:52>> question.
1:10:52>> Yes.
1:10:53>> Um are there any restrictions on the
1:10:57dynamics of the beliefs? If it's brief
1:11:00on theta, is it going to be martingale
1:11:02or something?
1:11:03>> No. Well, well, of course it's b yeah
1:11:05this b base plausibility.
1:11:07>> Yeah.
1:11:07>> Yeah. Yes. Of course. There are
1:11:09everything all the base all the
1:11:11restrictions are come from base rule. So
1:11:13it didn't happen that in period one I
1:11:16believe that
1:11:18theta prime is true with probability one
1:11:21and suddenly in second period I started
1:11:24start to believe that the true theta is
1:11:27theta double prime with probability one
1:11:29>> that will not be possible you brow
1:11:32>> so there should be okay so b rule okay
1:11:35>> yes
1:11:36>> dynamics oh no there
1:11:38>> won't be equilibrium pass
1:11:39>> yes so so I'm going to talk about all of
1:11:41these things in a minute this. So um
1:11:45these mechanisms because they are
1:11:48eventually decomposed into mappings and
1:11:50this is like an experiment. This is if
1:11:53you think about kamisa what we think we
1:11:55think about an experiment mapping the
1:11:57state space of distribution over
1:11:59posteriors. Now our mechanism has an
1:12:02experiment and has an allocation or that
1:12:04maps the belief to our locations and
1:12:08that's why we call them direct blackwell
1:12:09mechanisms.
1:12:14Yeah. The if there were no uh no out
1:12:18messages then the belief is a private
1:12:22information of the principal. Is that
1:12:23right?
1:12:25>> If there
1:12:26>> if there's no output messages in the uh
1:12:29second point
1:12:31>> yes
1:12:31>> then the belief is the private
1:12:34information of the uh principal. Is that
1:12:37right? But here we are assuming that yes
1:12:40that's true I see it is possible it is
1:12:43yes. So that's an that's a version of
1:12:46the problem we did not analyze and we
1:12:49also you know did that
1:12:52because we wanted to yes so you could
1:12:55have a mechanism where the there is an
1:12:57output message observed by the principal
1:12:59and an output message observed by the
1:13:01agent potential.
1:13:02>> So then the agent does not know what are
1:13:05the leaks of the principal by himself.
1:13:07That's a very natural thing to to think
1:13:09about. We have not thought about this.
1:13:12The reason we haven't thought about this
1:13:13is not that we didn't think it's
1:13:15interesting is that we didn't want to
1:13:17start writing a problem where we had to
1:13:19deal with unique commitment and we
1:13:20private informed principle. So in bond
1:13:23if you're interested in form principle
1:13:25last week I gave four lectures in bond
1:13:28on the topic. They're also on video and
1:13:31you know informed principle is not an
1:13:34easy problem either. So we wanted when
1:13:37we started with Laura we said we we're
1:13:39going to do one thing at a time. So this
1:13:41is the one thing we did and that's why
1:13:44the public me the messages are public
1:13:47but it's also very natural like I don't
1:13:50want to defend this we did not do this
1:13:52because we thought this is simple we
1:13:54thought this is a relevant benchmark.
1:13:56>> I see. So in a in in in a situation
1:14:00where you're thinking about an auction.
1:14:02So when I started working on my job
1:14:04market paper commitment, I had a very
1:14:06specific motivation. I was not a very
1:14:10high brow theorist as a student. I was a
1:14:12student who went to seminars and thought
1:14:14saw this game theorist presenting about
1:14:17designing at the FCC auctions. Mgram was
1:14:20coming. Peter Crampton. They were really
1:14:22thinking about how the FCC should sell
1:14:24the spectrum and they were thinking
1:14:26about efficiency. So I was thinking why
1:14:30do they really care about efficiency so
1:14:31much? If in a market, you know, you sell
1:14:34something and if someone values it more
1:14:36than the buyer, they will just buy it.
1:14:39There's going to be trade. So I was
1:14:41thinking about resale. So initially I
1:14:43started thinking about resale. But then
1:14:44I thought well there is an a step before
1:14:48resell. Suppose I am the FCC and I start
1:14:50to sell it but nobody buys it. What
1:14:53happens then?
