a. Which combination or combinations of X and Y would make a situation in which The Old Familiar does not advertis and The Beehive advertises a Nash equilibrium? A Nash equilibrium occurs if X = $2,500; Y = $2,000. X = $1,200; Y = $1,500. X = $3,300; Y = $1,250. X = $1,500; Y = $2,500. b. Which combination or combinations of X and Y would cause this game to have multiple Nash equilibria? Multiple Nash equilibria occur if X = $2,500; Y = $2,000 X= $3,300; Y = $1,250 X = $1,200; Y = $1,500 X = $1,500; Y = $2,500 c. Based on your answer in part b, when this game has multiple Nash equilibria the game will be

• (1,0) (1,1) Player 2 G D H (2,0) Action Player 1 Player 1 Action B Player 1 (3,-1) E Player 2 (0,1) (-1,-1) (a) Find all the Subgame Perfect Nash equilibria in this game. (b) Find all the Nash equilibria in this game. (Hint: write the game in strategic form.)

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7. Solving for dominant strategies and the Nash equilibrium Suppose Felix and Janet are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Felix chooses Right and Janet chooses Right, Felix will receive a payoff of 7 and Janet will receive a payoff of 4. Felix Left Right Left 6,3 3,3 Janet Right 6,4 7,4 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Felix chooses and Janet chooses

8. Solving for dominant strategies and the Nash equilibrium Suppose Tim and Alyssa are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Tim chooses Right and Alyssa chooses Right, Tim will receive a payoff of 3 and Alyssa will receive a payoff of 6. Tim Left Left 5, 6 Right 4, 2 Alyssa Right 5,5 3,6 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Tim chooses and Alyssa chooses

For problem 1 and 2, consider the following game : с D A (0,5) (5,-5) B (3, 1) (1, 0) 1. Show that the game above doesn't have a pure Nash equilibrium. 2. Show that the game above has a unique mixed Nash equilibrium, and describe the playing strategy.

Player 2 Middle Left P1: $45 P1: $70 Up P2: $45 P2: $50 Player 1 P1: $50 P1: $60 Middle P2: $50 P2: $60 P1: $60 P1: $50 Down P2: $60 P2: $70 In the game shown above, list all of the Nash Equilibrium (please check ALL that apply) (up, left) (up, middle) (up, right) (middle, left) (middle, middle) (middle, right) (down, left) (down, middle) (down, right) No equilibrium Right P1: $45 P2: $60 P1: $50 P2: $70 P1: $60 P2: $60

The bimatrix represents a simultaneous move game between Rowena and Colin. Rowena's payoff is the left number in each cell. ROWENA Up Down 0.25Left +0.75Right 0.65Left +0.35Right Find Colin's mixed strategy that makes Rowena indifferent between a pure strategy of playing Up and a pure strategy of playing Down. 0.5Left +0.5Right Left 1,16 2,20 0.45Left +0.55Right COLIN Right 4,6 3,40

7. Solving for dominant strategies and the Nash equilibrium Suppose Carlos and Deborah are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Carlos chooses Right and Deborah chooses Right, Carlos will receive a payoff of 5 and Deborah will receive a payoff of 1. Carlos Deborah Left Left 4,4 Right 2,4 Right 6,7 5,1 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Carlos chooses and Deborah chooses

Superstore (S) and Megastore (M) are two competing discount supermarkets.  If neither store advertises, S makes $115 in profits and M makes $165.  If both stores advertise, S makes $80 and M makes $95.  If M advertises and S does not, they make $305 and $55 respectively.  If S advertises and M does not, they make $285 and $65 respectively.  Based on Game Theory, what will the two firms do?  Hint: illustrate the Nash equilibrium to support your answer.