1 Introduction↩︎

Recently, large language models (LLMs) have made remarkable improvements and demonstrated a high level of expertise in comprehending and producing human-like natural languages [1]. Given the remarkable text generation capabilities of LLMs, social deductive games, such as Werewolf [2], [3] and Avalon [4], are suitable scenarios for studying the social preference of LLMs [5]. Some studies have indicated that LLMs, such as GPT-4 [6], exhibit strategic behaviors including cooperation, confrontation, deception, and persuasion in these games [2]–[4], [7]. However, the potential opinion leadership of LLMs has been overlooked and confounded by the existing frameworks and evaluation metrics [8]. Opinion leaders are individuals who exert personal influence upon a certain number of other people in certain situations [9]. Contrary to the aforementioned interaction strategies, opinion leadership is the composite ability of opinion leaders to comprehensively employ these strategies to influence the decisions of their followers and shape public opinion [10].

Investigating the extent of opinion leadership in LLM-based agents is crucial for interaction design (IxD), decision optimization, and public regulation. In multi-agent systems, such as smart manufacturing, a few opinion leaders can significantly impact task efficiency and outcomes [11]. As AI assistants and customer service agents, LLMs can shape user experience and business decisions. In social media and forums, the opinion leadership of LLMs risks influencing social discourse and public decisions. Despite its significance, the literature on this topic remains scarce. Unlike externally designated traditional leaders, opinion leaders emerge from collective consensus. While various methods exist for detecting opinion leaders in human populations [10], [12], there is a notable absence of systematically evaluating AI agents’ opinion leadership, particularly for LLMs.

Examining opinion leadership among LLM-based agents is non-trivial. Opinion leaders are often concealed [10], making it challenging to identify and distinguish them from other agents and humans in general tasks. Additionally, assessing their opinion leadership is also a daunting task due to the complexity of decision-making tasks and the impracticality of conducting large-scale real-task randomized controlled trials to evaluate the diverse effects on human and AI followers. Fortunately, as a common interactive scenario for studying the emergence of AI agent behaviors  [8], the Werewolf game provides a promising testing ground to address these challenges. The Sheriff role in the Werewolf game is jointly elected by other players and can summarize the statements and give decision-making suggestions. This role represents the collective will and enables the player to provide information and exert influence, making it a credible proxy for an opinion leader. By assessing the reliability and spillover of the Sheriffs’ decisions, we can evaluate their opinion leadership. Furthermore, incorporating diverse LLM and human players allows us to compare the impact of LLMs’ opinion leadership across different followers. Our main contributions can be summarized as:

1. To the best of our knowledge, this is the first in-depth analysis of opinion leadership within LLMs. We clarify the opinion leadership of diverse LLMs in various contexts.

2. We introduce the setting of the Sheriff and implement a Werewolf game framework, which can seamlessly integrate diverse LLM-based agents and human players. Besides, We devise two novel metrics to evaluate the opinion leadership of different LLMs.

3. We conduct simulations and human evaluations to assess LLMs’ opinion leadership, and we collect a Werewolf question-answering (WWQA) dataset for further analysis.

2 Related Work↩︎

LLM-based Agent has risen to prominence in recent research, spurred by advancements in LLMs. These agents are capable of reasoning and decision-making based on their own states [13], as well as interacting with the environment [14], [15]. Some studies indicate that LLMs also exhibit a certain level of cognitive abilities [16], [17] and demonstrate potential in simulating believable human behavior [18]. This has led to the development of collaborative frameworks for multi-agent interactions such as AgentVerse [19], AutoGen [20], MetaGPT [21], ChatEval [22], etc. LLM-based agents have shown promise in simulating complex social dynamics in various scenarios, including courtroom [23], game development [21], auctions [24], etc. To further introduce adversarial dynamics between agents, LLMs are also being utilized in gameplay scenarios such as Texas Hold’em poker [25], complex video games [26], [27], and multi-player social deduction games (such as the Werewolf and Avalon game) [2]–[4], [7], [28], [29], which heavily rely on natural language interactions. These games provide prototypes of mixed cooperative-competitive social environments similar to human society, allowing researchers to observe whether these agents can exhibit humanlike behavior patterns and social principles  [30], as well as explore approaches to incentivize the agents to perform better or more human-like. However, existing LLM-based frameworks for the Werewolf game have regrettably overlooked the inclusion of a pivotal character, the "Sheriff", resulting in an underestimation of the influence of opinion leaders. Unlike the passive leaders in the Avalon game, the "Sheriff" in the Werewolf game is actively elected through agent voting, making it fundamentally distinct.

Opinion leadership is an important concept in the process of social learning in human society [31]. It was first developed in [32]‘s study of the 1940 presidential election. When faced with uncertainty, humans tend to learn from "opinion leaders" to enhance the wisdom of their decisions [33]. Opinion leaders can be individuals with extensive knowledge in a specific subject area (experts) or individuals with extensive social connections (social connectors) [34]. They come from the collective will and differ from their followers in information sources, social participation, social status, etc. [9]. To exert influence, opinion leaders must be trustworthy [35], and the extent to which their opinions are adopted by recipients relies on their credibility [36], [37]. Trust diminishes conflicts, alleviates the necessity for information verification [38], and enhances people’s inclination and ability to attentively and efficiently embrace others’ perspectives [37], [39]. As we strive to create credible simulations of human society, it seems intuitive to extend the concept of opinion leaders to the interactions among LLMs. The agent who garners the most trust and is selected by the majority of agents naturally emerges as the opinion leader. This analysis delves deeper into the social ties among multiple agents, transcending the narrower confines of discussions solely focused on individual agents’ strategic behaviors.

