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\begin{document}

\title{Endogenous Sanctioning Networks}



\author[1]{Philipp Chapkovski}%
\affil[1]{University of Zurich}%


\vspace{-1em}



  
  \date{May 24, 2024}


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\vspace{-2cm}
\noindent\textbf{RQ} : How does anonymity influence the relative popularity of whistleblowing and peer punishment?

\section*{Basic Setting}
Subjects interact in groups of $n=4$. Each period a common pool resource $G=36$ is exogenously provided.

\subsection *{Stage 1}

\begin{itemize}
\item One subject (\textit{``Thief''}) is randomly selected and provided with an opportunity to take $x=9$ from the common pool (\textit{``steal''}). Taking $x=9$ reduces the amount in the common pool to $G=36-2x = 18$. The remaining common pool $G$ is subsequently split equally among the four group members, resulting in a payoff vector of $(4.5; 4.5; 4.5; 13.5)$ if stealing is chosen and $(9; 9; 9; 9)$ otherwise. 
\item One other subject (\textit{``Observer''}) is randomly chosen to observe the ID of the thief and may choose between three action alternatives. The decision is taken under the strategy method (conditional on the thief's decision to steal). 
\begin {itemize}
\item Abstain: The observer refrains from taking an action.
\item Whistleblowing: The observer reveals the ID of the thief to all group members. The cost of whistleblowing to the observer is $c=1$.
\item Heroic Punishment: The observer punishes the thief directly at a cost-effectiveness ratio of $3:9$ resulting in a payoff vector of $(4.5; 4.5; 1.5; 4.5)$. No information on the thief's identity is revealed to the other group members.
\end{itemize}

\subsection*{Stage 2}

\begin{itemize}
\item The subjects engage in a peer rewarding/ punishing game. Each subject (including the thief and the observer) may for each other group member select whether to 1) take no action, 2) send a reward point, or 3) send a punishment point at cost-effectiveness ratios of $1:3$, respectively. 
\end{itemize}

\subsection*{Treatments}

\begin{itemize}
\item[T1] (Private Info): The ID of the observer is hidden from all group members.
\item[T2] (Public Info): The ID of the observer is revealed to all group members. 
\end{itemize}

\section*{Incentives (Stage 1)}

\begin{center}
\begin{tabular}{c|cc}
 & Costs to Observer & Benefits to Group  \\
\hline
Abstain & No Costs & None  \\
& $\wedge$ & $\vee$ \\
Whistleblowing & \makecell{Cost of WB ($=1$) \\ + Cost of PP} & \makecell{Norm Enforcement \\ delegated to the group \\ Cooperation Problem}  \\
& $\wedge$ & $\vee$ \\
Heroic Punishment & Most Costly ($c=3$) & Norm is Enforced  \\
\end{tabular}
\end{center}

\section*{Expectations on the relative popularity of the observer's actions}

\begin{center}
\begin{tabular}{c|ccc}
 & T1 (Private Info) & & T2 (Public Info)  \\
\hline
Abstain & & $<$ &  \\
Whistleblowing & & $?$ &  \\
Heroic Punishment & & $>$ &   \\
\end{tabular}
\end{center}

The first and the third relation are based on the assumption that punishing the thief is rewarded and hence incentivized by making information public. Concerning the $?$, we have diverging predictions. On the one hand, compared to Abstain, Whistleblowing is socially beneficial and expected to be incentivized if information is public. We would thus expect relatively more whisteblowing in T2 than in T1. On the other hand, whistleblowing is also a sign that the observer was unwilling to bear the higher costs of heroic punishment, worsening the observer's prosocial reputation and reducing its relative popularity in T2 compared to T1. Additionally, T2 also lowers the profitability of whistleblowing if counter-punishment dominates reputational rewards.

\section*{Open Questions}

\begin{itemize}
\item Should we have a repeated game in order to study dynamics or look at a one-shot repetition? If repeated game, a stranger matching seems more adequate to prevent counter-punishment across periods. Also, do variations need to be between-subjects or within-subjects?
\item Should we provide additional endowment prior to Stage 2, in order to facilitate action in this stage? Alternatively, we might have to choose a higher $G$ to prevent negative incomes in a period.  
\item An alternative would be to have each of the three non-thiefs take the observer's decision under the strategy method and randomly select one of the decisions.
\item Do our parameters create sufficient variation between the three action alternatives? Are some actions disincentivized entirely (e.g. rewarding)?

\end{itemize}

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