1:14:55So in an auction when you are say
1:14:58government privatizing a company like my
1:15:00country was asked to sell all its assets
1:15:02a few years ago because Greece was
1:15:04almost back. It's not like you know
1:15:06Japan where strong economy. So if you're
1:15:09a seller and you are selling a company
1:15:12and you run the auction,
1:15:14you see what happened. You know there is
1:15:16some common knowledge of what happens in
1:15:18that auction and then you come back and
1:15:20you want to design another one because
1:15:22nobody won the competition. At that
1:15:24point we assume that if there was only
1:15:27one one firm trying to acquire the the
1:15:31the asset, it's very natural that the
1:15:33firm and the seller know what happened
1:15:36and it's common knowledge. So that's
1:15:38what we're trying to capture here. Sorry
1:15:40for that parenthesis. All right. So
1:15:42let's go back to theory now.
1:15:46Okay. So let me recapitulate what things
1:15:48have been already saying. So as I said a
1:15:50direct back mechanism consists of a
1:15:52disclosure policy mapping types to
1:15:55distributions over posteriors and
1:15:57allocation rule mapping posteriors to
1:16:00distributions over allocations.
1:16:02And as I was already as I already said
1:16:05direct revelation mechanisms are a
1:16:07subset of direct blackwell mechanisms.
1:16:10Why? Because a direct revelation
1:16:13mechanism simply maps each type to the
1:16:15direct measure. So that beta beta
1:16:18mapping is basically the identity
1:16:22theta goes to delta theta. And we also
1:16:26get canonical behavior in our with our
1:16:29result. The agent's behavior is reduced
1:16:32to participation, true telling and base
1:16:34plausibility constraints and our result
1:16:38replicates all equilibrium outcomes of
1:16:41any mechanism selection game among the
1:16:43class we wrote down not different
1:16:46classes and this is crucial for infinite
1:16:48horizon settings. So now I want to kind
1:16:52of contrast a little bit mechanism
1:16:55design
1:16:56mechanism selection games versus
1:16:59mechanism design. So in standard
1:17:01mechanism design the pre the principal
1:17:05is a designer who commits to the rules
1:17:07of the game and commits to the
1:17:08equilibrium selection. So if you read
1:17:11Meerson's 82 paper and Meerson's book,
1:17:14he says mechanism design is a sort of
1:17:16blend cooperative appro non-ooperative
1:17:20and cooperative approach is cooperative
1:17:22because we are sort of selecting this
1:17:25the um the equilibrium as well. Um I
1:17:32mean that's what he writes. I know that
1:17:33a lot some of some of you work on
1:17:36cooperative solution concepts and
1:17:39hopefully the common is not very
1:17:40confused. But any case, in in Sar
1:17:43mechanism design, we commit to the rules
1:17:45of the game and and the equilibrium. In
1:17:47a mechanism selection game, the
1:17:50principle is a strategic player. So
1:17:52there are three consequences. First, a
1:17:54smaller choices can help.
1:17:57So restricting the menu of admissible
1:18:00mechanism ties the principal's hand and
1:18:03can improve outcomes. So I don't know if
1:18:05you know about Odyssey which is a a
1:18:09Greek ancient Ulysus tied himself on the
1:18:12boat so he doesn't get attracted to the
1:18:15singing of the sirens. So this when you
1:18:18have union commitment is is good to tie
1:18:20your hands. Uh
1:18:24optimizing in every beard can hurt. So
1:18:26the principal optimal PV may require
1:18:29continuation play that does not
1:18:31myopically maximize her payoff and
1:18:34that's why characterizing all
1:18:36equilibrium outcome matters and not just
1:18:38the efficient ones and that's the big
1:18:40contrast with the best trials papers as
1:18:42well. So not it's not just that we get
1:18:45the output messages we also allow to we
1:18:48replicate all equilibrium outcomes and
1:18:52sometimes I someone asked about that
1:18:54already less information is better noise
1:18:57and pulling in the current mechanism
1:18:59reduces the principal's future
1:19:01temptation to reoptimize
1:19:04so if I know you're you're going to
1:19:06basically exploit me I am not going to
1:19:08behave in a revealing way today but if I
1:19:12If there is a way to prevent myself from
1:19:14being revealed too much then that gives
1:19:17me more makes me makes it makes my
1:19:20behavior loosen up a bit. So not noise
1:19:24or pulling in the current mechanism
1:19:25reduces the principal's future
1:19:27temptation to reoptimize and alleviate
1:19:30the ratchet effect. So contrast with
1:19:32commitment with in standard mechanism
1:19:35design larger design set is weekly
1:19:38better more information is weekly better
1:19:41and design is equivalent to optimization
1:19:44that's not necessarily the case in a
1:19:46game however after we have a relation
1:19:49principle we can sort of turn the the so
1:19:54the the search of the optimal of the
1:19:56optimal VB into a constraint
1:19:58optimization problem ultimately so we
1:20:01you get. Yes, please.
