This is Exercise 35.2 in Rubinstein's "A Course in Game Theory". This problem is very difficult. The lecturer gave us the answer but it's very hard to understand.
I paste the problem here:
Two investors are involved in a competition with a prize of $1$. Each investor can spend any amount in the interval $[0,1]$. The winner is the investor who spends the most; in the event of a tie each investor receives $0.50$. Formulate this situation as a strategic game and find its mixed strategy Nash equilibria.
The payoff function of player $i$ is $$u_i(b_1,b_2)=\left\{\begin{matrix} -b_i & b_i < b_j\\ 0.5-b_i & b_i = b_j\\ 1-b_i & b_i > b_j \end{matrix}\right.$$ I can understand it's mixed Nash equilibrium if both players's probability distribution are uniform on $[0,1]$. But I don't know how this mixed Nash equilibrium is unique.