Solvency games are a gambling problem on infinite-state MDPs where the
investor's fortune n ∈ N is the state.
In every round, the investor chooses an action from a finite action
set, and every action yields a distribution over integer-valued gains
in an interval {-l, …, m}. The risk-averse investor wants to
minimize the probability of eventual ruin (reaching a fortune ≤ 0).
It was shown in [BKSV08]
that memoryless deterministic optimal strategies exist,
but they do not have a pure tail in general.
Even in the special case of gains in {-2, …, 1}, the optimal strategy
may need to make use of two different actions at arbitrarily high fortunes.
We show that in the case of gains in {-2, …, 1} there exists an optimal
strategy that either has a pure tail or strictly alternates between just two actions.
Moreover, already in the slightly more general case of gains in
{-3, …, 1}, it is possible
for the optimal strategy to be unique but aperiodic.
Finally, we show that the optimal strategy is computable if it is unique.
Moreover, (some) optimal strategy can always be computed
in the case of gains in {-l, …, 1} for any
l ∈ N.
Submitted, 2025. 27 pages.
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© 2025 Kousha Etessami, Quentin Guilmant, Florian Luca, Richard Mayr,
Joël Ouaknine, and James Worrell.