1:20:03>> Oh, so
1:20:04>> no no please.
1:20:05>> Um Oh, so at this point uh what's the
1:20:08intention for that result? So is it
1:20:10possible to get
1:20:12>> I'm going to show you an a very
1:20:14contrived example. So this is an
1:20:16example. So there was so there's some
1:20:18history behind this example. So that I
1:20:20had also some discussions for the volume
1:20:22that you hear as for the paper and we
1:20:24had a lot of back and forth about
1:20:27whether there is whether at least in
1:20:30this optimization in the second period
1:20:32is without loss blah blah blah and what
1:20:34it's a so here I wrote an I I wrote an
1:20:37example that is a bit contrived because
1:20:39I wanted to make
1:20:42a point of how naively thinking about
1:20:45the principle optimizing in the second
1:20:47period how it can be bad and how
1:20:50restricting choices can be a good thing.
1:20:52Okay, so I'm going to illustrate the
1:20:54three points in the previous slide with
1:20:57this trivial example. So the example has
1:21:01two types. So has two periods and two
1:21:03types one and three. So the period one h
1:21:07outcome is a probability is the
1:21:10probability or equality of trade Q and a
1:21:12transfer.
1:21:14And the second period allocation is some
1:21:16number between zero and one. So you can
1:21:19think of the like a some harassment
1:21:22possible harassment. So the agent's
1:21:24payoff is
1:21:27theta times q - x if the harassment in
1:21:3002 is going to be zero otherwise it's
1:21:35uh zero. So I think about I'm going to
1:21:38choose a gym and I'm going to consume
1:21:40cure and pay X today but I don't want to
1:21:43be bothered every period next period
1:21:45about buying personal training blah blah
1:21:47blah blah something extra I just want to
1:21:50join and not be harassed and if I'm
1:21:53expecting harassment I'm not going to
1:21:55join that's sort of like a a now the
1:21:58principal payoff he gets a money in the
1:22:01first period and then he gets a payoff
1:22:03that depends on the level of harassment
1:22:05the expectation about the agent's type
1:22:07and some divided by some number k which
1:22:09is greater or equal than three or
1:22:11something. So now if the agent joins in
1:22:15per has no commitment he's going to
1:22:19choosing the the thing that so under
1:22:23commitment first of all I did not write
1:22:25on the slide under commitment the
1:22:26optimal thing to do is to um select a to
1:22:31equal zero and then max because no
1:22:34matter what you do you're never going to
1:22:35make a lot of money the second period
1:22:37and under uniform prior it's better to
1:22:40ask a price of 1.5 and sell the good
1:22:44only to type three because that's the
1:22:46commitment. You have two types, one or
1:22:48three, they're uniform. So if you sell
1:22:51to both types, you're going to make one.
1:22:54If you sell only to the high type, you
1:22:56can sell with at price three with
1:22:58probability half is 1.5 and you are
1:23:01selling at price three and you're
1:23:03committing not to harass. Suppose now we
1:23:06have naive sequential rationality and
1:23:09the Bristol maximizes in period two. So
1:23:11in period 2 you have forgotten about
1:23:13period 1 because now we're in period 2.
1:23:16Now you're going to choose a2 equals 1
1:23:18here. No matter what is your expectation
1:23:21about type you're going to just choose
1:23:23the a2 that maximizes this term. So
1:23:25there's a unique
1:23:29um naive optimal choose the highest
1:23:33possible level of harassment once we're
1:23:34in period two.
1:23:36>> Question.
1:23:37>> Yes. What's the law of expectation of
1:23:40theta multiplied by a2?
1:23:43>> Expectation is always positive, right?
1:23:46>> Yes. This is
1:23:47>> why do you need this expectction
1:23:50expectation term?
1:23:52>> Oh, because I just said no matter what
1:23:53you learn in the first period,
1:23:55>> it's always positive, right?
1:23:56>> Yes, it's positive. Exactly. Doesn't
1:23:58matter.
1:23:59>> So, just cross out the
1:24:01>> you can you can cross it out.
