>> Yuval Peres: Good afternoon. Today Balasubramanian Sivan will tell us about economics of
repeated sales.
>> Balasubramanian Sivan: Thanks, Yuval. This talk is about a really basic economic setting of
repeated interaction. The setting is very simple. You have a single seller who is offering a fresh
copy of an item everyday to the same buyer. Think of a fishmonger who is offering fresh fish
every day. The buyer goes in and buys fish, and the buyer has some value, just private, only he
knows it for consuming the fish, and the value is the same every day. It doesn't change.
Initially this value is drawn from a publicly known distribution, meaning that the seller also
knows the distribution. Once it is drawn, it remains the same every day. The seller has a cost
which is public and here we normalize to zeros. It doesn't change things. The setting is very
simple. This happens for n days. Each day the seller comes, talks to the buyer, so what are the
rules of the game? The seller can post a price on every day. That's all. The buyer can take it or
leave it. If he takes it he pays the price. We call it the fishmonger's problem. Amos Fiat
suggested this name for us. Let's look at the most basic performance problem, one day
interaction. You have a seller and a buyer. The buyer’s value is private. He knows it but for the
seller drawn from this distribution. And what happens? The seller posts a price. Clearly, the
seller posts a price P. The buyer lacks separate only his values about the price. He should do it
if and only if. The seller is maximizing the quantity price times the probability the value exceeds
the price, which for the uniform distribution it's clear that it should be a price of half and he will
accept it or reject it and get the revenue of exactly a quarter.
>>: That's it the buyer doesn't use the hammer. [laughter].
>> Balasubramanian Sivan: Yeah. What happens if you have a two-day interaction between the
seller and the buyer? Again, the value remains the same for the two days. Otherwise there is
nothing to here. The value is getting freshly drawn every day. The seller obviously is interested
in collecting as much information as possible about the buyer’s value from the first sale. The
buyer, on the other hand is inclined to hide his value because the seller will exploit it if he
knows the value. So what are transfers in two days, right? Something didn't happen on the
first day; nothing happened on the second day, so what's the revenue? That's the question to
be asked in this talk. Clearly, what the problem is, what is the buyer afraid of? The buyer is
afraid of the fact that the seller will exploit the knowledge of his value. Suppose the seller can
pacify the buyer by saying that I'm not going to exploit it. The seller can commit that in all the
future raises of the price and he reveals it in the first day. If he has the power to commit then
one thing that the seller could do is the following. The first day he puts the price up of a half
and he commits on the first day itself that regardless of whether you accept or reject I'm going
to put the price up of a half on the second day also. Then it's clear that the revenue for two
days is just two times the revenue for one day, so half is the same token revenue for n days is
just n over 4. I'm going to call this as the Myerson optimal revenue benchmark, so a single day
optimal. It's a very special case of Myerson's mechanism which is word of the single optimal
thing and that's why I call this the Myerson optimal revenue file. If the seller were able to give
this commitment that I'm not going to exploit you, the situation is very clear. You have a nice
revenue linearly growing revenue. But realistically, the seller is often not able to commit to this
kind of a price sequence. The seller is very difficult to resist the temptation of lowering the
price if you think the buyer is not buying or to raise the price if you think he is capable of
buying. In any case, the buyer is not willing to believe that the seller will not succumb to this
temptation. So what will happen in this case? You have no commitment from the seller about
future prices, so what do you think will happen? How many people think that the seller
extracts the entire value to buyer?
>>: Eventually.
>> Balasubramanian Sivan: Eventually. He slowly learns, retracts and then at some point and
the number of rounds goes to infinity you get almost the entire value.
>>: And what can the buyer do?
>> Balasubramanian Sivan: The buyer can only say yes or no every round. That's it.
>>: [indiscernible]
>>: He can't do anything. You post any price I can say it's no arbitrary, there's no rules that I
should do something and he extracts a higher price, anything.
>>: What does the buyer’s value, what's the role of the buyer’s value in this? The buyer can do
anything?
>> Balasubramanian Sivan: I mean the buyer can, in principle he can do anything, but he wants
to maximize his utility at the end of the game, which is his value for consuming the fish every
day minus the price he paid and the utility adjective or.
>>: And there's no loss for not buying any fish?
>>: There is no loss, yes, but he gained something if he consumes and pays a price lesser than
its value. Right? So do you think at the end of this interaction the seller will eventually know
everything? It doesn't matter right whether they can commit or not, I'll get exactly Myerson
optimal revenue or that you'll get something even lesser than Myerson optimal revenue.
>>: So what is the buyer trying to maximize?
>> Balasubramanian Sivan: The buyer is trying to maximize his own happiness, which is showed
by his utility is added rounds and in each round the utility is the value for consuming the fish
minus the price he paid.
>>: When you say seller cannot commit to future price he can mark, signed the contract for
something?
>> Balasubramanian Sivan: There is no contract, correct.
