decision-making and opinion formation in simple ... - ariel

Knowl Inf Syst (2017) 51:691–718DOI 10.1007/s10115-016-0994-0

REGULAR PAPER

Decision-making and opinion formation in simplenetworks

Matan Leibovich1 · Inon Zuckerman2 ·Avi Pfeffer3 · Ya’akov Gal4

Received: 28 March 2016 / Accepted: 8 September 2016 / Published online: 19 September 2016© Springer-Verlag London 2016

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

Keywords Multi-agent systems · Cooperation and coordination · Decision-making innetwork

B Inon [email protected]

Avi [email protected]

Ya’akov [email protected]

1 Department of Computer Science, The Open University, Raanana, Israel

2 Department of Industrial Engineering and Management, Ariel University, Ariel, Israel

3 Charles River Analytics, Cambridge, MA, USA

4 Department of Information Systems Engineering, Ben-Gurion University, Beer-Sheva, Israel

123

http://crossmark.crossref.org/dialog/?doi=10.1007/s10115-016-0994-0&domain=pdf

http://orcid.org/0000-0002-9999-1750

692 M. Leibovich et al.

1 Introduction

In many networked decision-making settings, information about the world is distributedacross multiple agents, and agents’ success depends on their ability to aggregate and reasonover (partial) local information that is available to them. In such settings, the structure of thenetwork and the way agents convey information in the network are key determinants of howagents aggregate information over time.

As an example, consider a recent news article describing the strategies used by customersin a bar in reaction to anti-smoking regulations. The customers in the bar wish to smoke,but are unsure about whether an inspector from the municipality is in the premises. Initially,all customers have their own opinions about whether there is an inspector in the bar. Theysee whether other customers seated close to them choose to smoke, and use these decisionsto revise their opinions. Over time, information about the existence of the inspector wouldcascade through the bar.

This paper presents a computational model of information aggregation in networks inwhich information is distributed among different agents and agents make decisions overtime. Initially, agents receive a private noisy signal about the result of an unknown process,which affects their utility. At each decision point, they simultaneously select an action from agiven set, while observing their neighbors’ actions in the previous time steps. Agents use theirobservations of each other’s actions over time to try to infer the true value of the unknownprocess.

Past works within the general framework provided above have explored how rationalagents reach consensus in such settings using Bayesian learning [2,24], or directly consid-ering their utilities [6]. Our study focuses on notions of convergence in agents’ actions andknowledge given any behavioral function mapping states to actions. Our first result showsthat all networks regardless of their structure will converge in both knowledge and action.That is, for any network, there is a point in time in which none of the agents will learn newinformation from its observations, and consequently, the agents will not change their actionsin the future rounds (Theorem 1). While the above result is useful in some settings, we showthat the convergence might not be to the “correct” action, which is the action that would bechosen if all of the agents had full visibility of agents’ signals. Our second result characterizeswhich types of network structures that will converge to the correct action (Theorem 2). Ourthird (and main) result is to prove that all networks necessarily converge to the same action,though not necessarily to the correct action (Theorem 3).

These theoretical results are supported by a new open-source empirical test-bed for facil-itating the study of information aggregation in general networks. The software is flexible,in that it allows to vary aspects of the environment such as the network structure, agents’decision-making rules, and the number of agents. It allows to empirically study the effectsof complex network structures on agents’ behavior, show how agents’ knowledge about theworld changes over time, and provide explanations for why agents choose certain actionsgiven their observations about other agents. This system can be of use to network designersthat wish to measure the effects of different network structures on agents’ decision-making.

The contribution of this paper is threefold. First, it formalizes a newmodel for informationaggregation in general networks. Second, it uses the model to define the conditions underwhich agents converge in action in general networks. Third, it shows that for particularnetwork structures, agents’ behavior is guaranteed to converge to the correct action whenusing a large class of myopic decision rules.

123

Decision-making and opinion formation in simple networks 693

The rest of this paper is organized as follows: Sect. 2 presents relatedwork; Sect. 3 presentsthe computational model and shows that every system will eventually converge in action andknowledge. In Sect. 4, we describe the notion of behavior rules and present our first result thatall (finite) systems will eventually converge in knowledge and in action. Section 5 discussesstronger convergence criteria and moves on to Sect. 6 which presents convergence propertiesfor more general networks. This section also introduces our empirical test-bed for simulatingnetwork behavior and uses it to demonstrate the theoretical results.We also present a theoremin this section showing that in a connected network all agentswill converge to the same action.Section 7 concludes and presents future work.

2 Related work

Information aggregation in networks has been studied from economical, computational, andbehavioral perspectives. We expand on each of these in turn.

Work in economics studies how groups can reach agreement and consensus given thatparticipants have distributed noisy information about the state of the world. Rational learningrequires that all agents consider the beliefs of all other agents about the state of the world, andhow their choices impact the beliefs of themselves and their neighbors in subsequent periods.The seminal paper by DeGroot [14] models how groups reach agreement on a commonsubjective probability distribution over an unknown parameter value by aggregating theirindividual opinions. Each agent maintains a belief over the subjective distributions of theother agents, and its own belief about the unknown parameter is a weighted average of itsbeliefs over these distributions. This process repeats over time, with each member in thegroup updating its beliefs about the distribution, assuming that all other agents are using thesame aggregation technique. Other works have since extended this model [3,15].

Meuller-Frank [24] models learning in networks by representing each agent’s knowledgeas an information set, the smallest subset of the state space in which the agent knows the truestate of the world is contained. Learning is based on the inference agents make about theinformation sets of their neighbors, given the observations they receive from them. The workshows that any two agents that receive the same observations can disagree about their finalactions only if they are indifferent between these actions. They also show that incompletenetworks (where agents’ histories of actions are not common knowledge)may convergemorequickly than complete networks in certain conditions. Our own work is novel in showing thata general class of myopic strategies can be optimal for many types of networks. Further, weprovide a theoretical guarantee of convergence in our model.

Other works in economics have modeled the spread of contagion in social networks,inspired by applications like the transmission of infectious disease and viral marketing cam-paigns. Most of these works model communication in the network as a stochastic processin which nodes become active with an independent probability or a threshold that stochas-tically depends on the activation of other nodes [23,25]. Dodds and Watts [16] suggested acontagion model which incorporates memory of past interactions. A node becomes activewhen it engages with enough active nodes within a predefined time window. They show theapplicability of this mode to capture large contagion events in a population, and also forassessing strategies for inhibiting or facilitating them. Kempe et al. [21] study the problemof choosing the most influential individuals to seed a contagion process in a social network.They provide the first provable approximation for the NP-hard problem. In their model, thedegree of activity of each node is stochasitcally determined by the degree of activity of itsneighbors.

123


Several works have emphasized the types of dysfunctional outcomes that can ensue whenpublic information eclipses the agents’ private knowledge. Bikhchandani et al. [9] providetwo types of models for the spread of information in a group of agents. In the observable-signals scenario, the information signals form a pool of public information as they arrive.Because all past signals are publicly observed, information keeps accumulating so that allagents eventually settle on the correct action. In contrast, in the observed-actions scenario,information may not accumulate when the pool of public information becomes more infor-mative than the private signal of agents. They show that when communicating actions, theagents may reach an information cascade, in which public information stops accumulatingand agents ignore their signals. This phenomenon is called herding and is also captured bythe model we present. Blume et al. [10] studied the spread of cascading failure in networksin which each node in the network has a threshold that is generated stochasitcally, and thenode fails if the number of failed neighbors exceed the threshold.

Game theoretic approaches to communication in networks have focused on coordinatedstrategies, defining a coordinated equilibrium (with multiple equilibria) over agents’ actionsgiven their private signals [2]. In this model, each agent observes the choices of others ina network setting that is unknown to the agents and is generated stochastically. Acemogloet al. [1] defined conditions under which agents’ actions converge to equilibrium given thatagents are fully rational, Bayesian decision makers. Banerjee and Fundenberg [6] computea Bayesian Nash equilibrium which is a mapping from history (signals and other’s actions)to actions for each agent. Bikhchandani, Hirshleifer and Welch [8], Banerjee [7] and Smithand Sorensen [27] show the herding phenomenon can occur in the context of the (perfect)Bayesian equilibrium of a dynamic game. Work in cellular automata specified models inwhich cells change states according to a set of rules that depend on the states of their neighborsin the networks [22]. Recent work has augmented cellular automata with memory to performaggregation tasks such as density estimation [28], but they do not reason about the true stateof the world or incorporate knowledge, as in our model.

