Language Understanding as a Constraint on Consensus Size in LLM Societies
Giordano De Marzo Claudio Castellano David Garcia


    
    
    giordano.de-marzo@uni-konstanz.de
    
    
    ^1University of Konstanz, Universitaetstrasse 10, 78457 Konstanz, Germany
    ^2Centro Ricerche Enrico Fermi, Piazza del Viminale, 1, I-00184 Rome, Italy.
    ^3Complexity Science Hub, Josefstaedter Strasse 39, 1080, Vienna, Austria
    ^4Istituto dei Sistemi Complessi (ISC-CNR), Via dei Taurini 19, I-00185 Rome, Italy
     
    
	§ ABSTRACT 
The applications of Large Language Models (LLMs) are going towards collaborative tasks where several agents interact with each other like in an LLM society. In such a setting, large groups of LLMs could reach consensus about arbitrary norms for which there is no information supporting one option over another, regulating their own behavior in a self-organized way. In human societies, the ability to reach consensus without institutions has a limit in the cognitive capacities of humans.
To understand if a similar phenomenon characterizes also LLMs, we apply methods from complexity science and principles from behavioral sciences in a new approach of AI anthropology. We find that LLMs are able to reach consensus in groups and that the opinion dynamics of LLMs can be understood with a function parametrized by a majority force coefficient that determines whether consensus is possible.
This majority force is stronger for models with higher language understanding capabilities and decreases for larger groups, leading to a critical group size beyond which, for a given LLM, consensus is unfeasible.
This critical group size grows exponentially with the language understanding capabilities of models and for the most advanced models, it can reach an order of magnitude beyond the typical size of informal human groups.
	

	Language Understanding as a Constraint on Consensus Size in LLM Societies
    David Garcia^1,3
    September 9, 2024
=========================================================================






§ INTRODUCTION



Large Language Models (LLMs) have proven capabilities for particular applications, such as summarization <cit.> or sentiment analysis <cit.>, but can also be used in group settings where several heterogeneous agents interact with each other to tackle more complex collaborative tasks <cit.>. This goes beyond ensembles of models for one task <cit.>, for example in collaboration setups where multiple LLMs have different roles and tasks, such as AutoGPT [https://github.com/Significant-Gravitas/AutoGPT] and more recently Microsoft's AutoGen <cit.>. In a more organic way, AI-powered devices and assistants, such as Siri [https://openai.com/index/openai-and-apple-announce-partnership/] or the Humane AI Pin, can perform everyday tasks in interaction with each other, for example coordinating events or negotiating prices.

As we move towards a society where intelligent machines interact with each other, it becomes important to understand their ability to agree with each other in large groups. This can motivate new applications but also identify risks stemming from undesired collective behavior of machines. For example, trading bots interacting through the stock market can lead to flash crashes <cit.>.  Current research on the behavior of LLMs has mostly focused on their behavior in isolation <cit.> and collective behaviors have been explored less <cit.>, so far with a focus on social simulation of network structures <cit.>, opinion and information spreading <cit.> and online interaction <cit.>. To prepare for large numbers of interacting LLMs, we need to understand if they can display collective alignment, such as emerging consensus, what determines the abilities of LLMs to coordinate, and at what scale that can happen.


Social groups can reach consensus on behavioral norms even when there is no preference or information supporting one option over another. Animals and early human groups develop and sustain those norms when each individual in the group knows the identity and behavior of all other members of the group. This leads to a scaling of group size with brain structure, where human groups reach sizes between 150 and 300 <cit.>. Human societies have built institutions and other ways of decision making to reach higher scales, but the cognitive limit of keeping a scale of about 250 contacts remains even in an online society <cit.>. This insight can be translated to LLM societies, where consensus could emerge in arbitrary norms and where the cognitive abilities of LLM agents could play a role in the size of consensus. These are new questions of AI anthropology where the insights and methods for the previous study of human societies can be applied to study the size and complexity of LLM societies.


In this article, we investigate the ability of groups of LLMs to reach consensus about norms for which there is no information supporting one option over another. The emergence of consensus is a foundational aspect of social systems, where individual interactions lead to the formation of a unified agreement or shared understanding without the need for a central authority or structure <cit.>. We develop a framework to test if groups of LLMs can reach consensus and use it to analyze a benchmark of proprietary and open-source models. We apply insights from previous opinion dynamics research to understand the emergence of consensus in LLM societies, which allows us to measure a majority force that enables consensus in groups of LLMs. This majority force is a function of language understanding capabilities of models and group size, where consensus might not emerge beyond a critical size for a given LLM. Furthermore, we test if the most capable LLMs are able to reach consensus at scales that go beyond the spontaneous consensus formation of human groups.



