ECOS3012 Strategic Behaviour
Quiz for Lecture 10 solution
1. A seller is recommending a product to a buyer. The price for this product is 5. If the buyer
buys, the true value she gets from this product is either v = 2 or v = 10. Suppose that the
buyer’s prior belief is Pr(v= 2) = 0.7 and Pr(v= 10) = 0.3.
Without additional information, will the buyer buy this product?
A: No.
Her expected gain from this purchase = expected value - price = (2×0.7+10×0.3)−
5 =−0.6 < 0.
She strictly prefers not to buy this product.
2. (Continue)
Suppose that the seller knows the true value. Based on this knowledge, he can "highly
recommend" this product or "stay silent".
What can be the probability that this seller will highly recommend the product when v = 2
in a perfect Bayesian equilibrium?
A: Any value between 0 and 1 (including 0 and 1) is correct.
This is a "cheap talk" scenario because it doesn’t cost the seller anything to say that he
"highly recommends" the product.
In the lecture, we discussed the following PBE: the seller always highly recommends
the product regardless of its true value (i.e., the probability that this seller will highly
recommend the product when v= 2 is 1).
However, this is not the unique PBE. To see why, let p be a value in [0,1]. Let the
seller’s strategy be “always recommend the product with probability p regardless of v.
”.
When p= 1, we discussed in class that this is a PBE strategy.
When p= 0, the seller never recommends. The buyer’s posterior belief is equal to her
prior belief if the seller stays silent, and therefore she does not buy. Assume that in the
zero-probability event “seller recommends”, the buyer’s posterior belief is still such
that she does not buy. Given that the buyer never buys in any event, the seller does not
strictly profit from deviating to a different strategy, so this is a PBE.
When p ∈ (0,1), the seller recommends with the same probability regardless of v.
This makes his recommendation (or the absence of recommendation) white noise: the
buyer’s posterior belief is equal to her prior belief with or without recommendation,
and she never buys. Again, given that the buyer never buys in any event, the seller does
not strictly profit from deviating to a different strategy, so this is a PBE.
3. (Continue)
In a perfect Bayesian equilibrium, what can be the probability that the buyer buys the product
when it’s highly recommended?
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A: 0.
In all PBE discussed in Q2, the seller’s recommendation (or absence of recommenda-
tion) is completely uninformative. The buyer relies on only her prior belief and chooses
not to buy the product.
4. A seller is selling a product to a buyer at price = $50 (this means that the seller’s payoff is
50 if the product is sold and 0 otherwise).
If the buyer buys, the true value she gets from this product is either v = 0 or v = 100. The
buyer buys the product only if Pr(v= 100)≥ 0.5. Her prior belief is Pr(v= 100) = 0.4.
Suppose that the seller knows the true value. Based on this knowledge, he can ask a pre-
vious customer, Jiemai, to recommend this product to the new buyer. If the true value is
v = 100, Jiemai is happy to give her recommendation for free. However, if v = 0, Jiemai is
unwilling to give the recommendation unless she receives a $20 bribe from the seller. Ev-
eryone (including the buyer) knows Jiemai can be bribed with $20.
Is there a perfect Bayesian equilibrium in which the seller always pays Jiemai to recommend
the product when v= 0?
A: No. If Jiemai always recommends the product, her recommendation is completely unin-
formative. The buyer relies only on the prior belief and chooses not to buy the product.
Since Jiemai’s recommendation is useless, the seller does not have any incentive to pay
for it when v= 0 => a profitable deviation.
5. (Continue) Is there a perfect Bayesian equilibrium in which Jiemai recommends the product
if and only if v= 100?
A: No. If Jiemai recommends the product only when v = 100, then the buyer buys the
product when Jiemai recommends it. However, if this is the case, the seller has an
incentive to pay Jiemai the bribe when v = 0, because the value of her recommendation
($50) is higher than the cost ($20) => a profitable deviation.
6. (Continue) Suppose that the seller pays for Jiemai’s recommendation with some probability
0 < p< 1 when v= 0.
What is Pr(buyer buys product | Jiemai recommends product) in a perfect Bayesian equilib-
rium?
A: 0.4. If q= Pr(buy | Jiemai’s recommendation), then the value of Jiemai’s recommenda-
tion is 50×q.
The seller is randomising, so he must be indifferent about paying $20 to gain $50q. In
other words, 50q−20 = 0 => q= 0.4.
7. (Continue) If 0< Pr(buyer buys product | Jiemai’s recommendation) < 1, what is Pr(Jiemai’s
recommendation | v= 0)?
A: 0.6667. The buyer must be indifferent when she buys the product, which means that
Pr(v= 100 | Jiemai’s recommendation) = 0.5.
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Applying Bayes’ rule, let Pr(Jiemai’s recommendation | v= 0) = q, then
Pr(v= 100 | Jiemai’s recommendation) = Pr(Jiemai’s recommendation | v= 100)×Pr(v=
100)/Pr(Jiemai’s recommendation)
= 0.4/(0.4+0.6q) = 2/(2+3q)
Let 2/(2+3q) = 0.5.
2+3q= 4 => q= 2/3 = 0.6667.
8. (Continue) Suppose that Jiemai is unavailable. The seller considers hiring another previous
customer, Jaimie, to rate the product online. Jaimie can choose a rating from zero to five
stars. If v= 0, Jaimie charges the seller $20 for each star she gives (e.g., she charges $40 to
give two stars). If v= 100, she charges $15 for each star she gives.
