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Unformatted text preview: Chapter 28 NAME Game Theory Introduction. In this introduction we offer two examples of twoperson
games. The ﬁrst game has a dominant strategy equilibrium. The second
game is a zerosum game that has a Nash equilibrium in pure strategies
that is not a dominant strategy equilibrium. Example: Albert and Victoria are roommates. Each of them prefers a
clean room to a dirty room, but neither likes housecleaning. If both clean
the room, they each get a payoff of 5. If one cleans and the other doesn’t
clean, the person who does the cleaning has a utility of 2, and the person
who doesn’t clean has a utility of 6. If neither cleans, the room stays a
mess and each has a utility of 3. The payoffs from the strategies “Clean”
and “Don’t Clean" are shown in the box below. Clean Room—Dirty Room Victoria
Clean Don‘t Clean Alb Clean
6” Don’t Clean In this game, whether or not Victoria chooses to clean, Albert will get
a higher payoff if he doesn’t clean than if he does clean. Therefore “Don’t
Clean” is a dominant strategy for Albert. Similar reasoning shows that
no matter what Albert chooses to do, Victoria is better off if she chooses
“Don‘t Clean.” Therefore the outcome where both roommates choose
“Don’t Clean” is a dominant strategy equilibrium. This is true despite
the fact that both persons would be better off if they both chose to clean
the room. Example: This game is set in the South Paciﬁc in 1943. Admiral Imamura
must transport Japanese troops from the port of Rabaul in New Britain,
across the Bismarck Sea to New Guinea. The Japanese ﬂeet could either
travel north of New Britain, where it is likely to be foggy, or south of
New Britain, where the weather is likely to be clear. U.S. Admiral Ken
ney hopes to bomb the troop ships. Kenney has to choose whether to
concentrate his reconnaissance aircraft on the Northern or the Southern
route. Once he ﬁnds the convoy, he can bomb it until its arrival in New
Guinea. Kenney’s staff has estimated the number of days of bombing 344 GAMETHEORY (Ch. 28) time for each of the outcomes. The payoﬂs to Kenney and lmamura from
each outcome are shown in the box below. The game is modeled as a
“zerosum game:” for each outcome, Imamura’s payoff is the negative of
Kenney’s payoff. The Battle of the Bismarck Sea Imamura
North South North
Kenney South This game does not have a dominant strategy equilibrium, since there
is no dominant strategy for Kenney. His best choice depends on what Ima
mura does. The only Nash equilibrium for this game is where Imamura
chooses the northern route and Kenney concentrates his search on the
northern route. To check this, notice that if Imamura goes North, then
Kenney gets an expected two days of bombing if he (Kenney) chooses
North and only one day if he (Kenney) chooses South. Furthermore, if
Kenney concentrates on the north, Imamura is indifferent between go
ing north or south, since he can be expected to be bombed for two days
either way. Therefore if both choose “North,” then neither has an incen
tive to act differently. You can verify that for any other combination of
choices, one admiral or the other would want to change. As things actually
worked out, Imamura chose the Northern route and Kenney concentrated
his search on the North. After about a day’s search the Americans found
the Japanese ﬂeet and inﬂicted heavy damage on it.* 28.1 (0) This problem is designed to give you practice in reading a game
matrix and to check that you understand the deﬁnition of a dominant
strategy. Consider the following game matrix. A Game Matrix Player B
Left Right Player A TOP m.
Bottom e,f * This example is discussed in R. Duncan Luce and Howard Raiffa’s
Games and Decisions, John Wiley, 1957, or Dover, 1989. We recommend
this book to anyone interested in reading more about game theory. NAME —.—__._... 345 (a) If (top, left) is a dominant strategy equilibrium, then we know that a) 6 ,b) d , C >g,and f >11. (5) If (top, left) is a Nash equilibrium, then which of the above inequalities
must be satisﬁed? a. > 8; b > d . (c) If (top, left) is a dominant strategy equilibrium must it be a Nash equilibrium? Why? Yes. A dominant strategy equilibrium is always a Nash equilibrium. 28.2 (0) In order to learn how people actually play in game situations,
economists and other social scientists frequently conduct experiments in
which subjects play games for money. One such game is known as the
voluntary public goods game. This game is chosen to represent situations
in which individuals can take actions that are costly to themselves but
that are beneﬁcial to an entire community. In this problem we will deal with a twoplayer version of the voluntary
public goods game. Two players are put in separate rooms. Each player
is given $10. The player can use this money in either of two ways. He can
keep it or he can contribute it to a “public fund.” Money that goes into
the public fund gets multiplied by 1.6 and then divided equally between
the two players. If both contribute their $10, then each gets back $20 x
1.6/2 : $16. If one contributes and the other does not, each gets back
$10 >< 1.6/2 : $8 from the public fund so that the contributor has $8
at the end of the game and the non—contributor has $18—his original $10
plus $8 back from the public fund. If neither contributes, both have their
original $10. The payoff matrix for this game is: Voluntary Public Goods Game Player 13
Contribute Keep Player A conmbute $16: $16 $8, $18
Keep $18, $8 $10, $10 (a) If the other player keeps, what is your payoff if you keep? $10 . If the other player keeps, what is your payoff if you contribute? $8 . 346 GAMETHEORY (Ch. 28) (b) If the other player contributes, what is your payoff if you keep? $18 . If the other player contributes, what is your payoff if you con— tribute? $ 1 6 . (c) Does this game have a dominant strategy equilibrium? Yes .' If so, what is it? Both keep . 28.3 (1) Let us consider a more general version of the voluntary public
goods game described in the previous question. This game has N players,
each of whom can contribute either $10 or nothing to the public fund.