There are several works in AI that have modeled aggregation of information in networksof agents. Prior work in the multi-agent systems literature that has modeled agents’ decision-making in networks using a limited set of rules did not analyze the evolution of their behaviorover time [17]. Other work has modeled information propagation in networks over time, butassumed agents’ beliefs depend only on their neighbors’ previous actions rather than on theentire history [18], and did not provide theoretical guarantees of convergence. Our work isdistinct from logics of reasoning about knowledge [19,26] in two ways. First, in these worksagents’ actions are descriptive, that is they are defined as propositions within the logic. Inour formalism, agents’ behavior in the network is inferred over time. Second, in their worksknowledge is expressed as propositions that can be true or false with a given probability,whereas in our formalism knowledge is restricted to the states that the agent thinks possible.

Lastly, we turn to a multitude of laboratory studies that have examined how people com-municate information in networks and have compared their behavior to theoretical models.Both the Bayesian-based economic models and DeGroot’s learning were verified in differentlaboratory experiments [11] and through large-scale data analysis [4]. Kearns et al. [20] haveempirically studied the way humans coordinate and convey information in different types ofgraphs. Banerjee et al. [5] showed how information about a new microfinance program wasinjected in several important nodes (with high eigenvector centrality) of a 43 villages socialnetwork in South India, and the information was successfully diffused around the network.Choi et al. [12,13] empirically studied learning in simple three-person, directed networks.The data show that the symmetry of the network structure and the way information is distrib-uted among individuals significantly influences individual and group behavior. In addition,

123


subjects converge quickly to a uniform action that is often consistent with the correct actionand on aggregate conform to a quantal response equilibrium.

3 The model

In our model, agents are organized in an undirected network. Initially (at time 0), each agentin the network receives a signal about an unknown outcome. The signal represents subjectiveinformation about the event that is available to the agents and is probabilistically related tothe actual outcome. At each future round, all of the agents take actions simultaneously afterobserving the actions taken by their neighbors in the previous round. The real outcome ofthe event itself is never revealed to the agents.

3.1 Definitions

Before presenting the model, we present the following definitions.

Definition 1 (System) A system is a quintuple Σ = (A, S, C, N , W ) where (1) A is a set ofagents; (2) S is a set of signals; (3) C is a set of actions; (4) N is a reflexive binary relation onA. If (i, j) ∈ N , we say that j is a neighbor of i . (N need not be symmetric); and (5) finally,W is a set of possible worlds, where each world w ∈ W consists of a signal s ∈ S for eachof the agents.

There are |S||A| possible worlds in a system. The signal of agent i in a possible world w willbe denoted as wi ∈ W . Consider the following running example of an election. The signalof each agent at time 0 is its own vote. The signal wi is the local observation of agent i (itsvote), and w represents the votes of all agents. At time 0, each agent can only see its ownvote (and not the votes of the other agents). At each time step, each agent declares the winnerof the election according to its knowledge at that time (which will be defined in Definition 5)and observes the declarations of its neighbors.

To describe the behavior of agents in the system, including the original signals and theactions played by agents at every round, we introduce the concept of a realization.

Definition 2 (Realization) A realization of a system (A, S, C, N , W ) consists of a pair(w, F) where w is a world in W and F is a function that assigns an action c ∈ C forevery agent i ∈ A in every round t ≥ 1. The action assigned by F to agent i in round t isdenoted Ft

i . If I is a set of agents, the actions played by them in round t are denoted FtI .

A realization of our election example will include all agents’ votes in the election and theirdeclaration of the winner at each time step.

We can now define the observation history of agents in the system, including their originalsignals, using the notion of context. Let N (i) denote the function that returns the neighborsof agent i .

Definition 3 (Context) Given a realization (w, F), the context for agent i in round t is(wi , F1

N (i), . . . , Ft−1N (i)), which includes the signal for i and its own actions as well as its

neighbors’ actions through round t − 1. The context captures all of the agent’s observationsthrough round t .

In our example, the context for an agent at round t includes its own vote in the election (itssignal) and its declaration of the winner (its action), as well as its neighbors’ declarations,up to round t − 1.

123


4 Rules of behavior and convergence

In order to determine how a system is realized, we need a rule that specifies how eachagent behaves at every round, given its context. This rule can only depend on the agent’sobservations, which are its own signal and the actions of its neighbors at previous rounds.

Definition 4 A behavior rule for agent i for system Σ = (A, S, C, N , W ) is a function Bfrom contexts of agent i to actions in C .

For example, an intuitive behavior rule for an agent in the voting example is to declare thewinner as the candidatewho received themajority of declarations from itself and its neighborsin the previous round. Once we have a behavior rule, we can inductively define the realizationof the system that results from applying the rule to an original set of signals. We can thendefine the realization resulting from B in w as the pair (w, F), where the individual action Ft

i

of agent i is given by the behavior rule Fti = B(wi , F1

N (i), . . . , Ft−1N (i)). The above definition

assumes that all the agents use the same behavioral rule. For ease of presentation, we willuse Ft

i to denote the action of agent i in round t in the set of action assignments F .Now,what sort of behavior rules shouldwe consider?Anatural possibility is to envision the

agents reasoning about which worlds are possible given their observations. In this approach,an agent’s decision depends only on what it knows about the current set of possible worldsand ignores how it arrived at that knowledge. This intuition is captured by the followingdefinitions.

Definition 5 (Indistinguishable worlds) Let K ti (w) denote the context of agent i at round t in

world w. Two worlds w′, w

′′are indistinguishable to agent i at round t if K t

i (w′) = K t

i (w′′),

that is the context for agent i in t is the same for both worlds w′, w

′′.

Indistinguishable worlds describe a relationship between two possible worlds, in whichthe agent cannot tell the difference between them, because at the current round (t) theysuggest the same behavior, and in all previous rounds (1, . . . , t − 1) its neighbors behavedexactly the same. We define the knowledge of an agent in a given context to include the setof possible worlds that it cannot tell apart.

Definition 6 (Knowledge) Given a system Σ = (A, S, C, N , W ), a world w ∈ W and acontext K t

i (w) of an agent i at round t , the set of worlds the agent considers possible at w isdenoted as the knowledge of agent i , E(K t

i (w)).

The knowledge of an agent describes the set of possible worlds that are valid from the agent’sperspective, and this knowledge evolves over time as the agent collects observations from itsneighbors.

The following Lemma shows that the size of the agent’s knowledge set cannot increaseover time.

Lemma 1 The size of the agent’s knowledge set cannot increase over time. Formally, forany world w we have E(K t+1

i (w)) ⊆ E(K ti (w)).

Proof By Definition 6, if w′ and w′′ are indistinguishable at time t + 1, then the context ofthe agent at t + 1 given w′ is equal to the context given w′′.

w′ ∈ E(K t+1i (w′′)) ⇒ K t+1

i (w′) = K t+1i (w′′) (1)

123


By Definition 3, it holds that the context of an agent at time t +1 also includes the contextat time t .

K ti (w

′) ⊆ K t+1i (w′) (2)

We get that if the context is equal given w′ w′′ at time t + 1, then it is also equal at time t :

K t+1i (w′) = K t+1

i (w′′) ⇒ K ti (w

′) = K ti (w

′′) (3)

By Definition 6, we know that

K ti (w

′) = K ti (w

′′) ⇒ w′ ∈ E(K t (w′′)) (4)

Therefore, we can conclude that E(K t+1(w′′)) ⊆ E(K t (w′′)) for any worldw′′, as required.

We use a running example to illustrate the model in which agents’ signals represent theirvote in an election. There are three agents configured in a network corresponding to a line.There are two candidates, Red and Green. The signal at time 0 for each agent is simply theirown vote in the election. The actual results of the election are not revealed. At each round,agents declare who they believe won the election.

