Continuous distributions

Chapter IV: Weak and Strong Ties in Social Network

B.2 Continuous distributions

From the proof of Proposition 16 we know that when𝑞is fixed𝑁behaves as𝑂(ln𝛾). Now, let us transform the first term into a more clear form

𝑢(𝑁) −𝑞 =

𝑞^𝑁

𝑞^𝑁 + (1−𝑞)^𝑁 −𝑞

−

𝑞^𝑁−(1−𝑞)^𝑁 𝑞^𝑁+(1−𝑞)^𝑁

1−𝑞 𝑞

^𝑁 1−𝑞

𝑞

^𝑁 +1

−𝑂(𝛾 𝑁)

As𝑁 → ∞and𝑞 > ¹

2, ( (1−𝑞)/𝑞)^𝑁 →0. Suppose( (1−𝑞)/𝑞)^𝑁 =𝜈then 𝑞^𝑁

𝑞^𝑁 + (1−𝑞)^𝑁 ≥ 1−𝜈

and

𝑞^𝑁−(1−𝑞)^𝑁 𝑞^𝑁+(1−𝑞)^𝑁

1−𝑞 𝑞

^𝑁 1−𝑞

𝑞

^𝑁 +1

≤ 𝜈

𝜈+1

≤ 𝜈 . From the proof of Theorem 16𝜈 =𝑂(𝛾). Hence,

𝑢(𝑁) −𝑞 ≥ 1−𝑞−2𝜈−𝑂(𝛾 𝑁) =1−𝑞−𝑂(𝛾ln𝛾).

It means that as𝑁 → ∞,𝑢(𝑁)goes to its maximal value as𝐶( (1−𝑞)/𝑞)^𝑁·𝑁 → 0, which is quicker than ( (1−𝑞)/𝑞)^{𝑁 𝑘}, for any𝑘 < 1. Also, as optimal𝑁^∗increases then absorbing beliefs are further away from 1/2.

the world is never fully revealed if signals have bounded strength. To prove this, we look at what are the expected loss and gain, when the social planner chooses a price 𝑘_𝑡 ≠ 1. As we do not know the exact formula for the expected utility function, we first bound the expected gain in the public belief and then translate it to the expected utility gain. The latter one uses the fact that𝑢(𝑝) is convex and the absolute value of its derivative is less than 1.

Bounded private signals

Suppose without loss of generality that the modified LR satisfies 𝑞/(1− 𝑞) ≥

𝑝𝑡

(1−𝑝𝑡)𝑘𝑡

≥ 1/2, so the 𝐻𝑖𝑔 ℎ state is more likely and we are still in the learning period (less than𝑞/(1−𝑞)). Given this, the total modified belief of agent𝑡 will be in the interval

𝑝_𝑡 (1− 𝑝_𝑡)𝑘_𝑡

· 1−𝑞 𝑞 ;

𝑝_𝑡 (1− 𝑝_𝑡)𝑘_𝑡

· 𝑞 1−𝑞

.

In order to get the final modified likelihood-ratio equal to𝑦in the interval above we need

𝑔_𝐻(𝑥)

𝑔_𝐿(𝑥) =𝑦· (1− 𝑝_𝑡)𝑘_𝑡 𝑝_𝑡

·

Before we calculate the expected loss and the expected gain from the price 𝑘_𝑡 we need a few more facts.

Note 40. The public belief and the corresponding likelihood ratio satisfy the follow- ing relation

𝑝_𝑡 = 𝑙_𝑡 1+𝑙_𝑡

·

Furthermore, if the total belief 𝜇_𝑡 > 1/2 and agent𝑡 takes the action 0, then the expected loss is 𝜇_𝑡− (1−𝜇_𝑡) =2𝜇_𝑡−1.

The following lemma helps us understand how the change in the LR translates to the change in the public belief.

Lemma 41. If𝑙_𝑡+

1=𝑙_𝑡(1+𝛿) then 𝑝_𝑡+

1− 𝑝_𝑡 =𝛿 𝑝_𝑡(1− 𝑝_𝑡) +𝑜(𝛿(1−𝑝_𝑡)).

