1. (35 points) Magic: The Gathering (MTG) is card game in which players develop different types of card decks prior to a competition. MTG players usually use one of three archetypes deck strategies in forming their decks control, aggro, or combo. Aggro decks tend to beat control decks, control decks tend to beat combo decks, and combo decks tend to beat aggro decks'. The table below estimates the probability that a Player beats an Opponent using one of the three archetypes. Player Deck Opponent Deck Archetype Archetype Aggro Combo Control Aggro 0.50 0.10 0.88 Combo 0.90 0.50 0.09 Control 0.12 0.91 0.50 The Player must decide his deck archetype prior to seeing the Opponent's deck archetype. a) (5 points) Determine whether the Player should use a pure strategy (ie, only one archetype in every match). If so, what is the probability that the Player beats the Opponent? If not, what are the lower and upper bounds on the probability that the Player beats the Opponent? b) (15 points) Formulate, but do not solve, a linear program that determines an optimal strategy for the Player to select his deck archetype. e) (15 points) Formulate, but do not solve, a linear program that determines an optimal strategy for the Opponent to select his deck archetype