>>: Like you think I can only tell you the price but, today I don't know what will happen
tomorrow?
>> Balasubramanian Sivan: Anything could happen tomorrow.
>>: I think solution seller will be [indiscernible]
>>: Given that you are asking the answer is probably c.
>> Balasubramanian Sivan: Yeah, okay. [laughter]. So it is c but the proof is very simple and
cute. Suppose there was an equilibrium where equilibrium was a pair of strategies for the seller
and the buyer where each one is mutually best just pointing to the other. If you give me this
equilibrium and tell me that there is an equilibrium where you can extract more than Myerson
optimal revenue, and I can use it to get one round mechanism where I get more than a quarter.
Here's how it works. You give me the mechanism and here is a one round mechanism. One
round allows the buyer to report its true value to me. Say that I assimilate this perfect Bayesian
equilibrium. I'll define what an equilibrium is, but for now I simulate the equilibrium and pick a
day at random and whatever transpired on that day is basically going to be the result of today,
which means if you bought on that day you better buy it today and pay the price that went on
that day paid. Clearly, if you got revenue more than n over 4 in the equilibrium you give me
strictly, then in the pick a date uniformly at random you are going to get a revenue strictly more
than a quarter which is not possible. That is known. So that's it. But the surprising fact is not
just that you cannot get more than Myerson optimal revenue. You get far, far lesser than this.
For most distributions the revenue doesn't even grow with n. It doesn't even grow with the
number of days of extraction, any equilibrium and we will see what equilibrium is. This is
known. This is not our research here. To get a sense of what is going on with this let's begin its
two round and play and see what happens. Again, the seller cannot commit to future prices, so
the first day let's say he puts the price that P1. And the second and is my notation. The
subscript is the day and zero means that the buyer rejected on the first day. One means he
accepted. What should the buyer do? Think of a simpleminded buyer, for now. The buyer will
reject if his value is smaller than some threshold t and will accept if the value is larger. If he
rejects, then the seller, if he follows this strategy…
>>: V is the buyer, right?
>> Balasubramanian Sivan: V is the buyer’s value.
>>: And what is t?
>> Balasubramanian Sivan: T is some threshold.
>>: So it doesn't depend on the price whether he rejects or not?
>> Balasubramanian Sivan: No, it will depend on the price. T is the function of all of the prices.
T is the function of the price on the first day and there's a whole strategy here. The buyer has
all of the tree and it might as well his strategy is basically the price in the first round and what
he would do upon rejection or acceptance or whatever. The buyer’s strategy is to say what I
will do in the first round. You are analyzing it piece by piece. Do you have a question?
>>: The decision on day one, I don't see how it can depend on the prices on day two because
he doesn't know them.
>> Balasubramanian Sivan: No, but the point of equilibrium is after having played this game for
a long time, basically, the buyer has a tree. He can see a tree, basically. There's not returned
somewhere, but he can see it.
>>: I understand that [indiscernible]
>>: The thing for a long time is true so in searching for equilibrium you can think that the buyer
knows the seller’s randomized strategy and response to that. So the seller, in other words, if
the buyer, given the seller’s strategy, the buyer could do something better than you are not in
equilibrium.
>>: What does equilibrium mean for the two-day thing?
>>: Equilibrium doesn't matter if it's two. It's the whole game.
>>: What's the whole game?
>> Balasubramanian Sivan: So I'm going to define, I'm going to come to that actually. I'm going
to far more define what equilibrium is, but for now, basically, let's just say that, just assume
that the buyer can see the entire strategy of the seller. He can see the price. Of course he can
see this price before reacting and let's say that he can also see these things, which I'll say what
the game is, but he can see those things. Then he will come up with some threshold t and then
if the value is smaller he will reject and if the value is larger he will accept.
>>: On day one and on day two?
