decisions in each, and he may safely maximize each period’s utility independently of
all the others. A stationary strategy is appropriate here. Since the strategy space for B
consists of the amount of money paid to M,
, and a rule for when to retain M,
which we will call , this implies that each is constant over time.
0
w
q
The retention rule,
q
, arises from the circumstances of the interaction. In our case,
a successful attack always results in another job for that particular M, so the rule
applies only to failures.
Since the only available information is the occurrence of a
failure itself, the rule can only take the form: after the
failure, retain with
probability
. Note that this allows for steadily increasing (or decreasing) penalties.
Since we are considering stationary strategies, however, there is no need to go beyond
the most basic of such rules: upon each failure, retain with probability . In this case,
as M responds to B’s actions within a completely static environment, M’s equilibrium
strategy–how much to skim, –will also be time independent.
th
z
z
q
q
x
We further make the reasonable assumption that M cannot borrow money to add
to the success of an attack. Of course, M also cannot skim more than is given to him.
Under these assumptions we are presented with a constrained optimization problem in
which B maximizes his utility, conditional on M’s maximizing his utility, subject to
the constraint that
0
[0
]
x
w
∈ ,
. The full optimization breaks the problem up into three
parts: the case where all the money is skimmed, the “honest" case where no money is
skimmed, and the interior or “skimming" solution, which generally obtains for greedy
middlemen and rich bosses focused on attacks. In this subsection we derive solutions
for the equilibrium actions of both Boss and Middleman, and briefly discuss how
changing the assumptions of the model would alter this equilibrium. In the next we
examine how the equilibrium outcome changes with the model’s parameters. Finally,
19
Convention