In a gambling game, Player A and Player B both have a $$1$ and a $$5$ bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B$\text{'}$s bill. If the bills match, Player B wins Player A$\text{'}$s bill.

a$.$ Develop the game theory table for this game. The values should be expressed as the gains $($or losses$)$ for Player A$.$

b$.$ Is there a pure strategy? Why or why not?

c$.$ Determine the optimal strategies and the value of this game. Does the game favor one player over the other?

d$.$ Suppose Player B decides to deviate from the optimal strategy and begins playing each bill $50\%$ of the time. What should Player A do to improve Player A$\text{'}$s winnings? Comment on why it is important to follow an optimal game theory strategy.