1:24:03>> This is a simp maybe I could have make
1:24:05it the example nicer. First of all,
1:24:07sorry.
1:24:08>> Okay.
1:24:09>> Or maybe in the P maybe the chapter is
1:24:11nicer. I don't remember what I have now.
1:24:13But yes, it doesn't matter. Your payoff
1:24:15is some a a
1:24:17>> increase
1:24:18>> increase. Yes, you're right. Thank you.
1:24:21So,
1:24:23uh
1:24:25I had a deadline.
1:24:28I'm joking.
1:24:30Yes. So, maybe I should have made that.
1:24:33So in any case, so notice because the
1:24:36agent now once we're in 02, the agent
1:24:39doesn't really care about the A2, he
1:24:42um at this point rejecting any mechanism
1:24:46is also a configuration equilibrium for
1:24:47the agent in per 2. So if the so um
1:24:54once the agent is in he the A2 does not
1:24:57matter for the agent in in this in this
1:25:00specification. rejection is the best
1:25:02response which is equivalent to
1:25:04selecting in my modeling a2 equals zero.
1:25:08So um what when we specify a
1:25:11continuation equilibrium where the agent
1:25:13rejects the second period mechanism no
1:25:15matter what is the proposal um the
1:25:18principal payoff is 1.5 and we get the
1:25:21commitment
1:25:23h optimum back so what are the lessons
1:25:26from this example naive sequential
1:25:29rationality is not the same as QB
1:25:31feasibility because if it it imposes a
1:25:35marco friction which is with loss.
1:25:38Multiplicity is a feature when we have a
1:25:41continu a continuation equilibrium where
1:25:43the buyer where the buyer rejects no
1:25:45matter what.
1:25:48This the principal can select this
1:25:51continuation equilibrium, this rejection
1:25:53because it works as a as a restrain as a
1:25:57um because it essentially works as a
1:26:00restraint and the smaller choice set
1:26:04helps. If the choice set of the
1:26:06harassment policy was a priority
1:26:09restricted to just zero, then we would
1:26:11get the commitment outcome even under
1:26:14this myopic optimization.
1:26:17because we would tie the designer's
1:26:19hands in the second pair not to harass.
1:26:22So that's the things I wanted to
1:26:24communicate with this kind of example.
1:26:27>> Awesome.
1:26:29>> This one the naive sequential
1:26:32rationality. So assuming that the the
1:26:34principal maximizes payoff in second in
1:26:36the second period. So in the best stra
1:26:40approaches
1:26:41>> that's what that's what you have that's
1:26:43what you're going to do, right? And
1:26:44>> that's what I was That's what Spencer
1:26:46Straws were doing.
1:26:47>> And then that means the agent's never
1:26:48going to join the jail.
1:26:49>> Exactly.
1:26:50>> And so you're stuck with 1.5.
1:26:52>> You're stuck with zero because if
1:26:55>> Oh, yes. You're stuck at zero. Exactly.
1:26:56>> Yes. Because if the agent anticipates
1:26:58harassment, they never join.
1:26:59>> Yes. So you're stuck. So the agent and
1:27:01if but if if it if we have if we think
1:27:05about the agent
1:27:07if the if we have if we specify
1:27:09continuation play
1:27:11uh that the the agent rejects no matter
1:27:15what is the mechanism and so the agent
1:27:17then the principal will set a a= zero
1:27:21because
1:27:22>> because that's that's an equilibrium
1:27:24then we can sustain commitment under the
1:27:27specification of continuation
1:27:28equilibrium.
1:27:30And then the PB feasibility. So that's
1:27:33>> the PB feasibility is that it's a
1:27:34continuation equilibrium for the agent
1:27:36to reject anything. So then I'm going to
1:27:39offer a toal zero and rejection as a
1:27:41best response and by meaning by
1:27:44specifying that play tomorrow we are
1:27:47basically achieving commitment as a PB.
1:27:52Do you does it make sense? So I guess
1:27:54the if you have naive sequential
1:27:58rationality the principle can only have
1:28:00zero.
1:28:01>> Yes.
1:28:01>> And then you're saying that first bullet
1:28:04point is saying that's not equal to PB
1:28:06feasibility.
1:28:07>> Yes.
1:28:08>> Sorry what was PB
1:28:09>> vis PBS because PB feasibility is to
1:28:12think about all PB all continuation
1:28:15plays and then select the one that is
1:28:18best from the exander perspective
1:28:20>> for the principal
1:28:21>> for the principal.