§ RESULTS


   
		

 §.§ Opinion Dynamics of Large Language Models

                To investigate the opinion dynamics of large language models, we perform simulations using agents guided by various LLMs, as for instance models from the GPT, Claude, and Llama families. Simulations run as follows. Each agent is assigned an initial opinion randomly chosen from a binary set (e.g., "Opinion A" and "Opinion B"), where one is chose as the norm for reference. At each time step, a single agent is randomly selected to update their opinion. The selected agent receives a list of all other agents and their current opinions and is then prompted to choose their new opinion based on this information.
                This approach mirrors binary opinion dynamics models such as the voter model or Glauber dynamics <cit.>, where agents update their opinions based on peer interactions only. However, unlike traditional Agent-Based Models (ABMs) with predefined, hard-coded opinion update rules, here agents are allowed to autonomously decide their opinions. More details on this simualtion framework can be found in the Methods section.
   
                In order to follow the evolution of the system, we define the average group opinion m as 
                
    m=1/N∑_is_i=N_1-N_2/N.

                Here s_i is the opinion of agent i and we adopt the convention that the first opinion corresponds to s_i=+1, while the second to s_i=-1 (i.e. in favor and against the norm). We also introduce N_1 and N_2 as the number of agents supporting opinion 1 and 2, respectively, while N is the total number of agents. In these terms we can define the consensus level C=|m| that quantifies the level of agreement among individuals. Full consensus corresponds to C=1, while C=0 means that the system is split in two groups of equal size and different opinion. Three scenarios can happen in the evolution of C(t):
			
                    
  * C can converge to 1 (m converges to ±1), meaning that all agents are aligned and consensus is reached;
                    
  * C can oscillate around a value greater (smaller) than 0 without ever reaching 1. In this case a partial consensus is reached, but not as a collective.
                    
  * if C keeps fluctuating around zero, consensus is completely absent and the group is constantly in a disordered state.
			



                
                
                We show in Fig. <ref> the evolution of the consensus level for five different models and a group size of N=50 LLM agents. We also show the boxplot of C(t=10) over 20 realizations. The two most advanced models we considered, Claude 3 Opus and GPT-4 Turbo, reach consensus in all simulations, which corresponds to |m|=1 in the boxplot. Conversely, smaller models (Llama 3 70b, Claude 3 Haiku and GPT-3.5 Turbo) do not reach consensus in any of the simulations and only reach partial levels of agreement with |m|<0.5.
			
		
                We can get a deeper understanding of the underlying opinion dynamic process by looking at the adoption probability P(m), defined as the probability of an agent to support the norm as function of the average group opinion m. The right panel of Fig. <ref> shows P(m) for ten of the most popular LLMs (we consider N=50 and we set the model temperature to T=0.2). The most advanced models (GPT-4 family, Llama 3 70b, Claude 3 Sonnet and Opus) show a stronger tendency to follow the majority, with more pronounced S-curves. The adoption probability is an increasing function of m that saturates to high values (low values) when m=1 (m=-1). On the other hand, smaller models (GPT-3.5, Claude 3 Haiku), have a less pronounced tendency to follow the majority, with GPT-3.5 going against the majority for small values of m. 

                Adoption probabilities can be approximated by the function
			
    P(m)=1/2*tanh*β· m+1.

                The parameter β, that we call majority force, regulates the level of randomness in the system. For β=0 each agent behaves fully randomly (the new opinion is selected by coin-tossing) and no consensus can be reached. For β=∞ agents always align with the global majority and consensus is reached very quickly.
                The agreement with (<ref>) is made fully evident by fitting the parameter β empirically for each model and plotting P(m^*) as a function of the rescaled average opinion m^*=m·β, where all models except GPT3.5 have a good agreement with the function.
                As shown in the inset of Fig. <ref>, all adoption probabilities collapse on the same curve. 
                The adoption probability of (<ref>) is analogous to the case of the simplest spin system in physics of complex systems, the Curie-Weiss model <cit.>, where atoms interact and their spins can align in a magnet as opinions can align in consensus. 
                