3 Framework and Proposed Metrics↩︎

We consider the Werewolf game with \(N=7\) players, including two Werewolves, three Villagers, one Seer, and one Guard. There is a moderator to move the game along, who will only pass on information and not participate in the game. At the beginning of the game, each player is assigned a hidden role which divides them into the Werewolves and the Villagers (Seer, Guard, Villager). Then the game alternates between the night round and the day round until one side wins the game. Let \(X_i^{r}\) be the \(i\)-th player. We use the superscript \(r\in\mathcal{R}\) to represent the role and \(\mathcal{R} = \{\text{W, V, Se, G}\}\) denotes the Werewolf, Villager, Seer, and Guard, respectively. For example, \(X_3^{\text{V}}\) denotes player_3 and it is a Villager. We might omit the index \(i\) or the role \(r\) when there is no ambiguity.

We refer to one night-day cycle as a round, indexed by \(t\). At night, the Werewolves recognize each other and choose one player to kill. The Seer chooses one player to check its hidden tole. The Guard protects one player including themselves from the Werewolves. The Villagers do nothing. All night actions in the \(t\)-th round are formally defined as follows. We use \(A_{night,t}\) to represent all night actions. \[\begin{align} \begin{array}{ll} \text{The Werewolf X^{\text{W}} kills } X_i \text{:} & X_i = A_{{kill},t}\left(X^{\text{W}}\right)\\ \text{The Seer X^{\text{S}} sees } X_i \text{:} & X_i = A_{{see},t}\left(X^{\text{Se}}\right)\\ \text{The Guard X^{\text{G}} protects } X_i \text{:} & X_i = A_{{protect},t}\left(X^{\text{G}}\right) \\ \end{array} \end{align}\] During the day, three phases including an announcement phase, a discussion phase, and a voting phase are performed in order. First, the moderator announces last night’s result to all players. We denote the result of the \(t\)-th night round as \(D_t\). \(D_t = X_i\) means \(X_i\) was killed in the \(t\)-th night round, while \(D_t = X_{\infty}\) means no player was killed. Then, each player takes turns to make a statement in an order determined by a special role, the Sheriff. Only the first day has an election phase for the Sheriff. The Sheriff is a special role that does not conflict with the players’ roles. The elected Sheriff has the authority to determine the statement order and provide a summary to persuade other players to vote in alignment with him. We use the superscript \(^*\) to denote the Sheriff, e.g., \(X^{\text{V},*}_3\). Then the actions of determining the statement order and making a statement are defined as follows. \[\begin{align} \begin{array}{ll} \text{X^* determines the statement order}\text{:} & [X_i, X_{i+1}, ... ,X_N, X_1, ..., X^{*}] = A_{{order},t}\left(X^{*}\right)\\ \text{X_i makes a statement}\text{:} & S_{i,t} = A_{{state},t}\left(X_i\right) \\ \end{array} \end{align}\] At last, each player votes to eliminate one player or chooses not to vote, e.g., \[\begin{align} \begin{array}{ll} \text{X_i votes to eliminate } X_j\text{:} & X_j = A_{{vote},t}\left(X_i\right) \end{array} \end{align}\] After the voting phase, the moderator announces the result \(E_t\) and the game continues to the next night’s round. \(E_t = X_i\) means \(X_i\) received the most votes and was eliminated during the day of round \(t\), while \(E_t = X_{\infty}\) means no player was eliminated. The Werewolves win the game if the number of remaining Werewolves is equal to the number of remaining Seer, Guard, and Villagers. The Seer, Guard, and Villagers win if all Werewolves are eliminated.

Figure 1 depicts our framework extended from [3]. The players in our framework can be LLMs or humans. All notations are listed in Table [table:notation] in Appendix 7.

3.1 LLM-based Player↩︎

Players of different roles are required to perform a series of actions, including night actions (to kill, protect, or see) and day actions (to speak and vote). Each LLM-based player is implemented by an LLM-based agent, which can store the related information during the game and perform actions according to the game situation.

The LLM-based players perform actions through prompting. The prompt consists of 3 major components: (1) the game rules and the assigned role; (2) the contextual information; and (3) task description. The prompt template can be found in Appendix 6.2. Communication and action history play a crucial role in the Werewolf Game. However, due to the limited context length of LLMs, it might be infeasible to integrate all information into the prompt. Thus, following [3], we design a strategy to manage the contextual information.

To begin with, we divide the contextual information for \(X_i^r\) into the following two parts.

1. The set of facts \(\mathcal{F}_{i,t}\) consists of all factual information before round \(t\), including the role of \(X_i^r\), the announcement made by the moderator, etc. For example, \(\mathcal{F}_{i,t}\) of an Seer \(X_i^{\text{Se}}\) can be expressed as

\[\begin{align} \begin{array}{ll} \mathcal{F}_{i,t} = & \underset{{\text{ID \& role}}}{ \underline{\{i, \text{Se}\}} } \bigcup \underset{\text{night results}} {\underline{\big\{D_{\tau}\big\}_{\tau = 1}^{t-1}}} \bigcup \underset{\text{day results}}{\underline{\big\{E_{\tau}\big\}_{\tau = 1}^{t-1}}} \bigcup \underset{\text{X_i's statements}}{\underline{\big\{S_{i,\tau}\big\}_{\tau=1}^{t-1}}} \bigcup \underset{\text{X_i's night actions}}{\underline{\big\{ A_{see, \tau}(X_i^{\text{Se}})\big\}_{\tau = 1}^{t-1}}} \\ & \underset{k\in\texttt{Alive}_d(\tau)}{\bigcup} \underset{\text{voting results}}{\underline{\big\{A_{vote,\tau}(X_k)\big\}_{\tau=1}^{t-1}}} \end{array} \end{align}\]

1. where \(\texttt{Alive}_d(\tau)\) is the IDs of alive players before the day of round \(\tau\).

