05-XX COMBINATORICS (For finite fields, see 11Txx)
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Institute
We introduce the class of spot-checking games (SC games). These games model
problems where the goal is to distribute fare inspectors over a toll network.
Although SC games are not zero-sum, we show that a Nash equilibrium
can be computed by linear programming.
The computation of a strong Stackelberg equilibrium is
more relevant for this problem, but we show that this is NP-hard.
However, we give some bounds on the \emph{price of spite},
which measures how the
payoff of the inspector
degrades when committing to a Nash equilibrium.
Finally, we demonstrate the quality of these bounds for a real-world application,
namely the enforcement of a truck toll on German motorways.
Network Spot Checking Games: Theory and Application to Toll Enforcing in Transportation Networks
(2014)
We introduce the class of spot-checking games (SC games). These games model
problems where the goal is to distribute fare inspectors over a toll network.
In an SC game, the pure strategies of network users correspond to
paths in a graph, and the pure strategies of the inspectors
are subset of edges to be controlled.
Although SC games are not zero-sum, we show that a Nash equilibrium
can be computed by linear programming.
The computation of a strong Stackelberg equilibrium is
more relevant for this problem, but we show that this is NP-hard.
However, we give some bounds on the \emph{price of spite},
which measures how the
payoff of the inspector
degrades when committing to a Nash equilibrium.
Finally, we demonstrate the quality of these bounds for a real-world application,
namely the enforcement of a truck toll on German motorways.
Let the design of an
experiment be represented by an $s$-dimensional vector
$\vec{w}$ of weights with non-negative components. Let the quality of
$\vec{w}$ for the estimation of the parameters of the statistical model
be measured by the criterion of $D$-optimality defined as the $m$-th root
of the determinant of the information matrix $M(\vec{w})=\sum_{i=1}^s
w_iA_iA_i^T$, where $A_i$, $i=1,...,s$, are known matrices with $m$ rows.
In the paper, we show that the criterion of $D$-optimality is second-order
cone representable. As a result, the method of second order cone
programming can be used to compute an approximate $D$-optimal design with
any system of linear constraints on the vector of weights. More
importantly, the proposed characterization allows us to compute an
\emph{exact} $D$-optimal design, which is possible thanks to high-quality
branch-and-cut solvers specialized to solve mixed integer second order cone
problems.
We prove that some other widely used criteria are also second order cone
representable, for instance the criteria of $A$-, and $G$-optimality, as
well as the criteria of $D_K$- and $A_K$-optimality, which are extensions
of $D$-, and $A$-optimality used in the case when only a specific system of
linear combinations of parameters is of interest.
We present several numerical examples demonstrating the efficiency and
universality of the proposed method. We show that in many cases the mixed
integer second order cone programming approach allows us to find a provably
optimal exact design, while the standard heuristics systematically miss the
optimum.