Proof of Lemma 41. By the definition of the likelihood ratio 𝑝_𝑡

1− 𝑝_𝑡

=𝑥 𝑝_𝑡+

1

1− 𝑝_𝑡+

1

=𝑥(1+𝛿), then

𝑝_𝑡+

1− 𝑝_𝑡 = 𝑥+𝑥 𝛿 1+𝑥+𝑥 𝛿

− 𝑥

1+𝑥

= 𝑥 𝛿

(1+𝑥) (1+𝑥+𝑥 𝛿)

= 𝛿 𝑝_𝑡(1−𝑝_𝑡)

(1− 𝑝_𝑡+𝑝_𝑡) (1− 𝑝_𝑡+𝑝_𝑡(1+𝛿))

=𝛿 𝑝_𝑡(1−𝑝_𝑡) +𝑜(𝛿(1− 𝑝_𝑡)).

Furthermore, the following relation is satisfied between the PDFs of the likelihood ratio (L. Smith and Sørensen, 2000)

𝑓_𝐻(𝑥) =𝑥 𝑓_𝐿(𝑥). Therefore,

𝐹_𝐻(𝑥) =

∫ 𝑥

1−𝑞 𝑞

𝑓_𝐻(𝑦)𝑑𝑦

=

∫ ^𝑥

1−𝑞 𝑞

𝑦 𝑓_𝐿(𝑦)𝑑𝑦

≤ 𝑥 𝐹_𝐿(𝑥)

(B.1)

These facts help us prove one of the main results, that if private signals are bounded then the full revelation of the underlying state is not possible, unless𝛿 =1.

Proof of Theorem 18. Let us start with the expected loss which is equal to

∫ 𝑏 𝑎

2 𝑥

𝑝_𝑡 1−𝑝_𝑡

1+𝑥 ^𝑝𝑡

1−𝑝𝑡

−1

!

(𝑝_𝑡𝑓_𝐻 (𝑥) + (1− 𝑝_𝑡)𝑓_𝐿(𝑥))𝑑𝑥

≥ 𝑐

1𝑝_𝑡

𝐹_𝐻

1−𝑝_𝑡 𝑝_𝑡

𝑘_𝑡

−𝐹_𝐻

1−𝑞 𝑞

+ (1− 𝑝_𝑡)

𝐹_𝐿

1− 𝑝_𝑡 𝑝_𝑡

𝑘_𝑡

−𝐹_𝐿

1−𝑞 𝑞

=𝑐

1

𝑝_𝑡𝐹_𝐻

1− 𝑝_𝑡 𝑝_𝑡

𝑘_𝑡

+ (1− 𝑝_𝑡)𝐹_𝐿

1−𝑝_𝑡 𝑝_𝑡

𝑘_𝑡 ,

where𝑎 = (1−𝑞)/𝑞, 𝑏 = (1− 𝑝_𝑡)𝑘_𝑡/𝑝_𝑡 and𝑐

1gets arbitrary close to 1 as 𝑝_𝑡 goes to 1. Notice that we can make(1−𝑝_𝑡)𝑘_𝑡/𝑝_𝑡as close to(1−𝑞)/𝑞as we want, which is the lowest value for the likelihood ratio.

Now let us find the expected gain. Remember that the modified public belief is 𝑝_𝑡/( (1− 𝑝_𝑡)𝑘_𝑡). Hence we get the following expressions for 𝑝_𝑡+

1/(1− 𝑝_𝑡+

1) 𝑝_𝑡+

1

1− 𝑝_𝑡+

1

= 𝑝_𝑡 1− 𝑝_𝑡

·

1−𝐹_{𝐻 (}

1−𝑝𝑡)𝑘𝑡

𝑝𝑡

1−𝐹_{𝐿 (}

1−𝑝𝑡)𝑘𝑡

𝑝𝑡

if player𝑡 buys the new product (random walk goes up) and

𝑝_𝑡+

1

1−𝑝_𝑡+

1

= 𝑝_𝑡 1−𝑝_𝑡

· 𝐹_𝐻

₍

1−𝑝𝑡)𝑘𝑡

𝑝𝑡

𝐹_{𝐿 (}

1−𝑝_𝑡)𝑘_𝑡 𝑝_𝑡

if she does not (random walk goes down). Here, underline and overline represent possible LRs that can result from the current public belief𝑝_𝑡and price𝑘_𝑡depending on the action of player𝑡.

