Introduction
Asymptotic Speed of Social Learning
Introduction
Here, we precisely calculate the asymptotic behavior of the log-likelihood ratio of public belief. Chamley (Chamley, 2004) provides an estimate for the evolution of public belief for a class of private signal distributions with thick tails.
Model
𝐹+ and if 𝜃 = −1, they have CDF 𝐹−.4 We assume that 𝐹+ and 𝐹− are absolutely continuous with respect to each other, so that private signals never fully reveal the state. This is the rear held by an outside observer after recording the actions of the first officers.
The evolution of public belief
We then turn to estimate more precisely the long-term behavior of the public log-likelihood ratioℓ𝑡. 7By "the left tail of𝐺−is convex and differentiable", we mean that there are some𝑥0 which are restricted to(−∞, 𝑥 . 0),𝐺−are convex and differentiable.
Conclusion
But before we do that, we will state a general property of the expected utility function. In the following, we will analyze the behavior of the optimal 𝑁 and the corresponding 𝑢(0.5) when the social planner becomes patient, i.e.
Subsidizing Lemons for Efficient information aggregation
Model
This is an application of Bayes' rule, given that the player buys (does not buy) the new product if her total likelihood ratio is above (below) 𝑘𝑡. The high complexity of this problem is due to the complexity of the random walk in public belief.
Binary signals
Similarly, the first multiplier of the second term is a difference between our accuracy and public confidence at the −𝑁 level. In this case, instead of stopping at 𝑞 =0.7, we will stop at 𝑝𝑡 =0.9674 conditional on public confidence.
Continuous signal distribution
The main result in this case is similar to the one in the classical literature. To prove this statement, we first bind the expected gain in the public belief and then translate it to the expected gain in the utility. To recall what the asymptotic speed of learning is, recall that in the classical model with unbounded signals, people eventually start choosing the correct action, 1 if the condition is𝐻𝑖𝑔 ℎand 0 otherwise.
In the next section, we try a different approach to solve the utility function and optimal pricing policy.
Numerical calculations
The reason for this is that we impose a significant non-zero price (𝑘𝑡 ≠ 1) for high public beliefs, which results in utility gain. To bring this utility growth back to 𝑝𝑡 = 1/2 of those high public beliefs, we need to take a large number of steps. This means that the optimal prices significantly increase the asymptotic learning rate and social welfare.
Conclusion
However, the expected number of friends increases as we grow𝐾 as we will show in the next section. As in establishing Theorem 6, assume that the conditional distributions of the private log-likelihood ratio satisfy. We know that the expected number of strong and weak ties is bounded by 𝐵 and 𝐵/𝑐𝑤 𝑒 𝑎 𝑘 respectively due to the budget constraint.
Now we calculate the probability of the event that there is at least one weak-weak path back to village V𝑖.
Weak and Strong Ties in Social Network
Model
In the first period, this player sends information to all her friends and acquaintances. People in the same village can be friends with each other and players in different villages can potentially be acquaintances. Therefore, we can treat this objective as if we have equality in the budget constraint.
In the next section we prove that there is an equilibrium close to the unique equilibrium of this game.
Analytical results
The second and third terms of 𝑈e𝑁 calculate how many strong and weak ties we have at distance 1. The fifth calculates the number of people who are at distance 2 such that there is either a weak-strong or strong-weak path between us . The last term calculates the number of people who are at distance 2 from us, connected through weak-weak path7.
For example, if it is more beneficial by distance 1 to have a friend, we may still not want to spend all our time on the strong ties.
Quantitative results
So in a utopian world, where everyone can coordinate what to do, people should have more friends than in equilibrium. In other words, there are more friends in the optimal network than in the equilibrium network. As we saw in the previous section, we care about their relative value to each other.
As we increase the number of people in the village and keep the budget fixed, the friend graph becomes sparser.
Conclusion
The Paradox of Weak Ties in 55 Countries." In: Journal of Economic Behavior and Organization, 133, pp. Work, Friendship, and Media Use for Information Exchange in a Network Organization." In: Journal of the American Society for Information Science, 49(12), pp. An Overview of Social Networks and Economic Applications." In: Handbook of Social Economics, Vol.
Social Interactions and Well-Being: The Surprising Power of Weak Ties.” In: Personality and Social Psychology Bulletin, 40(7), pp.
Sub-linear learning
Then ℓ𝑡 is positive with probability 1 from some time, and all agents take action+1 from this point. Given 𝑟𝑡, we will construct private signal distributions such that lim inf𝑡|ℓ𝑡|/𝑟𝑡 > 0 with probability one. As a result, we have that regardless of the action chosen by the agent, as long as the sign of the action is equal to that of ℓ𝑡 (which happens from a certain point on w.p. 1).
Intuitively, if we can choose private signal distributions that decay 𝐷+(𝑥) very slowly, ℓ𝑡 will be almost linear.
Long-term behavior of public belief
0; this will indeed be the case for 𝑡large enough, since 𝐴and 𝐵 are positive and continuous, and so both𝑎𝑡 and 𝑏𝑡 are monotonically increasing and tend to infinity. Here we show that if (A.3) holds for large enough𝑡 then it holds for all𝑡0> 𝑡. Notice that 𝑎(𝑡) and 𝑏(𝑡) are eventually monotonically increasing and tend to infinity as 𝑡 tends to infinity.
