A rational and selfish environment may have an incentive to cheat
the system it interacts with. Cheating the system amounts to
reporting a stream of inputs that is different from the one
corresponding to the real behavior of the environment. The system
may cope with cheating by charging penalties to cheats it detects.
In this paper, we formalize this setting by means of weighted
automata and their resilience to selfish environments. Automata
have proven to be a successful formalism for modeling the on-going
interaction between a system and its environment. In particular,
weighted finite automata (WFAs), which assign a cost to each input
word, are useful in modeling an interaction that has a
quantitative outcome. Consider a WFA $\A$ over the alphabet
$\Sigma$. At each moment in time, the environment may cheat $\A$
by reporting a letter different from the one it actually
generates. A penalty function $\eta:\Sigma \times \Sigma
\rightarrow \Rp$ maps each possible false-report to a penalty,
charged whenever the false-report is detected. A
detection-probability function $p:\Sigma \times \Sigma \rightarrow
[0,1]$ gives the probability of detecting each false-report. We
say that $\A$ is $(\eta,p)$-resilient to cheating if
$\zug{\eta,p}$ ensures that the minimal expected cost of an input
word is achieved with no cheating. Thus, a rational environment
has no incentive to cheat $\A$.

We study the basic problems arising in the analysis of this
setting. In particular, we consider the problem of deciding
whether a given WFA $\A$ is $(\eta,p)$-resilient with respect to a
given penalty function $\eta$ and a detection-probability function
$p$; and the problem of achieving resilience with minimum
resources, namely, given $\A$ and $\eta$, finding the minimal
(with respect to $\sum_{\s,\s'}\eta(\s,\s')\cdot p(\s,\s')$)
detection-probability function $p$,
%i.e., $\sum_{\s,\s'}\eta(\s,\s')p(\s,\s')$)
such that $\A$ is $(\eta,p)$-resilient. While for general WFAs
both problems are shown to be PSPACE-hard, we present
polynomial-time algorithms for deterministic WFAs.