In the example, a possible world corresponds to the signal (the vote) of all agents, anaction c ∈ C at round t is a declaration of an agent about who won the election. Figure 1illustrates this concept. As can be seen in the left sub-figure, the problem at hand contains8 (23) possible worlds. From the first agent’s perspective (the leftmost square), at t = 0 itsknowledge includes its own signal, Red in our case. Therefore, the set of possible worldswill be reduced to the worlds in which its action is Red.1 At t = 1, after observing thatits neighbor’s action (middle square) was Green, the set of possible worlds is reduced to 2worlds. Now, the missing information is what the rightmost agent did. If it chose Green, themiddle agent would stick to Green and the upper world would be the correct one, whereas ifit chose Red, the middle agent would change its action to Red (as this is the default action incase of ties), and the bottom world would be the correct one.

4.1 Convergence properties

Following the definition of the system and its agents, our goal is to study what happensin our setting for various kinds of networks. We are particularly interested in convergenceproperties. Do the agents converge to the same action or to the same beliefs? The followingdefinitions formalize convergence properties and conditions in our setting.

Definition 7 (Convergence in action) Given a system Σ = (A, S, C, N , W ) and a behaviorrule B, we say that agent i converges in action to action c in world w by round T if for allt ≥ T , Ft

i = {c}.In our voting example, an agent converges in action at a round if it does not change its declaredwinner after the round.

An agent converges in knowledge at a round if it does not learn any more informationregarding agents’ signals after the round. Thus, the information available to the agent doesnot change over time, as described in the following definition.

Definition 8 (Convergence in knowledge) Given a system Σ = (A, S, C, N , W ) and abehavior rule B, we say that agent i converges in knowledge to a set of worlds V ⊆ W inworld w by round T if for all t ≥ T , E(K t

i (w)) = V .

1 For simplicity of presentation, the actions and signals are the same. In other words, agents’ actions at time1 repeat their own signal. In general, this may not be the case.

123


Fig. 1 An example of the concept of knowledge in our model. In this world, we have 3 agents (the squares)and two possible s (Red and Green). The network structure is a line (color figure online)

Note that when an agent’s knowledge has converged to a set V containing a single world, wecan say that the agent has complete knowledge as defined below.

Definition 9 (Complete knowledge)We say that an agent has a complete knowledge at roundT if E(K T

i (w)) = {w}.In our example, the correct action for agents is to declare the candidate that received the

most votes as the winner. An agent has complete knowledge when it knows all the otheragents’ votes. With the following definitions in mind, we can show that when an agentcoverages in knowledge, it will also converge in action.

Proposition 1 (convergence in knowledge ⇒ convergence in action) For any system Σ

and behavior rule B, convergence in knowledge to a set of worlds V by round T impliesconvergence in action to some action c ∈ C. Formally, ∀t ≥ T, ∃c ∈ C such that thefollowing holds: E(K t

i (w)) = V ⇒ Fti (B, w) = {c}.

Proof If an agent has converged in knowledge to a set V at time t , then by Definition 8 theagent will not be able to distinguish between any of the worlds in V after this time step.Therefore, it holds that K t ′

i (ω) = K t ′+1i (ω) for any t ′ > t , that is, that the context of the

agent for ω ∈ V does not change after t . By Definition 4 of the behavior rule, it directlyfollows that Ft

i = Fi (t ′ + 1).

From its definition, convergence in knowledge results in a set of indistinguishable worlds,which consequently results in taking the same action assuming that the behavior rule isdeterministic.

Definition 10 (Correct action) We denote the action as the one that would be taken if anagent has complete knowledge. Formally, c is the correct action for agent i if E(K T

i (w)) ={w} ⇒ B(w, i) = c.

In the voting example, if the agent had access to the ballots, it could declare the truewinner. The following proposition states that when an agent has a complete knowledge, itknows the exact world (w), and as such it will choose the correct action in future rounds.

Proposition 2 (complete knowledge ⇒ convergence to the correct action) If agent i con-verges in knowledge to a single possible world, then the agent knows the actual world and,consequently, the correct action to take. Formally, ∀t ≥ T, ∃c ∈ C such that the followingholds: E(K t

i (w)) = {w} ⇒ Fti = {c}, and c is the correct action to take.

123


The proof follows directly from the definitions of convergence in knowledge and correctaction.

We can now show our first result, stating that all (finite) systems will eventually convergeboth in knowledge and in action.

Theorem 1 (All systems converge in knowledge and action) Let Σ = (A, S, C, N , W ) bea system in which A and S are finite. Let B be a behavioral rule, and let w ∈ W be an initialworld. Then, there exists a round T such that Σ converges in knowledge and action by T .

Proof We first show that if all agents have the same knowledge at two successive rounds Tand T + 1, then Σ will have converged in knowledge and action by T . Assume, for eachagent i , that E(K T +1

i (w)) = E(K Ti (w)). Since knowledge at T fully determines action at

T + 1 we get thatFT +1

i (B, w) = FTi (B, w) (5)

That is, agent i inworldw behaves the same in both T and T +1 (using Prop. 1). Now, supposeE(K T +2

i (w)) = E(K T +1i (w)). Then, there must be a world w

′which was indistinguishable

to i from w in round T +1 which became distinguishable in round T +2. For this to happen,some neighbor j would have chosen a different action in w

′from what it actually choose in

w at time T + 1, but acted the same at time T . Therefore, we get that

FT +1j (B, w

′) = FT +1

j (B, w) (6)

FTj (B, w

′) = FT

j (B, w) (7)

Equation 1 is stated without loss of generality, so we can plug w′and j and get that

FT +1j (B, w

′) = FT

j (B, w′) (8)

From Eqs. 3 and 4, we get that

FT +1j (B, w

′) = FT

j (B, w) (9)

We plug in j in Eq. 1 and get

FTj (B, w) = FT +1

j (B, w) (10)

FromEqs. 5 and 6,we get FT +1j (B, w

′) = FT +1

j (B, w), but this contradicts Eq. 1. Therefore,

E(K T +2i (w)) = E(K T +1

i (w)). Now, since A and S are finite, W must also be finite. Sincethe set of worlds an agent considers possible is non-increasing, there must be two successiverounds at which the agents have the same knowledge. Therefore, the theorem holds.

This theorem states that all systems converge in knowledge and consequently in action.However, this action may not be correct, and agents may converge to different actions. Forinstance, in Figs. 2 and 3 we see two possible scenarios to the next step of the examplepresented before (at Fig. 1). In one of them, the neighbor of the left most agent took theGreen action, and in the other, it took the Red action.

5 Convergence properties in networks

In this section, we study whether we can show a stronger criteria than Theorem 1, mainlythat all agents converge to the same action, as we now define:

123


Fig. 2 In this example, we have 3 agents (the squares) and two possible actions (Red and Green). Looking atthe world from the first agent’s perspective (left square) at t = 0, we can see that there are 4 possible worlds.In all of these possible worlds, the first agent’s action is Red. At t = 1, after observing that its neighbor’ssignal (middle square) is Green, the set of possible worlds is now reduced to 2 worlds. Now assume that inthe next time step t = 2 the agent observes the middle agent still issues the green action, it can now reasonthat the (Red, Green, Red) world is not possible, and he will be left with only one possible world (color figureonline)

Fig. 3 In this example, we have 3 agents (the squares) and two possible actions (Red and Green). Looking atthe world from the first agent’s perspective (left square) at t = 0, we can see that there are 4 possible worlds,as it owns signal is Red. At t = 1, after observing that its neighbor’s signal (middle square) is Green, the setof possible worlds is now reduced to 2 worlds. Now assume that in the next time step t = 2 the agent observesthe middle agent changes its action to Red, and it can now reason that the (Red, Green, Red) world is the onlypossible option (color figure online)

Definition 11 (Uniform convergence) The system (A, S, C, N , W ) uniformly converges inaction under behavior rule B in world w by round T if all agents converge to the same actionby round T . We say a system uniformly converges to the correct action if all agents convergein action to the correct action.

We will characterize special classes of systems that include line configurations for whichwe can guarantee uniform convergence to the correct action. To facilitate this analysis, wewill make the following two assumptions. First, the set of actions and signals are equal.Second, all agents use the majority voting rule, in which agents choose the action that is mostcommon in all the worlds they consider possible. This decision rule is inspired by consensustasks, in which agents’ actions correspond to choosing the most likely candidate to win giventheir observations, with ties being consistently broken in favor of a default action f . Wedefine this voting rule as follows:

Definition 12 (Majority behavior rule) Let #w(s) be the number of times signal s appearsin world w, and #W (s) = Σw∈E(K t

i (w)#w(s). The majority behavioral will be defined as:Bmaj = argmaxs#w(s).

