consider the following economic model:

\section{The Model}

We analyse a dynamic model of norm adoption in a society comprised of two groups with conflicting interests. In each period, agents observe the aggregate behaviour from the previous period and myopically choose an action to maximize their own utility. The model is structured to reveal how individual incentives aggregate into collective social outcomes like widespread norm-adherence, social gridlock, or sudden social change.

\subsection{Agents and Groups}

The set of agents is a continuum, which we normalize to have measure one. Each agent $i$ belongs to one of two observable, exogenously assigned groups: a \textbf{Majority} group, denoted $maj$, or a \textbf{Minority} group, denoted $min$. We let $g_i \in \{Maj, Min\}$ be the group identity of agent $i$. The population share of the Minority group is $\lambda \in (0, 1/2)$, and the share of the Majority group is $1-\lambda$.

\subsection{Actions, Payoffs, and Preferences}

\paragraph{The Stage Game.} In each discrete time period $t = 0,1,2,...$, agents simultaneously choose one of two actions: adhere to the status quo ($SQ$) or adopt an alternative ($ALT$). After choosing their actions, agents are randomly matched into pairs. An agent receives a positive payoff only if their partner chose the same action; otherwise, the payoff is zero. The payoff matrix for an interaction between agent $i$ and agent $j$ is:

\begin{center}
\begin{tabular}{c|cc}
    & $SQ$ & $ALT$ \\
\hline
$SQ$ & $U_{g_i}^{SQ}, U_{g_j}^{SQ}$ & $0, 0$ \\
$ALT$ & $0, 0$ & $U_{g_i}^{A},U_{g_j}^{A}$ \\
\end{tabular}
\end{center}

The coordination payoff an agent receives is dependant on their group. We are interested in societies where groups have \textbf{conflicted interests}. While both groups prefer to coordinate, they disagree on the ideal norm. The Majority group derives greater utility from coordinating on the status quo equilibrium, while the Minority group prefers the alternative. This is formalized by the following assumptions:
\begin{align}
    U_{maj}^{SQ}\:&> \:U_{min}^{SQ} > 0, \label{eq:1}\\
    U_{min}^{A} \:&>\: \:U_{maj}^{A} \:\:\:> 0 \label{eq:2}
\end{align}

In subsequent sections, we normalise $U_{Maj}^{SQ,SQ}$ and $U_{Min}^{A,A}$ to 1 for tractability. As such, I denote majority resistance as $R$ where $R=1-U_{maj}^A$. A value of $R=0$ signifies that the majority is indifferent between the status quo and alternative norm. As $R$ increases, the majority will increasingly suffer from and resist the alternative norm. We also restrict analysis to parameters which satisfy:
\begin{equation}
        \lambda U_{min}^A + (1-\lambda)U_{maj}^A > \lambda U_{min}^{SQ} + (1-\lambda)U_{maj}^{SQ} \label{eq:welfarerestriction}
\end{equation}
This ensures that full adoption of the alternative norm provides a greater population welfare than the status quo norm. If this were not the case, there is little reason for a policy maker to intervene.

\paragraph{Agent Types and Utility.} In addition to the stage game payoff, each agent has a privately known, idiosyncratic preference for the status quo. This is captured by a utility shock, $\eta_i$, which is realized with certainty if the agent chooses action $SQ$. This term can be positive or negative and captures an individual's unique political leanings, personal tastes, or moral convictions that are independent of social influence.

The shocks $\eta_i$ are drawn independently from a normal distribution with a mean of zero and a group-specific variance: $\eta_i \sim N(0, \sigma_{g}^2)$. An agent's per period utility from choosing $a_i$ is the sum of the stage game payoff and this idiosyncratic shock:

\begin{equation}
    U_{i}(a_{i},a_{j})=u_{i}(a_{i},a_{j})+1_{\{a_{i}=SQ\}}\cdot\eta_{i}
\end{equation}


Where $u_{i}$ is the coordination payoff from the table above.

\subsection{The Agent's Decision Problem and Individual Thresholds}

Let $p_t$ represent the \textit{actual} proportion of agents choosing $ALT$ in period $t$. To make their decision, agents must form a belief about this proportion, which we denote by $p_t'$. Agents are \textbf{myopic}; they observe the proportion of the population who chose $ALT$ in the previous period ($p_{t-1}$), and assume that the same fraction will choose $ALT$ in the current period ($p_t' = p_{t-1}$). This prescribes a relatively low degree of sophistication to agents -- they do not predict or anticipate trends in behaviour. However, this has proven to be a reasonable assumption, particularly in large group settings.

Given this belief, the expected utility for an agent $i$ of group $g_i$ for each action is:
\begin{align}
    E[U_i(SQ)] = &\: (1-p_t')U_{g_i}^{SQ} + \eta_i \\
    E[U_i(ALT)] = &\:  p_t'U_{g_i}^{A}
\end{align}

Agents maximise their expected utility, choosing $ALT$ if $E[U_i(ALT)] > E[U_i(SQ)]$. This implies that for each agent, there is a critical proportion of $ALT$-adopters that makes them indifferent between the two actions. We define this as their \textbf{individual threshold}, $x_i$. By setting the expected utilities equal, we can solve for this threshold:
\begin{equation}
x_i = \frac{U_{g_i}^{SQ} + \eta_i}{U_{g_i}^{A} + U_{g_i}^{SQ}}
\end{equation}
Since the expected utilities are monotonic in $p'$, the agent's optimal strategy is a simple threshold rule: choose $ALT$ if enough people chose $ALT$ in the previous period to meet or exceed their personal threshold ($p_{t-1} \ge x_i$), and choose $SQ$ otherwise.

What will the distribution of thresholds in society be?