				
		

 §.§ Language understanding in consensus formation

                The fit of the function P(m^*), to the opinion dynamics of models highlights that their differences are captured by the majority force parameter β. The left panel of Fig. <ref> shows the values of β for N=50 versus the MMLU benchmark of each model, which measures the language understanding and cognitive capabilities of LLMs <cit.>. There is a clear monotonic relationship between MMLU and β, with a correlation coefficient of 0.75 (p-value 0.01). This means that models with higher language understanding capabilities tend to exhibit a stronger tendency towards consensus, but none of the models show consistent behavior against the majority.
                This is directly related to language understanding and not just context window length as a plain "memory size", as the context window length L has a weaker and less significant correlation with β (correlation 0.49 with p-value 0.15). More details are reported in the Appendix.

                In animal (human and non-human) societies, group size plays a crucial role, with a progressive loss of norm stability as the number of individuals increases. We hypothesize that a similar phenomenon may occur also in LLMs societies, where agents might have their ability to understand the norm limited by the amount of information they have to process on the rest of the group.
                The right panel of Fig. <ref> shows the estimated β as a function of N for different LLMs. Given the cost of performing the simulations with proprietary models, we selected a subset of the LLMs we analyzed before to probe the space of possible values of β as a function of N. Independently of the model, we observe a general tendency where the larger the group, the lower the β and the weaker is the majority force. Models with a low MMLU tend to reach low values of β already for N of the order of few tens of agents, while the most advanced models still present a substantial majority force even for N=1000.
                
            

 §.§ Critical Consensus Size

                

                As we discussed above, the adoption probability curves of the models are the same as the Curie-Weiss model on a fully connected network. A well known result for this model is that β_c=1 represents a transition point: for β<1 the system shows no sign of consensus, while when β>1 order emerges and the agents can coordinate and reach consensus <cit.>. As a consequence, we expect a change in the behavior of the LLM society depending on the group size and also on the specific LLM driving the agents. Such a result is valid in the limit of very large systems, while here we are dealing with relatively small sizes. However, we can still observe the effects of this transition by inspecting how the consensus time i.e., the average time needed to reach consensus starting from a random state, grows for larger N. If β were infinite, the consensus time would grow logarithmically with N. Instead, as shown on Fig. <ref> using Llama 3 70b as an example, the consensus time shows two different regimes. When N is small the consensus time increases slowly, while for larger N, consensus time grows extremely fast. The transition between the two regimes occurs for a critical size N_c, that we define as β(N_c)=β_c=1. For Llama 3 70b N_c≈50. Note that as we detail in the methods, this value of N_c is actually an upper bound. On the other hand, a more capable model like GPT-4 Turbo does not present any deviation from the linear regime, as shown on Fig. <ref>, with the same scaling pattern as the Curie-Weiss model.
                
                By studying the values of β as a function of N we can determine, for each model, the critical consensus size N_c where β crosses the line at β=1 and that determines the size above which a model does not reach consensus. You can see some examples of this on the right panel of 
                Fig. <ref>. In the case of the most powerful models, provide a lower bound for this quantity as we were not able to find cases with β<1 even for N=1000. 
                Anthropology provides an expectation on how β depends on the language understanding capabilities of models, as primates exhibit a monotonic relationship between neocortex ratio and typical group size <cit.>.
                
                We report the values of the critical consensus size N_c on the right panel of Fig. <ref>, where we plot N_c as a function of the MMLU benchmark. We also plot, in the same figure, the critical consensus size for humans as Dunbar's limit N_c≈ 150 <cit.>.
                LLMs display a similar trend as observed in anthropology, where language understanding capability predicts the limit of consensus size, i.e. the size of groups above which consensus becomes unlikely. Remarkably, the simplest models and humans are well aligned along an exponential growing N_c, suggesting that the capacity to reach consensus in large groups is connected to cognitive capabilities both in humans and in LLMs, when measure as language understanding ability. However, the most modern and advanced LLMs go beyond this exponential scaling, reaching a super-human coordination capability despite having a human-level MMLU performance. For instance both GPT-4 and GPT-4 Turbo are well above β_c also for N=1000, the largest system we considered and substantially beyond Dunbar's number.
			