All money that is contributed to the public fund gets multiplied by some
number B > 1 and then divided equally among all players in the game
(including those who do not contribute.) Thus if all N players contribute
$10 to the fund, the amount of money available to be divided among the
N players will be $IOBN and each player will get $10BN/N = $103 back
from the public fund. (a) If B > 1, which of the following outcomes gives the higher payoff to
each player? a) All players contribute their $10 or b) all players keep their me. All players contribute their $10. (b) Suppose that exactly K of the other players contribute. If you
keep your $10, you will have this $10 plus your share of the public
fund contributed by others. What will your payoff be in this case? + . If you contribute your $10, what will be the total number of contributors? K+l . What will be your payoff? $10B(K + 1)/N. (c) If B = 3 and N = 5, what is the dominant strategy equilibrium for this
game? All keep their 55 10 . Explain your answer. If K other players contribute, a player’s payoff
will be $10aSOIY/5 if he keeps the $10 and
$30QRT+GU/5 if he contributes it. Where .3 2:3 and PV== 5, the difference between
these two payoffs is $10—$6 =I$4 > 0, so
we see that for all possible If, the payoff from keeping the money is always higher. NAME—— 347
(d) In general, what relationship between B and N must hold for “Keep”
tobeadmnmmmsuemgﬂ We ask when the payoff
from keeping is larger than the payoff from
contributing. Using the formulas above,
we want to know when 10(1—l—BK/N) > 10B(K—l—
l)/N, which reduces to B/N < l. (e) Sometimes the action that maximizes a player’s absolute payoff, does
not maximize his relative payoff. Consider the example of a voluntary
public goods game as described above, where B = 6 and N = 5. Suppose
that four of the ﬁve players in the group contribute their $10, while the
ﬁfth player keeps his $10. What is the payoff of each of the four contribu— tors? X = . What is the payoff of the player who keeps
his $10? + $60 X 4/ = $58 . Who has the highest payoff in _ the group? The player who keeps his $10 . What
would be the payoﬁ to the ﬁfth player if instead of keeping his $10, he contributes, so that all ﬁve players contribute. X 5 / 5 = .
If the other four players contribute, What should the ﬁfth player to max— imize his absolute payoff? contribute . What should he do to maximize his payoff relative to that of the other players? Keep his $10. If B 2 6 and N = 5, What is the dominant strategy equilib rium for this game? All contribute . Explain your answer.
Where B = 6 and N = 5, if K other
players contribute, a player’s payoff will
be $10t601f/5 if he keeps the $10 and
$60(K+ 1)/5 if he contributes it. For
every possible K, the difference between
these two payoffs is$10 — $12 2 —$2, so
the payoff from contributing the $10 is 348 GAMETHEORY (Ch. 28) always higher. 28.4 (1) The Stag Hunt game is based on a story told by Jean Jacques
Rousseau in his book Discourses on the Origin and Foundation of In
equality Among Men (1754). The story goes something like this: “Two
hunters set out to kill a stag. One has agreed to drive the stag through
the forest, and the other to post at a place where the stag must pass. If
both faithfully perform their assigned stag—hunting tasks, they will surely
kill the stag and each will get an equal share of this large animal, During
the course of the hunt, each hunter has an opportunity to abandon the
stag hunt and to pursue a hare. If a hunter pursues the hare instead of the
stag he is certain to catch the hare and the stag is certain to escape. Each
hunter would rather share half of a stag than have a hare to himself.” The matrix below shows payoffs in a stag hunt game. If both hunters
hunt stag, each gets a payoff of 4. If both hunt hare, each gets 3. If one
hunts stag and the other hunts hare, the stag hunter gets 0 and the hare
hunter gets 3. The Stag Hunt Game Hunter B
Hunt Stag Hunt Hare Hunt Stag
Hunt“ Hunt (at) If you are sure that the other hunter will hunt stag, what is the best thngﬁrﬂmtodd? Hunt stag. (b) If you are sure that the other hunter will hunt hare, what is the best thing for you to do? Hunt hare . (0) Does either hunter have a dominant strategy in this game? No . If
so, what is it? If not explain why not. The best action
to take depends on What the other player does. NAME _.._......—_.__.. 349 (d) This game has two pure strategy Nash equilibria. What are they?