As utility function on [0.5,1] is convex and symmetric around 0.5 we know that expected gain is less than𝑢(𝑝_𝑡+

1) −𝑢(𝑝_𝑡)(as if we go up with probability 1). Which in its turn is less than𝑝_𝑡+

1−𝑝_𝑡as𝑢is also above 45 degree line and at 1 is equal to 1. In order to calculate the latter we should find(𝑝_𝑡+

1/(1− 𝑝_𝑡+

1)). We are thinking about(1− 𝑝_𝑡)𝑘_𝑡/𝑝_𝑡 as a point close to the left end of the domain of𝐹_𝐻 and𝐹_𝐿, so in the following calculation we are using notation𝜀= (1− 𝑝_𝑡)𝑘_𝑡/𝑝_𝑡.

𝑝_𝑡+

1

1− 𝑝_𝑡+

1

= 𝑝_𝑡 1− 𝑝_𝑡

©

­

­

« 1−𝐹_𝐻

₍

1−𝑝_𝑡)𝑘_𝑡 𝑝_𝑡

1−𝐹_{𝐿 (}

1−𝑝𝑡)𝑘𝑡

𝑝𝑡

ª

®

®

¬

≤ 𝑝_𝑡 1−𝑝_𝑡

1−𝑐

2𝐹_𝐿(𝜀)) (1+𝐹_𝐿(𝜀) +𝑂(𝐹₋²(𝜀))

= 𝑝_𝑡 1− 𝑝_𝑡

(1+𝐹_𝐿(𝜀) (1−𝑐

2) +𝑂(𝐹²

𝐿(𝜀))) where𝑐

2 > 0 is a constant comes from (B.1). And hence, 𝛿(𝑝_𝑡+

1− 𝑝_𝑡) ≤𝛿

𝐹_𝐿(𝜀) (1−𝑐

2) (1− 𝑝_𝑡) 𝑝_𝑡

+𝑂(𝐹²

𝐿(𝜀))

= 𝐹_𝐿(𝜀) (𝛿(1− 𝑝_𝑡+𝑝_𝑡𝑐

2−𝑐

2)) 𝑝_𝑡

+𝑂(𝐹²

𝐿(𝜀)). Comparing it to𝐹_𝐿(𝜀) ( (1−𝛿) (1−𝑝_𝑡+𝑝_𝑡𝑐

2))𝑐

1tells us that for 𝑝_𝑡 →1 coefficient 𝛿(1−𝑝_𝑡+ 𝑝_𝑡𝑐

2−𝑐

2)/(𝑝_𝑡)goes to 0 whereas the other one goes to(1−𝛿)𝑐

2 > 0.

Unbounded private signals

Now we are going to consider an example of unbounded private signals. This is the only situation, when it is appropriate to talk about the asymptotic speed of learning, as in other cases the public belief does not converge to 0 or 1. The second main result is Theorem 19, which says that the optimal prices,𝑘_𝑡, are bounded away from 0 and

∞ and that it is optimal to choose a positive price, 𝑘_𝑡 > 1, when the public belief is high enough. We need to assume that 𝑝_𝑡 is high due to the lack of information about𝑢 and its derivative. As we mentioned above, if we use the results from the numerical calculation, it can be generalized for 𝑝_𝑡 ≥ 1/2.

One of the corollaries is that we indeed have the asymptotic learning and the asymptotic speed of learning is also significantly increased, due to taxation of the good that is more likely to be better. The situation when 𝑝_𝑡 < 1/2 is symmetric.

Recall that the CDFs of the likelihood ratios are 𝐹_𝐻(𝑦)=P

𝑔_𝐻(𝑠) 𝑔_𝐿(𝑠) ≤ 𝑦

𝜃 =𝐻𝑖𝑔 ℎ

= 𝑦² (1+𝑦)² 𝐹_𝐿(𝑦) =P

𝑔_𝐻(𝑠) 𝑔_𝐿(𝑠) ≤ 𝑦

𝜃 =𝐿 𝑜𝑤

= 𝑦2+2𝑦 (1+𝑦)², for𝑦 ∈ [0,∞].