	

§ DISCUSSION

	
        Human societies are characterized by emergent behavior that cannot be understood just by studying individuals in isolation. Consensus is one of such emergent group capabilities that has proven crucial in the development of languages, widely accepted norms and religions. For humans, the Dunbar number N_c≈ 150 gives the maximal number of personal relations we can cultivate and thus also the maximal group size in which consensus, intended as the emergence of common social norms, can exist. Studying humans and other primates, researchers have identified a power-law scaling connecting the neocortex ratio to the average group size <cit.>, thus proving the link between cognitive capabilities and the development of large societies. 
		
		LLMs are attracting a growing interests in the social sciences for their ability to mimic humans, both at the individual and, as recent studies suggest, at the group level. Indeed, like humans, LLMs show emergent group properties that were not directly coded in their training process. As we argue in this paper, consensus is one of these properties, with LLMs showing striking similarities with primates including humans. As a first result, we showed that all most advanced LLMs are characterized by a majority-following tendency described by a universal function with a single parameter β, the majority force. Remarkably, this function depends on the specific model and is the same describing magnetic spin systems. Different models typically have a different β, with the less sophisticated ones showing a smaller β, i.e., a less pronounced majority force and thus a more stochastic behavior that prevents consensus. The majority force depends not only on the model, but also on the group size: it tends to be larger in smaller groups. This evidence and the equivalence with the Curie-Weiss models allowed us to compute the “Dunbar number” of LLMs, a threshold above which societies composed by these artificial animals can no longer reach a consensus. While the less sophisticated models show a human-like scaling, the most sophisticated LLMs are capable to reach consensus in groups of size that go beyond what humans can do without explicit rules, despite having human-like cognitive capabilities in language-based tasks.
		
           
	These results are important due to the relevance of collective behavior and coordination in social contexts. More research is needed to understand other conditions that lead to the emergence of LLM consensus, especially when different models coexist or when some agents have privileged data access. The ability of LLMs to reach consensus can be beneficial, for example when looking to coordinate group activities of LLMs where incentives might not be aligned. When there is no information to guide how to behave, LLM societies could reach their own social norms that regulate their behavior to be predictable by other LLMs despite that lack of objectivity. However, this also poses a threat, as these norms might not be aligned with human values or could pose situations of coordinated behavior that threaten the integrity of a system, such as the case of flash crashes due to trading bots. Future research on AI anthropology can understand better how this kind of norms emerge in actionable scenarios, going beyond the idealized situation we studied here and illustrating both the promise and peril of LLMs agents collaborating within our society.
 
 
 
		
		
		
		
		
	

§ METHODS

		

 §.§ Opinion dynamics simulations

                In order to simulate an opinion dynamics process we implement a voter model-like process with the only difference being the use of LLMs. 
                
                    
  * At each infinitesimal time step dt=1/N an agent is randomly selected;
                    
  * the agent is given the full list of all agents in the system, each identified by a random name, and the opinions they support. Note that the opinion of the agent itself, like in the voter model, is not relevant;
                    
  * the selected agent is then asked to reply with the opinion it wants to support and its opinion is updated correspondingly; 
                    
  * the process is then iterated till consensus is reached or till the maximal number of updates is performed.
                
                Note that in one time step t→ t+1 we thus perform N updates (being dt=1/N), so that, on average, each LLMs is selected at least once. 
                In all our simulations we set the initial collective opinion m to zero, meaning that there are initially the same number of agents supporting the two opinions. In order to practically perform the opinion simulations, following the framework introduced in <cit.>, we exploit the prompt below 

                8cm
                                            Below you can see the list of all your friends together with the opinion they support.

                                            You must reply with the opinion you want to support.

                                            The opinion must be reported between square brackets.

                                            X7v A

                                            keY B

                                            91c B

                                            gew A

                                            4lO B

                                            ...

                
                
                Here A and B are the opinions names, which, in general, do not play any role. In all our simulations we used the opinion names k and z and we tested the effect of using different opinion names, obtaining no major difference.

                It's important to remark that most LLMs present an opinion bias, tending to prefer an opinion name over the other. This behavior is particularly strong when the opinion names have an intrinsic meaning, like for instance “Yes” and “No”. In this case LLMs have a strong preference toward the more “positive” opinion, tending to strongly prefer “Yes” over “No”. For this reason it is important to use random letters or random combinations of letters as opinion names. Even doing so, small biases are typically always present. However they are not as pronounced as in the situations we described above and they can easily removed. We do so by performing a random shuffling of the opinion names at each iteration. So for instance at t=0 the first opinion may be called k and the second z, while at t=dt these names may swap with probability 0.5, meaning that the first opinion will now be called z and the second k. More details about this procedure are reported in the Appendix.
		