Before proceeding to analyze how agents make decisions in different networks, we needto make the following definitions:

123


Edge agent An agent that has only one neighbor.Border agent Agent i (in world w at time t) is a border agent if there is a path from

i to an edge agent in which each agent j = i takes the same actionFt

j (B, w), and Ftj (B, w) = Ft

i (B, w).Minority agent Agent i (in world w at time t) is a minority agent given a set of agents

G if i’s action is chosen by a strict minority of agents in G. For twoagents, this corresponds to #(Ft

i (B, w)) <|G|2 .

Agent cluster An agent cluster is a set of agents that take the same action. Formally, anagent cluster of size n at time t in world w is a set of agents ai , . . . , an

such that Fta1(B, w) = Ft

a2(B, w) = · · · = Ftan

(B, w)

Consecutive line a set of c agent clusters of sizes n1, n2, . . . , nc such that the agents ineach adjacent cluster take a different action.

For example, the following configuration is a consecutive line at round t with 2 clustersof size n1 = 2, n2 = 3. The agent a2 (in bold) at round t is both a minority and a borderagent (termed “minority border” agent). Agent a3 is also a border agent, while agents a1 anda5 are edge agents.

T1 − T2 − F3 − F4 − F5

The Boolean function constn1,n2,...,nc

(w) returns “true” if a network configuration in worldw at time t is a consecutive line of c clusters. We can now state convergence for a simplegraph which includes consecutive lines of two clusters.

5.1 Simple line networks

We begin our analysis with networks forming a simple line of 3 agents, as in the runningexample described earlier. We assume the signal for agents can be G or R, and use notationRi to represent the fact that the signal for agent ai was R (and similarly for Gi ). For example,a line configuration of 3 agents can be G1 − R2 − G3.

Proposition 3 (Line of 3 agents)Let Σ be a system comprising a consecutive line of 3 agents,binary signals, and actions. Then, the system uniformly converges to the correct action in 3rounds.

Proof At round 0, the context for each agent includes solely its own signal. Therefore, thebest agents can do in this round given our decision rule, is to repeat their signal. At round 1,the middle agent a2 observes the actions of both of its neighbors, and therefore, its contextincludes the actions representing the signals of all of the agents. As a result, it has fullknowledge at round 2, and by Theorem 1, it has converged to the correct action at round 2.Both agents a1 and a3 will necessary copy this action at round 3 onwards, because there doesnot exist a world in which this agent chooses the wrong action at this round.

However, it is not the case that all of the agents in a 3-agent configuration converge to fullknowledge. For example, take the two line configuration G1 − G2 − R3 and G1 − G2 − G3.In both of these configurations, agent a2 will take action G at round 2. Therefore, agent a1will not be able to distinguish between the two worlds corresponding to these configurations.

Moreover, in general, line configurations do not converge to the correct action. Considerthe following example of 4 agents who observe the following signals: F1 − T2 − T3 − F4.The context of agent a3 at round 1 includes the signals w1 = F, w2 = T and w3 = T atround 1, and it will thus play T at round 2. This action will reveal to agent a1 that w3 = T ,

123


or else agent a2 would have chosen F at round 2. Therefore, a1 changes to T in time step3. Now, agent a3 will play T at round 2 for both cases w4 = T, w4 = F . Therefore, agenta2 does not discover the value of w4 at round 3 and will continue to play T at round 3. Inaddition, a2 will not change its action at round 3 and will continue to play T . This is becausea3 will play T at round 3 regardless of whether a4’s signal is T or F . In both of these cases,it will have observed more T ’s than F’s. A symmetric argument can be made to show thata4 will play T at round 3. Therefore, both agents a1, a4 will play T at round 4. It is easy tosee that no agent will change its behavior after round 3; thus, this configuration converges tothe wrong action T .

We next look at class of systems in which the network includes consecutive lines of twoclusters.

Proposition 4 (Convergence of two clusters consecutive line) Let Σ be a system comprisinga line of agents, with binary actions and signals, and in which the possible worlds W thatagents consider correspond to a consecutive line of 2 clusters with sizes n1 and n2. That is∀w ∈ W : cons0n1,n2(w). Then, the system uniformly converges to the correct action in w

after n1 rounds when n1 < n2.2

Wewill demonstrate the process of convergenceusing a running example of a configurationconsisting of a consecutive line of two clusters n1 = 4, n2 = 5:

T1 − T2 − T3 − T4 − F5 − F6 − F7 − F8 − F9

The basis of the proof relies on the Lemma below. Note that the condition cons0n1,n2(w)

means that agents know they are on a line with two clusters (but not their size).

Lemma 2 (Convergence: line of two clusters) The following statements specify the conver-gence process for a line of two clusters.

1. At round t ≥ n1 in world w, it holds that constn1−t,n2+t (w). The minority border agent

at round t in world w is an1−t

2. At round t + 1, we have Ftj (B, w) = Ft+1

j (B, w) for any agent a j that is not a minorityborder agent at round t.

3. The minority border agent at round t has complete knowledge at round t + 1.4. At round t + 1, the minority border agent at round t takes the correct action.

Statement 1 guarantees that at each round t, there is a consecutive line with 2 clusters withsizes n1 − t , n2 + t . Applying this statement to the configuration shown above shows that atround t = 2, for example, we have the following line configuration with two clusters n

′1 = 2,

n′2 = 7:

T1 − T2 − F3 − F4 − F5 − F6 − F7 − F8 − F9

The minority border agent at round 2 (in bold) is a2. Statement 2 states that at round t +1,all agents that are not the minority border agent at round t do not change their action. Inour example configuration, this means that the only agent to change action at round 3 is theminority border agent at round 2, which is a2. Statement 3 states that the minority borderagent learns the world w at time t + 1. According to Statement 4, it will choose to changeto the correct majority action at round 3, which is F (the default action in case of ties). Thiscomplete proof with the inductive process can be found in 7.

2 When n1 = n2, it can be shown that the configuration converges to the default action.

123


Fig. 4 a Progression of behavior in a consecutive line of two clusters. At the end of this process, agent a4has complete knowledge of all agents’ signals, b and agent a4 never has full knowledge

We illustrate convergence in the network using a simulation tool called ONetwork, anempirical test-bed that is designed to study information aggregation in networks using dif-ferent rules of behavior.3 We will use ONetwork to demonstrate several configurations thatwere introduced in the last section. Agents are aligned along the x-axis, and time is alignedalong the y-axis. Bright green nodes represent agents that take action G (Green) in all worldsthat they consider possible (and similarly for bright red nodes). Agents’ behavior at round 0corresponds to their signals.

ONetwork can be used to simulate how agents’ behavior changes over time. Figure 4ashows the progression of agents’ behavior in a consecutive line of two clusters of size n1 =7, n2 = 8. As shown by the figure, the system converges uniformly (in action) to R aftern1 = 7 rounds, as stated by Theorem 4. Specifically, at round 3, agent a5 changes its action toR. As a result, agent a4 learns the signal of all other agents in the network and has completeknowledge.

Figure 4b describes the knowledge of agent a4 at round 3 given a context in which agenta5 stays G at rounds 0–3 (regardless of the signals of agents a9,...,13). In this case, agent a4

3 ONetwork is free software and is available for download under the GNU public license at the followingURL: https://github.com/inonzuk/InfoAggrSimul.git.

123

https://github.com/inonzuk/InfoAggrSimul.git


cannot gain knowledge of the signals of these agents. The system converges in action to anincorrect action G. Pale green and pale red-colored nodes indicate agents that take action G inthe majority (but not all) worlds that they consider possible. These agents are uncertain aboutwhich action to take. White-colored nodes indicate agents for whom there is a tie betweentaking actions G and R.

5.2 Consecutive lines of three clusters

In this section, we extend Proposition 4 to consecutive lines of three clusters.

Proposition 5 (Convergence of three clusters consecutive line) Let Σ be a system compris-ing a line of agents, with binary actions and signals, and in which the possible worlds Wcorrespond to a consecutive line of 3 clusters with size n1, n2 and n3, that is ∀w ∈ W , wehave that cons0n1,n2,n3(w). Then, the system uniformly converges to the correct action after⌈ n2

2

⌉rounds given that n2 <

⌈ n12

⌉ + ⌈ n32

⌉.