2. The set of public statements \(\mathcal{P}_{i,t}\) consists of the public statement of other players before the \(t\)-th round. \[\begin{align} \mathcal{P}_{i,t} =\underset{k\in \texttt{Alive}_d(\tau), k\neq i}{\bigcup} \big\{S_{k,\tau} = A_{state,\tau}(X_k)\big\}_{\tau=1}^{t-1} \end{align}\]

Note that the statement made by other players might be deceptive. In light of this, we design a reasoning step to deduce the reliability of other players, and based on this reliability, we categorize public statements \(\mathcal{P}_{i,t}\) into potential truth \(\mathcal{P}_{i,t}^{true}\) and potential falsehoods \(\mathcal{P}_{i,t}^{false}\). Concretely, we implement an additional action, called reliability reasoning, before making night or day actions. In this step, we prompt the LLM to infer the identities of other players based on historical information and provide confidence levels. For example, at the night of round \(t\), \(X_i\) reasons the identity \(\hat{r}_{i,j,t}^n\) of \(X_j\) and provides the confidence level \(c_{i,j,t}^n\) before making actions as follows: \[\begin{gather} \label{eq3} \begin{array}{ll} \text{X_i reasons the identity of X_j:} & c_{i,j,t}^n, \hat{r}_{i,j,t}^n = A_{{reason},t}\left(X_i, X_j\right) \end{array} \end{gather}\tag{1}\] The superscript \(\{n,s,v\}\) denote the reasoning results before the night actions, making a statement and voting, respectively. The confidence level is a scalar from \(5\) to \(10\), then the reliability is \[\begin{align} \label{eq4} m_{i,j,t}^n = \left\{ \begin{array}{ll} 11-c_{i,j,t}^n, & \text{if } \hat{r}^n_{i,j,t} = \text{W}\\ c_{i,j,t}^n, & \text{otherwise} \end{array} \right. \end{align}\tag{2}\] Given the reliability of other players, \(X_i\) will divide the statements in \(\mathcal{P}_{i,t}\) into two parts. It regards the statements of players with reliability larger than \(\alpha = 6\) as potential truths, otherwise are potential falsehoods. \[\begin{align} \label{eq5} \mathcal{P}_{i,t}^{n,true}, \mathcal{P}_{i,t}^{n,false} = M(\mathcal{P}_{i,t} \mid \alpha,\{ m_{i,j,t}^n\mid {j\in \texttt{Alive}_n(t), j\neq i}\}) \end{align}\tag{3}\] where \(M(\cdot\mid \alpha, \{\cdots\})\) is an operator that splits a set of public statements into two disjoint subsets according to the reliability threshold \(\alpha\) and \(\texttt{Alive}_n(t)\) is the IDs of alive players before the night of round \(t\).

The whole process during round \(t\) is illustrated in Figure 3 in Appendix 7. On the night of round \(t\), we have the set of facts \(\mathcal{F}_{i,t}\) and the set of public statements \(\mathcal{P}_{i,t}\). \(\mathcal{P}_{i,t}=\emptyset\) if \(t=1\) (at the beginning of the game). The LLM-based player will divide the set of public statements \(\mathcal{P}_{i,t}\) into \(\mathcal{P}_{i,t}^{true}\) and \(\mathcal{P}_{i,t}^{false}\) based on the previous reasoning results. Before taking night actions, \(X_i\) reasons the reliability of other players. \[\begin{gather} \label{eq6} c_{i,j,t}^n, \hat{r}_{i,j,t}^n = A_{{reason},t}\left(X_i, X_j\mid \mathcal{F}_{i,t}, \mathcal{P}_{i,t}^{true}, \mathcal{P}_{i,t}^{false}\right),\quad j\in \texttt{Alive}_n(t), j\neq i \end{gather}\tag{4}\] where \(\texttt{Alive}_n(t)\) is the IDs of alive players before the night of round \(t\). Then \(X_i\) takes a night action as follows. \[\begin{gather} A_{night,t}\left(X_i\mid \mathcal{F}_{i,t}, \mathcal{P}_{i,t}^{n,true}, \mathcal{P}_{i,t}^{n,false}\right ) \end{gather}\] where \(\mathcal{P}_{i,t}^{n,true}, \mathcal{P}_{i,t}^{n,false}\) are sets of potential truth and potential falsehoods obtained as Eq. (3 ).