 §.§ Details on the LLMs

                In all the simulations reported in the main text we used as model temperature T=0.2. As we details in the Appendix, there are no major differences using different values for T, but a low model temperature ensures reliability in the output format. We also report in Table <ref> the detailed list and model version of all the LLMs we exploited.
                
               
	 	

 §.§ Curie-Weiss Model

                The Curie-Weiss (CW) Model is arguably the most simple spin model. It describes a system of N atoms that interact with ferromagnetic interaction i.e. that tend to align, and that can only have two states either s_i=+1 (up) or s_i=-1 (down). Moreover, these spins interact on a fully connected network, meaning that each of them is influenced by all the other spins. The CW model is therefore the mean field limit of the well know Ising model. The order parameter of this model is the magnetization m, the equivalent of our collective opinion, defined as the average of the spin values m=*s_i. The mapping between our LLM based opinion simulations and the CW model derives from the transition probability defined by Eq. (<ref>). This expression is indeed equivalent to the Glauber dynamic <cit.>, which allows to simulate the CW model by means of a Markov chain Monte Carlo approach.

                It is relatively easy to compute the equilibrium value of the magnetization in the CW model. This is done by deriving the so called self-consistency equation, that reads 
                
    m=tanh(β m).

                This equation has a different behavior depending on the value of the majority force β (that in the CW model is called inverse temperature). For β<1 it only admits the solution m=0, while for β>1 m=0 stops to be a stable solution, and two new solution m=± m^* appear. The point β=1 is a critical point characterized by a second order phase transition. This means that as soon as β>1, m^* starts to grow gradually, till reaching the value m^*=1 for large values of β. It is important to remark that in finite size systems, the ability to reach consensus depends both on the value of m^* and on the fluctuations around this value. In general, order will emerge as soon as β>1, but the system may still not reach a full consensus due to statistical fluctuations being too small. For this reason the condition β(N_c)=1 is actually an upper bound to the maximal group size where consensus can be reached, since for values of the majority force close to one, the fluctuations may still be not enough for a full consensus to be reached. 


        
    
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§ BIAS REMOVAL

        As we mention in the main text, in order to remove opinion biases we have to shuffle the opinion names at each iteration. However, this only works when the initial bias is not extremely pronounced. We show in Fig. <ref> the adoption probability with and without shuffling for two opinion names combinations: “yes, no” and “k, z”. As it is possible to see, the former has a very strong bias toward “yes” and therefore the shuffling procedure results in an adoption probability not described by a tanh function. On the other hand, “k, z” only present a very mild bias and the shuffling procedure allows such a bias to be removed without altering the shape of the adoption probability.
    

§ ROLE OF OPINION NAMES

        
        In order to test the stability of our results we investigate the shape of the adoption probability under different opinion names. We report in Fig. <ref> (top row) the results of this procedure for four possible combinations of opinion names and three different LLMs, representative of the three families of models we studied in our work. As it is possible to see there are differences only in the case of Llama and just for one of the opinion name pairs we considered. In any case, the functional form of the adoption probability is always the same and therefore the general picture is not affected by these minor variations. Moreover, the most advanced models, GPT-4 Turbo and Claude 3.5 Sonnet, show no differences at all, suggesting that as the model become more capable, biases and differences due to the opinion names disappear. 
    

§ MODEL TEMPERATURE

        Another aspect we tested is the effect of the model temperature T. This parameter sets the level of creativity or randomness of the LLM. For T=0 the model behaves deterministically, always producing in output the token (word) with the highest probability. Instead, when T>0, randomness starts to play a role and also other tokens can be observed in the output. As shown in Fig. <ref> (bottom row), we considered three different temperatures T=0.2, 0.6, 1.0 observing no substantial difference in the adoption probability.
   
    

§ ROLE OF CONTEXT WINDOW

        
        In order to understand if the majority force parameter β is influenced by the language understanding and cognitive capabilities of the LLMs or rather by their context window length, we repeat the analysis performed in Fig. <ref> (left panel). In this case, however, we compare the majority force with the context windows of the ten models we analyzed. As shown in Fig. <ref> there is a much weaker and less significant correlation (0.49 with a p-value of 0.15) with respect to the MMLU benchmark, suggesting that the context window length only plays a marginal role. 
        


