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Unformatted text preview: u. __, .‘u Aaluuk IUUA 1U]. d _  u equinonum in mixed strategies, r
the analysis of which is presented in Chapters 7 and 8. y:
a Rs? KEY TERMS i assurance game (107)
battle of the sexes (108)
belief (89)
best response (87)
bestresponse analysis (99)
cellbycell inspection (89)
chicken (109)
constant—sum game (85)
convergence of expectations
(107)
coordination game (105)
dominance solvable (96)
dominant strategy (92)
dominated strategy (92)
enumeration (89)
focal point (106) game matrix (84) example of a game that illustrates your an EXERCISES game table (84)
iterated elimination of
dominated strategies (96) maximin (100) minimax (100) minimax method (99)
mixed strategy (84)
Nash equilibrium (87)
normal form (84) payoff table (84) prisoners' dilemma (90) pure coordination game (105) pure strategy (84) strategic form (84) successive elimination of
dominated strategies (96) zero—sum game (85) SWEI’. (c) COLUMN 3 1 (d) 3. Find all Nash equilibria in pure strategies in the following nonzero—sum
games. Describe the steps that you used in ﬁnding the equﬂibna. 5 [CH. 4] SlMULTANEOUSMOVE GAMES WITH PURE STRATEGIES COLUMN (bl (c) m ‘nd all of the purestrategy Nash equilibria for the following game. De
ribe the process that you used to ﬁnd the equilibtia. Use this game to ex EXERCISES 117 plain why it is important to describe an equilibrium by using the strategies
employed by the players, not merely by the payoffs received in equilibrium. COLUMN 6. The game known as the battle of the Bismarck Sea (named for that part of the southwestern Paciﬁc Ocean separating the Bismarck Archipelago from
Papua—New Guinea) summarizes a wellknown game actually played in a
naval engagement between the United States and Japan in World War II. In
1943, a Japanese admiral was ordered to move a convoy of ships to New
Guinea; he had to choose between a rainy northern route and a sunnier
southern mute, both of which required 3 days sailing time. The Americans
knew that the convoy would sail and wanted to send bombers after it. but
they did not know which route it would take. The Americans had to send re
connaissance planes to scout for the convoy, but they had only enough re
connaissance planes to explore one route at a time. Both the Japanese and
the Americans had to make their decisions with no knowledge of the plans
being made by the other side. If the convoy was on the route explored by the Americans ﬁrst. they could
send bombers right away; if not, they lost a day of bombing. Poor weather on
the northern route would also hamper bombing. 1f the Americans explored
the northern route and found the Japanese right away. they could expect only
2 {of 3) good bombing days; if they explored the northern route and found
that the Japanese had gone south, they could also expect 2 days of bombing.
If the Americans chose to explore the southern route ﬁrst, they could expect 3
full days of bombing if they found the Japanese right away but only 1 day of
bombing if they found that the Japanese had gone north. (a) Illustrate this game in a game table. (b) Identify any dominant strategies in the game and solve for the Nash
equilibrium. 7. An old lady is looking for help crossing the street. Only one person is needed to help her; more are okay but no better than one. You and I are the
two people in the vicinity who can help; we have to choose simultaneously [CH. 4] SIMULTANEOUSrMOVE GAMES WITH PURE STRATEGIES ether to do so. Each of us will get pleasure worth a 3 from her success
0 matter who helps her). But each one who goes to help will bear a cost of
this being the value of our time taken up in helping. Set this up as a
me. Write the payoff table, and ﬁnd all purestrategy Nash equilibria. 0 players, lack and Iill, are put in separate rooms. Then each is told the
les of the game. Each is to pick one of six letters; G, K, L, Q, R, or W. If the
o happen to choose the same letter, both get prizes as follows: Letter G K L Q R W
Ilack’s prize 3 2 5 3 4 5
Iill’s prize 6 5 4 3 2 1 they choose different letters, each gets 0. This whole schedule is re
aled to both players, and both are told that both know the schedules,
(1 so on. ) Draw the table for this game. What are the Nash equilibria in pure
strategies? ) Can one of the equilibria be a focal point? Which one? Why? ppose two players, A and B, select from three different numbers, 1, 2, and
Both players get dollar prizes if their choices match, as indicated in the
llowing table. What are the Nash equilibria of this game? Which, if any, is likely to
emerge as the (focal) outcome? Explain. ) Consider a slightly changed game in which the Choices are again just
numbers but the two cells with (15, 15) in the table become (25, 25).