Proof of Theorem 19. Recall that if we apply non zero price 𝑘_𝑡 then we have the expected loss due to non optimal actions and the expected gain from bigger expected increase in public belief 𝑝_𝑡+

1. We start with the former one.

Suppose we have a public belief 𝑝_𝑡and a price at this period is 𝑘_𝑡. As was stated in the Section 3.1, agent𝑡is going to buy the new productiff

𝑝_𝑡 (1− 𝑝_𝑡)𝑘_𝑡

· 𝑔_𝐻(𝑥) 𝑔_𝐿(𝑥) ≥ 1.

Notice, that a player 𝑡 will take a non optimal action only if her likelihood belief with price 𝑘_𝑡 is less than 1 and her likelihood belief without the price is above 1.

More formally private likelihood ratio can take the following values (1− 𝑝_𝑡)𝑘_𝑡

𝑝_𝑡

≥ 𝑔_𝐻(𝑥)

𝑔_𝐿(𝑥) ≥ 1−𝑝_𝑡 𝑝_𝑡

If a person𝑡 gets a private likelihood belief in this interval and her total likelihood (without accounting for the price) is equal to 𝑦 then the loss, due to taking a non

optimal action, is equal to

2𝑦 1+𝑦

−1.

Therefore, expected loss𝑙(𝑝_𝑡, 𝑘_𝑡) has the following form 𝑙(𝑝_𝑡, 𝑘_𝑡) =

∫ ^𝑏

𝑎

©

­

«

2𝑥^{(1−𝑝𝑡)}

𝑝𝑡

𝑥^{(1−𝑝𝑡)}

𝑝𝑡

+1

−1ª

®

¬

(𝑝_𝑡𝑓_𝐻(𝑥) + (1−𝑝_𝑡)𝑓_𝐿(𝑥))𝑑𝑥

≤ 2𝑘_𝑡

𝑘_𝑡+1

−1 ∫ ^𝑏

𝑎

(𝑝_𝑡𝑓_𝐻 (𝑥) + (1− 𝑝_𝑡)𝑓_𝐿 (𝑥))𝑑𝑥

=

𝑘_𝑡−1 𝑘_𝑡+1

𝑝_𝑡

𝐹_𝐻

(1−𝑝_𝑡)𝑘_𝑡 𝑝_𝑡

−𝐹_𝐻

1−𝑝_𝑡 𝑝_𝑡

+ + (1− 𝑝_𝑡)

𝐹_𝐿

(1−𝑝_𝑡)𝑘_𝑡 𝑝_𝑡

−𝐹_𝐿

1−𝑝_𝑡 𝑝_𝑡

! ,

where𝑎= ^{(1−𝑝𝑡)}

𝑝𝑡 and𝑏 = ^{(1−𝑝𝑡)𝑘𝑡}

𝑝𝑡 . After pluging in the expressions for𝐹_𝑖 we get 𝑙(𝑝_𝑡, 𝑘_𝑡) = ( (−1+𝑘_𝑡)²𝑝²

𝑡(1− 𝑝_𝑡)²) (𝑘_𝑡+ 𝑝_𝑡 −𝑘_𝑡𝑝_𝑡)² ·

Now let us go to the expected gain term. Suppose that in period𝑡 the public belief 𝑝_𝑡 > 1/2 is high enough. If there is no price, 𝑘_𝑡 = 1, then the public belief in the next period goes to either𝑎or𝑏depending on the action of player𝑡(buying the new and the old products correspondingly). And if we apply a price 𝑘_𝑡 then the public belief in the next period goes to𝑎⁰and𝑏⁰correspondingly

𝑎= 2− 𝑝_𝑡 3−2𝑝_𝑡

, 𝑏= 𝑝_𝑡 (1+2𝑝_𝑡), 𝑎⁰= (2𝑘_𝑡(1−𝑝_𝑡) +𝑝_𝑡)

(2𝑘_𝑡(1−𝑝) +1) , 𝑏⁰= 𝑘_𝑡𝑝_𝑡 (𝑘_𝑡 +2𝑝_𝑡)

As the public belief is a martingale then in expectation it does not move. This means, that the expected gain is equal to the distance between two chords that connect𝑢(𝑎), 𝑢(𝑏) and𝑢(𝑎⁰), 𝑢(𝑏⁰) at point 𝑝_𝑡. This is depicted in Figure B.1. In other words, the expected gain is equal to the distance between 𝑑 and 𝑑⁰, where𝑑 belongs to the chord𝑢(𝑎), 𝑢(𝑏)and𝑑⁰to the chord𝑢(𝑎⁰)𝑢(𝑏⁰)and their𝑝-coordinate is 𝑝_𝑡. So the distance between𝑑⁰and𝑑 is alongside the𝑦-coordinate.

