please show work

2. Alice, being bored, decides to shake things up a bit. She declares that she is adding dynamite to her arsenal. Dynamite, she explains, will beat both rock and paper (for reasons that should be obvious); however, scissors will beat dynamite since they can cut the fuse. Bob is a little bitter that he can't use dynamite in his play, so he makes a deal: Alice can use dynamite, and if she wins a round with it her payoff will be 1 dollar. If Bob manages to beat Alice when she plays dynamite, his payoff will be M dollars (Bob is thinking fast, and hasn't quite decided what M ought to be yet). Aside from these modifications, the payoff scheme is as before. (a) Write down a linear program whose optimal solution determines Alice's best strategy. Be sure to leave M as an undetermined parameter in this LP. (b) Write the dual of Alice's LP from part (a). 7 (c) Bob decides that M ought to be 1, since all the other payoffs in the game are equal to one dollar. Does setting M = 1 make the game fair? (d) Determine optimal strategies and the corresponding optimal payoffs for Alice using sev- eral values of M. You may, for example, try values of M in the interval (0,3), such as 0,0.25, 0.5, 0.75, ...,2.25, 2.5, 2.75, 3. Based on your findings, are there any values of M for which Bob can expect long-term gain from this game? If yes, what are these values of M? If no, argue (or prove!) that the game does not favor Bob for any value of M. (e) Find, with justification, all values of M such that the game is fair