arXiv subjects

Burkhard Monien

Publications and source records attributed to Burkhard Monien.

3 recordsLinked to original sources

arXiv2020

(In)Existence of Equilibria for 2-Players, 2-Values Games with Concave Valuations

We consider 2-players, 2-values minimization games where the players' costs take on two values, $a,b$, $a b$, then there exists a normal 2-players, 2-values, 3-strategies game without $\mathsf{F}$-equilibrium. To the best of our knowledge, this work is the first to provide an (almost complete) answer on whether there is, for a given concave function $\mathsf{F}$, a counterexample game without $\mathsf{F}$-equilibrium.

cs.GT↗

arXiv2015

The Complexity of Equilibria for Risk-Modeling Valuations

We study the complexity of deciding the existence of mixed equilibria for minimization games where players use valuations other than expectation to evaluate their costs. We consider risk-averse players seeking to minimize the sum ${\mathsf{V}} = {\mathsf{E}} + {\mathsf{R}}$ of expectation ${\mathsf{E}}$ and a risk valuation ${\mathsf{R}}$ of their costs; ${\mathsf{R}}$ is non-negative and vanishes exactly when the cost incurred to a player is constant over all choices of strategies by the other players. In a ${\mathsf{V}}$-equilibrium, no player could unilaterally reduce her cost. Say that ${\mathsf{V}}$ has the Weak-Equilibrium-for-Expectation property if all strategies supported in a player's best-response mixed strategy incur the same conditional expectation of her cost. We introduce ${\mathsf{E}}$-strict concavity and observe that every ${\mathsf{E}}$-strictly concave valuation has the Weak-Equilibrium-for-Expectation property. We focus on a broad class of valuations shown to have the Weak-Equilibrium-for-Expectation property, which we exploit to prove two main complexity results, the first of their kind, for the two simplest cases of the problem: games with two strategies, or games with two players. For each case, we show that deciding the existence of a ${\mathsf{V}}$-equilibrium is strongly ${\mathcal{NP}}$-hard for certain choices of significant valuations (including variance and standard deviation).

cs.GT↗

arXiv2008

How Many Attackers Can Selfish Defenders Catch?

In a distributed system with {\it attacks} and {\it defenses,} both {\it attackers} and {\it defenders} are self-interested entities. We assume a {\it reward-sharing} scheme among {\it interdependent} defenders; each defender wishes to (locally) maximize her own total {\it fair share} to the attackers extinguished due to her involvement (and possibly due to those of others). What is the {\em maximum} amount of protection achievable by a number of such defenders against a number of attackers while the system is in a {\it Nash equilibrium}? As a measure of system protection, we adopt the {\it Defense-Ratio} \cite{MPPS05a}, which provides the expected (inverse) proportion of attackers caught by the defenders. In a {\it Defense-Optimal} Nash equilibrium, the Defense-Ratio is optimized. We discover that the possibility of optimizing the Defense-Ratio (in a Nash equilibrium) depends in a subtle way on how the number of defenders compares to two natural graph-theoretic thresholds we identify. In this vein, we obtain, through a combinatorial analysis of Nash equilibria, a collection of trade-off results: - When the number of defenders is either sufficiently small or sufficiently large, there are cases where the Defense-Ratio can be optimized. The optimization problem is computationally tractable for a large number of defenders; the problem becomes ${\cal NP}$-complete for a small number of defenders and the intractability is inherited from a previously unconsidered combinatorial problem in {\em Fractional Graph Theory}. - Perhaps paradoxically, there is a middle range of values for the number of defenders where optimizing the Defense-Ratio is never possible.

cs.GT↗