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Unformatted text preview: 756 Chapter 18: Mlultﬂateml Approaches to Trade Policy l (6) Why might environmentalists and trade pr'otectionists form political
alliances in developed countries? I (7) Consider the following parable There are five individuals They can
form into groups of different sizes. A group of size 1 gets a total income of
$100; of size 2, $250; of size 3, $600; of size 4, $750; and of size 5, $900. Appendix 1
X
Elementary Game Theory (a) Which collection of groupings maximizes ”world income” in this world
of ﬁve people? (b) if all group income must be divided equally among group members,
which grouping would you expect to arise under spontaneous group forma
tion? All. Introduction The theory of games is a useful way to capture interactions across individ
uals when each mdrvrdual’s actions have effects on the utilities or monetary returns of other agents The purpose of this appendix is to provide you with
an introduction to game theory as we use it in this book. The emphasis is
on the concepts that are useful in a study of develo . pment. In particular, this
chapter no way substitutes for a comprehensive introduction to game theory for which you will have to consult specialized texts.1 (c) Explain why the answers to (a) and (b) are different. Can you tell an al—
ternative story that attains world income maximization if people are allowed
to propose unequal divisions of group income? What additional assumptions
on group formation would be required for this alternative story to work? (d) Can you use this parable to throw light on regional groupings in inter
national trade? A12. Basic concepts The simplest way to describe a strategic situation or a game is also possibl — the most abstract, so a little patience is needed to get through this part if you want to properly understand what follows. Suppose that there is a set I of mdrvrduals, sometimes called agents or players. Let’s give them names by referring to each player by his/ her index in the set I . lhus player i will refer i tpya playgrh w1:1h the index 1' Subsequently, we shall often subscript various o JEC 5 W1 in ices such as l, 2, i, or ‘, to remind o ”
to” players 1, 2’ I" j! _ ] urselves that they belong
For each player i there is a set of stmte ' ' I gzes A typical strate from this set may be denoted 3,. A strategy proﬁle is a list of strategies (5133/82 5 ) I I " 'I n I one £01 each PlaYer. To Shorten our notation let us 
. 1 refer to
proﬁle by the boldface letter s. a typical strategy d Given any strategy profile s, each player 1' receives a payoff n1, which may
epend on the entire vector of strategies within that profile To emphasize
this dependency, we sometimes refer to it, as Iii(S), to stress the fact that u
is a function of s, for every player i. I
Believe it or not, We are done with our description of a game! Of course what we have so far looks deceptively simple. There are many varieties of
games that fit into this framework, and soon we will see some of them 1 See Binmor'e {1992], Gibbons 1992 , d ‘  .
tions to the subject { 1 an 05130111.? and Rubmsmm [1994] 1:01 excellent mtroduc‘ 758 Appendix 1: Elementary Game Iheory Before that, let us take some time to understand the various concepts that
we have introduced so far. First, a strategy can mean many things, and it may or may not be de—
scribable as a single number A strategy may stand for a simple action or
a list of actions dependent on various contingencies Second, a payoff may
mean several things It could mean the satisfaction or utility of an individ—
ual, it may stand for the monetary reward she receives; it could stand for the expected value of monetary reward or utility in an uncertain situation Here
are some examples to illustrate these matters. Fertility decisions Two couples live in a joint family, and each is deciding how many children to
have. There is a cost of bringing up children if they are your own. Moreover,
because the family is joint, the children of other couples also impose a cost (in
terms of occasional child care) upon you Suppose that the cost of bringing
up your own child is e per child, and the cost imposed by the children of
the other couple is 11 per child. (Think of this as an unavoidable cost of joint
family life.) There are also benefits to having children, but to make things
simple assume that each couple derives benefits only from its own children Assume that the total beneﬁt of having n children of your own is A(n) Let’s
suppose that no couple can have more than N children (1) Here the ”individuals” or ”players” are couples. Describe the strategy
set for each couple and the payoff from each strategy profile. (2) Describe the special case that would arise if each couple was living completely isolated from the others. How would you describe the new payoff
functions? Adopting a new technology In a country there are two computer systems available, A and B Each person
may adopt one of systems A or B. The cost of adoption is given by c for either
system. However, the payoff from adoption of a system depends positively
on the number of people who adopt that system as well. If n is this number,
then the payoff is some function f(n). (3) Describe this situation as a game In the examples discussed so far, a strategy is a simple action, such as
the decision to have x children or adopt a particular technology. In many
cases, a strategy is a description of an action under various contingencies.