>> Balasubramanian Sivan: No. On date to he basically has to accept or reject just based on the
value and the price. If the value is greater than the price he accepts, otherwise he rejects. This
is for day one. If this were an equilibrium, for instance, one of the conditions that has to be
true is that if the buyer follows the strategy of rejecting when the value is smaller than t then
the seller should update his belief of the distribution to be uniform zero t and believes the price
on this day should be t over 2. You can deduce this. Similarly, if he accepts a v greater than t
you better update the distribution to uniform t1 and it's not difficult to calculate what the
optimal prices to gain its max half of t and calculate. These are all necessary conditions for
inclusion, not sufficient, but these are necessary. If this is true, what can we say from this
alone? One thing that is true is the buyer whose value is exactly equal to the threshold should
be indifferent between rejecting and accepting. If he rejects he gets to buy just on day two at
the price of t over 2 so that is this utility. P minus t over 2’s value is t; price is t over 2. If he
accepts he gets to buy on both of these days and his utility is t minus P1 on the first day and the
second day he doesn't get any utility because the price is at least at threshold t. The seller
knows this information. This left-hand side utility after rejection should be equal to the righthand side utility so this will give p1 as t over 2. I'm not saying this is the equilibrium. This is one
we are guessing, basically. And you can right the expression for revenue purely as a function of
t. Optimize it and it will come out to be .6. In other words, I'm saying this is a guess that we
have come up with for an equilibrium for two rounds. The price on the first date is .3 and the
buyer rejects if his value is smaller than .6, not .3, but .6. And he accepts if the value is greater
than .6. Then you have this and you can calculate the revenue to be 9 over 20. If you
remember the Myerson optimal revenue was two times a quarter which is 1/2. It's ten percent
smaller than the Myerson optimal revenue of half. I didn't say we are at equilibrium yet, but it
is something. But this still doesn't look too bad. This is only 10 percent smaller. But things
change radically from three rounds onward. Before going to three rounds I now formalize
what? A perfect Bayesian equilibrium. If there is no commitment, then how do you reason
about the game? He said that the game's outcome will be governed by this perfect Bayesian
equilibrium, so what is perfection? The perfectness means that the strategies of the seller and
the buyer are in equal agreement, that is they are in mutual best response for every sub game.
One round has transpired. Something has happened in that round. Then there is a new
situation. The belief has changed. For this belief the strategy, again, should be an equilibrium.
That is perfection and Bayesian means that the seller updates his belief consistently according
to buyer’s strategies. I give you two objects in hand, buyer strategy and seller strategy. Given
the buyer strategy your belief updates about the distribution should be consistent with the
buyer's strategy. And the seller’s strategy itself should be optimal given his beliefs. In other
words, in full glory seller strategies as follows, it should specify the price to be posted on the
first round for two rounds and basically like this price on the first round. Then for every
possible price he could have posted on the first round he should tell you what is the price you'll
post in the second round if the buyer rejects this price, what is the price of x in the second
round if the buyer accepts this price of x. Let's think about this. Clearly a strategy should
specify the price in the first round and the second round prediction. Why is it that we should
specify the price in the second round for every possible price on the first round? For doing this
you have to put yourself in the shoes of the verifier. I give you a pad of objects called strategies
and your goal is to verify that these are mutual best responses. What all information do you
need to do this? If I tell you that the price on the first round was P1, you have to see why can't
he put a price different from P1, some x? To see this you have to know how the buyer will react
to a price of x for every x. Then the buyer’s strategy should tell you for every possible price of x
in every possible history what he will do. This is the strategy the buyer gave you where the
buyer is best responding by the strategy, which means the buyer is saying at the price of x I will
react like this. You should ask him why do you react like this, because then he will tell you I
reacted like this because if the seller sets the price of x in the first round, then this is the price
on the second round for acceptance interdiction. Then that better be a part of the seller
strategy. In full glory this is the strategy. From this the backward induction Myerson can verify
that this is the perfect Bayesian equilibrium. The last round I will verify that the buyer’s
strategy are the best response with the seller's strategies. Seller believes constantly updated
given buyer strategy and seller strategies are optimal given buyer's beliefs. There is a circularity
here. Beliefs are consistent given strategies and strategies are optimal given beliefs. So you
can't compute a perfect Bayesian equilibrium; you can only verify it. You can only verify it. If I
give you something you can go and verify it. You can verify here.
>>: [indiscernible] the buyer and the seller would be better off. Is that correct? Would they be
better off?
>> Balasubramanian Sivan: If they commit, yes.
>>: Not a best response.
>> Balasubramanian Sivan: Right, exactly. Basically, if the buyer is not willing to believe that
the seller will not succumb to temptation.
>>: Is a clear like [indiscernible] or something? Do you have to go to random?
>> Balasubramanian Sivan: I'm going to talk about… There is a big hammer here which suggests
that for every game there always exists a PBE, but the question is the nicety of the PBE. The
perfect Bayesian equilibrium could be horrible, but how nice is it? That is basically the starting
point of this work. This is what a perfect Bayesian…
>>: [indiscernible]
>> Balasubramanian Sivan: I'm going to cover that. Exactly. This is what previous work says.
They analyzed finite horizon n rounds and considered to point distribution, buyer's values low
or high. For this they said for all except the last few rounds, by few I mean a constant number
of rounds independent of n the price, the first initial n minus constant rounds is the lowest
point of the distribution. Here it's L. Now Schmidt basically described this property as discrete
distributions. Let L be the lowest point in the support of the distribution. What he showed is
that the PBE is guaranteed to exist according to some big gamma theorem and every PBE…
>>: Are you saying that if L and H are known then the seller is going to price at a low price
except at the end, which means he won't even find out whether he could have sold at the
higher price until the end? I don't understand. That doesn't seem like a logical thing.
>> Balasubramanian Sivan: Yeah. But the thing is this is basically what equilibrium depicts. If
you want all of the conditions that are specified in the previous slide you go through…
>>: Oh, and then reject.