On the day of round \(t\), the moderator announces \(D_t\). Without loss of generality, we assume that \(X_i\) was not killed on the night \(t\) and the order of statement is the same as the IDs of players. Then \(X_{i}\) will receive a set of public statements made by previous players, i.e., \[\begin{align} \mathcal{H}^s_{i,t} = \{S_{j,t}\mid j\in \texttt{Alive}_d(t), j<i\} \end{align}\] Now the available public statements for \(X_{i}\) become \(\mathcal{P}_{i,t}^{s} = \mathcal{P}_{i,t} \cup \mathcal{H}^s_{i,t}\), where the superscript \(s\) denotes the timestamp before \(X_i\) makes a statement. Similar to Eq. (4 ), \(X_i\) first performs reliability reasoning and then makes a statement. \[\begin{align} & c_{i,j,t}^s, \hat{r}_{i,j,t}^s = A_{{reason},t}\left(X_i, X_j\mid \mathcal{F}_{i,t}, \mathcal{P}_{i,t}^{n,true}, \mathcal{P}_{i,t}^{n,false}, D_t, A_{night,t}(X_i), \mathcal{H}^s_{i,t}\right), j\in\texttt{Alive}_d(t),j\neq i \\ & \mathcal{P}_{i,t}^{s,true}, \mathcal{P}_{i,t}^{s,false} = M\left(\mathcal{P}_{i,t}^s \mid \alpha,\{ m_{i,j,t}^s = 11 - c_{i,j,t}^s \mid {j\in \texttt{Alive}_d(t), j\neq i}\}\right) \\ & S_{i,t} = A_{state, t}\left(X_i\mid \mathcal{F}_{i,t}, \mathcal{P}_{i,t}^{s,true}, \mathcal{P}_{i,t}^{s,false} , D_t, A_{night,t}(X_i)\right) \end{align}\] Subsequently, \(X_{i}\) will collect a set of public statements after its statement, i.e., \[\begin{align} \mathcal{H}^v_{i,t} = \{S_{j,t}\mid j\in \texttt{Alive}_d(t) , j>i\} \end{align}\] When the discussion phase ends, each player begins to vote individually. Now the available public statements for \(X_{i}\) become \(\mathcal{P}_{i,t}^{v} = \mathcal{P}_{i,t}^s \cup \mathcal{H}^v_{i,t}\), where the superscript \(v\) denotes the timestamp before \(X_i\) votes. Then \(X_i\) reasons the reliability of other players again and votes. \[\begin{align} & c_{i,j,t}^v, \hat{r}_{i,j,t}^v = A_{{reason},t}\left(X_i, X_j\mid \mathcal{F}_{i,t}, \mathcal{P}_{i,t}^{s,true}, \mathcal{P}_{i,t}^{s,false} , D_t, A_{night,t}(X_i), S_{i,t}, \mathcal{H}^v_{i,t}\right), \nonumber \\ & \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: j\in \texttt{Alive}_d(t), j\neq i \tag{5}\\ & \mathcal{P}_{i,t}^{v,true}, \mathcal{P}_{i,t}^{v,false} = M\left(\mathcal{P}_{i,t}^v \mid \alpha,\{ m_{i,j,t}^v = 11 - c_{i,j,t}^v \mid {j\in \texttt{Alive}_d(t), j\neq i}\}\right) \tag{6} \\ & A_{vote, t}\left(X_i\mid \mathcal{F}_{i,t}, \mathcal{P}_{i,t}^{v,true}, \mathcal{P}_{i,t}^{v,false} , D_t, A_{night,t}(X_i), S_{i,t}\right) \tag{7} \end{align}\] After the voting phase, the moderator announces the result \(E_t\). \(X_i\) updates the set of facts by adding all its actions, other players’ voting actions, and the night and day results. \[\begin{align} \mathcal{F}_{i,t+1} = \mathcal{F}_{i,t} \bigcup \{D_t\}\bigcup\big\{A_{night,t}(X_i)\big\}\bigcup \big\{S_{i,t}\big\} \underset{k\in \texttt{Alive}_d(t)}{\bigcup} \big\{A_{vote,t}(X_k)\big\}\bigcup \big\{E_t\big\} \end{align}\] For the set of public statements, we require the LLM to output evidence when performing reasoning in round \(t\), and then remove unmentioned statements in \(\mathcal{P}_{i,t}^{v,true}, \mathcal{P}_{i,t}^{v,false}\) to get \(\widetilde{\mathcal{P}}_{i,t}^{v,true}\) and \(\widetilde{\mathcal{P}}_{i,t}^{v,false}\). Finally, we have \[\begin{align} \mathcal{P}_{i,t+1}^{true} = \widetilde{\mathcal{P}}_{i,1}^{v, true}, \quad \mathcal{P}_{i,t+1}^{false} = \widetilde{\mathcal{P}}_{i,t}^{v,false}, \quad \mathcal{P}_{i,t+1} = \mathcal{P}_{i,t+1}^{true} \bigcup \mathcal{P}_{i,t+1}^{false} \end{align}\] The game moves to the next round.

3.2 Opinion Leadership Metrics↩︎

Now we introduce the special role, i.e., the Sheriff into the game framework. On the day of round \(t=1\), the moderator will announce the selection of the Sheriff after the election phase. Let \(L(t)\) be the operator that returns the ID of the Sheriff during the day of round \(t\). Without the loss of generality, we assume \(X_l\) is selected as the Sheriff, i.e., \(L(1)=l, X_l = X^*\). All alive players will receive a special message after the election phase:

After discussion and a vote, player_\(l\) was selected as the Sheriff, who can determine the order of statements, summarize the discussion, and provide advice for voting at last.

In this way, combined with the description in the game rule, we establish the authority of the Sheriff. During the day of round \(t\), \(X_{L(t)}\) can determine the order of statements. It first performs the reasoning step to yield \(\mathcal{P}_{L(t),t}^{d,true}\) and \(\mathcal{P}_{L(t),t}^{d,false}\) similar to Eq. (1 )-(3 ) and then selects its left- or right-hand side player to speak first by \[\begin{align} \mathcal{O}(t) = A_{order,t } \left( X_{L(t)} \mid \mathcal{F}_{{L(t)},t}, D_t, \mathcal{P}_{L(t),t}^{d,true}, \mathcal{P}_{L(t),t}^{d,false} \right) \end{align}\] The statement order \(\mathcal{O}(t)\) will slightly influence the collection of public statements, i.e., \(\mathcal{H}^s\) and \(\mathcal{H}^v\). But the whole process of making statements and voting is the same.

In different scenarios and research contexts, various definitions of opinion leaders have been proposed [9], [40]–[42]. Despite diverse definitions, the common characteristics of opinion leaders are mainly reflected in two aspects: (1) opinion leaders are generally more trustworthy; (2) opinion leaders influence the views and even decisions of others. Based on these, we design two evaluation metrics to measure the opinion leadership of LLM-based players.

Ratio is to measure the reliability of the Sheriff. When all players finish their statements, the voting phase starts. We collect all the reasoning results before each player’s voting action, i.e., \(m_{i,j,t}^v\). Then we calculate the average mutual reliability of all players except the Sheriff. \[\begin{align} \bar{m}_1(t) = \frac{1}{(N_d(t)-1)(N_d(t)-2)}\sum_{i\in \texttt{Alive}_d(t), i\neq L(t)}\sum_{j\in\texttt{Alive}_d(t), j\neq L(t), j\neq i} m_{i,j,t}^v \end{align}\] where \(N_d(t) = |\texttt{Alive}_d(t)|\) be the number of alive players on the day of round \(t\). Similarly, the average reliability of other players towards the Sheriff is \[\begin{align} \bar{m}_2(t) = \frac{1}{N_d(t)-1} \sum_{i\in\texttt{Alive}_d(t), i\neq L(t)} m_{i,L(t),t}^v \end{align}\] Then the Ratio is defined as: \[\begin{align} \mathrm{Ratio} = \frac{1}{T}\sum_{t=1}^T\frac{\bar{m}_2(t)}{\bar{m}_1(t)} \end{align}\] where \(T\) is the total number of rounds. A Ratio greater than 1 indicates that the Sheriff’s trustworthiness is higher than that of other players; the larger the Ratio value, the greater the opinion leadership.

