Let $v_{i,j}\in [0,1], i,j = 0,\ldots,n$ solve the following system of equations for $r>0$: \begin{align} 0 &= \tfrac 1 2 e_{i,j}^2 + e_{j,i} v_{i,j+1} - (r+e_{j,i}) v_{i,j}, \tag{*} \\ e_{i,j} &= v_{i+1,j}-v_{i,j}, \\ v_{n,j} &= 1, \\ v_{i,n} &= 0, \end{align} for $i,j = 0,\ldots,n-1$.
Claim:
- a) for fixed $i\in \{0,\ldots,n-1\}$ the function $j\mapsto e_{i,j}$ is increasing for $j\leq i$ and decreasing for $j\geq i$.
- b) for fixed $j\in \{0,\ldots,n-1\}$ the function $i\mapsto e_{i,j}$ is increasing.
Specifically, in case of $n=2$, we want to prove that $e_{0,1} < e_{0,0} < e_{1,0} < e_{1,1}$.
Interpretation: These equations represent the competition in a patent race: Two players compete for a patent. It takes $n$ discoveries to be able to patent, the player that patents first wins the race (reward 1), whist the other loses (reward 0). A player with $i$ discoveries knowing that his rival has made $j$ discoveries makes effort $e_{i,j}\geq 0$, which translates into the hazard rate $e_{i,j}$ of making the next discovery, and cost flow $\tfrac 1 2 (e_{i,j})^2$. The time is continuous and future payoffs are discounted at the rate $r>0$. Continuation value $v_{i,j}$ is the expected discounted future payoff flow (probability of winning minus the integral of effort cost over time). The claim is that: Players exert higher effort when being neck and neck; having a lead or lagging discourages effort.
The problem has been intensely studied by economist since the 80', but most studies consider a general convex effort cost function, whist the equations (*) are obtained under a more restrictive assumption of quadratic effort cost functions. In the general case the monotonicity of $v_{i,j}$ does not necessarily hold. (Cf. Grossman and Shapiro (1987))
I have solved the problem for $n=2$, but it is a several page long proof that is very technical (I find upper and lower boundary on the variables step by step, there is not much intuiting behind it).