>> Balasubramanian Sivan: Yes.
>>: I see. Okay.
>> Balasubramanian Sivan: And basically this is tantalized not just for two point distributions
but for arbitrary discrete distributions. Again, the seller would be posting the small price of L
for all except the last few rounds. As I said, this is an extremely bad deal for the seller, but this
is a natural and you won't expect this in practice also. Why? Here is one possible explanation
for this. We showed the following. We posit that the threshold equilibria is what constitutes
natural equilibrium, but threshold equilibrium I mean what we just saw for the two round of it.
The buyer rejects if the value is smaller than something and accepts if the value is larger than
something. Non-threshold equilibrium means that except for some set of values and then I
reject and then I except things like this. It seems unnatural for several reasons, first, from the
buyer’s point of view and also from the seller. This would mean that his belief is basically
supported on fragmented intervals. If the buyer rejects then L is set for value between 0 1/5 of
some property half and one third and two thirds of some property have this kind of belief will
happen if it's not threshold. If we agree that threshold equilibria are naturally [indiscernible]
then we characterize exactly when the threshold equilibrium exists. We show threshold
equilibrium exists only for those distributions for which the two round game has an equilibrium
where the first round has a price of L. By the way, the two round game always has a threshold
equilibrium. It's not just for the uniform zero. L is the low point of the distribution, lowest
point in support of the distribution, arbitrary distributions, but the lowest point in support of
the distribution is L. If you have an equilibrium for the two round game where the first round
price is L only for precisely such distributions is there a PBE in the short strategies. Otherwise a
PBE doesn't exist, basically. That's the problem in this case. There is no natural equilibria
exists. And this kind of almost never happens, I believe. But there are some distributions you
can constrict for which this is true. For instance for the uniform 0, 1, we just saw and it is
essentially the unique threshold equilibrium, unique equilibrium also. The first round price was
.3, not zero. Zero is a low point. This is not quite satisfying, so you look at the natural
equilibrium doesn't exist. There's no commitment game. We looked at what could happen if
there is low partial commitment. The seller says that I will not raise the price, that much
guarantee I can give you. I will not exploit. But I could lower the price in the future.
Decreasing the price is beneficial both to the buyer and the seller. The seller is doing it because
he is benefiting. The buyer in any case is happy to accept a lower price. Basically, what we're
saying is liquidation sales are allowed. One side is fine. Other side you can't raise the price. If
this were the case this commitment can be given credibly. Then we show that threshold
equilibria is guaranteed to exist. Unique threshold equilibrium for most distributions and for
the uniform 0, 1 revenue we can analytically calculate what the revenue is. It goes from
constant to square root n over 2, in the unique threshold equilibrium that that takes us for
uniform 0, 1. So you can compute this for any distribution. It's easy to write a program, but
analytically it's not possible to do for every distribution like 2 uniform 0, 1. You will get this as
opposed to a constant.
>>: But still much lower.
>> Balasubramanian Sivan: Still much lower than the Myerson optimal revenue.
>>: So why would the buyer be more willing to accept such a commitment than the other
commitment that he wouldn't accept?
>> Balasubramanian Sivan: I'm taking that as a rule of the game and then driving what
happens, so this is, suppose this is a rule of the game.
>>: So the problem is the seller can give a coupon to the buyer that guarantees that you can
come, if you buy it today at this price, here is a coupon that allows you to buy at this price every
day until the next day. But the commitment that I won't lower the price is less credible because
it's a kind of promise which, you know, people would tend to say now I have to give you a
special deal and that's -- you don't see people giving coupons promising in a legally binding way
that they won't lower the price. People do announce this price and then next day it's an even
better price and no one ever sues them for lowering the price. But if they promise certain
threshold on the price they, that's a commitment that can be more reason to make.
>>: I am not sure I am tracking what it is you ask. In case I keep the same price, of course,
that's exactly what the Myerson thing is. In this case you are telling me if I try to do something
like that the seller would hurt me, sorry, the buyer would hurt me. I would not be able to do
[indiscernible]
>> Balasubramanian Sivan: Right. You are not in equilibrium because you learned that this guy
rejected. You have to lower the price. The distribution is to be updated let's say uniform 0, 1.
Then you have to lower the price. You can't put the same price of a half. Half is optimal only
for uniforms are all, one. It is a two leg game, so you can't put the price of a half again. You
could lower the price, but on this branch, the acceptance branch the price will remain where
you are bound to put the same price by the rules of the game.
>>: [indiscernible] the seller retains the ability to lower the price, the buyer will want to inspire
that and so he will reject the first day even if the price is below the value.
>>: So this means the price and monotonic might not have the first price as the maximum.
>> Balasubramanian Sivan: In this branch of the price remains a constant. In acceptance
branch the price remains a constant.
>>: [indiscernible] the first day is the maximum.