Let us calculate this distance. Denote by𝑦_𝑎and𝑦_𝑏-𝑦-coordinates of𝑎and𝑏. Notice, that the𝑦-coordinate of𝑎⁰and𝑏⁰are equal to 𝑦_𝑎+𝑎

1(𝑎⁰−𝑎) and𝑦_𝑏−𝑏

1(𝑏⁰− 𝑏) where𝑎

1, 𝑏

1are some constants that are in [𝑢⁰(𝑎), 𝑢⁰(𝑎⁰)]and [−𝑢⁰(𝑏⁰),−𝑢⁰(𝑏)].

Where𝑑and𝑑⁰are the𝑦-coordinates of point𝑝_𝑡on the cords𝑢(𝑎)𝑢(𝑏)and𝑢(𝑎⁰)𝑢(𝑏⁰)correspondingly.

Figure B.1: The expected gain induced by the cost.

𝑑 =𝑦_𝑏+ 𝑦_𝑎−𝑦_𝑏 𝑎−𝑏

(𝑝_𝑡−𝑏) 𝑑⁰=𝑦_𝑏−𝑏

1(𝑏⁰−𝑏) + 𝑦_𝑎− 𝑦_𝑏+𝑎

1(𝑎⁰−𝑎) +𝑏

1(𝑏⁰−𝑏) 𝑎⁰−𝑏⁰

(𝑝−𝑏⁰). Pluging in the expressions for𝑎, 𝑏, 𝑎⁰, 𝑏⁰gives us the following formula

expected gain(𝑝_𝑡, 𝑘_𝑡) =𝑑⁰−𝑑 = ( (2𝑎

1(−1+𝑘_𝑡) (−1+ 𝑝_𝑡)²𝑝²

𝑡) ( (3−2𝑝_𝑡) (𝑘_𝑡+𝑝_𝑡−𝑘_𝑡𝑝_𝑡)²))−

− (2𝑏

1(−1+𝑘_𝑡)𝑘_𝑡(−1+𝑝_𝑡)²𝑝2 𝑡) ( (1+2𝑝_𝑡) (𝑘_𝑡+𝑝_𝑡−𝑘_𝑡𝑝_𝑡)²) + + ( (−1+𝑘_𝑡) (−1+ 𝑝_𝑡)²𝑝2

𝑡(−1−3𝑘_𝑡+2(−1+𝑘_𝑡)𝑝_𝑡) (𝑦_𝑎−𝑦_𝑏)) (𝑘_𝑡+ 𝑝_𝑡−𝑘_𝑡𝑝_𝑡)² =

= (−1+𝑘_𝑡) (1−𝑝_𝑡)²𝑝2 𝑡

(𝑘_𝑡+ 𝑝_𝑡 −𝑘_𝑡𝑝_𝑡)²

2𝑎

1

3−2𝑝_𝑡

− 2𝑏

1𝑘_𝑡 1+2𝑝_𝑡

− (1+3𝑘_𝑡−2(−1+𝑘_𝑡)𝑝_𝑡) (𝑦_𝑎−𝑦_𝑏))

.

Denote by𝑃the terms in the parenthesis 𝑃(𝑝_𝑡, 𝑘_𝑡)=

2𝑎

1

3−2𝑝_𝑡

− 2𝑏

1𝑘_𝑡 1+2𝑝_𝑡

− (1+3𝑘_𝑡−2(−1+𝑘_𝑡)𝑝_𝑡) (𝑦_𝑎−𝑦_𝑏))

.

Notice, that 𝑦_𝑎 ≥ 𝑦_𝑏 as𝑎is closer to 1 than𝑏is to 0 for𝑝_𝑡 ≥1/2 2− 𝑝_𝑡

3−2𝑝_𝑡

−

1− 𝑝_𝑡 1+2𝑝_𝑡

= 2𝑝_𝑡−1

(3−2𝑝_𝑡) (1+2𝑝_𝑡) ≥ 0.