Decision Change (DC) measures how the sheriff’s statements influence the voting decisions of other players. At the end of the discussion on day \(t\), all players except the Sheriff have made a statement. Now all players are required to reason the reliability of the Sheriff as Eq. (5 ) and make a pseudo-voting decision as Eq. (6 -7 ). Note that in Eq. (7 ), the available set of public statements is \(\mathcal{P}_{i,t}^v \setminus \{S_{L(t),t}\}\). We denote the pseudo-voting decision without the public statement from the Sheriff as \(A_{vote,t}'(X_i)\). Then the Sheriff makes a statement and all players move into the voting phase to yield \(A_{vote,t}(X_i)\).

Based on this, we can calculate the proportion of players that change their decision to be in line with the Sheriff. \[\begin{align} \mathrm{DC} = \frac{1}{T}\sum_{t=1}^T \frac{\sum\limits_{i\in\texttt{Alive}_d(t),i\neq L(t)}\mathbb{I}\left\{ \left(A_{vote,t}'(X_i) \neq A_{vote,t}(X_{L(t)} )\right) \text{ AND } \left(A_{vote,t}(X_i) = A_{vote,t}(X_{L(t)}) \right) \right\}}{N_d(t)-1} \end{align}\] where \(\mathbb{I}(\cdot)\) is the indicator function. The larger the DC value, the better the Sheriff’s ability to influence other players’ decisions, indicating greater opinion leadership.

The two proposed metrics are motivated by the key characteristics of opinion leaders. Although the specific calculation of these two metrics is coupled with our game framework, the underlying concepts are universal and can be easily applied to other similar game frameworks, or even adapted to other types of social deductive games, because reasoning and decision-making are the core steps in such games. For example, the framework in [3] is similar to ours and also contains reasoning steps. The early attempts in LLMs for the Werewolf game [2] also explore the trust relationships between players. Similar settings can be found in [29]. Besides, reasoning and decision-making are common steps within the framework of the Avalon game [4], [7], [28], and the two proposed metrics can naturally adapt to such scenarios.

3.3 Human Player↩︎

Our framework allows human players to participate in the game through text input. The process for human players to engage in the game is similar to that of LLM-based players, as they also need to reason about the roles of other players before taking action. The Moderator will review the game state and historical information to assist players in reasoning and decision-making. The interface for human players is depicted in Figure 4 of Appendix 7.

4 Experiments↩︎

4.1 WWQA Dataset↩︎

To assess and enhance LLMs’ understanding of the game rules, we create a Werewolf Question-Answering (WWQA) dataset. WWQA contains rule-based and situation-based questions. Rule-based questions directly target the game rules, while situation-based questions construct a virtual game state for LLMs, requiring simple reasoning to derive answers. In addition, we collect a binary QA dataset for rapid evaluation, where the standard responses are ‘Yes’ or ‘No’. Please refer to Appendix 8 for more details.

4.2 Simulation↩︎

We consider the simulation first, where all players are LLM-based. To fairly evaluate and compare the opinion leadership of different LLMs, we select a powerful model that can better understand the rules of the Werewolf game as our baseline model. Specifically, we select GLM-3 [43] as the backbone for all players except for the sheriff, due to its performance on binary QA dataset and its great instruction-following ability. GLM-3 can generate outputs in a specified format, which ensures the game progression. We provide more details on resolving issues with non-compliant output formats in Appendix 7.1.

For the player selected as the Sheriff, we implement it by LLMs of different sizes from different organizations, including ChatGLM3-6B (C3-6B) [43], Mistral-7B (M-7B) [44], Baichuan2-13B (B-13B) [45], InternLM-20B (In-20B) [46], Yi-34B [47], GLM-3 [43], GLM-4 [48], and GPT-4 [6]. We simulate 30 Werewolf games for each model; the maximum number of rounds is 6.

The results are shown in Table 1. Except for C3-6B, all models achieve decent performance on the binary QA dataset. Overall, the larger the model, the better the results, suggesting a more accurate understanding of the rules. However, in terms of opinion leadership, a larger scale does not mean better results, and open-source models smaller than 20B generally struggle to produce ideal results, with Ratios all below 1 and smaller DC values. Even with the special role of Sheriff, open-source LLMs are unable to gain higher credibility, and only M-7B can change the decision of one player on average. Due to the limited context length (4096), Yi-34B might fail to generate responses as the game deepens, leading to poor performance. Commercially available large-scale LLMs (larger than 100B)¹ perform comparatively better. GLM-3 gains more trust by utilizing the Sheriff’s identity, and GLM-4 achieves an accuracy of 0.846 on the binary QA dataset, with a Ratio value reaching 1.167. However, the ability of LLMs to change other players’ decisions is still relatively weak. We conclude that only a few large-scale LLMs demonstrate a certain degree of opinion leadership. More details and results are provided in Appendix 9.

Intuitively, the opinion leadership LLMs might be improved by enhancing their understanding of the rules. Thus, we use the LoRA [50] method with the rank of 8 to fine-tune open-source LLMs with the rule-based and situation-based data from WWQA for 4 epochs. More details are provided in Appendix 9.

The evaluation results are shown in Table 2, where models with (FT) are LLMs after fine-tuning. Fine-tuned LLMs generally achieve better results on the binary QA dataset, except for C3(FT). Whereas, a better grasp of the rules does not necessarily equate to better opinion leadership, which is consistent with results in Table 1. In-20B enjoys a 7.2% increase in Ratio and a 61.8% growth in DC. B-13B gains an 8.6% improvement in Ratio, but its DC value significantly deteriorates. It’s non-trivial to improve LLMs’ opinion leadership, especially the ability to influence others’ decisions.

