Local Majority Rule Research Articles

Speed-accuracy trade-offs in best-of-n collective decision making through heterogeneous mean-field modeling.

To succeed in their objectives, groups of individuals must be able to make quick and accurate collective decisions on the best option among a set of alternatives with different qualities. Group-living animals aim to do that all the time. Plants and fungi are thought to do so too. Swarms of autonomous robots can also be programed to make best-of-n decisions for solving tasks collaboratively. Ultimately, humans critically need it and so many times they should be better at it! Thanks to their mathematical tractability, simple models like the voter model and the local majority rule model have proven useful to describe the dynamics of such collective decision-making processes. To reach a consensus, individuals change their opinion by interacting with neighbors in their social network. At least among animals and robots, options with a better quality are exchanged more often and therefore spread faster than lower-quality options, leading to the collective selection of the best option. With our work, we study the impact of individuals making errors in pooling others' opinions caused, for example, by the need to reduce the cognitive load. Our analysis is grounded on the introduction of a model that generalizes the two existing models (local majority rule and voter model), showing a speed-accuracy trade-off regulated by the cognitive effort of individuals. We also investigate the impact of the interaction network topology on the collective dynamics. To do so, we extend our model and, by using the heterogeneous mean-field approach, we show the presence of another speed-accuracy trade-off regulated by network connectivity. An interesting result is that reduced network connectivity corresponds to an increase in collective decision accuracy.

Deviations from the majority: A local flip model

We study the effect of probabilistic distortions to the local majority rules used in the Galam model of opinion dynamics and bottom-up hierarchical voting. A different probability for a flip against the local majority within the discussion group is associated with each ratio of majority/minority. The cases of groups of sizes 3 and 5 are investigated in detail. For hierarchical voting, the local flip corresponds to a ‘faithless elector’, a representative who decides to vote against the choice of their electing group. Depending on the flip probabilities, the model exhibits a rich variety of patterns for the dynamics, which include novel features in the topology of the landscape. In particular, for size 5, we uncover for the first time an interplay between five fixed points, which split into either three attractors and two tipping points or two attractors and three tipping points, depending on the flip probabilities. Larger groups are also analysed. These features were absent in the former versions of the Galam model, which has at maximum three fixed points for any group size. The results shed a new light on a series of social phenomena triggered by one single individual who acts against the local majority.

Local Majority-with-inertia Rule Can Explain Global Consensus Dynamics in A Network Coordination Game

We study how groups reach consensus by varying communication network structure and individual incentives. In 342 networks of seven individuals, single opinionated “leaders” can drive decision outcomes, but do not accelerate consensus formation, whereas conflicting opinions slow consensus. While networks with more links reach consensus faster, this advantage disappears under conflict. Unopinionated individuals make choices consistent with a local majority rule combined with “inertia” favouring their previous choice, while opinionated individuals favour their preferred option but yield under high peer or time pressure. Simulations show these individual rules can account for group patterns, and allow rapid consensus while preventing deadlocks.

TWO-OPINIONS-DYNAMICS GENERATED BY INFLEXIBLES AND NON-CONTRARIAN AND CONTRARIAN FLOATERS

We assume a community whose members adopt one of two opinions [Formula: see text] or [Formula: see text]. Each member appears as an inflexible, or as a non-contrarian or contrarian floater. An inflexible sticks to its opinion, whereas a floater may change into a floater of the alternative opinion. The occurrence of this change is governed by the local majority rule: members meet in groups of a fixed size, and a floater then changes its opinion provided it is a minority in the group. Subsequently, a non-contrarian floater keeps the opinion as adopted under the local majority rule, whereas a contrarian floater adopts the alternative opinion. Whereas the effects of on the one hand inflexibles and on the other hand non-contrarians and contrarians have previously been studied separately, the current approach allows us to gain insight in the effect of their combined presence in a community. Given fixed proportions of inflexibles [Formula: see text] for the two opinions, and fixed fractions of contrarians [Formula: see text] among the [Formula: see text] and [Formula: see text] floaters, we derive the update equation [Formula: see text] for the overall support for opinion [Formula: see text] at time [Formula: see text], given [Formula: see text]. The update equation is derived respectively for local group sizes 1, 2 and 3. The associated dynamics generated by repeated local updates is then determined to identify its asymptotic steady configuration. The full opinion flow diagram is thus obtained, showing conditions in terms of the parameters for each opinion to eventually win the competing dynamics. Various dynamical scenarios are thus exhibited, and it is derived that relatively small densities of inflexibles allow for more variation in the qualitative outcome of the dynamics than higher densities of inflexibles.