Here are two examples to make this point Al 3 Nash Equilibrium 759 Protecting an industry A monopolistic industry is protected by a tariff. It must decide whether or
not to cut costs and become internationally competitive. After this decision
is made, the government observes whether the industry has cut costs or not,
and then decides whether or not to remove the tariff that protects the indus—
try. Following these actions, certain payoffs are received by both. government
and industry. Later we will describe these payoffs in more detail. For now, I
would like to describe the strategy sets. There are two ”players”: the industry and the government The strategy
set of the industry is simple enough: it consists of two options, "cut costs” or
”do not cut costs .” At ﬁrst glance, a similar description appears to be the case
for the government as well: ”keep the tariff” or ”remove the tariff.” However,
this is not true, and the reason it is not true is because the government gets
to observe the move of the industry before making its decision. Thus in the
description of the government’s strategy we must allow for the possibility
that the government’s choice can depend on what it observed. It follows that
the government really has four strategies, not two] These are (1) remove the
tariff regardless of what the industry does, (ii) keep the tariff regardless of
what the industry does, (iii) keep the tariff if the industry cuts costs and
remove it otherwise, and (iv) remove the tariff if the industry cuts costs and
keep it otherwise. As hinted at before, a strategy (for the government) here
is not a simple action but a series of ”conditional rules,” depending on what it obser ves Landlord—tenant relationship A landlord leases out land to a tenant. Ihe tenant chooses labor L and pro—
duces an output Y according to a production function Y = PU.) Labor costs
him to per unit. The landlord chooses a contract for the tenant, which is a
scheme dividing up the produced output. The scheme consists of srmply an
output share a and a fixed payment 1". The interpretation is that the tenant
keeps a share a of the output and makes a fixed payment l: to the landlord
for the right to farm the land Let c = (a, 1:) be the shorthand notation for
the contract thus offered After the contract is offered, the tenant chooses the amount of labor to devote to the land. (4) Describe the strategies available to landlord and tenant, and the pay
offs that arise from them. A13. Nash equilibrium The fundamental concept of an "equilibrium” in a game comes from the
work of John Nash. The concept, known as a Nash equilibrium, can be 760 Appendix 1: Elementary Game Theory Al 3 Nash Equilibrium 761 Nash equilibrium of this game This is curious, to say the least, but we can observe various reallife situations that correspond, roughly speaking, to the
Prisoners’ Dilemma. ‘ described as follows. Imagine a strategy profile that‘has the property that
no individual can do better by choosing an alternative strategy, assuming
that all other players are choosing the strategy described in the strategyuptrp
file. Understand well that this property must Simultaneously apply to all d e
players at the some strategy profile. Then such a strategy profile is ca e a Nash e ur'librr‘um. . ' . ‘
lt'sq best to illustrate the concept of a Nash equilibrium through the use of examples i ing to ﬂy to convince you that this has the essential feature of the Prisoners’
Dilemma; namely, that the two couples might depart from an outcome that
maximizes their joint utility See also Chapter 9 where this and other related
problems are discussed.
It will be useful to work with a specific numerical example Suppose that
N = 2, so that each couple can have no more than two children Now sup—
pose that cost of bringing up each of your own children is $200, whereas
the cost imposed by the other couple’s children is $100 per child Spec
ify the beneﬁt function as follows: the first child is valued at $350, whereas
the second child is valued at $250. Thus AU) = 350, whereas A(2) = 600.