1:28:22>> Yes. So I'm saying that when you're
1:28:24thinking about this more as a strategic
1:28:26setting and you allow for these
1:28:27multiplicity continuations, this can be
1:28:29a feature because it can help you get
1:28:32the commitment which was something that
1:28:34Bon Stra's work work did not show
1:28:37because they were always assuming that
1:28:39the Y of M is optimal given the M.
1:28:41>> I see. And this PB feasibility involves
1:28:44setting this this A2 or the action that
1:28:48the principal can take in the second
1:28:49period to be equal to Z
1:28:51>> zero. and I combined with rejection. So
1:28:54a to zero and rejection are best mutual
1:28:57best responses. So when the buyer says
1:28:59I'm rejecting no matter what, a toals 0
1:29:02is a best response
1:29:04for the principle.
1:29:06Yes.
1:29:08Okay. So now let's talk about let's look
1:29:11uh when do I how much
1:29:14>> uh you have 15 minutes. Yeah.
1:29:16>> Okay. Perfect. Good. So um let's talk
1:29:19about okay let's talk about the theorem
1:29:23um
1:29:25so the theorem says for any collection
1:29:27of messages calligraphic I and any PV
1:29:30assessment of this me of this game
1:29:33indexed by this collection of messages
1:29:37an outcome equivalent public so there's
1:29:39something I did not stress so far public
1:29:42and I will explain in a minute of a game
1:29:45where the messages are theta and g and
1:29:50output messages of our beliefs exist
1:29:52such that after every history on and off
1:29:56the path I know you like off the path
1:30:00the principal offers a direct black
1:30:02mechanism in response to the principal's
1:30:04equilibrium offer of a mechanism if data
1:30:07is in the support of the principal's
1:30:08belief at HD then theta always
1:30:11participates
1:30:13conditional on participating the agent
1:30:15truthfully reports her type.
1:30:18If the mechanism outputs belief mu then
1:30:22the marginal on theta because remember
1:30:25the beliefs are about data and the
1:30:28history
1:30:30but the marginal on theta of the
1:30:32principles believe coincide with mu and
1:30:36we call a BB assessment that satisfies
1:30:38the above properties a canonical BB.
1:30:41So canonical BB of the canonical gain.
1:30:44So we have a canonical PB which means
1:30:47this behavior
1:30:49the canonical gate is extensive form
1:30:51game where now we don't have many
1:30:53messages many collections we have one
1:30:56collection inputs are type reports
1:30:58output messages are beliefs so this game
1:31:03is much coarser than the classical game
1:31:05we started and this play is also much
1:31:09coarser and public so canonical TV and
1:31:12the canonical game replicate all
1:31:14equilibrium outcomes of all mechanism
1:31:16selection games in this family
1:31:19and like the standard revelation
1:31:21principle it reduces the agent's
1:31:23behavior and its impact on the
1:31:25principal's belief to a series of
1:31:27constraints that the mechanism must
1:31:29satisfy. of truthtelling and
1:31:32participation which is the standard ones
1:31:34plus base plausibility constraint which
1:31:37is the constraint that keeps track of
1:31:39the designer sequential rationality.
1:31:44All right. So that's sort of the
1:31:45theorem. I'm letting you pause for a
1:31:48second.
1:31:51>> I know by the way. So so we are just
1:31:54interested in the given outcomes which
1:31:56is just like you know the whole
1:31:59>> sequence of the AP we we are not
1:32:02concerned with like how the information
1:32:04is revealed through the past.
1:32:06>> Is this right?
1:32:09>> Yeah. Ultimate outcome. So, so like we
1:32:12might also be interested in like how the
1:32:14information is gradually like you know
1:32:16revealed to the like the principle
1:32:18potentially for the different
1:32:19mechanisms.
1:32:21>> Um
1:32:21>> probably that can be replicated.
1:32:25>> It could. Yes, probably. But I guess you
1:32:27care about the information because
1:32:29essentially you care about the
1:32:30allocation.
1:32:30>> Yeah. Right. Right. Right.
