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*There are <math>n\;</math> identical firms, indexed by <math>i\;</math>*Each firm invests <math>x_i\;</math> to buy a random variable <math>\tau(x_i)\;</math> which gives a completion date*The firms with the earliest realised completion date wins <math>V\;</math>*<math>\tau \sim F_{\tau}(h(x_i))\;</math> where <math>F_{\tau}\;</math> is the CDF for the exponential distribution: <math>F_{\tau}(h(x_i)) = 1 - e^{-h(x_i)t}\;</math>*<math>h(x_i)\;</math> is the rate parameter, or the instantaneous probability of the innovation occuring.

<math>h(x_i)\;</math> is assumed to have the following properties:*<math>h(0) = 0 = \lim_{x \to \infty} h'(x)\;</math>*For some <math>\overline{x} \ge 0\;</math>, <math>h''(x) \ge0\;</math> for <math>x \le \overline{x}\;</math>, and <math>h''(x) \le0\;</math> for <math>x \ge \overline{x}\;</math>*<math>\tilde{x}\;</math> is defined as the point where <math>\frac{h(x)}{x}\;</math> is greatest

Let <math>\hat{\tau_i}\;</math> be an random variable giving the date of the earliest other firm:

<math>\hat{\tau_i} = \min_{j \ne i} \{ \tau(x_j) \}\;</math>

Assuming iid tau's (no externalities in innovation!), then we can use a [http://en.wikipedia.org/wiki/Exponential_distribution#Distribution_of_the_minimum_of_exponential_random_variables nice feature of the exponential distribution] which is that if <math>X_1,\ldots,X_N\;</math> are iid exponential with rates <math>\lambda_1,\ldots,\lambda_N\;</math>, then <math>\min(X_1,\ldots,X_N)\;</math> is distributed exponential with rate <math>\sum_1^N \lambda_i\;</math>.

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

For convenience we denote <math>a_i= \sum_{j\ne i} -h(x_j) \; </math>