Both hunters hunt stag. Both hunters hunt hare. (:3) Is one Nash equilibrium better for both hunters than the other? Yes Ifm,whmhstmebamremmmnmn? Both hunt
stag. ( f ) If a hunter believes that with probability 1/2 the other hunter will
hunt stag and with probability 1/2 he will hunt hare, what should this hunter do to maximize his expected payoff? Hunt hare . 28.5 (1) Evangeline and Gabriel met at a freshman mixer. They want
desperately to meet each other again, but they forgot to exchange names
or phone numbers when they met the ﬁrst time. There are two possible
strategies available for each of them. These are Go to the Big Party or
Stay Home and Study. They will surely meet if they both go to the party,
and they will surely not otherwise. The payoff to meeting is 1,000 for
each of them. The payoff to not meeting is zero for both of them. The
payoffs are described by the matrix below. Close Encounters of the Second Kind Gabriel
Go to Party Stay Home Evan eline Go to Party 1000,1000 m
g Stay Home “m (a) A strategy is said to be a weakly dominant strategy for a player if the
payoff from using this strategy is at least as high as the payoff from using
any other strategy. Is there any outcome in this game where both players are using weakly dominant strategies? The only one is (Go to party, Go to party). (0) Find all of the pure~strategy Nash equilibria for this game. There are two: (Go, Go) and (Stay, Stay). 350 GAME THEORY (Ch. 28) (c) Do any of the pure Nash equilibria that you found seem more rea— sonable than others? Why or why not? Although (Stay,
Stay) is a Nash equilibrium, it seems a silly one. If either player believes that there is any chance that the other will go to the party, he or she will also go. (0!) Let us change the game a little bit. Evangeline and Gabriel are still
desperate to ﬁnd each other. But now there are two parties that they
can go to. There is a little party at which they would be sure to meet
if they both went there and a huge party at which they might never see
each other. The expected payoff to each of them is 1,000 if they both go
to the little party. Since there is only a 5050 chance that they would ﬁnd
each other at the huge party, the expected payoff to each of them is only
500. If they go to diﬂ’erent parties, the payoff to both of them is zero.
The payoff matrix for this game is: More Close Encounters Gabriel
Little Party Big Party Evan eline Little Party 1000:1000 m
g BigParty “ 500,500 (6) Does this game have a dominant strategy equilibrium? NO . What
are the two Nash equilibria in pure strategies? (1) Both go to the little party. (2) Both go to the big
party. { f ) One of the Nash equilibria is Pareto superior to the other. Suppose
that each person thought that there was some slight chance that the other
would go to the little party. Would that be enough to convince them both to attend the little party? NO . Can you think of any reason why the NAME —__— 351 Pareto superior equilibrium might emerge if both players understand the
game matrix, if both know that the other understands it, and each knows that the other knows that he or she understands the game matrix?
If both know the game matrix and each
knows that the other knows it, then each
may predict the other will choose the
little party. 28.6 (1) The introduction to this chapter of Workouts, recounted the
sad tale of roommates Victoria and Albert and their dirty room. The
payoff matrix for their relationship was given as follows. Domestic Life with Victoria and Albert Victoria
Clean Don’t Clean Alb Clean
6” Don’t Clean Suppose that we add a second stage to this game in which Victoria
and Albert each have a chance to punish the other. Imagine that at the
end of the day, Victoria and Albert are each able to see whether the
other has done any housecleaning. After seeing what the other has done,
each has the option of starting a quarrel. A quarrel hurts both of them,
regardless of who started it. Thus we will assume that if either or both of
them starts a quarrel, the day’s payoﬂ’ for each of them is reduced by 2.
