Decision-making and opinion formation in simple networks

Abstract

In many networked decision-making settings, information about the world is distributed across multiple agents and agents’ success depends on their ability to aggregate and reason about their local information over time. This paper presents a computational model of information aggregation in such settings in which agents’ utilities depend on an unknown event. Agents initially receive a noisy signal about the event and take actions repeatedly while observing the actions of their neighbors in the network at each round. Such settings characterize many distributed systems such as sensor networks for intrusion detection and routing systems for Internet traffic. Using the model, we show that (1) agents converge in action and in knowledge for a general class of decision-making rules and for all network structures; (2) all networks converge to playing the same action regardless of the network structure; and (3) for particular network configurations, agents can converge to the correct action when using a well-defined class of myopic decision rules. These theoretical results are also supported by a new simulation-based open-source empirical test-bed for facilitating the study of information aggregation in general networks.

Appendix: Proof of Theorem 4

We now present the complete proof for Theorem 4. We restate it here for convenience:

Convergence of two clusters consecutive line—Let $\varSigma$ be a system comprising a line of agents, with binary actions and signals, and in which the possible worlds W that agents consider correspond to a consecutive line of 2 clusters with sizes $n_1$ and $n_2$. That is $\forall w \in W : cons^0_{n_1,n_2}(w)$. Then, the system uniformly converges to the correct action in w after $n_1$ rounds when $n_1 < n_2$.⁵

Also, recall that the basis of the proof relies on Lemma 2, restated here as well.⁶ The following statements specify the convergence process for a line of two clusters.

• 1. At round $t\le n_{1}$ in world w, it holds that $cons_{n_{1}-t,n_{2}+t}^{t}(w)$. The minority border agent at round t in world w is $a_{n_1-t}$.

• 2. At round $t+1$, we have $F_{j}^{t}(w)=F_{j}^{t+1}(w)$ for any agent $a_j$ that is not a minority border agent at round t.

• 3. The minority border agent at round t has full knowledge at round $t+1$.

• 4. At round $t+1$ the minority border agent at round t takes the correct action.

We prove Lemma 2 by use of induction. Initially, we assume that all statements are true for all rounds up to t. We use Statements 1 and 3 to prove Statement 2 as follows.

Proof

By the induction hypothesis of Statement 1, we have a consecutive line for any round $t'<t+1$ and the index of the border agent in the minority at time t is $n_1-t$. Let j be the index of a non-majority agent. There are 2 cases. In the first case, we have that $j<n_1-t$ or $j>n_1+1$. Here, $a_j$ represents all agents that were not a minority border agent at time $t'$ (for example, agent $a_1$ in our configuration). In this case it holds that $F_{j-1}^{t'}(w)=F_{j-1}^{0}(w), F_{j+1}^{t'}(w)=F_{j+1}^{0}(w)$, that is, j observed the same action from its neighbors at each round $t'$. As a result it will not change its action at round $t'+1$.

In the second case, we have that $j\ne n_1-t$ and $j\le n_1+1$. Here, $a_j$ represents all agents that were a minority border agent at some point $t''<t$. By the induction hypothesis of Statement 3, agent $a_j$ has full knowledge at $t''+1$. By Theorem 1, $a_j$ will have converged to the correct action at round $t''+1$. Because $t''+1<t$, $a_j$ will not change its action at round $t'+1$.

We use Statement 1 to prove Statement 3.

Proof

By the induction hypothesis of Statement 1, at round $t+1$ we have $cons_{n_{1}-t,n_{2}+t}^{t}(w)$. The index of the minority border agent at round t is $n_1-t$. Therefore, we have $F_{n_{1}-t}^{t}\ne F_{n_{1}-t+1}^{t}$, that is the action of the border agent is different than the action of the adjacent agent. There is a single world w corresponding to a consecutive line that meets these criteria; therefore, the border agent has full knowledge at time $t+1$.

The proof of Statement 4 follows immediately from Statement 1 and Theorem 1. Consider our example configuration at round 2. The context of the minority border agent $a_2$ at round 2 includes the F action of agent $a_3$ which is different than the T action of $a_2$. There is a single world w satisfying $cons_{n_{1},n_{2}}^{0}(w)$ in which this can occur which is $w=(T,T,T,T,F,F,F,F,F)$. Therefore, agent $a_2$ has full knowledge and changes its action at round 3 to F.

We now use Statements 1, 2, and 4 to prove Statement 1 for round $t+1$.

Proof

Following the induction hypothesis of Statement 1, the border agent at round t is $a_{n_1-t}$. According to Statement 2, no agent other than the border agent at round t will change value at round $t+1$. By Statement 4 the border agent at round t will choose the correct action at round $t+1$. Therefore, at round $t+1$ we get a consecutive line with 2 clusters of size $n_{1}-(t+1),n_{2}+(t+1)$.

Finally, we prove Theorem 4 using Lemma 2

Proof

According to Statement 1, at round $t=n_1$, we have a consecutive line with one cluster of size $n_1+n_2$ in which all agents take the correct majority action. There are no border agents in the line configuration. Statement 2 states that non-border agents do not change their action at any round t. This statement holds for all rounds greater than $n_1$. Therefore, the system has converged uniformly.