Under the conditions specified above, the action represented in the middle cluster is inthe minority, and the configuration uniformly converges to the majority action representedby the first and third cluster.

Lemma 3 The following statements specify the convergence process of a consecutive lineof 3 clusters.

1. At round t ≤ n22 in world w, it holds that

constn1+t,n2−2t,n3+t (w). There are two border agents in the minority at round t, namely

an1+t and an1+n2−t .2. Let I be the group of border agents in the minority at round t in world w. At round t + 1,

for each agent j ( j /∈ I ) it holds that Ftj (w) = Ft+1

j (w).3. Let I be the group of border agents in the minority at round t in world w. All agents in

I will have full knowledge at round t + 1 and behave correctly at round t + 1.

Statement 1 states that at time t there is a consecutive line of 3 clusters with size n1 + t, n2 −2t, n3 + t . Statement 2 states that all agents other than the border agents in the minority willnot change their action, and Statement 3 states that the border agents will behave correctly.

Interestingly, when the size of the middle cluster no longer satisfies n2 <⌈ n1

2

⌉ + ⌈ n32

⌉,

the line configuration may not converge to the correct action. An example using ONetworkis shown in Fig. 5a, in which a consecutive line with 3 clusters of sizes n1 = 5, n2 = 6 andn3 = 3 uniformly converges to the wrong action represented in the middle cluster. Here,A “fan effect” occurs in which the action represented by the first cluster propagates to themiddle cluster and back. We provide an intuitive explanation for this progression. Consider,for example, the minority border agent a8 at round 2. It observed its neighbor, a7, change itsaction to G at the previous round 1. Now, because agents only consider line configurations of3 clusters, agent a8 learns that the actions for agents a1, . . . , a6 at round 0 were G. Also, a8observes that the agent a9 did not change its action at round 2. Thus, it learns that the actionof agent a10 at round 0 was R. Now, there are more G than R signals in the worlds that a8considers possible, and therefore, it changes to G at round 2. In a similar fashion, agent a9changes to G at round 3. This “left-to-right” propagation terminates when agent a10 learnsat round 4 there are more Rs than Gs in the worlds it considers possible. The knowledgeof agent a10 at round 4 is shown in Fig. 5b. As shown by the figure, the agent knows thesignals for agents a1, . . . , a12 (the brightly colored squares in round 0) and is uncertain aboutthe actions of agents a13, . . . , a15 (the pale, outlined, colored squares in round 0). Because

123


Fig. 5 Progression of behavior in a line of three clusters. In a, the system is shown to uniformly converge tothe incorrect action. In b, the knowledge of agent a10 is shown in round 4

there is an equal amount of Gs and Rs in the world it considers possible, agent a10 choosesthe default action and continues to play R at round 4. This process reveals the actions ofagents a11 and a12 at round 0 to agent a9, which causes it to choose the action R at round5. This “right-to-left” pass propagates this knowledge to agents a1,...,a8 and they all choose

123


Fig. 6 A network for illustratingleadership a1 a2 a3 a4

a5

R, without ever discovering the actions at round 0 of agents a13, . . . , a15. A symmetric “faneffect” occurs with agents a11, . . . , a15.

6 Convergence in clique-based networks

In this section, we show that it is possible to guarantee convergence for a general class ofnetwork structures characterizing many distributed decision-making settings. We will proveconvergence in a network constructed of n-cliques and a constraint class of behavior rules,and through that continue to prove that in a fully connected network, all agents will alwaysconverge to the same action.

6.1 Additional definitions

We start with defining the notion of “leadership” in a network.

Definition 13 (leader) Agent i is a leader of agent j if the following hold: (1) Agent i is aneighbor of agent j , and (2) any neighbor of j’s neighbor (other than i) is also a neighbor ofi . Formally, (∀y, x ∈ A)N ( j, x) ∧ N (x, y) → N (i, y).

To illustrate, we observe that in the network shown in Fig. 6, agent a2 is a leader of agenta1. This is because the sole neighbor of agent a1 (other than a2) is agent a5, which is aneighbor of agent a2. Similarly, it is easy to see that a2 is also a leader of a5. However, agenta2 is not a leader of agent a3. This is because agent a4 is a neighbor of agent a3 but not aneighbor of agent a2.4 The following proposition states that all agents “follow their leader”in a general network.

Proposition 6 (“follow the leader”) If i is a leader of j , then Ft+1j (B, w) = Ft

i (B, w) forany behavioral rule B, any world w, and any round t > 1.

Proof By definition of a leader, if i is a leader of j , all new information available to j willbe given solely by observing i . Therefore, if j issues a new action in time t + 1, it will onlybe due to new observation on i at time t .

A leader can observe at a current round t all of the actions of the observations that theagents it leads will see at the next round t + 1. For example, in the network shown in Fig. 6,agent a2 will be able to observe the signal from agent a3 before making an action that isobserved by agent a5. In particular, all information from agents a3 and a4, both of which arenot directly connected to a5, is conveyed to a5 by a2. In addition, all agents that are directlyconnected to a5 (i.e., agent a1 and a2 itself) are also connected to a2. Consequently, any

4 Leadership is not a symmetric property, as exemplified by the fact that agent a5 is not a leader of agent a2.

123


Fig. 7 A 2-clique network.Agents a2 and a5 are thecommon members

a1 a2 a3

a4 a5 a6

information that is observed by agent a5 at round t was already observed by agent a2 in theprevious round t − 1. Therefore, the best action agent a5 can do under any behavior rule isto follow its leader a2.

We can use the notion of leadership to characterize agents’ behavior in a family of networksthat comprise unions of (possible intersecting) complete graphs. Each of these completegraphs corresponds to a set of agents who can observe each others’ actions. These mayrepresent sensor networks deployed at an installation, a group of friends that coordinate theiractivities together on a social network, or work peers at an institution who share information.As agents may be members of many groups, these complete graphs may overlap. Formally,we make the following definitions.

Definition 14 (n-clique network) LetC1, . . . , Cn be a set of complete graphs and an n-cliquenetwork as a union of n complete graphs C1, . . . , Cn . The common members of an n-cliquenetwork are defined as C1 ∩ · · · ∩ Cn .

For example, a 2-clique network can describe sensors arranged on two sides of a borderor two groups of friends with a (possibly empty) set of common members. The commonmembers of the network may represent those sensors that can observe all of the other sensors,or those friends that are common to both groups. An example of a 2-clique network withcommon members is shown in Fig. 7.

6.2 Convergence of 1-clique network

To formalize convergence in this class of networks, we begin with a system comprising asingle complete graph (a 1-clique network). In such a configuration, all agents can observeeach others behavior. The following proposition states that this system uniformly convergesto the correct action in 2 rounds. The proposition holds for a general class of rules of behaviorthat only requires agents to repeat their signals in the first round of interaction. We denote therules that belong to this class as “first-action-signaling” rules of behavior. This is a reasonableassumption to make as the context for each agent only includes its own signal. Note that thisclass does not restrict agents’ strategies for any future rounds.

Proposition 7 (convergence of 1-clique network) Let Σ be a system comprising a 1-clique.Then, for any rule of behavior of the first-action-signaling class, the system will uniformlyconverge to the correct action at round 2.

Proof At round 0, all agents will repeat their signal because they use a first-action-signalingrule. At round 1, each agent observes the signals of all other agents. Thus, at round 2, allagents have complete knowledge of each others signals. Therefore, each agent will take thesame correct action at round 2. Now, because the agents are configured in a clique, there isan edge from ai to all other agents and, in particular, to the neighbors of ai . By Definition 13,any agent ai in the system is a leader of ai itself. By Proposition 6, it follows that all agents

123


will take the same correct action in all rounds following round 2, and thus, the system willuniformly converge to the correct action.

6.3 Convergence of n-clique network

The following theorem claims that all n-cliques networks necessarily converge to the correctaction.

Theorem 2 (n-clique networks convergence to the correct action) Let Σ be a system inwhich the graph G is a n-clique network C1, . . . , Cn and the set of common members isnot empty. Then, for any behavior rule of the first-action-signaling class, the system willuniformly converge to the correct at round 2.

