  We introduce a deterministic analogue of Markov chains that we call the
hunger game. Like rotor-routing, the hunger game deterministically mimics the
behavior of both recurrent Markov chains and absorbing Markov chains. In the
case of recurrent Markov chains with finitely many states, hunger game
simulation concentrates around the stationary distribution with discrepancy
falling off like $N^{-1}$, where $N$ is the number of simulation steps; in the
case of absorbing Markov chains with finitely many states, hunger game
simulation also exhibits concentration for hitting measures and expected
hitting times with discrepancy falling off like $N^{-1}$ rather than
$N^{-1/2}$. When transition probabilities in a finite Markov chain are
rational, the game is eventually periodic; the period seems to be the same for
all initial configurations and the basin of attraction appears to tile the
configuration space (the set of hunger vectors) by translation, but we have not
proved this.
