Changes

:<math>F_{\hat{\tau}} = 1 - e^{-\left( \sum_{j\ne i} -h(x_j) \right) t}\;</math>.

:<math>a_i= \sum_{j\ne i} h(x_j)\; </math> The firm discounts the future reciepts at a rate <math>r\;</math> (note that using continuous compounding, <math>PV = FV \cdot e^{-rt})\;</math>. The firm wins the prize at time <math>t\;</math> with probability: :<math>pr(\tau(x_i) \le \min(\hat{\tau_i},t) = e^{-a_i t}(1-e^{-h)x_i)t}) + a_i \int_0^t (x_j1-e^{-h(x_i)s})e^{-a_i s})ds\;</math> :<math>\therefore pr(\tau(x_i) \le \min(\hat{\tau_i},t) = \frac{h(x_i)}{a_i + h(x_i)} (1-e^{-(a_i+h(x_i))t})\;</math>  This is directly comparable to a contest success function: :<math>pr(\tau(x_i) \le \min(\hat{\tau_i},t) = \underbrace{\frac{h(x_i)}{\sum_{i=1}^{n} h(x_i)}}_{\mbox{Firm i's relative effort}} \cdot \underbrace{\left ( 1-e^{-\left(\sum_{i=1}^{n} h(x_i)\right)t}\right )}_{Prob of innov. at t}\;</math>  ===Solution concept=== The model is not actually solved, but comparative statics can be performed on an implicit solution. The implicit solution is arrived at by noting that:#If a firms expectations are rational then the beliefs about the fastest competing firm are indeed formed using <math>\hat{\tau_i}\;</math>#<math>a_i\;</math> can be taken as constant by firm <math>i\;</math> (i.e. in equilibrium <math>a_i\;</math> will be correct)#<math>V\;</math> and <math>r\;</math> are exogenously given#As the firms are identical we can look for a symmetric solution! Each firm maximizes profit: :<math>\max_x \Pi (a_i,x,V,r) = \max_x \left (\frac{V h(x_i)}{r(a_i + r +h (x_i)} - x \right)\;</math>  This is presumably constructed by taking: :<math>\Pi = \int_0^{\infty} \left \underbrace{pr(\tau(x_i) \le \min(\hat{\tau_i},t)}_{\mbox{Prob of winning at t}} \cdot \underbrace{PV_t (V)}_{\mbox{PV of V at t}} \right )dt - \underbrace{x}_{\mbox{cost}}\; </math>