Proof As before, all agents will repeat their signal at round 0 because they use a first-action-signaling rule. Let A = C1 ∪ C2 ∪ · · · Cn be the set of common nodes in the network. Byconstruction, there is an edge from agent that is a common member to all other agents. Thus,by Definition 13, any common agent is a leader of all agents in the network. At round 2, allcommon agents will necessarily have full knowledge. Therefore, they will take the correctaction at round 2. By Proposition 6, it follows that all agents will take the same correct actionin all rounds following round 2, and thus, the system will uniformly converge to the correctaction.

6.4 Convergence of cliques without common members

Lastly, we introduce networks comprising of cliques that do not have common members.An abstraction of such a network is shown in Fig. 8. There are three clique groups in thefigure, represented by the yellow (Y), green (G), and red (R) circles. Each node in the networkrepresents a complete graph of agents. As shown in the figure, the set of agentsC2 is commonmembers of the 2-clique network containing G and Y, while the set of agents C4 is commonmembers of the 2-clique network containingR andG.All agents inC2 are leaders of all agentsin Y and C3. However, there is no agent in the graph that is a leader of all other agents, andthe convergence properties of the network cannot be determined in advance using Theorem2. In these types of settings, we were not able to prove convergence to the correct action.

An example configuration of such networks is agents that are situated in two roomsseparated by a wall with a window. The first room corresponds to the Y clique group andcontains the agents in C1 and C2. The second room corresponds to the R clique and containsthe agents in C4 and C5. All agents in C2 are leaders of the Y clique group, and all agents inC4 are leaders of the R clique group. The leaders in C2 and C4 are the only agents that canobserve each others action through the window (in this example, the set of agents C3 is theempty set).

C4

Fig. 8 A 2-clique with no common members

123


Although convergence cannot be proven when the network has no common members,we can simulate agents’ actions in the two-room example using the ONetwork simulation.Figure 9a shows 10 agents. Agents a1, . . . , a5 are in the first room and led by agent a5. Agentsa6, . . . , a10 are in the second room and led by a6. Bright green nodes represent agents thattake action G (Green), while bright red nodes represent agents that take action R (Red).Agents’ behavior at round 0 (the first line) corresponds to their signals. At time step 1, allagents repeat their signal. At time step 2, the leader agents stay red. Therefore, at time 3 allthe agents turn to red and the system converges to the incorrect action (as there is a greenmajority in agents’ signals at round 0). Figure 9b shows a different configuration of the two-room example. Here the leader a5 turns to red in time step 1 so all agents it leads stay red intime 2. At time step 4, this agent turns back to green (after observing that the other leader a6has not turned to red in 2 time steps). Consequently, all of the agents led by a5 turn to green inthe next time step. This process continues until agents converge to the correct default actiongiven an equal number of red and green signals.

6.5 All agents will act the same in a connected network

In the following subsection, we will present a theorem stating that when we have a connectedgraph of agents, that is, there is a path from each pair of agents, after converging in knowledge,all agents will issue the same action. This will not necessarily be the correct action, but allwill act the same. We will start with additional definitions and built our proof on them.

The following relation connects every pair of indistinguishable worlds by some agent i attime t .

Definition 15 (Indistinguishable worlds relation (Cluster)) I nd(i, t) = {(w,w′) ∈

I nd(i, t) | K ti (w) = K t

i (w′)}

The set of all indistinguishable worlds relation for some agent i at time t will be denotedas I N D(i, t), and individual relations in it will be denoted as I ndn(i, t) ∈ I N D(i, t).

The following proposition states that the indistinguishable worlds relation is an equiva-lence relation. In other words, each world can be in only a single individual relation in the setI N D(i, t). We will denote it as a “cluster,” and the proposition states that any intersectionof such clusters is empty.

Proposition 8 (Equivalence relation) I nd(i, t) is an equivalence relation

Proof We need to show that the relation is reflexive, symmetrical, and transitive. (1)Reflexivity—identical worlds are in the relation by setting w

′ = w in the definition. For-mally, (w,w) ∈ I nd(i, t) ⇔ K t

i (w) = K ti (w). (2) Symmetry—if two worlds are in the

relation, their symmetric pair is also in the relation by settingw′ = w andw = w

′. Formally,

(w,w′) ∈ I nd(i, t) ⇔ (w

′, w) ∈ I nd(i, t) as by definition K t

i (w) = K ti (w

′) ⇔ K t

i (w′) =

K ti (w). (3) Transitivity—(w,w

′) ∈ I nd(i, t), (w

′, w

′′) ∈ I nd(i, t) ⇒ (w,w

′′) ∈ I nd(i, t).

True by definition of K .

The theorem we present will be true under any behavior rule as long as it adheres to thefollowing additive criteria.

Definition 16 (Additive behavior rule) An additive behavior rule is a behavior rule thatprovides the same behavior under union operation of a set of indistinguishable worlds(clusters). In other words, if a behavior rule results in a similar behavior in two clus-ters, it will result in the same behavior in a union of these clusters. Formally, when B is

123


a1 a2 a3 a4 a5 a6 a7 a8 a9 a10

(a)

(b)

Fig. 9 The two-room example, convergence to the incorrect action (a) and correct action (b)

123


a behavior rule and Fti (B, I ndn(i, t)) extends the original definition of F to denote the

action assigned to agent i in round t in any of the indistinguishable worlds belongingto I ndn(i, t), B will be called an additive rule when the following holds: (∀i ∈ A, t ∈T, I ndn(i, t), I ndm(i, t) ∈ I N D(i, t), n = m)Ft

i (B, I ndn(i, t)) = Fti (B, I ndm(i, t)) =

Fti (B, (I ndn(i, t) ∪ I ndm(i, t))).

In our voting example throughout the paper, we have used the majority selection votingrule as our behavior rule. It is easy to see intuitively that if, for example, in two clusters ofworlds the selected action is Red (as there are more possible worlds in which the Red is thewinner), the union of these two clusters will still result in Red as the majority selected action.We now formally show in the following property that this behavior rule is indeed additive.

Proposition 9 The majority behavior rule is additive.

Proof Without loss of generality, let s be the selected action according to the majority behav-ioral rule (while the other action will be denoted as s̄).

Fti (B, I ndn(i, t)) = Ft

i (B, I ndm(i, t)) = s

Fti (B, I ndn(i, t)) = s ⇔ Σ#w∈I ndn(i,t)(s) − Σ#w∈I ndn(i,t)(s̄) > 0

Fti (B, I ndm(i, t)) = s ⇔ Σ#w∈I ndm (i,t)(s) − Σ#w∈I ndm (i,t)(s̄) > 0

Fti (B, (I ndn(i, t) ∪ I ndm(i, t))) = s

⇔ Σ#w∈(I ndn(i,t)∪I ndm (i,t))(s) − Σ#w∈(I ndn(i,t)∪I ndm (i,t))(s̄) > 0

⇔ (Σ#w∈I ndn(i,t)(s) − Σ#w∈I ndn(i,t)(s̄)) + (Σ#w∈I ndm (i,t)(s) − Σ#w∈I ndn(i,t)(s̄)) −(Σ#w∈(I ndn(i,t)∩I ndm (i,t))(s) − Σ#w∈(I ndn(i,t)∩I ndm (i,t))(s̄))

⇔ Fti (B, (I ndn(i, t) ∪ I ndm(i, t))) = 0+ + 0+ − 0

⇔ Fti (B, (I ndn(i, t) ∪ I ndm(i, t))) = s (11)

The above proposition uses in Eq. 7 the fact that when uniting two clusters relations, thedisjunction of them will be empty (they are an equivalence relation); thus, the additional

Fig. 10 In the upper figure a, we can see two clusters. In the left hand one, we see twoworlds that will result ina Red vote (the bottom world is a tie which will be broken toward Red as the default action). In the right-handside, we see two worlds: one which will result in a Green vote (upper) and another in the Red vote (lower).So in both clusters the Red vote will prevail. In figure b, we can see the union operation on the clusters, whichwill also result in a Red vote from the joint cluster (as two worlds will be Red and one will be Green) (colorfigure online)

123


Fig. 11 A shared knowledge graph between agents 2 and 3. Each agent has 4 possible worlds that are dividedinto two clusters in which the worlds within them are indistinguishable. The worlds on the edges are thosewhich exist as possible worlds for both agents clusters

information will not change the behavior. Figure 10 presents an example of the additivebehavior rule concept.