(For example if Victoria cleans and Albert doesn’t clean and if Victoria,
on seeing this result, starts a quarrel, Albert’s payoff will be (i — 2 : 4
and Victoria’s will be 2 7 2 = 0.) (a) Suppose that it is evening and Victoria sees that Albert has chosen not
to clean and she thinks that he will not start a quarrel. Which strategy will give her a higher payoff for the whole day, Quarrel or Not Quarrel? Not Quarrel. 352 GAME THEORY (Ch. 28) ( b ) Suppose that Victoria and Albert each believe that the other will try
to take the actions that will maximize his or her total payoff for the day. Does either believe the other will start a quarrel? N 0 Assuming
that each is trying to maximize his or her own payoff, given the actions
of the other, what would you expect each of them to do in the ﬁrst stage of the game, clean or not Clean? Not Clean . (c) Suppose that Victoria and Albert are governed by emotions that they
cannot control. Neither can avoid getting angry if the other does not
clean. And if either one is angry, they will quarrel so that the payoﬁ” of
each is diminished by 2. Given that there is certain to be a quarrel if
either does not clean, the payoff matrix for the game between Victoria
and Albert becomes: Vengeful Victoria and Angry Albert Victoria
Clean Don’t Clean Alb Clean
6” Don’t Clean (d) If the other player cleans, is it better to clean or not clean? Clean
If the other player does not clean, is it better to clean or not clean. Not
Clean.Emnmn If one player does not clean,
the other will be angry so there will be a
quarrel whether or not the first player
cleans. Given that there will be a quarrel anyway, the second player is better off not Cleaning than cleaning. (6) Does this game have a dominant strategy? N0 . Explain The
best response depends on what the other does. NAME ——_..._ 353 (f) This game has two Nash equilibria. What are they? Both
Clean the room and both don’t Clean the room. (9) Explain how it could happen that Albert and Victoria would both be
better off if both are easy to anger than if they are rational about when
to get angry, but it might also happen that they would both be worse oﬁ If they both are easy to anger, there
are two equilibria, one that is better
for both of them and one that is worse
for both.of them than the equilibrium where neither gets angry and neither cleans. (h) Suppose that Albert and Victoria are both aware that Albert will get
angry and start a quarrel if Victoria does not clean, but that Victoria is
levelheaded and will not start a quarrel. What would be the equilibrium oMamm? Albert does not clean and Victoria does clean. Nobody starts a quarrel. 28.7 (1) Maynard’s Cross is a trendy bistro that specializes in carpaccio
and other uncooked substances. Most people who come to Maynard’s
come to see and be seen by other people of the kind who come to May—
nard’s. There is, however, a hard core of 10 customers per evening who
come for the carpaccio and don’t care how many other people come. The
number of additional customers who appear at Maynard’s depends on
how many people they expect to see. In particular, if people expect that
the number of customers at Maynard’s in an evening will be X, then
the number of people who actually come to Maynard’s is Y = 10 + .8X.
In equilibrium, it must be true that the number of people who actually
attend the restaurant is equal to the number who are expected to attend. (a) What two simultaneous equations must you solve to ﬁnd the equilib— rium attendance at Maynard’s? Y 2 10+ .8X and X = Y . (b) What is the equilibrium nightly attendance? 50 . 354 CAMETHEORY (Ch. 28)
(c) On the following axes, draw the lines that represent each of the two equations you mentioned in Part (a). Label the equilibrium atten— dance level.
V 80 60 40 20 (d) Suppose that one additional carpaccio enthusiast moves to the area.
Like the other 10, he eats at Maynard’s every night no matter how many
others eat there. Write down the new equations determining attendance at Maynard’s and solve for the new equilibrium number of customers.
y=11+.8:13 and yzzr, so 13:31:55. (e) Use a different color ink to draw a new line representing the equa—
tion that changed. How many additional customers did the new steady customer attract (besides himself)? 4: . ( f ) Suppose that everyone bases expectations about tonight‘s attendance
on last night’s attendance and that last night’s attendance is public know1~
edge. Then Xi = Yt_1, where X: is expected attendance on day t and
Yt_1 is actual attendance on day t — 1. At any time t, Yt = 10 + .SXt.
Suppose that on the ﬁrst night that Maynard’s is open, attendance is 20. What will be attendance on the second night? 26 .
(9) What will be the attendance on the third night? 30 . 8 . (h) Attendance will tend toward some limiting value. What is it? 50 . 28.8 (0) Yogi’s Bar and Grill is frequented by unsociable types who hate
crowds. If Yogi’s regular customers expect that the crowd at Yogi’s will
be X, then the number of people who show up at Yogi’s, Y, will be the
larger of the two numbers, 120 — 2X and 0. Thus Y = max{120 — 2X, 0}. NAME — 355 {a} Solve for the equilibrium attendance at Yogi’s. Draw a diagram de—
picting this equilibrium on the axes below. Y
80 60 40 20 (b) Suppose that people expect the number of customers on any given
night to be the same as the number on the previous night. Suppose that
50 customers show up at Yogi’s on the ﬁrst day of business. How many stMwupmimewmmddw? 20.7HmﬂMddw? 80. Tm
fourth day? 0. The ﬁfth day? 120. The sixth day?
0 . The ninetyninth day? 120 . The hundredth day? 0 . (c) What would you say is wrong with this model if at least some of ibgrs cushnners have nunnory spans of nuns than a day or hho?
They’d notice that last night’s attendance
is not a good predictor of tonight’s. If
attendance is low on odd—numbered days and
high on evennumbered days, it would be
smart to adjust by coming on odd—numbered days. ...
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