What is the expected (average) payoff to each player if each ﬂips a coin
to decide whether to play 2 or 3? Is this better than focusing on both
choosing 1 as a focal equilibrium? How should you account for the risk
that A might do one thing while B does the other? . _....._'_,.._... _ ._ ___.. ._.. ._... n .t...,..... .. _.. .. EXERCISES 119 10. In Chapter 3, the three gardeners, Emily, Nina, and Talia, play a sequential 11. 12. version of the streetgarden game in which there are four distinguishable outcomes (rather than the six different outcomes speciﬁed in the example of Section 4.6). For each player, the four outcomes are: (1) player does not contribute, both of the others do [pleasant garden,
saves cost of own contribution) (ii) player contributes, and one or both of the others do (pleasant garden,
incurs cost of contribution) (iii) player does not contribute, only one or neither of the others does (sparse garden, saves cost of own contribution) (iv) player contributes, but neither of the others does (sparse garden, incurs
cost of own contribution) Of them, outcome i is the best (payoff 4) and outcome iv is the worst (payoff 1). if each player regards a pleasant garden more highly than her own contribution, then outcome ii gets payoff 3 and outcome iii gets payoff 2. (a) Suppose that the gardeners play this game simultaneously, deciding
whether to contribute to the street garden without knowing what
choices the others will make. Draw the threeplayer game table for this
version of the game. (13) Find all of the Nash equilibria in this game. (c) How might this simultaneous version of the streetgarden game be
played out in reality? Consider a game in which there is a prize worth $30. There are three contes—
tants, A, B, and C. Each can buy a ticket worth $15 or $30 or not buy a ticket
at all. They make these choices simultaneously and independently. Then,
knowing the ticketpurchase decisions, the game organizer awards the
prize. If no one has bought a ticket, the prize is not awarded. Otherwise, the
prize is awarded to the buyer of the highestcost ticket if there is only one
such player or is split equally between two or three if there are ties among the highestcost ticket buyers. Show this game in strategic form. Find all
purestrategy Nash equilibria. in the ﬁlm A Beautiful Mind, Iohn Nash and three of his graduate school
colleagues ﬁnd themselves faced with a dilemma while at a bar. There are
four brunettes and a single blonde available for them to approach. Each
young man wants to approach and win the attention of one of the young
women. The payoff to each of winning the blonde is 10: the payoff of win
ning a brunette is 5; the payoff from ending up with no girl is 0. The catch is
that, if two or more young men go for the blonde, she rejects all of them and
then the brunettes also reject the men because they don’t want to be second .. .n__..___ EXERClSES 151 SUMMARY When players in a simultaneousmove game have a continuous range of actions
to choose, bestresponse analysis yields mathematical best—response rules that
can be solved simultaneously to obtain Nash equilibrium strategy choices. The
best»response rules can be shown on a diagram in which the intersection of the
two curves represents the Nash equilibrium. Firms choosing price or quantity
from a large range of possible values and political parties choosing campaign
advertising expenditure levels are examples of games with continuous strategies. The results of laboratory tests of the Nash equilibrium concept show that a
common cultural background is essential for coordinating in games with multiple
equilibria. Repeated play of some games shows that players can learn from expe
rience and begin to choose strategies that approach Nash equilibrium choices.