We conclude this section with a lemma showing that under some technical conditions on the left tail of 𝐺−, the function 𝑢+(𝑥) = 𝑥 + 𝐷+(𝑥) (i.e., the function that determines how the public log -probability ratio updated when action +1 is performed) is eventually monotonically increasing.
Gaussian private signals
Denote by 𝐸𝑡 the event that 𝑎𝜏 = +1 for all 𝜏 ≥ 𝑡; that is, over time there are no more errors𝑡. Since 𝜇𝑡 =𝑞 is equivalent toℓ𝑡 = log𝑞/(1−𝑞), what we have shown implies that there is a 𝜀 > 0 such that for all𝑥 ≥ 1 (here the choice of 1 is arbitrary and can be replaced by any positive number ) . Furthermore, since the probability that agents𝑡 through𝑡 +𝑛𝐿 −1 all take action+1 conditional onℓ𝑡 =𝑥 is continuous in𝑥, there is a 𝑝𝐿 > 0 such that.
Furthermore, it follows from a routine application of L'Hopital's rule (or from the standard asymptotic expansion for the CDF of a normal distribution) that for all sufficiently large𝑥,.
Upsets and runs
In particular, the probability that Ξ𝑡 is logarithmic in 𝑡 tends to zero as 𝑡 tends to infinity. One important consequence of Corollary 34 is that with high probability there is at least one maximal run ahead of time𝑡 that is long relative to𝑡. Then there exists a 𝑧 > 0 such that, if there is a good length 𝑠of𝑡, thenℓ𝑡+𝑠 ≥ ℓ∗𝑠−𝑧.
Furthermore, since 𝑢+(𝑥) > 𝑥 for all𝑥, it follows that whenever there is a series of lengths 𝑁 from 𝑡, ℓ𝑡+𝑁 > ℓ∗.
Distributions with polynomial tails
This is the event that the number of disturbances so far is small and the majority of agents have so far taken the correct action. Furthermore, since there are at least 12𝑡 agents taking action+1, there is at least one maximal good length of at least𝑡/(𝛽log𝑡). This is the case where the number of perturbations so far is small and the majority of agents so far have taken the wrong action.
Mark with
Binary distributions
When we condition on statehood𝐿 𝑜𝑤 we have the same drift but in the opposite direction, towards level−𝑁. Suppose that if we increase the stopping limits from𝑁 to𝑁+1 and from-𝑁to-𝑁−1 accordingly so that the utility at level𝑁,𝑢(𝑁), increases. Note that the utility at level 0, given this strategy, is equal to the utility at the frontier levels multiplied by the expected discount factor plus some utility we collect on the way there.
Recall that the expected utility when we start with a prior of 1/2 (at level 𝑁) and stop the random walk upon arrival at level 2𝑁 or 0.
Continuous distributions
Since we do not know the exact formula for the expected utility function, we first plug the expected gain into public trust and then translate it into the expected utility gain. Before we calculate the expected loss and expected profit from the price 𝑘𝑡 we need some more facts. This means that the expected gain is equal to the distance between the two chords connecting 𝑢(𝑎), 𝑢(𝑏) and
In other words, the expected gain is equal to the distance between 𝑑 and 𝑑0, where 𝑑 belongs to chord 𝑢(𝑎), 𝑢(𝑏) and 𝑑0 to chord 𝑢(𝑎0)𝑢(𝑏0) and their 𝑝-coordinate is 𝑝𝑡.
Agents’ objective and its approximation
But because all their corresponding probabilities converge to 1 the probability limit of the unconditional event 𝐴. The probability that player 𝑗 is not connected to 𝑖 via person 𝑘(𝑖, 𝑗 , 𝑘are in the same villageV𝑖) is (1− Subtrahend is the probability that 𝑖 is not connected to 𝑗 via any player 𝑘 in the same village.
Thus, for ¯𝑁 large enough so that𝑈∞ −𝑈𝑁 < 𝛿/3 for 𝑁 > 𝑁¯ there are no equilibria of the initial utility.
Equilibrium and comparative statics
Under the assumptions of the theorem, 𝐹 has the following form: it is 0 at 𝑝 = 0, has a positive derivative there (when𝑐𝑤 𝑒 𝑎 𝑘 > 𝜋𝑤), it is negative atp. Let's replace this with 𝜕 𝐹0(𝑝)/𝜕 𝐾, but skip the term with the logarithm and the 𝑝 term outside the parentheses. Now let's remember a well-known fact about Erdős - Rényi random graphs: the gorbal clustering coefficient for the network of friends is equal to 𝑝∗2+𝑂 (𝐾 𝑁)−0.5.
Using this fact and the inequality we found above, we can conclude that when 𝑐𝑤 𝑒 𝑎 𝑘 is not too close to 𝜋𝑤 or when 𝑁 is large enough (i.e. 𝑝∗ > 0 and𝑞∗ < 𝑝∗ for large enough 𝑁 ) the𝐶 𝐶 of the friend network is larger than that of the acquaintances.
Socially optimal network
We obtained the last inequality from the same condition as we had in Theorem 23, which is necessary for the existence of equilibrium, 𝑝∗. Now let's see if the solution to 𝜕𝑈𝑜 𝑝𝑡𝑖 𝑚 𝑎𝑙/𝜕 𝑝 =0, 𝑝𝑜 𝑝𝑡𝑖 𝑚 𝑎𝑙, is greater than 𝑝∗, is a non-trivial solution for