Besides, we further control the identity of the Sheriff. The results in Figure 2 show that LLMs exhibit different levels of opinion leadership under different roles. When the Sheriff is a Werewolf, it’s difficult to gain the trust of other players because more players are on the Villager team. Therefore, all LLMs have poor opinion leadership when they are the Werewolf. In contrast, LLM has stronger opinion leadership with the roles of Seer and Guard, as they can take action at night and gather more information.

4.3 Human Evaluation↩︎

We invite 8 college students proficient in English and familiar with the Werewolf game to join in the human evaluation experiment. Each human player will play 5 games with 6 LLM-based players. The process and details can be found in Appendix 9.3. We choose GLM-3 as the backbone for LLM-based players, track the human player’s reasoning and decision-making before and after the Sheriff’s statement, and calculate the Spearman correlation coefficient between the human player’s reliability scores of the LLM-based player and the scores among other LLM-based players. Results in Table 3 indicate that GLM-3 can gain the trust of human players and demonstrate opinion leadership in human-AI interaction scenarios. Nonetheless, swaying human players with more stringent logic and formidable reasoning skills proves challenging for GLM-3. Besides, we observe a positive correlation between the reliability scores provided by human players and those inferred by LLM. This partially supports the use of reliability scores provided by LLM-based players during the simulation to calculate opinion leadership. Additionally, participants reported hallucination issues with GLM-3 during the experiment, with specific details presented in Appendix 9.3.

5 Conclusion↩︎

This study investigates the opinion leadership of LLMs by employing the Werewolf game as the simulation environment. We implement a framework that incorporates a Sheriff role, catering to both LLMs and human players. Besides, we propose two metrics to measure the Sheriff’s opinion leadership: the first metric measures their reliability and the second assesses their influence on other players’ decisions. Through extensive simulations, we evaluate LLMs of different scales and find that only a few LLMs show a certain degree of opinion leadership. Furthermore, we collect the WWQA dataset to enhance LLMs’ grasp of the game rules by fine-tuning. Initial attempts indicate that it’s non-trivial to promote the opinion leadership of LLMs. Finally, human evaluation experiments suggest that LLMs can gain human trust but struggle to influence human decisions.

6 Game Rules and Prompt Templates↩︎

6.1 Game Rules↩︎

The detailed game rules are as follows. We use to represent them hereafter.

You are playing a game called Werewolf with some other players. This game is based on text conversations and involves seven players: player_1, player_2, player_3, player_4, player_5, player_6, and player_7. Here are the game rules:

Roles: The moderator, who is also the host, organizes the game, and you must follow their instructions correctly. Do not communicate with the moderator. There are seven roles in the game, including two Werewolves, three Villagers, one Seer, and one Guard. At the beginning of the game, each player is assigned a hidden role, which categorizes them as either a Werewolf or a member of the Village (which includes the Seer, the Guard, and the Villagers). The game then proceeds through alternating night and day rounds until one side emerges victorious.

During the night round: The Werewolves identify each other and select one player to eliminate; the Seer chooses one player to determine if they are a Werewolf; the Guard selects one player, including themselves, to protect without knowing who the Werewolves have chosen; the Villagers take no action.

During the day round: There are three phases–announcement, discussion, and voting–conducted in sequence. In the announcement phase, the results of the previous night are disclosed to all players. If player_i was killed and not protected, the announcement will be "player_i was killed." If a player was killed but protected, the announcement will be "No player was killed."

Only on the first day is there an election phase. In this phase, you can decide whether to nominate yourself for Sheriff based on your role. Players nominated for Sheriff explain their reasons in turn. Each player then votes for one Sheriff nominee or chooses not to vote. The Sheriff is a unique role that does not interfere with a player’s primary role. The elected Sheriff has the power to decide which player begins the next discussion round and, as the final speaker, can summarize the discussion to persuade others to vote in agreement with them. In the discussion phase, each remaining player speaks once, in the order determined by the Sheriff, to debate who might be a Werewolf.

In the voting phase, each player votes to eliminate one player or chooses not to vote. The player with the most votes is removed, and the game progresses to the next night’s round.

The Werewolves win if their number equals that of the remaining Seer, Guard, and Villagers. The Seer, Guard, and Villagers win by eliminating all the Werewolves.

6.2 Prompt Templates↩︎

We construct our prompt templates following the design in [3]. The prompt template used to instruct LLMs for reasoning and decision-making consists of three parts.

1. illustrated in Appendix 6.1.

2. [Contextual Information]. Contextual information includes facts and public statements. We categorize public statements into potential truths and potential falsehoods based on the LLM’s reasoning results. Table 4 depicts an example that organizes contextual information for the model.

   Table 4: An example of contextual information

   All information you can leverage is listed below.

   Remaining Players: player_1, player_2, player_4

   The following information is true.

   You are player_1.

   You are a Werewolf.

   ...

   The following information might be true.

   In day 1 round, player_5 said: "Good day, everyone. As the first speaker, I believe it’s crucial to approach this game with a clear head...".

   ...

   The following information might be false.

   In day 1 round, player_2 said: "Let’s carefully consider the information and behavior of each player during the voting phase to make an informed decision. Let’s work together to eliminate the Werewolves."

3. [Action Instruction] specifies what the LLM needs to accomplish, defining the format of responses to facilitate parsing of the actions. We will provide further explanation subsequently.

Therefore, we leverage [Game Rule] + [Contextual Information] + [Action Instruction] to prompt the LLM to generate responses.

To parse the feedback from LLMs more accurately, we require them to output results in JSON format and provide corresponding output examples. To further reduce the occurrence of invalid actions in LLM outputs (such as selecting a player who has already been eliminated during a night action), we list all available options in the [Action Instruction]. Considering that our experiment involves multi-lingual LLMs, we explicitly request the language of LLMs’ outputs at the end. Besides, we require LLMs to reason about the current situation before taking action.

The template of [Action Instruction] for the Werewolf’s night action is shown in Table 5.