Let us now compare the expected gain and the expected loss due to the price𝑘_𝑡.We do this by subtracting the former one by the latter and taking the discount factor into account

𝛿

1−𝛿expected gain−expected loss= (−1+𝑘_𝑡) (1− 𝑝_𝑡)²𝑝²

𝑡

( (𝑘_𝑡 +𝑝_𝑡−𝑘_𝑡𝑝_𝑡)² · 𝛿

1−𝛿 2𝑎

1

3−2𝑝_𝑡

− 2𝑏

1𝑘_𝑡 1+2𝑝_𝑡

− (1+3𝑘_𝑡−2(−1+𝑘_𝑡)𝑝_𝑡) (𝑦_𝑎−𝑦_𝑏)

− (𝑘_𝑡−1)

=

= 𝛿

1−𝛿

𝑃(𝑝_𝑡, 𝑘_𝑡) −𝑘_𝑡+1

· (−1+𝑘_𝑡) (1− 𝑝_𝑡)²𝑝2 𝑡

( (𝑘_𝑡+𝑝_𝑡−𝑘_𝑡𝑝_𝑡)²

We can see that the sign of this expression is defined by the parenthesis and 𝑘_𝑡−1 terms.

Suppose that the expression in the parenthesis is positive for some 𝑘_𝑡 > 1 (and 𝑝_𝑡 ∉{0,1})

𝛿 1−𝛿

2𝑎

1

3−2𝑝_𝑡

− 2𝑏

1𝑘_𝑡 1+2𝑝_𝑡

− (1+3𝑘_𝑡−2(−1+𝑘_𝑡)𝑝_𝑡) (𝑦_𝑎−𝑦_𝑏)

− (𝑘_𝑡−1) > 0 (B.2) Then𝑑⁰−𝑑is positive as the sign here is defined by this parenthesis and (𝑘_𝑡−1). If we now make𝑘less then 1, the expression above will increase and𝑘_𝑡−1 will become negative, which will make the expected gain also negative. Therefore, if (B.2) holds for some𝑘_𝑡 > 1 then the optimal𝑘^∗

𝑡 is not less than 1.

Now let see whether there exists 𝑘_𝑡 > 1 which gives higher utility than 𝑘_𝑡 =1 for high enough 𝑝_𝑡. As we said before, it is enough to show that (B.2) holds for some 𝑘_𝑡 > 1. The main challenge here is that we do not the utility function𝑢 and how convex it is.

Still, we can say the following: 𝑦_𝑎− 𝑦_𝑏 < 𝑎

1(𝑎 −𝑏) < 𝑎

1/3 and 𝑏

1 < 𝑎

1. This means that for 𝑝_𝑡 close enough to 1

𝑃(𝑝_𝑡, 𝑘_𝑡) >

2𝑎

1

3−2𝑝_𝑡

− 2𝑎

1𝑘_𝑡 1+2𝑝_𝑡

− (1+3𝑘_𝑡−2(−1+𝑘_𝑡)𝑝_𝑡)𝑎

1

3 )

.

For𝑘_𝑡 =1 this function is increasing in 𝑝_𝑡 and at𝑝_𝑡 =1 it is 0. The expected loss is also 0 for𝑘_𝑡 =1. But as the bounds for𝑦_𝑎−𝑦_𝑏and𝑏

1are not tight,𝑃 >0 at 1. As 𝑃and−(𝑘_𝑡−1) are also continuous functions of𝑘_𝑡 there exists a threshold 𝑝such that for any 𝑝_𝑡 > 𝑝the expected gain is positive for some𝑘_𝑡 =1+𝜀. In other words, there exists 𝑘_𝑡 > 1 such that it is better to choose than 𝑘_𝑡 = 1 for 𝑝_𝑡 high enough.

Moreover, if we use tighter constraints that we get from section 3.4 we can see that it holds for 𝑝_𝑡 > 1/2.

Furthermore, there exists neighborhoods of 0 and∞ such that the optimal 𝑘^∗

𝑡 does not lie in them for any𝑝_𝑡.

