III.B. Stage Three: Theft and Enforcement
At stage three, the CA decides whether to monitor the agent or not and
the agent decides whether to steal or not. The payoffs determining the
monitoring and theft decisions are shown in Figure 1.
Central Authority
Agent
|
Monitor |
Not Monitor | |
|
Theft |
П — t, t — C |
πτ, 0 |
|
No Theft |
π(θ)Nτ, θq(θ) - c |
∏(θ)Nτ, θq(θ) |
Figure 1. The Game between the Agent and the Central Authority
We assume that the agent has some wealth in addition to the revenues
he earns from stealing πT , i.e., t>πT and that t>cso there are no
pure strategy equilibria of the game. We solve for the mixed strategy
equilibrium that would make the CA indifferent between monitoring
and not monitoring, and the agent indifferent between stealing and not
stealing. The threshold probability of theft that would make the CA
indifferent between monitoring and not monitoring, μ, becomes
μ=C. (7)
In equilibrium, the higher the monitoring cost, the higher is the proba-
bility of theft.
The threshold probability of monitoring that would make the agent
indifferent between theft and no theft, γ , is equal to
2a - θ
γ = θ
γ 4tb
The CA’s expected profits from fees, πF , are in equilibrium equal to
πF = μγ(t - c) + (1- μ)γ(θq(θ)- c) + (1- γ)(1 - μ)θq(θ) =
= (1 - μ)θq(θ).
(9)
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