The templates of [Action Instruction] for the Guard’s and the Seer’s night action are shown in Table 6 and 7.

During the day round, the Werewolf and the Villagers (Villager, Seer, and Guard) share the same template of [Action Instruction] to make a statement, which is shown in Table 8. For the Sheriff, the template is slightly different, as shown in Table 9.

In the voting phase, the Werewolf and the Villagers (Villager, Seer, and Guard) leverage different templates of [Action Instruction] to vote, which remind LLMs to maximize the benefit of their team. They are shown in Table 10 and 11, respectively.

For the Sheriff, we use the template in Table 12 to determine the order of statements. For simplicity, LLMs only need to decide which player to the left or right speaks first.

All LLM-based players utilize the template in Table 13 to perform reliability reasoning. Due to the fact that some small-scale LLMs cannot follow the predefined JSON template to reason about the reliability of all players at once, the template in Table 13 only requires the LLM to reason about the reliability of one specific player at a time, which enhances the stability of the simulation.

Additionally, during the election phase of the first day, we design templates as shown in Tables 14 and 15 to prompt the LLM to make a campaign statement and vote for the Sheriff.

7 Details of Our Framework↩︎

Table [table:notation] summarizes the notations in our model. Figure 3 depicts the sequence of events in Round \(t\). During the night, the Werewolf, Seer, and Guard take turns making decisions. During the day, the Moderator announces the results of the previous night \(D_t\), and the Sheriff decides the order of speaking. Subsequently, each player makes a statement and votes in turn. Note that the Sheriff always speaks and votes last. Finally, the Moderator announces the voting results and the game proceeds to the next round. Each player needs to perform the reliability reasoning step before taking any actions.

7.1 Resolving Invalid Output↩︎

We require LLMs to output results in the specified JSON format as shown in Appendix 6.2 so that we can quickly parse the corresponding content. However, in some cases, LLMs fail to follow the format instructions, and we refer to such outputs as invalid outputs. To ensure simulations do not terminate in such a situation, we design different mechanisms at various stages of the game to handle invalid outputs. Additionally, if the model cannot generate an output (such as when the prompt exceeds the context window or unsuccessful API calls), or the model selects an unavailable option, we handle them similarly.

• During the Night. If an LLM-based player who needs to take action at night generates an invalid output, we will randomly select one of all available options as the decision for this player.

• Discussion Phase. If an LLM-based player generates an invalid output when making a statement, we handle it by considering the player to keep silent in the discussion, and other players will receive the public information “player_{} said nothing”.

• Voting Phase. If an LLM-based player generates an invalid output during the voting phase, we consider that the player chooses not to vote.

• Reliability Reasoning. If an LLM-based player produces an invalid output when inferring the identity of \(X_i\) identity, the reliability score of \(X_i\) will not be updated.

7.2 Logging System↩︎

Our framework includes a comprehensive logging system, which facilitates monitoring the game process during the simulation and checking whether LLM-based players are functioning properly. Specifically, our framework automatically saves three types of logs:

1. Error logs. Error logs primarily store errors that occur during the model generation process, including failures to generate results, outputs that do not meet format requirements, and cases where the model selects an unavailable option.

2. Game logs. Game logs record the progress of the game, including the identities of all players, night actions, the order of speaking during the day, the statements, and voting decisions.

3. Player logs. Player logs record the identities and numbers of players, the contextual information used for decision-making, their statements, decisions (including night actions and voting), and the historical records of their reasoning. Note that we also design a similar player log for human players, which helps us monitor the human evaluation process.

7.3 Interface During Human Evaluation↩︎

Figure 4 displays the interface during human evaluation. We use different colored fonts to help users distinguish between different types of content. Yellow font is used for prompting user input, red font for announcing night results or voting results, and green font for displaying other players’ statements. The input from the human player is displayed in blue on the right side of the screen. Besides, we use different icons to highlight the source of the messages.

The prompt for requesting input from human players is similar to the ones displayed in Appendix 6.2, and human players are required to input content in the specified JSON format. Participants are allowed to quickly select the input template using the up and down keys.

8 WWQA Dataset↩︎

In previous related work, we discovered that some LLMs may exhibit subpar performance in the context of the Werewolf game [2], [3], [29]. This can be attributed, in part, to their limited understanding of the game rules and the inherent challenge of fully unleashing their potential strategic capabilities with simple prompts alone. Additionally, the multitude of versions and rules of different Werewolf games can pose a challenge, as large language models tend to heavily rely on prior knowledge of specific game rules, leading to strategic behaviors that may not align with the given context or even violate the rules.

To address these limitations, we propose a self-instructional framework [51] to generate a dedicated Werewolf Question-Answer (WWQA) dataset that directly caters to our Werewolf game tasks. This WWQA dataset serves as a valuable resource for effectively assessing a large language model’s grasp of the fundamental game rules. Furthermore, we utilize this dataset for supervised fine-tuning (SFT) of the language models, enhancing their foundational abilities within our game and enabling us to explore the model’s potential for strategic leadership and gameplay within an empirical setting. The following Figure 5 provides an overview of our data generation process.

Question Generation Phase. In the first phase, our objective is to leverage the generative capabilities of the LMs to generate a series of game-related questions based on our given game rules. Our questions can be categorized into two types based on their content: rule-based and situation-based. Rule-based questions directly pertain to the game rules. Situation-based questions, on the other hand, require the LMs to construct a simple virtual game situation, and then engage in a certain level of reasoning within that situation in order to obtain the answer.

We initialize separate example pools, each with a size of 5, for rule-based and situation-based questions. In the first three iterations, we utilize all initial examples along with the game rules to prompt the LMs to generate 10 questions. The generated questions are then reviewed and corrected in the second round by prompting the LMs for revisions. The revised questions are then fed into the Question Pool. Starting from the fourth iteration, we sample 2 human-written examples from the example pool and 3 LM-generated questions from the question pool separately as examples for prompting the LM in question generation."