Let star0 be the number of stars Jaimie gives in her rating if v= 0; let star100 be the number
of stars Jaimie gives in her rating if v= 100.
Is there a perfect Bayesian equilibrium in which star0 = star100 with probability 1?
(a) Yes, and the star rating (star0,star100) is unique in this type of perfect Bayesian equi-
librium.
(b) Yes, and there are multiple perfect Bayesian equilibria of this type with different star
ratings.
(c) No.
A: Yes, and the star rating (star0,star100) is unique in this type of perfect Bayesian equilib-
rium: star0 = star100 = 0.
When star0 = star100 with probability 1, the buyer does not gain any additional information
from the star rating. Therefore, the buyer simply relies on her prior belief and chooses not
to purchase the product.
If star0 = star100 > 0 with probability 1, the seller has paid a positive amount of money for
nothing. Therefore, he has a profitable deviation to not paying for any star regardless of v.
If star0 = star100 = 0 with probability 1, whether the seller has a profitable deviation or not
depends on the buyer’s belief in the zero-probability event “the product’s rating is one-star
or above”. If the buyer believes that some positive number of stars means Pr(v= 100)≥ 0.5,
then the seller indeed has a profitable deviation when v = 100: he will want to pay for that
positive star rating to sell the product. However, if the buyer believes that Pr(v= 100)< 0.5
no matter how many stars the seller gets, then the buyer never buys and the seller never
has an incentive to pay for a higher star rating regardless of v. This leads to a PBE with
star0 = star100 = 0.
9. (Continue) Is there a perfect Bayesian equilibrium in which star0 < star100 with probability
1?
(a) Yes, and the star rating (star0,star100) is unique in this type of perfect Bayesian equi-
librium.
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(b) Yes, and there are multiple perfect Bayesian equilibria of this type with different star
ratings.
(c) No.
A: Yes, and the star rating (star0,star100) is unique in this type of perfect Bayesian equilib-
rium: star0 = 0, star100 = 3.
When star0 < star100 with probability 1, this is a separating equilibrium and the star rating
perfectly reveals the true value of the product. This implies that the buyer never buys when
the rating is star0 and always buys when the rating is star100.
Therefore, if this separating equilibrium exists, star0 must be equal to 0, because the seller
will not pay for a positive rating that is ineffective.
Suppose that star100 = 1 or 2. This cannot be an equilibrium because the seller has a prof-
itable deviation when v= 0: he will pay $20 or $40 to mimic the same star rating in order to
sell the product and gain $50.
Suppose that star100 = 3. Also suppose that the buyer’s belief in the zero-probability event
“the rating is one or two stars” is “v= 0 for sure”, so that she does not purchase the product
when the star rating is below 3. A seller with v = 0 does not have a profitable deviation
because he must get three stars in order to sell the product, but the cost of mimicking three
stars ($60) is higher than the gain ($50). A seller with v= 100 does not have an incentive to
pay for fewer than 3 stars because he would lost the customer. He does not have an incentive
to pay for more than 3 stars because he cannot afford it.
There does not exist any PBE with star100 > 3 because the cost is higher than the benefit
even when v= 100.
Therefore, in a separating equilibrium with star0 < star100, we must have star0 = 0 and
star100 = 3.
10. (Continue) Suppose, instead, that Jaimie charges the seller $20 for each star she gives when
v= 0, but she now charges only $5 for each star if v= 100.
Is there a perfect Bayesian equilibrium in which star0 = star100 with probability 1?
(a) Yes, and the star rating (star0,star100) is unique in this type of perfect Bayesian equi-
librium.
(b) Yes, and there are multiple perfect Bayesian equilibria of this type with different star
ratings.
(c) No.
A: Yes, and the star rating (star0,star100) is unique in this type of perfect Bayesian equilib-
rium: star0 = star100 = 0. See answer for Q8.
11. (Continue) Is there a perfect Bayesian equilibrium in which star0 < star100 with probability
1?
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(a) Yes, and the star rating (star0,star100) is unique in this type of perfect Bayesian equi-
librium.
(b) Yes, and there are multiple perfect Bayesian equilibria of this type with different star
ratings.
(c) No.
A: Yes, and there are multiple perfect Bayesian equilibria of this type with different star
ratings. star0 = 0 and star100 = 3 or 4 or 5.
When star0 < star100 with probability 1, this is a separating equilibrium and the star rating
perfectly reveals the true value of the product. This implies that the buyer never buys when
the rating is star0 and always buys when the rating is star100.
Therefore, if this separating equilibrium exists, star0 must be equal to 0, because the seller
will not pay for a positive rating that is ineffective.
Suppose that star100 = 1 or 2. This cannot be an equilibrium because the seller has a prof-
itable deviation when v= 0: he will pay $20 or $40 to mimic the same star rating in order to
sell the product and gain $50.
Suppose that star100 = x for x = 3,4,5. Also suppose that the buyer’s belief in the zero-
probability event “the rating is y stars for 0 < y< x” is “v= 0 for sure”, so that she does not
purchase the product when the star rating is below x. A seller with v = 0 does not have a
profitable deviation because he must get x stars in order to sell the product, but the cost of
x stars (at least $60) is higher than the gain ($50). A seller with v = 100 can easily affort x
stars because of the low cost, and he does not have an incentive to pay for fewer than x stars
because he would lost the customer.
Therefore, multiple separating equilibria with star0 < star100 exist. star0 = 0 and star100 = 3
or 4 or 5.