Player $1$ and player $2$ are playing a simultaneous$-$move one$-$shot game, where player $1$ can move up or down and player $2$ can move left or right. The payoff that each player gets depends on what both players choose. If player $1$ moves up while player $2$ moves left, player $1$ receives a payoff of$-5,$ and player $2$ receives a payoff of$-5.$ If player $1$ chooses up and player $2$ chooses right, player $1$ receives a payoff of$-10,$ while player $2$ receives a payoff of$-3.$ If player $1$ instead chooses down and player $2$ chooses left, then player $1$ receives a payoff of$-4,$ and player $2$ receives a payoff of$-11.$ Finally, if player $1$ chooses down and player $2$ chooses right, then player $1$ receives a payoff of$-8,$ and player $2$ receives a payoff of$-10.$ Use this information to answer the questions below. Note that it may help to summarize the above information into a payoff matrix before tackling the questions.

$1$st attempt

Part $1(2$points$)$

See Hint

Which of the following statements is true regarding player $1?$

Choose one or more:A$.$Player $1$ does not have a dominant strategy.B$.$Player $1$ has a dominant strategy to play left.C$.$Player $1$ has a dominant strategy to play up$.$D$.$Player $1$ has a dominant strategy to play right.E$.$Player $1$ has a dominant strategy to play down.

Which of the following statements is true regarding player $2?$

Choose one or more:A$.$Player $2$ has a dominant strategy to play right.B$.$Player $2$ has a dominant strategy to play down.C$.$Player $2$ has a dominant strategy to play up$.$D$.$Player $2$ does not have a dominant strategy.E$.$Player $2$ has a dominant strategy to play left.

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Part $2(1$point$)$

See Hint

Which of the following outcomes is a pure strategy Nash equilibrium of this game?

Choose one or more:A$.$Player $1$ plays down; player $2$ plays left.B$.$Player $1$ plays up; player $2$ plays right.C$.$Player $1$ plays up; player $2$ plays left.D$.$There is no pure strategy Nash equilibrium in this game.E$.$Player $1$ plays down; player $2$ plays right.

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Part $3(1$point$)$

See Hint

Is there an equilibrium in this game where at least one player does not place a $100\%$ probability on an action?$$

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