The Trump phenomenon: An explanation from sociophysics

The Trump phenomenon is argued to depart from current populist rise in Europe. According to a model of opinion dynamics from sociophysics the machinery of Trump’s amazing success obeys well-defined counter-intuitive rules. Therefore, his success was in principle predictable from the start. The model uses local majority rule arguments and obeys a threshold dynamics. The associated tipping points are found to depend on the leading collective beliefs, cognitive biases and prejudices of the social group which undertakes the public debate. And here comes the open sesame of the Trump campaign, which develops along two successive steps. During a first moment, Trump’s statement produces a majority of voters against him. But at the same time, according to the model the shocking character of the statement modifies the prejudice balance. In case the prejudice is present even being frozen among voters, the tipping point is lowered at Trump’s benefit. Nevertheless, although the tipping point has been lowered by the activation of frozen prejudices it is instrumental to preserve enough support from openly prejudiced people to be above the threshold. Then, as infuriated voters launch intense debate, occurrence of ties will drive progressively hostile people to shift their voting intention without needing to endorse the statement which has infuriated them. The ongoing debate does drive towards a majority for Trump. The possible Trump victory at November Presidential election is discussed. In particular, the model shows that to eventually win the Presidential election, Trump must not modify his past shocking attitude but to appeal to a different spectrum of frozen prejudices, which are common to both Democrats and Republicans.

The Minimum Information Dominating Set for Opinion Sampling in Social Networks

We consider the problem of sampling a node-valued graph. The objective is to infer the values of all nodes from that of a minimum subset of nodes by exploiting correlations in node values. We first introduce the concept of information dominating set (IDS). A subset of nodes in a given graph is an IDS if the values of these nodes are sufficient to infer the values of all nodes. We focus on two fundamental algorithmic problems: (i) how to determine whether a given subset of nodes is an IDS; (ii) how to construct a minimum IDS. Assuming binary node values and the local majority rule for information correlation, we first show that in acyclic graphs, both problems admit linear-complexity solutions by establishing a connection between the IDS problems and the vertex cover problem. We then show that in a general graph, the first problem is co-NP-complete and the second problem is NP-hard. We develop two approaches to solve the IDS problems: one reduces the problems to a hitting set problem based on the concept of essential difference set , the other a gradient-based approach with a tunable parameter that trades off performance with time complexity. The concept of IDS finds applications in opinion sampling such as political polling and market survey, identifying critical nodes in information networks, and inferring epidemics and cascading failures in communication and infrastructure networks.

Consensus states of local majority rule in stochastic process

Abstract A sufficient condition for a network system to reach a consensus state of the local majority rule is shown. The influence of interpersonal environment on the occurrence probability of consensus states for Watts–Strogatz and scale-free networks with random initial states is analyzed by numerical method. We also propose a stochastic local majority rule to study the mean first passage time from a random state to a consensus and the escape rate from a consensus state for systems in a noisy environment. Our numerical results show that there exists a window of fluctuation strengths for which the mean first passage time from a random to a consensus state reduces greatly, and the escape rate of consensus states obeys the Arrhenius equation in the window.

Dynamics of opinion formation with strengthen selection probability

The local majority rule is extensively accepted as a paradigmatic model to reflect the formation of opinion. In this paper, we study a model of opinion formation where opinion update rule is not based on the majority rule or linear selection probability but on a strengthen selection probability controlled by an adjustable parameter β. In particular, our proposed probability function can proximately fit the two extreme cases–linear probability function and majority rule or in between the two cases under different values of β. By studying such model on different kinds of networks, including different regular networks and complex networks, we find that there exists an optimal value of β giving the most efficient convergence to consensus regardless of the topology of networks. This work reveals that, compared with the majority rule and linear selection probability, the strengthen selection probability might be a more proper model in understanding the formation of opinions in society.

