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Unformatted text preview: EXERCISES 131 Each girl would prefer to have a unique dress, so a girl’s utility is zero
if she ends up purchasing the same dress as at least one of her friends. All
three know that lulie strongly prefers black to both lavender and yellow, so
she would get a utility of 3 if she were the only one wearing the black dress,
and a utility of 1 if she were either the only one wearing the lavender dress
or the only one wearing or the yellow dress. Similarly, all know that Kris
tin prefers lavender and secondarily prefers yellow, so her utility would be
3 for uniquely wearing lavender, 2 for uniquely wearing yellow, and 1 for
uniquely wearing black. Finally, all know that Larissa prefers yellow and
secondarily prefers black, so she would get 3 for uniquely wearing yellow, 2
for uniquely wearing black, and 1 for uniquely wearing lavender. (a) Provide the game table for this three—player game. Make Julie the How
player, Kristin the Column player, and Larissa the Page player. (b) Identify any dominated strategies in this game, or explain why there
are none. (c) What are the pure—strategy Nash equilibria in this game? U10. Bruce, Colleen, and David are all getting together at Bruce's house on Fri U11. day evening to play their favorite game, Monopoly. They all love to eat sushi
while they play. They all know from previous experience that two orders of
sushi are just the right amount to satisfy their hunger. If they wind up with
less than two orders, they all end up going hungry and don't enjoy the eve
ning. More than two orders would be a waste, because they can’t manage to
eat a third order and the extra sushi just goes bad. Their favorite restaurant,
Fishes in the Raw, packages its sushi in such large containers that each in’
dividual person can feasibly purchase at most one order of sushi. Fishes in
the Raw offers takeout, but unfortunately doesn’t deliver. Suppose that each player enjoys $20 worth of utility from having
enough sushi to eat on Friday evening, and $0 from not having enough to
eat. The cost to each player of picking up an order of sushi is $ 10. Unfortunately, the players have forgotten to communicate about
who should be buying sushi this Friday, and none of the players has a cell
phone, so they must each make independent decisions of whether to buy
(B) or not buy (N) an order of sushi. (3) Write down this game in strategic form. (b) Find all the Nash equilibria in pure strategies. (c) Which equilibrium would you consider to be a focal point? Explain
your reasoning. Roxanne, Sara, and Ted all love to eat cookies, but there's only one left in
the package. No one wants to split the cookie, so Sara proposes the fol—
lowing extension of “Evens or Odds" (see Exercise $11) to determine
who gets to eat it. On the count of three, each of them will show one or two 132 [CH. 4] SIMULTANEOUSMOVE GAMES WITH PURE STRATEGIES ﬁngers, they’ll add them up, and then divide the sum by 3. If the remainder is zero Roxanne gets the cookie, if the remainder is 1 Sara gets it, and if it is 2 Ted gets it. Each of them receives a payoff of 1 for winning (and eating the cookie) and zero otherwise. (a) Represent this three»player game in normal form, with Roxanne as the
Row player, Sara as the Column player, and Ted as the Page player. (b) Find all the pure—strategy Nash equilibria of this game. Is this game a
fair mechanism for allocating cookies? Explain why or why not. U 12. (Optional) Construct the payoff matrix for your own twoplayer game that
satisﬁes the following requirements. First, each player should have three
strategies. Second, the game should not have any dominant strategies.
Third, the game should not be solvable using minimax. Fourth, the game
should have exactly two purestrategy Nash equilibria. Provide your game
matrix, and then demonstrate that all of the above conditions are true. U” SimultaneousMove Game “
with Pure Strategies [1:
Continuous Strategies and
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 Fall '08
 Charness,G

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