2. Two players have to divide a perfectly divisible cake of size 2. Half of the cake is made of chocolate, while the other half is made of marzipan. If a piece of cake contains a units of chocolate cake and y units of marzipan, player 1 gets utility U(x, y) = x + y, from this piece, while player 2 gets utility V(x, y) = 2x+y, from this piece. That is, player 1 cares only about the total size of the piece, while player 2 is values chocolate twice as much as marzipan. The players use the divide-and-choose protocol one player divides the cake into two, and the other chooses one of the two pieces. More precisely, the divider splits the cake into two pieces (x, y) and (1-2, 1-y) (where 0≤x≤ 1 and 0 ≤ y ≤ 1. The chooser may choose either the piece (x, y) or the piece (1-2, 1-y). a) Use backwards induction to prove (or argue for) the the following claim: "In any subgame perfect equilibrium, the chooser is indifferent between the two pieces." Solve by backwards induction, for the subgame perfect equilibrium when: b) player 2 divides. c) player 1 divides. d) Docs a player prefer to be a divider or a chooser? Explain why, briefly.

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Subgame Perfect Equilibrium. Stephen J. Seagull must decide whether or not to start a new movie project. If he decides not to, he and Clod VandeCamp both get a utility of 10. If he decides to begin the project, both he and Clod must simultaneously decide who the director should be: George Spellbinder, or Ed Tree. If they disagree on the director, the movie isn't made, but both have wasted time, so they get only a utility of 0. If they agree on George Spellbinder, the movie will be a roaring success, and each gets a utility of 20. If they agree on Ed Tree, the movie will be terrible. and they will only get a utility of 5. (a). Draw the extensive form of this game. (b). Write down the normal form. (c). Find all Nash equilibria. (d). Apply the theory of iterated elimination of weakly dominated strategies and state its prediction. (e). Find all the subgame perfect equilibria.

Mr. and Mrs. Ward typically vote oppositely in elections and so their votes “cancel each other out.” They each gain 6 units of utility from a vote for their positions (and lose 6 units of utility from a vote against their positions). However, the bother of actually voting costs each 3 units of utility. The following matrix summarizes the strategies for both Mr. Ward and Mrs. Ward.   Mrs. Ward Vote Don't Vote Mr. Ward Vote Mr. Ward: -3, Mrs. Ward: -3 Mr. Ward: 3, Mrs. Ward: -6 Don't Vote Mr. Ward: -6, Mrs. Ward: 3 Mr. Ward: 0, Mrs. Ward: 0   The Nash equilibrium for this game is for Mr. Ward to    and for Mrs. Ward to    . Under this outcome, Mr. Ward receives a payoff of   units of utility and Mrs. Ward receives a payoff of   units of utility.   Suppose Mr. and Mrs. Ward agreed not to vote in tomorrow's election. True or False: This agreement would decrease utility for each spouse, compared to the Nash equilibrium from the previous part of the question. True…

Suppose the utility possibility frontier for two individuals is given by   U_a+2U_b=200   Please plot the utility frontier on a graph.

Divya and Emily have to divide a bag of 200 pistachios between them. They derive enjoyment from pistachios according to the following hedonic utility functions: uD(x) = √x for Divya; uE (x) = 2√x for Emily (where x is the number of pistachios eaten). (a) Consider all possible ways of dividing the pistachios. Treating the utilities given above as the two agents" "levels," which division is best according to the utilitarian criterion? (b) Which division is best according to the leximin criterion? (c) Which is best according to the Nash criterion?

Emi's friend Xindi also likes tacos, but she doesn't like sandwiches. Instead, she always consumes exactly two tacos (good x) with one glass of horchata (y), and she doesn't enjoy either good without the other. Suppose prices are the same as initially in problem 1, so that horchata is twice as expensive as tacos (Px=$2, Py=$4), and Xindi has the same budget as Emi: I=$24 a) Propose a utility function that is consistent with these preferences and draw a couple of its isoquants. b) Solve Xindi's long run utility maximization to find the basket of goods she will purchase. c) What if, just as in problem 1, the good on the x axis goes on sale for $1/ unit?

- Individuals 1 and 2 are forming a company. The value of their relationship depends on the effort that each expends. Suppose that individual i's utility from the relationship is x² + x; − xixj, where x¿ is individual i 's effort and x ; is the effort of the other person (i = 1, 2). Assume x1, x2 ≥ 0. Which statement is true? There is a Nash equilibrium in which each player chooses effort level k > 0. This equilibrium is Pareto efficient. There is no mixed-strategy Nash equilibrium. There is a unique Nash equilibrium in which each player chooses no effort x¿ = x; = 0. The set of Nash equilibria in this game is finite. There is a Nash equilibrium in which xi = 0 and x ; = 1| =

Tinky Winky and Dipsy both choose to play an action, and the payoff from that choice is dependent on what the other player chooses. Both players choose their actions simultaneously and reveal their choice to each other at the same time. Tinky-Winky Jump Punch Tinky-Winky gets payoff Tinky-Winky gets payoff Kick Dipsy Dipsy gets payoff Dipsy gets payoff Tinky-Winky gets payoff Tinky-Winky gets payoff Duck 4. Dipsy gets Dipsy gets payoff payoff 10 Which of the following statements is true? O This game does not have any Nash equilibria. O There is more than one Nash equilibrium in this game. O The only Nash equilibrium of this game is Dipsy playing "Kick" and Tinky-Winky playing "Jump" O The only Nash equilibrium of this game is Dipsy playing "Duck" and Tinky-Winky playing "Punch O The only Nash equilibrium of this game is Dipsy playing "Kick" and Tinky-Winky playing "Punch" O The only Nash equilibrium of this game is Dipsy playing "Duck" and Tinky-Winky playing "Jump"

3. Christina loves pizza and hates burger. Her utility function is U(p – b) = p – 6", where is the number of pizzas she consumes and b is the number of burgers she consumes. John likes both pizza and burger. His utility function is U(p, b) = p + 2\b Christina has an initial endowment of no pizzas and 8 burgers. John has an initial endowment of 16 pizzas and 8 burgers. a. Graph the initial endowment and label it E b. If Christina hates burgers and John likes them, how many burgers can Christina and John be consuming at a Pareto optimal allocation? What is John's marginal utility for pizzas and burgers? Mark the locus of Pareto optimal allocations of pizzas and burgers between Christina and John on the grapsh above.