papers/zoo-impact-bonds/sections/04-incentive-analysis.tex

\section{Incentive Analysis}

\label{sec:incentives}
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\subsection{Implementer Incentives}

\begin{theorem}[Incentive Compatibility]
Under the BIB mechanism, an implementer's expected payoff is maximized
by allocating conservation effort to genuinely achieve outcomes, provided
the oracle network has detection probability $\delta > 0.5$ for outcome
manipulation.
\end{theorem}

\begin{proof}
Let $e \in [0, E]$ be the implementer's effort level and $c(e)$ the
convex cost function. The probability of achieving the outcome target
$O^*$ is $p(e)$, monotonically increasing in $e$. Under honest effort:
\begin{equation}
  U_H(e) = p(e) \cdot (\pi_{\text{base}} + \pi_{\text{bonus}}) - c(e)
\end{equation}

Under manipulation (faking outcomes):
\begin{equation}
  U_M = (1 - \delta) \cdot (\pi_{\text{base}} + \pi_{\text{bonus}})
  - \delta \cdot \psi - c_m
\end{equation}
where $\psi$ is the penalty for detected manipulation and $c_m$ is the
cost of manipulation.

For incentive compatibility, we require $\max_e U_H(e) > U_M$. This
holds when:
\begin{equation}
  \delta > \frac{\pi_{\text{base}} + \pi_{\text{bonus}} - c_m}
  {\pi_{\text{base}} + \pi_{\text{bonus}} + \psi}
\end{equation}

Since $\psi$ includes stake slashing and reputational cost, typically
$\psi \gg \pi_{\text{bonus}}$, making the right-hand side less than
0.5. Thus $\delta > 0.5$ suffices.
\end{proof}

\subsection{Investor Incentives}

Investors face outcome risk and market risk. We model participation as:
\begin{equation}
  U_I = \E[\text{Return}] - \gamma \cdot \text{Var}[\text{Return}]
  + \alpha \cdot \text{ImpactUtility}
\end{equation}
where $\gamma$ is risk aversion and $\alpha$ captures impact investor
``warm glow'' utility. The term $\alpha > 0$ allows BIBs to offer
below-market risk-adjusted returns while still attracting capital.

\subsection{Oracle Incentives}

Verifiers stake tokens and earn fees for accurate verification.
Inaccurate reporting results in stake slashing. The mechanism follows a
Schelling point design \citep{buterin2014schellingcoin}: verifiers are
rewarded when their assessments agree with consensus.


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