>> Balasubramanian Sivan: No. For instance, for a two-day game the first day price is larger
than the first day price for nine day game. Is that what you mean? Yes. It goes the reverse. If I
play a two-day game then what the first price is, that price is larger than the price for 1000 day
game. Is that what you are asking?
>>: The day’s price, the first, this commitment cannot increase the price. That means each day
is at most the previous day. It doesn't mean that each day is the most the first day? Right?
>> Balasubramanian Sivan: Right. Uh-huh. But the thing is once you accept, basically the price
will change it all. I can't raise the price but I won't lower the price. There is no meaning in
lowering the price. I know you were offering this price. Basically it is just fixed that branch.
Let’s go over this proof quickly. It says that Pn is the price of the first round of a n day game
and tn is the buyer’s threshold in the first round. That is I'll buy if I lose at least tn. I'll reject if
the value is smaller than tn. Un is the utility of the buyer with the highest value. Value equals
one, what utility he gets. And rn is the revenue in the n rounds game. Let's draw this diagram.
Put a price of Pn for the first day. If the buyer accepts to buy if value is at least tn then the
distribution gets updated but nothing really happens here. The price has to be Pn. You can't
change the price. It'll stay the same forever. If the buyer accepts in the first-round the point is
that the price is to remain fixed at the same Pn and from this you can write what utility of the
buyer’s value one is. The buyer’s value one, the highest value. He'll buy in all of the rounds and
his utility is n times the utility in one round which is one minus Pn. If you reject value is then Pn
then the distribution is uniform between zero and Pn so what do you think the price should be
here? So uniform zero tn is basically just a scaled version of uniform 0, 1. And you are in an none n round game. The price better be P times P sub n -1 for just scaled by tn. This is the price
on the n round game, n-1 is the game and then this goes on. Given this, you can say that we
write the same [indiscernible] before. The buyer at the threshold is indifferent between buying
and rejecting. If he buys and the price is a constant, his value is n times tn minus Pn. If he
rejects, then his utility, this buyer is basically the n point of the support of the distribution just
like one, so his utility is just use of n -1 but scaled by tn. This utility upon rejection is. This is the
utility upon accept. Then you can write the revenue, so you now have this four variable
recursion and you solve this you get [indiscernible] ns, okay? And the price comes out to be a
very small price to begin with already, but the wires threshold is very close to one. The price is
extremely tiny but the stingy buyer still is bargaining and saying no for some time before he
accepts.
>>: It's not exactly what he is saying. He is already scared scheduled it's not [indiscernible] to
get the point across he still goes and accepts it.
>> Balasubramanian Sivan: Yeah, exactly.
>>: [indiscernible] value is [indiscernible]
>>: The reason you don't see this in practice is because you don't have any losses for the seller
for not selling. Okay. You said it was normalized but…
>> Balasubramanian Sivan: It's normalized to some point. You could take the investment with
something like…
>>: So this really means just about his cost?
>> Balasubramanian Sivan: Uh-huh. That's the summary for two rounds. Basically, the point is
that a threshold equilibrium is at least guaranteed to exist. We have arrived back to that
particular and it’s also kind of unique for more distributions and you can compute it for any
distribution, but analytically we do it for uniform 0, 1. But as he pointed out this is still far from
n. N is still much further. Now go to an infinite horizon game where the game repeats infinitely
but to keep utilities finite we do a time discounting which means both the seller and the buyer
will discount tomorrow’s utility by a factor I minus delta. I will explain what Markovian
strategies is in a bit, but for now don't worry about it. Time discounting is really equivalent to
saying the game could stop at any given time with probability delta. For this week calculate the
PBE revenue for the uniform 0, 1 distribution. This is at least 69 percent of Myerson optimal
revenue.
>>: [indiscernible] that you want them to be independent? What's going on?
>> Balasubramanian Sivan: Sorry. What is the difference?
>>: Something has to be dependent on the history.
>>: But this is independent.
>> Balasubramanian Sivan: Yes.
>>: Right. So it's not just a stopping time? I think it was meaning a stop.
>> Balasubramanian Sivan: For uniform 0, 1 distribution you get a 69 percent of Myerson
optimal revenue. What is Myerson optimal revenue? Again, you get a quarter in every in every
town. But now it's quarter times this 1 +1 minus delta and so on. It's just 1 over 4 delta. You
get a quarter of, you know, you get 69 percent of 1 over 4 delta. That's the optimal benchmark.
This is basically the time that the seller was able to get back all this money in this model. The
main point is that delivered then goes to, n approaches infinite in the finite horizon game. It's
very different from the infinite horizon game.
