HealthLife and Voltality are competing in a patent race for a new generation of anti-ageing products. Each one of them needs to pick one of two research approaches. The first approach (X1), is fully deterministic and it will take 10 years to complete. The second one, (X2) is probabilistic and it has a probability of 60% to complete in 5 years and a probability of 40% to complete in 20 years. X2 is statistically independent, i.e. if both companies select it, then the outcome for one company is independent of the outcome for the other. Moreover, if one company achieves the innovation before the other then it will enjoy the advantage of being a monopolist for some period of time and it will generate expected profit of 40bn. The other company gets nothing in this case. Finally, if the two companies complete the innovation at the same time, they will have to share the market and, due to price competition, each one will generate expected profit of 18bn.

Suppose that the two companies also compete in a different patent race, which is fully probabilistic and there is only one available methodology. HealthLife invests and Voltality invests amount of money in the race. The winner gets a market worth of 100 bn and the loser gets nothing. The probability that each company has to win the patent race depends on the investment of both companies and it is given by:

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Find how much the two companies will invest in this patent race, their chances of winning it, and their expected profit from participating in this patent race.