Answer Generation Phase. In the second phase, our objective is to leverage the combined semantic understanding and generative capabilities of the LMs to generate concise and logically coherent answers for all the questions in the question pool. Similar to the first phase, we prompt the LMs for post-processing corrections and filtering to ensure higher-quality answers. Finally, we pair the questions with their corresponding answers to create the final Q&A Pair Pool. We have provided some examples of our question-and-answer data in the supplementary materials.

In addition, we also prompt the LMs to generate a binary QA dataset, where the standard responses are ‘Yes’ or ‘No’. For this dataset, we conduct additional human review and labeling, using it as a testing set to evaluate the LMs’ grasp of our game rules.

8.1 Examples of QA Pair Data↩︎

Below are some examples of our generated WWQA dataset, which includes three types of question-and-answer data: rule-based, situation-based, and binary QA pairs data.

9 Experiment Details↩︎

9.1 Simulation↩︎

We build a set of random seeds for initializing the IDs and roles of each player. For each random seed, we repeat the simulation three times. We consider the following two types of settings.

1. Heterogeneous simulation, which is also the setup reported in Section 4.2.

In this setup, we select one LLM as a reference model (e.g., GLM-3) to implement all players except the Sheriff. The Sheriff is implemented using the LLM whose opinion leadership is to be evaluated. It allows for a more fair comparison of the opinion leadership of different LLMs.

To achieve this, we skip the election phase. The moderator will secretly select the Sheriff during the initialization of the game roles and construct the player chosen as the Sheriff (i.e., \(X_l,l=L(1)\)) using the LLM to be evaluated. After the first night, the moderator would inform all players of this through the following message.

After discussion and a vote, player_\(l\) was selected as the Sheriff, who can determine the order of statements, summarize the discussion, and provide advice for voting at last.

If the individual chosen covertly as the Sheriff gets eliminated by the werewolves during the initial night, the current simulation round becomes void. In addition to terminating under the conditions defined in the game rules, a simulation round will immediately stop after the Sheriff is eliminated, because only the Sheriff is implemented by the LLM to be evaluated.

1. Homogeneous simulation, where all players are implemented using the same LLM. This setup allows the simulation to proceed strictly following the game setting.

Due to significant differences in the interaction objects of the Sheriff, it may be problematic to directly compare opinion leadership metrics under such a setting. Nevertheless, we also conduct homogeneous simulations on several LLMs.

On the first day, three players are randomly chosen to participate in the Sheriff’s election. After the three players speak in turn, all players vote to decide who will be the Sheriff. When the Sheriff is eliminated, we choose the player the Sheriff trusted the most as the next Sheriff. The game terminates under the conditions defined in the rules. The results after 10 iterations are shown in the Table 17.

9.2 Fine-tuning↩︎

We fine-tune four open-source LLMs on a 3*A100 server using the WWQA dataset. The training set contains 1453 records while the validation set contains 100 records. Specifically, we use the LoRA tuning method [50] with the rank of 8. The learning rate is \(1e-4\) and the batch size is \(4\). The base models are fine-tuned by causal language modeling tasks over 4 epochs. During the fine-tuning process, the loss on the validation set is shown in Figure [-fig. 6]². We can observe that the loss values of different models decrease and then increase, indicating that models have already overfitted the training data. The loss of LLMs smaller than 10B eventually stabilizes around 0.31, while LLMs larger than 10B can reach a better loss value of about 0.28.

The fine-tuning time for different models varies from 40 to 100 minutes. From the results in Table 2, we conclude that fine-tuning can improve LLMs’ understanding of the rules (as evidenced by their performance on the binary QA dataset), but it has a limited effect on enhancing opinion leadership. We need more sophisticated methods to enhance the opinion leadership of LLMs.

9.3 Human Evaluation↩︎

We invite 8 college students, including 3 graduate students, from a top-tier university in China to participate in the human evaluation. All students have passed the CET-6 English exam and have experience with Werewolf games either online or offline. Our experiment mainly consists of the following three steps.

1. Training. We first explain the purpose of the experiment and then confirm with each participant that they understand the settings and rules of the Werewolf game. We extract 5 records from the WWQA dataset to test the participants, ensuring they could participate in the game in English. We also demonstrate how to input content following the format instruction.

2. Experiment. Each participant participates in the game by typing text, as shown in Figure 4. Each participant plays 5 iterations with GLM-3. Before each iteration, they should input an integer to initialize the roles of all players. Human players do not participate in the election of the Sheriff and will not be selected as the Sheriff. An iteration of the game ends either according to the rules or immediately if the human player is eliminated.

3. Interview. After the experiment, we conduct a simple interview with the participants, asking them to evaluate the LLM’s performance in the Werewolf game, including whether the LLM can follow the game rules, whether its logic is rigorous, and whether it’s easy to distinguish between LLM-based players and the human players.

The experimental results are shown in Table 3. During the interview, all participants reported that it’s easy to recognize LLM-based players since they always urge other players to provide more information rather than analyzing the game situation. We display some examples in Appendix 10 and this can also be found in the game logs of [2]. Besides, human players also mentioned the hallucination issues, such as accusing player_6 of remaining silent before player_6‘s turn to speak. Combined with participants’ feedback, we find several examples of hallucination displayed in Table 18.

9.4 Cost↩︎

In addition to the open-source LLMs that can run on our local server, we also extensively utilize APIs provided by ZhipuAI and OpenAI, with the GLM-3 model being the most frequently called. Our experiment incurs approximately ¥ 1000 for calling GLM series models and $ 25 for calling GPT-4. A potential cost-saving method is to require the LLM to output the reliability scores of all players at once during the reliability reasoning step. However, this would increase the probability of invalid outputs, which might affect the simulation progress.

10 Game Logs↩︎

10.1 Simulation↩︎

We display an instance of the game log during the simulation in Table [table9519], with each LLM-based player implemented by GLM-3.

10.2 Human Evaluation↩︎

We display an instance of the game log during the human evaluation in Table [table9520], with each LLM-based player implemented by GLM-3.

References↩︎