Theorem 3 (Convergence in Knowledge ⇒ Uniform convergence) Let Σ = (A, S, C, N ,

W ) be a system, and B a behavioral rule. If ∃t ∈ T s.t. (∀i ∈ A)E(K t+1i (w)) = E(K t

i (w)),

then ∀(i, j ∈ A, t′> t) Ft

′i = Ft

′j .

Proof We construct a “shared knowledge graph” at time t between agents (i, j) ∈ N asfollows. For each agent, we build a node for each cluster in its indistinguishable worldsrelation. That is, we build a shared knowledge graph SK Gt

(i, j) = (V, E), where each nodev ∈ V is a cluster from I N D(i, t) or I N D( j, t). Let us denote nodes v, u as the followingclusters, v = I ndv(i, t), u = I ndu( j, t). Every edge e ∈ E is constructed between twonodes v, u ∈ V if and only if the clusters of these nodes are not mutually exclusive. Formally,{u, v} ∈ E ⇔ I ndv(i, t) ∩ I ndu( j, t) = φ.

An example of a shared knowledge is presented in Fig. 11. Next, we present a lemma thatshows that in a shared knowledge graph, at any time t , the neighbors of a node will providethe same behavior at t − 1.

Lemma 4 Given a shared knowledge graph, SK Gt(i, j), the neighboring nodes of every

node will behave the same at the previous time step. ∀(v = I ndv(i, t), {v, u, q} ∈SK Gt

(i, j), {u, v}, {q, v} ∈ E)Ft−1i (B, u) = Ft−1

i (B, q)

Proof By definition of a shared knowledge graph, SK Gt(i, j), at time t the worlds in node v

are indistinguishable by agent i . If at time t − 1 agent i could have distinguish between thebehavior of its current neighboring nodes, he would have divided his own cluster and theywould not be neighbors anymore.

In Fig. 12, we can see the cluster of agent 3 divides after learning the signal of agent 2. Thefollowing corollary, derived directly fromLemma 4, states that if the system had converged inknowledge, we will see the same behavior of the neighboring nodes in the shared knowledgegraph. This is due to the fact that the graph remains static after convergence.

Corollary 1 Given a shared knowledge graph, SK Gt(i, j), if the system had converged in

knowledge in round T , the neighboring nodes of every node will behave the same at all timesteps after T .

Wecontinue by considering a single connected component of the shared knowledge graph.Denote Ci to be the set of all worlds in all nodes belonging to agent i , and C j to be the setof all worlds in all nodes belonging to agent j . We now show that Ci = C j .

123


Fig. 12 The figure shows a snapshot of the shared knowledge graph between agents 2 and 3 (middle agents),in two consecutive time steps. In figure a, we can see that agent 3 cannot distinguish between the two worldsin his cluster. We can also see that agent 2 has 2 clusters: in the upper one, he will vote Red (there is a tieand Red is the default action), while in the lower cluster he will vote Green in both worlds. In figure b, afterobserving agent 2’s signal, agent 3 can now distinguish between the worlds, and the cluster divides into twoclusters (color figure online)

Lemma 5 Given a shared knowledge graph, SK Gt(i, j), after convergence in knowledge, in

any connected component the union of nodes for each agent contains the same set of worlds.Formally, Ci = C j .

Proof By definition of a shared knowledge graph, an edge exists only if there is a non-emptyintersection between two nodes (and each node belongs to each of the agents). As such,assume node v belongs to agent i and node u belongs to agent j . Given a world w found onthe {u, v} edge. When constructing the union of nodes for each agent, by definition of theshared knowledge graph, w will be included in both. This will be true for each world foundin Ci or C j as the nodes are part of a connected graph.

After showing that Ci = C j , the behavior function will return the same behavior in eachof the joint clusters. Moreover, having an additive behavior rule will also result in the samebehavior in the union of all the clusters in Ci or C j . Formally, Ft

I (B, Ci )) = FtI (B, C j )).

Finally, moving back from the shared knowledge graph to the graph of the agent themselves(recall, a shared knowledge graph is constructed only between two neighboring agents).Assuming a connected agent graph will allow us, by induction, to conclude that all agentswill behave the same after convergence.

7 Conclusions

Our study focused on notions of convergence in action and in knowledge for networks ofdecision-making agents. In our setting, the information available to each agent is given assignals, or observations of its neighbors’ behavior. Thus, information is being propagatedand aggregated over time.

123


We started by presenting a formal theoretical model of our setting. We then definedconvergence properties on that model and differentiated between convergence of knowledgeand action and showed that the first entails the second. In our first theorem, we showed thatany system, regardless of its structure, will eventually converge in both knowledge and action.

We then proceeded to inquire about a stronger convergence criteria, uniform convergence,inwhich all agents in the system converge to the same action, and an even stronger one, correctuniform convergence, in which all agents in the systems converge to the correct action. Asthose criteria are strong, we started by exploring different network structures. We started byshowing that different types of line networks converge to the correct action. We also verifiedour theoretic results with a ONetwork simulation software that was developed for that matter.

The next step was to move to more complex structures. We showed that clique-basednetwork (with various properties) that have common members will uniformly converge tothe correct action. Lastly, we proved a general claim that in any form of connected networkof agents, after converging in knowledge, all agents will issue the same action (though notnecessarily the correct one). These theoretical results correlate with laboratory studies doc-umenting herding phenomena in which people cease to accumulate information an convergeto wrong conclusions.

We are extending this work in two ways. First, we are extending our model to handlingmore than 2 possible action types and are examining the effects of decision rules from thesocial choice literature, such as Borda andCondorcet winners on the on network convergence.Second, we intend to conduct laboratory studies and compare our convergence results withhuman behavior. Based on the results of these studies, we intend to design new behavioralrules that are able to predict howpeople behave over time for different network configurations.

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 Σ be a system comprising a line of

agents, with binary actions and signals, and in which the possible worlds W that agentsconsider correspond to a consecutive line of 2 clusters with sizes n1 and n2. That is ∀w ∈W : cons0n1,n2(w). Then, the system uniformly converges to the correct action in w after n1

rounds when n1 < n2.5

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

1. At round t ≤ n1 in world w, it holds that constn1−t,n2+t (w). The minority border agent

at round t in world w is an1−t .2. At round t +1, we have Ft

j (w) = Ft+1j (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 truefor all rounds up to t . We use Statements 1 and 3 to prove Statement 2 as follows.

5 When n1 = n2 it can be shown that the configuration converges to the default action.6 Note that the condition cons0n1,n2 (w) means that agents know they are on a line with two clusters (but nottheir size).

123


Proof By the induction hypothesis of Statement 1, we have a consecutive line for any roundt ′ < t + 1 and the index of the border agent in the minority at time t is n1 − t . Let j be theindex of a non-majority agent. There are 2 cases. In the first case, we have that j < n1 − tor j > n1 + 1. Here, a j represents all agents that were not a minority border agent attime t ′ (for example, agent a1 in our configuration). In this case it holds that Ft ′

j−1(w) =F0

j−1(w), Ft ′j+1(w) = F0

j+1(w), that is, j observed the same action from its neighbors ateach round t ′. As a result it will not change its action at round t ′ + 1.

In the second case, we have that j = n1 − t and j ≤ n1 + 1. Here, a j represents allagents that were a minority border agent at some point t ′′ < t . By the induction hypothesisof Statement 3, agent a j has full knowledge at t ′′ +1. By Theorem 1, a j will have convergedto the correct action at round t ′′ + 1. Because t ′′ + 1 < t , a j will not change its action atround t ′ + 1.

We use Statement 1 to prove Statement 3.

Proof By the induction hypothesis of Statement 1, at round t +1 we have constn1−t,n2+t (w).

The index of the minority border agent at round t is n1 − t . Therefore, we have Ftn1−t =

Ftn1−t+1, that is the action of the border agent is different than the action of the adjacent

agent. There is a single worldw 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. Con-sider our example configuration at round 2. The context of the minority border agent a2at round 2 includes the F action of agent a3 which is different than the T action ofa2. There is a single world w satisfying cons0n1,n2(w) in which this can occur which isw = (T, T, T, T, F, F, F, F, F). Therefore, agent a2 has full knowledge and changes itsaction 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 isan1−t . According to Statement 2, no agent other than the border agent at round t will changevalue at round t +1. By Statement 4 the border agent at round t will choose the correct actionat round t + 1. Therefore, at round t + 1 we get a consecutive line with 2 clusters of sizen1 − (t + 1), n2 + (t + 1).