Further, predicted equilibria are accurate only when the experimenters’ assump
tions match the true preferences of players. Realworld applications of game the
ory have helped economists and political scientists, in particular, to understand
important consumer, ﬁrm, voter, legislature, and government behaviors. Theoretical criticisms of thteash equilibrium concept have argued that the
concept does not adequately accOunt for risk, that it is of limited use because
many games have multiple equilibria, and that it cannot be justiﬁed on the basis
of rationality alone. In many cases, a better description of the game and its pay
off structure or a reﬁnement of the Nash equilibrium concept can lead to better
predictions or fewer potential equilibria. The concept of rationaliznbility relies
on the elimination of strategies that are never a best response to obtain a set of
rationalizable outcomes. When a game has a Nash equilibrium, that outcome
will be rationalizable; but rationalizability also allows one to predict equilibrium
outcomes in games that have no Nash equilibria. KEY TER M S .;
bestresponse curves (127) rationalizability (145)
bestresponse rule [124) rationalizable (145)
continuous strategy (124) reﬁnement [143) never a best response (145) EXERCISES ? l. in the political campaign advertising game in Section LB, party L chooses
an advertising budget, x (millions of dollars}, and party R similarly chooses a
budget, y (millions of dollars]. We showed there that the bestresponse rules n that game are y = VE— x, for party R, and x = W — y for party L. USE
hese bestresponse rules to verify that the Nash equilibrium advertising
Judgets are x = y : 1/4, or $250,000. the bistro game of Figure 5.1 deﬁnes demand functions for Xavier’s (0,.) 1nd Yvonne’s (le as Qx = 44 — 2 P, + Py, and Q, = 44 e 2 P}, + PX. Proﬁts for :ach ﬁrm depend, in addition, on their costs of serving each customer. Sup— Jose, here, that Yvonne’s is able to reduce its costs to $6 per customer by re listributing the serving tasks and laying off several servers; Xavier’s :ontinues to incur a cost of $8 per customer. a) Recalculate the bestresponse rules and the Nash equilibrium prices for
the two ﬁrms, given the change in the cost conditions. b) Graph the two bestresponse curves and describe the differences be
tween your graph and Figure 5.1. In particular, which curve has moved
and by how much? Can you account for the changes in the diagram? Tuppietown has two food stores, La Boulangerie, which sells bread, and La
2romagerie, which sells cheese. It costs $1 to make a loaf of bread and $2 to
nake a pound of cheese. If La Boulangerie’s price is P1 dollars per loaf of
need and La Fromagerie’s price is P; dollars per pound of cheese, their re .pective weekly sales, Q1 thousand loaves of bread and Q2 thousand pounds
If cheese, are given by the following equations: QIZIO—Pl—0‘5P2! 02:1270.5P1_P2. a) Find the two stores’ bestresponse rules. Show the best—response
curves, and ﬁnd the Nash equilibrium prices in this game. b) Suppose that the two stores collude and set prices jointly to maximize
the sum of their proﬁts. Find the joint proﬁtmaximizing prices for the
stores. c) Provide a short intuitive explanation for the differences between the
Nash equilibrium prices and those that maximize joint proﬁt. Why is
joint proﬁt maximization not a Nash equilibrium? d) In this problem, bread and cheese are mutual complements. They are
often consumed together; that is why a drop in the price of one in
creases the sales of the other. The products in our bistro example in
Section 1.A are substitutes for each other. How does this difference ex
plain the differences between your ﬁndings for the bestreSponse rules,
the Nash equilibrium prices, and the joint proﬁtmaximizing prices in this question, and the corresponding entities in the bistro example in
the text? ‘wo carts selling coconut milk (from the coconut) are located at 0 and 1. 1
rule apart on the beach in Rio de Ianeiro. (They are the only two coconut milk carts on the beach.) The carts—Cart 0 and Can I—charge prices p0 and
p1, respectively, for each coconut. Their customers are the beach goers uni
formly distributed along the beach between 0 and 1. Each beach goer will
purchase one coconut milk in the course of her day at the beach and, in ad
dition to the price, each will incur a transport cost of 0.5 times d2, where dis
the distance (in miles) from her beach blanket to the coconut cart. In this
system, Cart 0 sells to all of the beach goers located between 0 and x, and
Cart 1 sells to all of the beach goers located between it and 1, where x is the
location of the beach goer who pays the same total price if she goes to 0 or 1.