>>: [indiscernible]
>> Balasubramanian Sivan: Even in the discounting I didn't say about that, but even if you
discount the previous results, the two results I pointed to, Hart and Tirole and Schmidt they
show even with discounting you get the same result, that for the first n minus constant around
you post a price of L and then in the last few rounds something exciting happens. You get a
constant revenue. This is a consequence of fixed horizon and backwards induction, but random
horizon is more natural. There is no backward induction effect here. Now I will motivate why
we do these Marcovian strategies and what Marcovian strategy is. Basically if these strategies
were as it was for the finite horizon, basically depending on the complete history, what all the
previous prices were and how the buyer rejected accepted. You really need infinite memory to
even exude the strategy. The seller has to [indiscernible] the entire history of prices and accept
reject sequences. [indiscernible] the game. That's not possible. You can't even check if a given
pair of strategies is in equilibrium. If it gives you something you can't even verify that these
guys are mutually best just responding in finite time, if you allow strategies to be arbitrarily
independent on history. Marcovian strategies are basically those which depend only on the
belief not on history. Belief means, belief is about the buyer’s distribution today. It doesn't
matter what path you took to come to this belief about the buyer’s value distribution. Belief, of
course not…
>>: What is disbelief dependent upon?
>> Balasubramanian Sivan: Belief is dependent on how the game progressed so far. The buyer,
I put a price of P; he rejected. Okay. I mean that is an initial distribution. It's a uniform zero
one and then I put a price, you reject it, then you accept it, then you reject it and so on and so
you constantly update my belief.
>>: [indiscernible] each time you just use the interval and one move to update the interval.
>>: So it is dependent on history but in a more limited way.
>> Balasubramanian Sivan: Yes. There are several history which could probably take you to the
same belief and you shouldn't say that for this history I will do something, that history I will do
something else. The belief is given that you said that's all I need for the game and you should,
yeah. So basically every subgame is identical to the original game except for the belief and
belief is defined as by these two numbers, the beginning and the end point of the interval. The
point is, again, this motivates why a threshold's strategies… I mean uniform is for uniform
distribution it could be anything but it's some interval. But this, again, motivates why threshold
started these are natural. Because of the strategies were not threshold, even the belief is not
finite. Belief will also grow because the belief will be as splintered collection of intervals and
the fragmented intervals and it just keeps on growing. But for threshold strategies it is clear
that that you all live in intervals with respect to how far the sequence was of accepts and
rejects. If you rested this alert to Marcovian strategies in the same notation and we say P is the
price in the first round and t is the buyer's threshold values higher than t [indiscernible] uses
utility of the buyer with the highest value namely 1 and R is revenue. You draw this diagram
and here, again, if the value is less than t you update the distribution to uniform 0, t. Then we
know that the price should be the same as before but scaled by Pt. But what do you do here?
Values larger than t then distribution is uniform t1 is different from uniform 0, 1 and earlier we
got out of this issue by saying that a price cannot be raised. Therefore, I just gave this price and
put this price here. But now I can't do this because I'm not putting any such constraint. The 0
can commitment now. How do you find out what happens to uniform t1 distribution? We
don't know, but there is only one case in which we would know. If it happens that this
threshold is at least a half, then for a uniform t1 distribution with t greater than half the optimal
is just to put the price of t on all rounds and calculate this. I won't get into that but that you can
do. If it's at least a half for the uniform 0 n distribution. Again going to be a guessing very fast
strategy. Let's assume that t is greater than a half and proceed. Get the buyer accepts in the
first round and the price remains fixed at t, because t is greater than a half so you get the utility
of one minus P in the first round and then he gets a one minus t in every subsequent round
discounted with the factor I minus delta. So that is that. And the buyer at value t is indifferent
to buying and rejecting. If he buys he gets t-P in the future he gets zero. If he rejects then he
gets a utility of the highest value U but scaled that t and then there's a one minus delta scaling
factor. Then the revenue expression can be returned. You saw this guy. As delta goes to zero
you take [indiscernible] you get exactly, so this is .69, 69 percent of Myerson optimal revenue
benchmark. t for delta goes to zero and t is approximately this. This also verifies what we
assumed that t should be at least a half. That's basically the summary for this infinite horizon
game. The main point of the finite horizon game as n approaches infinite is very different from
the infinite horizon game.
>>: Again, [indiscernible] this Markovian property assumption that you made?
>> Balasubramanian Sivan: Yeah, but Markovian basically I don't know what an equilibrium
means. I don't know how to verify. It's not just a question of analytical tractabilities. I don't
know what an equilibrium, I mean, I don't know how to verify it. So finally, in the infinite
horizon game we saw that even without any commitments, once there were commitments you
could get a constant fraction, 69 percent. Suppose I allow one-sided commitment now, which
means that I promised not to raise prices. Do you think that you can get even more than 69
percent?
>>: [indiscernible] prices in the previous one.