Modeling the Forming of Public Opinion: An approach from Sociophysics

This paper reviews a sociophysics two-state model for opinion forming that has proven heuristic power. The dynamics are driven by repeated small-group discussions; within each group, a local majority rule is applied to update the opinions of agents. Iterating the dynamics leads towards one of two opposite attractors at which every agent shares the same opinion. The successful attractor is a function of the initial support with respect to a certain threshold, the value of which depends on the size distribution of the local update groups. While odd-sized groups yield a threshold at fifty percent, even-sized groups, which allow the inclusion of doubt in the case of an opinion tie, produce a threshold shift toward either one of the two attractors, giving rise to minority opinion spreading. In addition, agents can be heterogeneous in their cognitive nature, obeying different rules to update their opinion. While floater agents are open to changing their mind, contrarians chose to oppose whatever opinion was held by the majority of agents in their vicinity, and inflexibles never change their mind. Contrarians and inflexibles have drastic and counter-intuitive effects on the opinion dynamics. Beyond certain critical proportions, contrarians trigger an upside change of the dynamics, making it threshold-less with only one attractor at precisely 50/50 regardless of the initial conditions. Inflexibles produce the same threshold-less dynamics, except with an asymmetric single attractor that favors a specific opinion, even when they start with very low support. The results are used to shed new and unexpected light on controversial issues such as global warming.

Activity patterns on random scale-free networks: global dynamics arising from localmajority rules

Activity or spin patterns on a random scale-free network are studied using meanfield analysis and computer simulations. These activity patterns evolve in timeaccording to local majority rule dynamics which is implemented using (i)parallel or synchronous updating and (ii) random sequential or asynchronousupdating. Our mean field calculations predict that the relaxation processes ofdisordered activity patterns become much more efficient as the scaling exponentγ of the scale-free degree distribution changes fromγ > 5/2 toγ < 5/2.For γ > 5/2, the corresponding decay times increase asln(N) with increasingnetwork size N whereasthey are independent of N for γ < 5/2. In order to check these mean field predictions, extensive simulations of the patterndynamics have been performed using two different ensembles of random scale-freenetworks: (A) multi-networks as generated by the configuration method, whichtypically leads to many self-connections and multiple edges, and (B) simplenetworks without self-connections and multiple edges. We find that the mean fieldpredictions are confirmed (i) for random sequential updating of multi-networks and(ii) for both parallel and random sequential updating of simple networks withγ = 2.25 and2.6. For γ = 2.4, the data for the simple networks seem to be consistent with mean field theory as well,whereas we cannot draw a definite conclusion from the simulation data for the multi-networks.The latter difficulty can be understood in terms of an effective scaling exponentγeff = γeff(γ, N) for multi-networks. This effective exponent is determined byremoving all self-connections and multiple edges; it satisfiesγeff ≥ γ and decreases towardsγ with increasingnetwork size N. For γ = 2.4, we find γeff ≳ 5/2 up to N = 217.

An equilibrium voting model of federal standards with externalities

This paper proposes a framework for studying policy making in a federal system in the presence of spillover externalities. Local jurisdictions choose local policies by majority rule subject to standards that are set by majority rule at the federal level. We characterize the induced preferences of voters for federal policies, prove existence of local majority rule equilibrium, provide an example of non-existence of global majority rule equilibrium, and explore the welfare properties of federal standards in the presence of spillovers.

Dynamic pattern evolution on scale-free networks.

A general class of dynamic models on scale-free networks is studied by analytical methods and computer simulations. Each network consists of N vertices and is characterized by its degree distribution, P(k), which represents the probability that a randomly chosen vertex is connected to k nearest neighbors. Each vertex can attain two internal states described by binary variables or Ising-like spins that evolve in time according to local majority rules. Scale-free networks, for which the degree distribution has a power law tail P(k) approximately k(-gamma), are shown to exhibit qualitatively different dynamic behavior for gamma < 5/2 and gamma > 5/2, shedding light on the empirical observation that many real-world networks are scale-free with 2 < gamma < 5/2. For 2 < gamma < 5/2, strongly disordered patterns decay within a finite decay time even in the limit of infinite networks. For gamma > 5/2, on the other hand, this decay time diverges as ln(N) with the network size N. An analogous distinction is found for a variety of more complex models including Hopfield models for associative memory networks. In the latter case, the storage capacity is found, within mean field theory, to be independent of N in the limit of large N for gamma > 5/2 but to grow as N(alpha) with alpha = (5 - 2gamma)/(gamma - 1) for 2 < gamma < 5/2.