Location x is then deﬁned by the expression pa + 0.5x2 = p, + 0.5[1 — x)? The two carts will set their prices to maximize their bottomline proﬁt ﬁg
ures, B; proﬁts are determined by revenue (the cart’s price times its number
of customers) and cost (the carts each incur a cost of $0.25 per coconut
times the number of coconuts sold). (a) Determine the expression for the number of customers served at each
cart. (Recall that Cart 0 gets the customers between 0 and x, or just x.
while Cart 1 gets the customers between xand 1, or 1 — x.) (b) Write out proﬁt functions for the two carts and ﬁnd the two best
response rules for their prices. (c) Graph the best—response rules, and then calculate (and show on your
graph) the Nash equilibrium price level for coconuts on the beach. . The game illustrated in Figure 5.3 has a unique Nash equilibrium in pure strategies. However, all nine outcomes in that game are rationalizable. Con
ﬁrm this assertion, explaining your reasoning for each outcome. . The game illustrated in Figure 5.4 has a unique Nash equilibrium in pure strategies. Find that Nash equilibrium, and then show that it is also the
unique rationalizable outcome in that game. . Section 4.B describes a ﬁshing game played in a small coastal town. When the response rules for the two boats have been derived, rationalizability can
be used to justify the Nash equilibrium in the game. In the description in the
text, we take the process of narrowing down strategies that can never be best
responses through three rounds. By the third round. we know that X (the
number of barrels of ﬁsh brought home by boat 1) must be at least 9, and
that Y (the number of barrels of ﬁsh brought home by boat 2) must be at
least 4.5. The narrowing process in that round restricted X to the range be
tween 9 and 13.75 while restricting Yto the range between 4.5 and 7.5. Take
this process of narrowing through one additional (fourth) round and show
the reduced ranges ()f X and Ythat are obtained at the end of the round. — 154 [CH. 5] SIMULTANEOUSeMOVE GAMES WiTH PURE STRATEGIES 9. Nash equilibrium through rationalizability can be achieved in games with up
wardsloping bestresponse curves if the rounds of eliminating neverbest
response strategies begin with the smallest possible values. Consider the
pricing game between Xavier's Tapas Bar and Yvonne’s Bistro that is illus«
trated in Figure 5.1. Use Figure 5.1 and the bestresponse rules from which it
is derived to begin rationalizing the Nash equilibrium in that game. Start with
the lowest possible prices for the two ﬁrms and describe [at least] two rounds
of narrowing the set of rationalizable prices toward the Nash equilibrium. 10. Optional, requires calculus Recall the political campaign advertising exam»
pie from Section LB concerning parties L and R. In that example, when i
L spends $x million on advertising and R spends $y million, L gets a share
x/ (x + y) of the votes and R gets y/ (x + y). We also mentioned that two types
of asymmetries can arise between the parties in that model. One panywsay,
R—may be able to advertise at a lower cost or R’s advertising dollars may be
more effective in generating votes than L’s. To allow for both possibilities,
we can write the payoff funchons of the two parties as ky
_ 3C 7 __ _
V].  x x and VR — k cy. These payoff functions show that R has an advantage in the relative effec
tiveness of its ads when k is high and that R has an advantage in the cost of
its ads when 6 is low. (a) Use the payoff functions to derive the bestresponse functions for R
(which chooses y] and L (which chooses x). (b) Use your calculator or your computer to graph these bestresponse
functions when k = 1 and c : 1. Compare the graph with the one for the
case in which k : 1 and c : 0.8. What is the effect of having an advan
tage in the cost of advertising? (c) Compare the graph from part b, when k : 1 and c : 1 with the one for
the case in which k 2 2 and c = 1. What is the effect of having an advan
tage in the effectiveness of advertising dollars? ((1) Solve the bestresponse functions that you found in part a, jointly for x
and y, to show that the campaign advertising expenditures in Nash
equilibrium are ck k = d : r.
x [c+k)2 an y (Ht)Z (e) Let k : 1 in the equilibrium spending level equations and show how the
two equilibrium spending levels vary with changes in c. The let c : 1
and show how the two equilibrium spending levels vary with changes in
k. Do your answers support the effects that you observed in parts b and
c of this exercise? ...
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 Spring '07
 Emre
 Game Theory, Iohn Nash

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