two firms $($form A and firm C$)$ compete for customers and are considering improving their operations. Each firm has one of two options: $($i$)$ to invest in a new AI$-$powered recommendation system or $($ii$)$ to invest in upgrading their Data Management Systems. Investing in the recommendation system carries a cost of $20$ for each firm and generates a benefit of v$>$ or equal to $0$ bgut as this system is patentable only one firm could reap this reward: if both firms invest in the recommendation system then each gets on expectation v$/2$ while if only one invests it receives v$.$ Upgrading data Management Systems generates an individual benefit of $20$ but also an individual cost that is increasing in the number of firms upgrading: if only one firm upgrades it bears a cost of $10$ while if both firms upgrade tehy each incur a cost of $19.$ Both firms simultaneously decide whether to invest in the recommendation system or upgrade Data Management

write is firm R$\text{'}$s best response as a function of the parameter v to firm C upgrading its data management system? What is F down he $2$x$2$ payoff matrix of this game. What is firm R$\text{'}$s best response as a function of the parameter v to firm C investing in the recommendation system?