A terrorist attack one of the two targets with some probability p to target1 and (1-p) probability to target2, and defender should allocate optimum amount of resources on these targets. The government has limited resource R and allocates T to target 1 and (R-T) to target2 where T≤R. The terrorist’s target valuation is private knowledge and he values target1 as V (and target two (1-v)), defender only knows it is uniformly distributed on the interval [0,1]. If the terrorist attack on target1 (target2) will be successful with probability (1-(T/R)) (on target2 this probability is (T/R)) the terrorist gets V (1-V) and the government loss will be L (K).If the attack will be unsuccessful they both get 0 on any of the targets. The government tries to minimizes her expected loss and the terrorist tries to maximize his expected gain. Both players are rational. According to my calculation in perfect Bayesian Nash equilibrium with probability p terrorist launches an attack on target 1 and optimal allocation of the government on target 1 and target2 are T* and (R-T)*, respectively. So at PBNE we have,$$P=1-\frac {T*}{R}$$ ,$$T*=\frac {LR}{(L+K)}$$
What I found is that L>K $$\frac {∂T*}{∂L} = \frac{KR}{(L+K)^2}$$ $$\lt$$ $$\frac {∂(R-T)*}{∂K} = \frac{LR}{(L+K)^2}$$
Can I interpret this result as follows: Although in both targets whether the losses are either high or low, increase in losses increases the defensive investment, when the losses are relatively low an increase in losses leads to higher defensive investment when the total resources are restricted by some amount. This is because when the losses are smaller with investing more on that target defender realizes more gains than investing more on the target where the losses are higher. If yes, could yu please give me any examples. If not, how can I interpret this result?
Thank you