Public Good Provision with a Governor

Abstract

We study a public good game with N citizens and a Governor who allocates resources from a common fund. Citizens may voluntarily contribute or be compelled to do so if audited, in which case shirkers face a penalty. The Governor decides how much of the fund to devote to public good provision, with the remainder embezzled. Crucially, the Governor's utility combines material payoffs from embezzlement with belief-dependent reputational concerns. We fully characterize the symmetric subgame perfect equilibria (SSPE) of the game. The model always admits at least one pure-strategy equilibrium, ranging from universal free-riding with complete embezzlement to full contribution with efficient provision. Mixed-strategy equilibria exist only in a narrow region of parameter values and may involve multiple equilibria. Our analysis highlights the roles of penalties, audits, and reputational incentives in sustaining contribution and provision, thereby linking public good provision with the broader literature on corruption, embezzlement, and psychological game theory.

Figures (6)

• Figure 1: Game tree for $N=2, k=1$.
• Figure 2: Feasible region for symmetric mixed-strategy SPE with $p \in (0,1)$, for $N=2$, $k=1$, $\sigma \geq 2$. The shaded region is bounded by $z = 1 - 2a$ (blue) and $z = 1 - a$ (red). A symmetric mixed-strategy SPE exists if and only if $(a,z)$ lies within this region, with equilibrium contribution probability $p^* = \frac{1-z}{a} - 1$.
• Figure 3: Visual characterisation of the pure-strategy equilibria in the $(\sigma, z)$ parameter space.
• Figure 4: Plot of the Net Gain function $f(p)$ for different parameter values. The blue curve shows $f(p)$, which measures the difference between the expected marginal benefit of contributing and the marginal cost. Mixed-strategy SSPE correspond to values of $p$ where the blue curve crosses the $x$-axis, i.e., $f(p)=0$. The vertical red dashed lines mark the kink points where citizens' expectations $\tau(p)$ cross an integer, delineating the distinct polynomial segments of $f(p)$. Panels (a) and (c) exhibit multiple mixed-strategy SSPE, while panels (b) and (d) show cases where $f(p)>0$ for all $p \in (0,1)$, implying no mixed-strategy SSPE exists.
• Figure 5: Feasible Regions for $N=3,k=1$

Theorems & Definitions (15)

• Lemma 1
• Lemma 2: Governor’s optimal rule
• Proposition 1: Free-Riding
• Proposition 2: Full-Contribution
• Proposition 3: Continuum of mixed equilibria
• Corollary 1
• Lemma 3: Continuity of the Net Gain Function
• Proposition 4: Existence of Mixed-Strategy Equilibria
• proof : Proof of Observation \ref{['prop:FR_N2']}
• proof : Proof of Observation \ref{['prop:FC_N2']}