>> Balasubramanian Sivan: I was not raising prices, but the price, no. I was raising prices in the
first round but here it turns out that you get exactly the same revenue as Myerson. So I was
raising prices. If you look at this the price was not P. The price was t. The t remains fixed
afterwards, but after the first round the price did raise. But it turns out that you get exactly the
same revenue even if you allow one sided commitment for the infinite horizon time
discounting. That's basically the summary. The finite horizon repeated sales game is not as
nice as was thought earlier. Threshold equilibria don't exist for most distributions. And onesided commitment is better, the revenue you gross with n, I think it seems like the infinite
horizon time discounting is the right model because it gives something that you would expect in
reality. This, as I said, is very basic setting of repeated interaction and could do more than this.
What happens if there are multiple buyers, multiple sellers. Seller has a cost which is not
publicly known, again, that is also drawn from a distribution and all those things. So how does
the [indiscernible] what is the equilibria in all these gains et cetera. What about the motivation
I didn't talk about. People actually do this. In search ads you set the reserve price for what the
minimum bid has to be for an ad to be shown and this depends on how you billed in the past,
things like this. Then there is behavior based price discrimination.
>>: The seller who sets the reserve prices depends on how the buyer did in the past?
>> Balasubramanian Sivan: Yes.
>>: You said how are you [indiscernible]
>> Balasubramanian Sivan: And there is behavior based price discrimination which firms
already do. They are known to do this kind of behavior based price discrimination depending
on your history. There are loyalty cards and once you basically swipe it I see your entire history,
what happened on Thursday and Friday and so on so I can do this. This is how economists
study this. They call it behavior based price discrimination. That's basically all I wanted to say.
Thank you [applause].
>>: Question. At the end you mentioned that in reality there are many sellers and many buyers
and that's perhaps a more important reason for why we don't see the behavior that is the
question you raised at the beginning. In terms of the commitment, one-sided commitment
you're saying that the commitment was monotone, not that the price be never raised in the
first day. What you might mention is you can get a coupon for the price that first day and you
could buy it for that price forever after. Then you might get upset if the price were lower. And
you, sorry if you got a commitment that the price didn't change and the price were lowered
that you might get it. But that just means the price being lower than the first day. I don't see
how that really monotonicity.
>> Balasubramanian Sivan: No. But the point is it never happens in equilibrium. There's a
possibility in the game, but in equilibrium once you accept there is no question of lowering the
price because the seller is not [indiscernible]. If I know that you can afford this price and I am
bound not to increase the price better quote the same price in all the states. Raising the price
is not allowed by the game. Lowering the price is not good for me.
>>: But you lower the price upon rejection.
>> Balasubramanian Sivan: Upon rejection, yes.
>>: But if you buy it it doesn't make any sense for me to lower it in the near future, right?
Because I know you are going to keep buying it, right?
>>: No. You buy it in the future as well.
>>: Your incentive is not to reject. It’s highest in the earlier rounds because then you have
more to benefit from the seller lowering the price. But once you the seller…
>> Balasubramanian Sivan: See, you won't buy and then come to a remorse that later this guy
lowered, right? You keep on rejecting and then, so basically the game proceeds like this.
Comes, reject, reject, reject, reject and accept, accept, accept accept. That's what happens in
the numbers I showed you, right? Tn was…
>>: Does it depend on what the rules of the game are?
>> Balasubramanian Sivan: Yes.
>>: So you are questioning the rules of the game.
>>: So your rules are that you are not allowed…
>>: So the [indiscernible] [multiple speakers]
>>: One commitment versus another commitment.
>>: The rules are that the seller always has the option to, I mean he can more generally just
give a coupon that says, that gives an upper bound on the price on a certain day. He can give
any kind of coupon. He can tell you he is giving you a coupon that says seven days you can
come to my shop and buy that, and get an item, its option to buy at this price. And he can give
any such number of coupons for any future days.
>> Balasubramanian Sivan: So what is exactly the…
>>: There was an objection to commitment not to change the price.
>> Balasubramanian Sivan: Right.
>>: But you said there wasn’t an objection to commitment that the price be monotone.
>> Balasubramanian Sivan: I mean it's in between not doing anything at all and giving full
commitment. It's something in between. And of the two sides of it you can do the side that
won't lower the price or I won't raise the price seems like it's more…
>>: It's a different kind of commitments that are, so it's easier to make your commitment not
to increase price because I can just give you a card promising not to increase the price, but it's
harder and less in practice to give a binding commitment in order to decrease profit.
>>: What?
>>: Because you need more like an outside monitor for that, while the buyer is himself a
monitor because, so later you increase the price then the, you know, generally, the buyer is less
likely to protest that, you know, you then…
>>: Then that is ignoring the rules of the game. The rules of the game were I… Was based on
the fact that you made this commitment and then you lied in the commitment and then he
would protest. You are saying buyers don't protest in the world. That's because they don't
have any such roles.
>> Balasubramanian Sivan: When I say the commitment is not…
>>: [indiscernible] are less likely to be…
>> Balasubramanian Sivan: No. For the original game when I said the seller is…
>>: You get a rebate. You get it free or something. If I were the price you get it free.