Assume that these data apply equally well to the other couple. Now we may
represent the net payoff (benefit minus cost) to each couple in the following
matrix, where the options ”one” and ”two” should be selfexplanator y: Ellilit? again, Let us look again at the foregoing fertility example. I am 80 The Prisoner‘s’ Dilemma This is our first instance of a game in which we will be able to frt mgny ofhoulr1
development examples. Two prisoners have committed a crrrne, otri “71123
they are being interrogated by the police. Each prisoner has two opdprn; or
can cooperate with his fellow prisoner and refuse to divulge ariyapprThu the
he can defect into the waiting clutches of the police. and revea .. d f8 t
strategy set of each prisoner consists of two strategies: cooperatcala or e ecb.e
Let us suppose that if both prisoners cooperate, then no ego gnce 1:131 is
brought against them, so that they go scot free to enjoy thjgrtr3 tiodyflt then
worth, say, ten units to each of them. On the other hand, (101 t e 53: the
they are put away for a while, after which they are parole I; Final13:5 to
payoff from this situation is worth five units to each of them. rei In this
describe the situation in which one cooperates, but the other squea s ‘ ‘ 1
case me latter turns state’s evidence and in return‘rs pardoned wrth milprma
punishment He also gets to enjoy the booty while the other langurs .es 1.1:
jail, and this is worth, say, ﬁfteen units to the squealer The cooperator is puC1
away as an unrepentant character who never confessed tolhi:1 crrgjps,f:pm
this gives him, say, zero units. The payoffs can he summarize m. e
of the following matrix, which is a common devrce in game theory. One Two Make sure you understand these entries (as you should if you solved
the previous exercises). For instance, the strategy pair (one, two) means that
the first couple has one child while the second couple has two The first
couple thus gets a benefit of 3.50, but then incurs a cost of 200 for its own
child, and a total cost of 200 for the two children of the other couple. The
net return is thus ~50. The couple with two children receives benefits of 600,
and although they incur costs of 400 for their two children and an indirect
cost of 100 from the other couple’s child, they receive a net payoff of 100. The relationship with the Prisoners’ Dilemma should now be very clear
indeed. Having one child each is the cooperative policy However, if each
couple fails to internalize the cost that it imposes on the other couple as a consequence of its fertility choice, then both couples may end up having two
children each, with a payoff of zero Cooper ate Defect It is very easy to see that there is only one Nash equilibrium in this ggm
that is, only one strategy combination that has the property that 8:1ij pis 3&1
is doing the best he can, given the actions of the other player. 8 coﬁl
combination in which both prisoners defect Look at any other stral’Eiegy t t
bination If it involves both players cooperating, one player wou war; _
turn witness for the state and defect. On the other hand, 1f1t1s an asyIpItn 7:7
combination, then the prisoner left out in the cold would also r1:j.¢;11n.u :1
fect (getting five is better than getting zero). 80 (defect, defect) is e Cooper ate
Defect The Commons. Suppose that groundwater is used for irrigation in a village
Overuse of groundwater can reduce the level of the water table, making it
more costly for all farmers to extract water. This is the typical problem of
the commons: groundwater is a common property resource, and the costs of
using it may not be fully internalized. A simple example can be provided to
make the point Suppose that water can be extracted at two levels—high and 762 Appendix 1: Elementary Game Theory low—and that there are two farmers. The revenue (from crop production) for
each farmer increases with the use of groundwater: say it is $2,000 if the high
level is applied and $1,000 if the low level is applied. The cost of extraction
depends on whether the other farmer’s use is high or low. Suppose that the
extraction cost for each farmer is $500 for low and $1,300 for high, but that an
additional fixed cost of $500 is incurred if the other farmer is extracting high
(deeper wells will have to be dug because the other farmer’s action reduces
groundwater levels). A table very similar to the fertility matrix represents
this situation: High L ow Again, make sure you understand the various entries. Now verify that
the situation indeed cor responds to a Prisoners’ Dilemma and that both farm—
ers will extract high, even though both extracting low would have 1oeen a
preferred outcome. These are only two of many, many examples that one could be put for
ward Environmental issues such as crossborder pollution or deforestation
can be addressed in this way. For instance, countries might impose taxes on
polluting firms insofar as their own citizens are affected by. pollution, but
they might fail to internalize the effects of pollution on the Citizens of other cormtries. Ihis may create overpollution. _ _
In the same vein, but in an entirely different context, several Industrial groups might lobby for protective tariffs so that they can have access to
