Added more exercises

mdps/monotone-mdps.qmd

@@ -649,6 +649,20 @@ c. $P^k$, for $k \in \integers_{> 0}$.
 Let $\mu_t$ denote the distribution of a Markov chain at time $t$. Suppose $\mu_0 \succeq_s \mu_1$. Then $\mu_t \succeq_s \mu_{t+1}$. 
 :::
 
+:::{#exr-submodularity-discrete}
+### Testing submodularity for functions defined on integers
+
+Suppose a function $f \colon \integers \times \integers \to \reals$
+satisfies
+$$ f(x+1, y+1)  - f(x+1, y) \le f(x, y+1) - f(x, y) $$
+for all $x, y \in \integers$. Then, show that $f$ is a submodular
+function.
+:::
+
+:::{#exr-properties-of-submodular-functions}
+Show that sum of submodular functions is submodular. Is the product of submodular functions submodular?
+:::
+
 :::{#exr-monotone-machine-replacement}
 Consider the example of machine repair presented in @exm-machine-replacement. Prove that the optimal policy for that model is weakly increasing.
 :::