1:32:31>> So if you have a if you have like a but
1:32:33you know there are some there's
1:32:35>> I'm not trying to criticize but right I
1:32:36just want to understand. Sorry if I
1:32:39responded in a way that it sounded like
1:32:40a criticism. I just wanted to add on
1:32:42what you said if you had a setting like
1:32:45a psychological gay in a psychological
1:32:47gay beliefs like if I'm the agent and I
1:32:50care about the perception of the
1:32:52principal about me. So like my daughter
1:32:54now that we are in this country and
1:32:56everyone is polite. Mom please don't
1:32:58embarrass us. You have to be you. She
1:33:00she was reading about the educate and
1:33:02she's so she cares about how you know
1:33:07people perceive. So that is a very
1:33:09natural environment. In this game we
1:33:12wrote down we did not assume people's
1:33:14payoffs depend on beliefs. Uh so by
1:33:18replicating a and thetas we replicate
1:33:20payoffs in a psychological game might
1:33:23you might want to actually replicate the
1:33:26beliefs because that actually something
1:33:28that is important. So here we replicate
1:33:30the allocations. Um okay and
1:33:36and I want to also say something about
1:33:38the publicity of the PPE. So at some
1:33:41point in the proof which I'm not I'm we
1:33:44show that with all of the generality the
1:33:47agents behavior and condition only on
1:33:50public histories. So the agent's
1:33:52behavior
1:33:55depends on the agent's history. the
1:33:57agents. He already have all these
1:33:59messages the agent has been saying all
1:34:01along.
1:34:02So um we show that actually we can
1:34:06replicate behavior when the the first
1:34:09proposition in the paper is to show that
1:34:11we can replicate everything with
1:34:13behavior that depends only on public
1:34:15information. And the intuition for that
1:34:18is that if we have suppose you have you
1:34:20are type two. Your type is two and you
1:34:24have two but two histories that include
1:34:26all these past messages and just to make
1:34:28it visible. One history is the red
1:34:30history, one is the yellow. Suppose all
1:34:33of those histories
1:34:35are your private histories they can they
1:34:38are projected on the same public
1:34:40history. So the allocations and what the
1:34:45principal has learned about you are the
1:34:47same in the red and then yellow which
1:34:49means that you have in the past consumed
1:34:51the same things.
1:34:53Now if you're using different behaviors
1:34:56for these private histories you must be
1:34:58different
1:34:59in the future because they're both best
1:35:02responses and the payoffs are the same
1:35:04in the past. So if you're doing
1:35:06different things you are indifferent. So
1:35:08we are using this indifference to build
1:35:10an outcome equivalent public history and
1:35:13that's how we get another property
1:35:19um which I did not stress so far which
1:35:21is the recursivity but let me say what
1:35:25our revelation principle allows us to do
1:35:28like in mechanism design the principal
1:35:31equilibrium choice of mechanism
1:35:32satisfies participation constraints and
1:35:34incentive constraints like in
1:35:36information design which a lot of us
1:35:38know consistency. We get consistency
1:35:41between output beliefs and equilibrium
1:35:43beliefs because we have we insisted that
1:35:45our messages have literal meaning. And
1:35:48then we have two implications. The
1:35:50principles belief plus the me the
1:35:53communication device of the mechanism
1:35:55induce a base plausible distribution of
1:35:57a posteriors and in the in the
1:36:00applications we can separately design
1:36:02the information and the allocation.
1:36:05So in in some settings we can with
1:36:09transfers for example we can get virtual
1:36:11circles representation optimize with
1:36:13respect to Q for to the allocation
1:36:17probability for each belief and then do
1:36:20information design two steps and the
1:36:24fourth thing is that uh that comes from
1:36:28the public PD we get recursivity and
1:36:33this is super nice when we have infinite
1:36:35horizon problems.
1:36:37So because of all of these things we
1:36:40have done some new applications h
1:36:42because we don't have cardinality
1:36:45on restrictions on cardinality of the
1:36:47type space and the length of the horizon
1:36:49we have we also have an extension to
1:36:52markoff settings in the terms of in the
1:36:53sense of marovian types
1:36:57information evolving in a marovian
1:36:59sense. we have been able to show that
1:37:03price are optimal in an infinite horizon
1:37:05binary type durable good model. So my in
1:37:08my job market paper I had finally many
1:37:12periods
1:37:13but a continuum of types. So actually
1:37:16that problem is still an open problem
1:37:18with more general mechanisms. The one I
1:37:20did in my job market
1:37:28and
1:37:30we are we are going to show you how we
1:37:32can use how we can do like um how we can
1:37:35do the product line design in the in the
1:37:37second lecture. So I we are able to show
1:37:39that limited commitment
1:37:43when we're thinking about a monopolist
1:37:45choosing varieties
1:37:47in um or choosing a nonlinear pricing
1:37:50like in a Musa Rosen setting. We are
1:37:53going to show that limited commitment
1:37:55actually coarsens the menu that of
1:37:57varieties that the seller um uses. And
1:38:01then to use this framework because we
1:38:03are we are introducing information
1:38:05design with additional constraints. We
1:38:08have developed a paper which was
1:38:10published in the mathematics of
1:38:11operations research which is called it
1:38:13constraint information design where we
1:38:16where we derive some results that one
1:38:18can use to solve um complex information
1:38:22design problems under arbitrary number
1:38:24of inequality or equality constraints.