>> Balasubramanian Sivan: No, no. When the seller says I am not able to commit…
>>: Just between the two parties, you come to me I say if you don't buy it I will give you this
coupon and you can always get price later on in the future. Don't worry.
>>: That's a maximum price, not monotone.
>>: No. Every day you, I will give you this coupon for the current day’s price. If you buy it, of
course, then I know I keep it at the same price. If you don't buy it you get this coupon so than
the price should be only, will never increase, right? It doesn't make sensitive increase, right?
>> Balasubramanian Sivan: No. When I said there is an objection to commitment, it's not
about objection. The objection is the buyer won't believe. The credibility is the problem, not
about whether they get upset or not. The buyer will not believe if I say anything. That's the
reason why we studied the no commitment setting, right?
>>: But if it's about credibility, then what about these coupons that you last mentioned?
>> Balasubramanian Sivan: The coupon is a legal commitment, right? If I tell you that you can
buy it at this price tomorrow then I have to honor it and so it turns out that…
>>: So you see a lot of people giving out coupons which legally bind them to an upper bound in
the price. Then you see sometimes people saying this is the lowest price, but you don't see
people making legally binding commitments that way.
>>: This is an issue in a market where there's many sellers and buyers and it's the real world
and that's all we really look at. We see how it applies.
>>: On the same point slightly different point, I tell you repeat the one about R option. What
was the like the results were like what?
>> Balasubramanian Sivan: Right. So you are a search engine and you have to set some
reserves on the minimum bid to be able to qualify for getting shown the, the sponsor starts,
then I can update these results based on how this particular, let's say chocolate, is a keyword
for what you are bidding. Auction is based on how chocolate has performed in the past. And,
of course, it includes not just you. It includes other Nestlé and other people to know, but
people do do this. They can count on how bidding has gone in the past and change the result
price. It's not clear how buyers will react knowing that this result is going to happen to the
result.
>>: Usually, people think of an isolated auction and then you should be truthful, but the point
is that in a repeated setting, even if it's a victory at a second prize auction you have a
motivation not to be truthful and you reject prices are below your value and those are not to
devalue information that will be used in reserve prices in the future.
>>: You should also know equilibrium in the [indiscernible]
>>: There are some situations where [indiscernible] is like equilibrium.
>> Balasubramanian Sivan: Well you read some model to reason an upgrade, so how do you
reason about what happens?
>>: [indiscernible] I made a statement [indiscernible]
>>: Somebody stupid at the time, they just moving and circling. Is that what you think? They
just, because it started [indiscernible]
>>: [indiscernible] optimizing.
>>: So in search engines [multiple speakers]
>> Balasubramanian Sivan: They do. For instance…
>>: [indiscernible] optimizing?
>> Balasubramanian Sivan: No. No. Let me tell you something. This is one of the most famous
things why people change from generalize first prize to…
>>: Yeah, but that's why [indiscernible] but there are many companies who just exactly do this.
GM won't go to Google and bid on ads for them. GM will actually ad on another company who
will make ads for them on Google and that company is optimizing.
>> Balasubramanian Sivan: So earlier, so the current rule is…
>>: [indiscernible] that's what I should do. I will have a done bid in the morning or by
afternoon I will go long, that's what I will go back.
>>: So big companies have enough research to do that.
>>: Yeah but, that's fair.
>> Balasubramanian Sivan: So for about one month this happened in nearly 2000s when the
[indiscernible] once to do a generalized first-place auction, not a second-place auction as it is
generally run, where if I get shown I pay what I did.
>>: Who was that?
>> Balasubramanian Sivan: That was Yahoo and Overture in the beginning, and Google came
and changed it. At that point, for one month they documented it. There were two bidders who
were basically competing, the subtle behavior, right? I bid something and you bid slightly a
little bit more than and then I bid some level and it dropped down to zero and I said I don't
care. And then once I don't care you realize that there's no point in bidding so high. Then you
bid so low. For one month they saw this behavior. The bots were bidding this.
>>: What did you say at the end?
>> Balasubramanian Sivan: The bots were bidding. Robots were bidding.
>>: Robots were bidding?
>> Balasubramanian Sivan: Yeah. So the point is if you don't take instances into account, these
kinds of things can happen. And they did happen, just giving you an example.
>>: What about these [indiscernible] effects?
>> Balasubramanian Sivan: Okay. They are exist and so everything is forever distribution
[indiscernible] one-sided coming from [indiscernible] and it holds for every distribution and I
just calculated down particularly for this. Before the infinite horizon we just proved for uniform
zero, one. It is not intended for other distributions [indiscernible].
>>: Even uniform discrete?
>> Balasubramanian Sivan: Even uniform discrete we haven't done it.
>> Yuval Peres: Let's thank Balasubramanian again. [applause]