The dynamics of minority opinions in democratic debate

A model for the dynamics of opinion forming in democratic public debate is presented. Using concepts and techniques from the physics of disorder the dynamics of social refusal spreading is studied within a perfect world, where the minority holds neither better arguments nor lobbying backing. The one-person-one-vote rule, together with local majority rules, are used to determine the outcome of local group discussions. In case of a local tie, the group decides on keeping the Status Quo. The geometry of social life shaped by offices, houses, bars, and restaurants is shown to determine the distribution size of these discussion groups. It is found to yield very asymmetric unstable thresholds to the total spreading of one opinion at the benefit of the refusal one. The associated dynamics is rather quick and completed within few days. This democratic paradox of public debate driven majority opinion reversal is discussed in light of some European construction issues. The model may apply to rumor and fear propagation.

Listen to Your Neighbors: How (Not) to Reach a Consensus

We study the following rather generic communication\slash coordination\slash computation problem: In a finite network of agents, each initially having one of the two possible states, can the majority initial state be computed and agreed upon by means of local computation only? We study an iterative synchronous application of the local majority rule and describe the architecture of networks that are always capable of reaching the consensus on the majority initial state of its agents. In particular, we show that, for any truly local network of agents, there are instances in which the network is not capable of reaching such a consensus. Thus, every truly local computational approach that requires reaching a consensus is not failure-free.

Democratic Consensus and the Local Majority Rule

In this paper we study a rather generic communication/ &lt;br /&gt;coordination/ computation problem: in a finite network of agents,&lt;br /&gt;each initially having one of the two possible states, can the majority&lt;br /&gt; initial state be computed and agreed upon (i.e., can a democratic&lt;br /&gt; consensus be reached) by means of iterative application of&lt;br /&gt;the local majority rule. We show that this task is failure-free only&lt;br /&gt;in the networks that are nowhere truly local. In other words, the&lt;br /&gt;idea of solving a truly global task (reaching consensus on majority&lt;br /&gt;) by means of truly local computation only (local majority rule)&lt;br /&gt;is doomed for failure.&lt;br /&gt;We also show that even well connected networks of agents that&lt;br /&gt;are nowhere truly local might fail to reach democratic consensus&lt;br /&gt;when the local majority rule is applied iteratively. Structural&lt;br /&gt;properties of democratic consensus computers, i.e., the networks&lt;br /&gt;in which iterative application of the local majority rule always&lt;br /&gt;yields consensus in the initial majority state, are presented.

社会的選好順序の選択に関する方略の発達

This study investigated developmental changes in strategies to determine a social preference order, given a set of individual preference orders.Experiment I examined how a social preference (decision as a group) was determined when each member's preference order was known. Seventy-four 5th graders and 86 undergraduates participated in this experiment. Subjects were shown patterns of individual preference orders of A, B, and C given by members of a group of three. One example is given below._??_The task was to decide upon a social preference order of three alternatives A, B, and C based on such preference order patterns.Experiment II aimed at examining more closely how children and undergraduates would make such decisions. Subjects were asked to describe in detail how they had decided such orders under two conditions. In PN (Positive-Negative) condition, each member's preference concerning the three alternatives were expressed as “Most desirable”,“Neutral”, and “Most undesirable”. In O (Order) condition, alternatives were all thought to be desirable, in a first, second, and third order. Seventy-two 5th graders, ninety 8th graders and eighty-six undergraduates participated in the experiment.The main results were as follows.1. Majority of decisions made by subjects were consistent when patterns of the individual preference order given by three members were the same.2. From analyzing subjects' own descriptions of the strategies of making decision, four underlying principles emerged:(1) minority-oriented majority rule (2) concession-oriented principle (3) majority rule (4) local majority rule.3. While most of 5th graders adopted the local majority rule, 8th graders and undergraduates adopted the variety of principles.4. As for 8th graders and undergraduates, the concession- oriented principle was adopted more often in the PN condition than in the O condition.Implications of these findings were discussed.

Iterative behaviour of generalized majority functions

Let us consider a society of n persons {P1,…,Pn} with interaction coefficient aij between Pi and Pj. Let {θ1,…,θp} be the set of opinions which may be assumed by any person. We study the dynamical behaviour of such a society under local majority rules. The results presented in this paper generalize many partial results previously obtained by several authors (Tchuente, 1977; Goles and Olivos, 1980; Poljak and Sura, 1982). Furthermore, by introducing algebraic invariants, analogous to those exhibited in Goles (1980) for the study of Boolean threshold functions, we provide a tool for a general approach of such dynamical models.