Finally, we prove Theorem 4 using Lemma 2

Proof According to Statement 1, at round t = n1, we have a consecutive line with one clusterof size n1+n2 in which all agents take the correct majority action. There are no border agentsin the line configuration. Statement 2 states that non-border agents do not change their actionat any round t . This statement holds for all rounds greater than n1. Therefore, the system hasconverged uniformly.

References

1. Acemoglu D, Dahleh M, Lobel I, Ozdaglar A (2008) Bayesian learning in social networks. MIT LIDsWorking Paper

2. Acemoglu D, Dahleh MA, Lobel I, Ozdaglar A (2011) Bayesian Learning in Social Networks. Rev EconStud 78(4):1201–1236

3. Acemoglu D, Ozdaglar A (2011) Opinion dynamics and learning in social networks. Dyn Games Appl1(1):3–49. doi:10.1007/s13235-010-0004-1

123

http://dx.doi.org/10.1007/s13235-010-0004-1


4. Alatas V, Banerjee A, Chandrasekhar AG, Hanna R, Olken BA (2012) Network structure and the aggre-gation of information: theory and evidence from Indonesia. Working Paper 18351, National Bureau ofEconomic Research. doi:10.3386/w18351. http://www.nber.org/papers/w18351

5. Banerjee A, Chandrasekhar AG, Duflo E, Jackson MO (2012) The diffusion of microfinance. WorkingPaper 17743, National Bureau of Economic Research. doi:10.3386/w17743. http://www.nber.org/papers/w17743

6. Banerjee A, Fudenberg D (2004) Word-of-mouth learning. Games Econ Behav 46(1):1–227. Banerjee AV (1992) A simple model of herd behavior. Q J Econ 107(3):797–8178. Bikhchandani S, Hirshleifer D, Welch I (1992) A theory of fads, fashion, custom, and cultural change as

informational cascades. J Polit Econ 100(5):992–10269. Bikhchandani S, Hirshleifer D, Welch I (1998) Learning from the behavior of others: conformity, fads,

and informational cascades. J Econ Perspect 12(3):151–17010. Blume L, Easley D, Kleinberg J, Kleinberg R, Tardos E (2011) Which networks are least susceptible to

cascading failures? In: 2011 IEEE 52nd annual symposium on foundations of computer science (FOCS),pp 393–402. doi:10.1109/FOCS.2011.38

11. Chandrasekhar AG, Larreguy H, Juan Xandri P (2012) Testing models of social learning on networks:evidence from a framed field experiment. Tech. rep, MIT

12. Choi S, Gale D, Kariv S (2005) Behavioral aspects of learning in social networks: an experimental study.Adv Appl Microecon 13:25–61

13. Choi S, Gale D, Kariv S (2012) Social learning in networks: a quantal response equilibrium analysis ofexperimental data. Rev Econ Des 16(2–3):135–157

14. DeGroot MH (1974) Reaching a consensus. J Am Stat Assoc 69(345):118–12115. DeMarzo P, Vayanos D, Zwiebel J (2003) Persuasion bias, social influence, and unidimensional opinions.

Q J Econ 118(3):909–968. http://EconPapers.repec.org/RePEc:tpr:qjecon:v:118:y:2003:i:3:p:909-96816. Dodds PS,Watts DJ (2004)Universal behavior in a generalizedmodel of contagion. Phys Rev Lett 92:218,

701. doi:10.1103/PhysRevLett.92.21870117. Gal Y, Kasturirangan R, Pfeffer A, Richards W (2009) A model of tacit knowledge and action. In:

International conference on computational science and engineering, 2009. CSE ’09., vol 4, pp 463–468.doi:10.1109/CSE.2009.479

18. Glinton R, Scerri P, Sycara K (2010) Exploiting scale invariant dynamics for efficient information prop-agation in large teams. In: Proceedings of the 9th international conference on autonomous agents andmultiagent systems: volume 1, AAMAS ’10, pp 21–30. International Foundation for Autonomous Agentsand Multiagent Systems, Richland, SC. http://dl.acm.org/citation.cfm?id=1838206.1838210

19. Halpern J, Moses Y (1992) A guide to completeness and complexity for modal logics of knowledge andbelief. Artif Intell 54(3):319–379

20. Kearns M, Suri S, Montfort N (2006) An experimental study of the coloring problem on human subjectnetworks. Science 313(5788):824

21. Kempe D, Kleinberg JM, Tardos É (2003) Maximizing the spread of influence through a social network.In: Getoor L, Senator TE, Domingos PM, Faloutsos C (eds) Proceedings of the ninth ACM SIGKDDinternational conference on knowledge discovery and data mining, Washington, DC, USA, 24–27 August2003, pp 137–146. ACM. doi:10.1145/956750.956769

22. Mitchell M, Crutchfield JP, Hraber PT (1994) Evolving cellular automata to perform computations:mechanisms and impediments. Phys D 75(1):361–391

23. Morris S (2000) Contagion. Rev Econ Stud 67(1):57–7824. Mueller-Frank M (2013) A general framework for rational learning in social networks. Theor Econ

8(1):1–4025. Murray JD (2001)Mathematical biology. II Spatial models and biomedical applications {Interdisciplinary

Applied Mathematics V. 18}. Springer, New York Incorporated26. Pacuit E, Salame S (2004) Majority logic. In: Principles of knowledge representation and reasoning, KR,

vol 4, pp 598–60527. Smith L, Sørensen P (2000) Pathological outcomes of observational learning. Econometrica 68(2):371–

39828. Stone C, Bull L (2009) Solving the density classification task using cellular automaton 184 with memory.

Complex Syst 18(3):329

123

http://dx.doi.org/10.3386/w18351

http://www.nber.org/papers/w18351

http://dx.doi.org/10.3386/w17743



http://dx.doi.org/10.1109/FOCS.2011.38

http://EconPapers.repec.org/RePEc:tpr:qjecon:v:118:y:2003:i:3:p:909-968

http://dx.doi.org/10.1103/PhysRevLett.92.218701

http://dx.doi.org/10.1109/CSE.2009.479

http://dl.acm.org/citation.cfm?id=1838206.1838210

http://dx.doi.org/10.1145/956750.956769


Matan Leibovich is a Masters student of Computer Science theOpen University, Israel. His research includes multiagent systems anddecision-making.

Inon Zuckerman is a faculty member of the industrial engineering andmanagement department at the Ariel University in Israel. He receivedhis Ph.D. from Bar-Ilan University in 2009. He has published works intop-tier conferences and journals, on adversarial interactions that tacklehuman fallacies as well as classical adversarial search algorithms. Thisincludes work on automated negotiation including the development ofan automated agent that won the second place in the 2015 ANAC com-petition. The interdisciplinary nature of his work includes adaptationof ideas from the social and behavioral sciences, including extensiveexperimental work (both online and offline) on human subjects.

Avi Pfeffer is Chief Scientist at Charles River Analytics. Dr. Pfef-fer is a leading researcher on a variety of computational intelligencetechniques including probabilistic reasoning, machine learning, andcomputational game theory. As an Associate Professor at Harvard, hedeveloped IBAL, the first general-purpose probabilistic programminglanguage, while at Harvard, he also produced systems for representing,reasoning about, and learning the beliefs, preferences, and decision-making strategies of people in strategic situations. Prior to joining Har-vard, he invented object-oriented Bayesian networks and probabilisticrelational models, which form the foundation of the field of statisticalrelational learning. Dr. Pfeffer has published many journal and confer-ence articles and is the author of a text on probabilistic programming.Dr. Pfeffer received his Ph.D. in computer science from Stanford Uni-versity.

123


Ya’akov Gal leads the human–computer decision-making researchgroup in the department of information Systems Engineering at Ben-Gurion University. He received his Ph.D. from Harvard Universityin 2006. He is a recipient of the Wolf foundation’s 2014 Krill prizefor young Israeli scientists, a Marie Curie International fellowship for2010, a two-time recipient of Harvard University’s Derek Bok awardfor excellence in teaching, as well as the School of Engineering andApplied Science’s outstanding teacher award. He has published over 50papers in highly refereed venues on topics ranging from artificial intel-ligence to the learning and cognitive sciences.

123

decision-making and opinion formation in simple ... - ariel

Documents