Let us start with 0

𝑃(𝑝_𝑡,0) = 2𝑎

1

3−2𝑝_𝑡

− (1+2𝑝_𝑡) (𝑦_𝑎− 𝑦_𝑏)

≥ 2𝑎

1

3−2𝑝_𝑡

− (1+2𝑝_𝑡)𝑎

1

3 , where the inequality comes from the fact that (𝑦_𝑎 − 𝑦_𝑏) ≤ 𝑎

1(𝑎 − 𝑏) ≤ 𝑎

1/3. Furthermore,

2 3−2𝑝_𝑡

> 1+2𝑝_𝑡 3

,

as(3−2𝑝_𝑡) (1+2𝑝_𝑡) ≤4.Thus, for𝑘 in some neighborhood of 0𝑃(𝑝_𝑡, 𝑘) > 0 and 𝑘−1< 0. Hence, there exists𝑘 >0 such that 𝑘^∗ > 𝑘.

Also, it is obvious that for high enough 𝑘_𝑡, 𝑃(𝑝_𝑡, 𝑘_𝑡) is negative. There is another case when 𝑏⁰ jumps to the right side of 1/2 such that 𝑦⁰

𝑏 becomes higher than 𝑦_𝑏. Then we have a bit different expression for𝑑⁰

𝑑⁰=𝑦_𝑏+𝑏

1(𝑏⁰− (1−𝑏)) + 𝑦_𝑎−𝑦_𝑏+𝑎

1(𝑎⁰−𝑎) −𝑏

1(𝑏⁰− (1−𝑏)) 𝑎⁰−𝑏⁰

(𝑝−𝑏⁰). And hence,

𝑑⁰−𝑑−expected loss(𝑝_𝑡, 𝑘_𝑡) = (−1+ 𝑝_𝑡)² (𝑘_𝑡 +𝑝_𝑡−𝑘_𝑡𝑝_𝑡)²

2𝑎

1(−1+𝑘_𝑡)𝑝2 𝑡

(3−2𝑝_𝑡) + + 𝑏1𝑘_𝑡(−2𝑝_𝑡(1+ 𝑝_𝑡) +𝑘_𝑡(−1+2𝑝²

𝑡)) 1+2𝑝_𝑡

+ (−1+𝑘_𝑡)𝑝²

𝑡(−1−3𝑘_𝑡+2(−1+𝑘_𝑡)𝑝_𝑡)×

× (𝑦_𝑎−𝑦_𝑏)) − (𝑘_𝑡−1)²𝑝²

𝑡

! .

Define the term in the big parenthesis by𝑃⁰(𝑝_𝑡, 𝑘_𝑡)and notice that 𝑃⁰(𝑝_𝑡, 𝑘_𝑡) ≤ 2𝑎

1(𝑘_𝑡−1)𝑝²

𝑡 + 𝑏

1𝑘_𝑡 ·𝑘_𝑡(2𝑝²

𝑡 −1) 2

− (𝑘_𝑡−1)²𝑝²

𝑡

! .

If we look at this as a function of𝑘_𝑡 then it is a downward looking parabola, which implies that for 𝑘_𝑡 high enough 𝑃⁰is negative and, as a consequence, the expected gain is smaller than the expected loss. Thus, 𝑘_𝑡does not go to∞as𝑝_𝑡 increases.

Therefore there exist𝑘 > 1 such that𝑘^∗ < 𝑘. This concludes the proof.

Proof of Corollary 20. This is an implication of two facts. The first one is that prices are bounded, so it is impossible to stop aggregating information. And second, public belief is a martingale and hence, converges. For more details see (L. Smith and Sørensen, 2000).

Proof of Corollary 21. Suppose at time 𝑡 the likelihood ratio is 𝑙_𝑡 and 𝜃 = 𝐻𝑖𝑔 ℎ. Let us calculate𝑙_𝑡+

1when agent𝑡 buys the new product with price𝑘 𝑙_𝑡+

1=𝑙_𝑡·

1−𝐹_𝐻

𝑘 𝑙𝑡

1−𝐹_𝐿

𝑘 𝑙𝑡

=𝑙_𝑡

2 𝑘 𝑙_𝑡

+1)

=2𝑘+𝑙_𝑡.

This means that when everybody start taking the same correct action instead of adding 2 to the𝐿 𝑅we going to add 2𝑘. Therefore, the convergence speed increases in𝑘 times.

A p p e n d i x C

APPENDIX TO CHAPTER 4