1:38:28Okay. So let me this someone shared this
1:38:31to us and I thought it was very funny.
1:38:33So this is best sts which is they were
1:38:37there you get lied but know your lie but
1:38:39continue where in in our case you tell
1:38:42the truth they say I I love listening to
1:38:44lies when I know the truth so I I hope
1:38:48you don't mind me adding this but when
1:38:50they shared it to me I thought this is
1:38:52so funny and I have to put in the slide
1:38:54um all right so this we did not do that
1:38:58Pablo can only do that all right so Um
1:39:03let's we have about 10 minutes before
1:39:05the break I think right?
1:39:07>> Yeah.
1:39:07>> So let me uh start telling you about how
1:39:11direct black hole mechanisms work and
1:39:14how we use them to replicate outcomes.
1:39:16I'm going to show you basically an a
1:39:18sketch of the proof and how to use the
1:39:22result in as simple as possible setting
1:39:24one imagine which is a durable good
1:39:27setting with two types and two periods.
1:39:31It doesn't get easier than this. So now
1:39:34we are going to specialize
1:39:37in many ways but please don't think that
1:39:41the the model only applies to durable
1:39:43goods because sometimes I we I read I
1:39:46read papers right in citing our words
1:39:47and says our paper you know relies on
1:39:50transfers whatever no so the durable
1:39:54good is just an illustration don't mold
1:39:57mod. Okay, so we have a buyer and a
1:40:00seller interacting over evenly many
1:40:03possible periods.
1:40:05Possibly we're going to do two periods
1:40:07in the example. The seller owns one unit
1:40:09of a durable good and assigns zero value
1:40:12to it. The buyer has private information
1:40:14that is either low or high and the
1:40:17probability that it's high is called
1:40:19new.
1:40:21um we are going to we also discuss in
1:40:24the paper the case where the tit
1:40:27continue and theta is drawn according to
1:40:29f1 but I'm I chose a different
1:40:32application for the lecture for the
1:40:34second lecture so a current allocation
1:40:37here a of t is just the pair which
1:40:40specifies
1:40:41q and x so q is either zero if we have
1:40:45no trade or one if we have trade and x
1:40:48is a real
1:40:52Um
1:40:54so Q indicates whether the good was sold
1:40:56or not and X is the payment from the
1:40:58buyer to the seller. If the good is sold
1:41:00in B or two the game ends and if the
1:41:03final allocation is A of T which
1:41:06specifies uh whether the good was sold
1:41:09or not and the transfers the buyer and
1:41:11the seller's payoff are now we have
1:41:13discounting and we're summing over
1:41:14periods.
1:41:16The buyers gets this and the seller gets
1:41:19the expected expected discounted payment
1:41:22and there is some discount factor which
1:41:24is common in this specification.
1:41:27Um so at the how does the game work? At
1:41:30the beginning of every period the seller
1:41:32offers a mechanism. The buyer accepts or
1:41:35rejects. If the buyer rejects we have no
1:41:38trade and no transfers. If the buyer and
1:41:41we move on to the second period or the
1:41:43subsequent period. If the buyer
1:41:45participates there is an allocation
1:41:47determined and if the allocation is
1:41:49trade the game ends. So in that sense
1:41:52the problem is um very simple because
1:41:56only if there is no trade we move on to
1:41:59the subsequent period. So limited
1:42:01commitment here binds only if no trade
1:42:04realizes.
1:42:08So under full commitment uh we have the
1:42:11standard revelation principle. So when
1:42:14we have the standard revelation
1:42:15principle
1:42:17um
1:42:19is the optimum is to repeat the static
1:42:21mechanism every period. So the the
1:42:26static optimum in every period is a
1:42:28direct mechanism. So now a direct
1:42:31mechanism is just the mapping from types
1:42:33to distributions of allocations. So in
1:42:36this space in this h model the
1:42:39allocations are these q's and x's but
1:42:42nobody writes mechanisms like this right
1:42:44when we write when we have transfers we
1:42:45say look instead of randomizing over the
1:42:48allocations because we have linear in
1:42:52because they also are linear in types
1:42:53and transfers we are going to um write
1:42:57down the phi instead as a probability of
1:43:01trade and and an expected transfer. So
1:43:04eventually this formulation here becomes
1:43:08a a probability of trade. So analog a
1:43:10rule that specifies for each data a
1:43:13probability of trade and a rule that
1:43:15specifies for each data an expected
1:43:17transfer. That's how we operationalize
1:43:21the revelation principle in everyday
1:43:23life. You never see these five things
1:43:27written. You only see the Q and the X.
1:43:29What is embedded here is that the
1:43:32lottery over those things has been
1:43:34already run and we have and we have
1:43:37already used the fact that everything is
1:43:38linear to get to this simplification.
1:43:42Sometimes people use a simplification
1:43:43when it's not actually justified.
1:43:46Um by the way but that's okay. Um uh so
1:43:52buyer the buyer reports the truth and
1:43:54participates and what's the optimum
1:43:56here? Well, if the here if the there is
1:44:00a belief which we call mu1 bar which is
1:44:03theta l over theta h that's the belief
1:44:06that below what below that if your prior
1:44:11is below that threshold the optimum is
1:44:13to sell at a low price and sell to both.
1:44:16So if the prior is low below that ratio
1:44:20it's it's better to sell at a low price
1:44:23and sell with probability one.
1:44:26If the prior on the other hand is high,
1:44:30it's above that threshold that
1:44:32threshold, then you're going to we it's
1:44:35better to sell only with with
1:44:36probability mu times theta h. Why?
1:44:38Because at mu1, you see that these two
1:44:42times this is theta. So you're just
1:44:44indifferent. So at mu1 you're
1:44:45indifferent between charging selling to
1:44:47everyone at theta l or selling to theta
1:44:50h only with probability theta l over
1:44:53theta h at at prior above mu1 m new *
1:44:58theta h is higher than theta l. So the
1:45:01full the commitment optimum is to charge
1:45:04the low price if mu1 is less than this
1:45:07threshold. So q of theta q of theta h
1:45:10equal one and x of theta l x of theta h
1:45:13equal theta. And if the prior is high so
1:45:17you're optimistic that you are facing a
1:45:19high value buyer the optimum is to
1:45:22charge the high price and then solve
1:45:24with probability one only to the high.
1:45:27So the low t does not get the good and
1:45:29does not pay anything. And then either
1:45:32the game so if we have a low prior the
1:45:36game will end for sure in period one
1:45:38because we're going to sell the good and
1:45:40we're going to go home and we're going
1:45:42to be happy no matter if there is
1:45:44commitment or there's no commitment. The
1:45:46problem with non-commmitment arises if
1:45:48the if the new is high because then
1:45:51we're going to ask a high price and if
1:45:53the buyer does not accept then the
1:45:55seller is stuck with the good.
1:45:58So either the game ends in period one so
1:46:02no issue or trade never occurs under
1:46:05commitment because under commitment you
1:46:08commit to charge theta h theta h theta h
1:46:13the commitment is to repeat the static
1:46:15optimum forever and that is of course
1:46:17hard to swallow no seller or no
1:46:21government will just say okay I'm never
1:46:22going to sell this asset forever because
1:46:25we not we did not nobody Got it? So the
1:46:29optimal mechanism is not sequentially
1:46:32rational and that's basically what got
1:46:34me started with my thesis with with posy
1:46:37probability is stuck with a good art two
1:46:39and is tended to optimize. We're gonna
1:46:42stop here and and and then we're going
1:46:45to continue after

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This page contains the full transcript of “Limited Commitment: Mechanism Design Meets Information Design” Professor Vasiliki Skreta #1 by UTMD 東京大学マーケットデザインセンター UTokyo Market Design center, generated from the public captions YouTube serves with the video. The transcript has 16,436 words across 2,524 segments, with the original timestamps preserved so you can click any line to jump to that moment in the embedded player.

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