domestic markets Of course, each group includes shareholders who are con—r
sumers of other products and would not like to see tariffs slapped on these Low High other goods. Thus they do not internalize the effects of their lobbying ac— . ' tions on other interest groups A Prisoners’ Dilemmalike situation may well
result Production cooperatives represent yet another example A group of farm _ Al 3. Nash Equilibrium 763 western India that once went by a more familiar name) and they have man
aged to lose each other. Ihey have made no plans to handle this kind of
emergency, but each suspects that the other will go to a familiar ”tourist” lo—
cation in the hope that her friend will think the same way. Indeed, the friend
does think the same way, but the problem is that there are two possible 10
cations (well, there are many, but let’s just say there are two): Chowpatti
Beach and the Apollo Bunder If they go to different places they will under—
standably be very unhappy, but if they go to the same place all will be well
Indeed, Chowpatti is a better option because they can eat excellent bhelpuri
there together We may represent all this by saying that each friend gets a
payoff of 0 if they go to different places, a payoff of i if they meet at the
Apollo Bunder', and a payoff of 2 if they meet at Chowpatti (and eat good
bhelpuri in the bargain) Here is the relevant matrix: Chowp atti Apollo Now it is easy enough to see that there are two Nash equilibria of this game‘2 1“ one equiﬁbrimn} both end up in Chowpatti. In the other, both — end up at the Apollo Bunder It all depends on what each expects the other
to (10.3 Ihe Tourists’ Dilemma creates what we might call a problem of coordi—
nation. It would be best for both to meet at Chowpatti, but if communication
is somewhat limited, it is hard to see how this outcome might be guaran—
teed. Ihe expectations that each holds regarding the other’s course of action
will crucially determine the ﬁnal decision that each tourist takes. like the Prisoners’ Dilemma, examples of coordination problems recur
through this textbook. ers working in a team have an incentive to undersupply effort, because'they .
do not internalize the full marginal product of additional effort (addltlonal :
output is divided among all farmers). The Nash equilibrium of tlus game is
typically worse for all relative to the fully cooperative outcome. _ ' Congestion on unregulated highways, the overuse of publicly provrded:
(or insured) health facilities, littering, low voter turnout—the lrst of potentra
Prisoner3’ Dilemmas is extensive and varied.  QWERTY As we have seen in Chapter 5, QWERTY refers to the keyboard system that
we are all familiar with. It is known to be an inefficient system, which orig
inally was designed to slow down the speed of typing on mechanical type
writers. Ihe problem, however, is that the beneﬁts of using QWERTY are
not defined in isolation, they depend on how many other people are using 2 More precisely, there are two Nash equilibria in pure strategies: we will not consider random;
ized strategies here 3There is also the problem of miscoordinstion, in which beliefs about each other’s actions are not commonly held by both players. These are important issues, but somewhat less relevant for the
reallife examples that we consider. Coordination game Here is a game with a somewhat different flavor We can call it the Touris '
Dilemma. Two friends are touring the Indian city of Mumbai (a large crty in 764 Appendix 11‘ Elementary Game Theory the same system. Ihus even if a better system is available, people may not
switch to it if they feel that other individuals are using the old system Ihe
same argument can be applied to a variety of technologies and systems: op
erating software for computers, television networks, the side of the road we
drive on (though enforced by law, a coordinated system would surely have
arisen spontaneously), and notions of what’s fashionable. lhe easiest way to represent any of these problems as a Tourists’
Dilemma is to choose a two«player version For instance, if two people
who share computing resources can use Macintoshes or PCs, then these
choices can be identified with Chowpatti or Apollo Bunder in the preceding
example. (5) Construct a table with the same features as the coordination game,
because the choice of the same type of computer permits greater sharing of
resources. One coordinated choice may yield a higher payoff than the other:
depending on your preferences, you may want to give this honor to the
strategy pair (PC, PC) or the strategy pair (Mac, Mac). Social norms. Coordination games arise somewhat less transparently in so—
cial situations Consider the social custom of having lavish weddings for
one’s sons and daughters, which (in my opinion) is reprehensible in any so;
ciety but particularly in developing countries. The reason is not that people
do not have the right to spend their money in any way they like—~they Clo——
but that such expenditures ultimately constitute a social norm to which the
less wealthy feel constrained to adhere Ihus the ”shame” attached to not
having a lavish wedding may be higher when the going tradition is to have
them. Such tr...
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 GustavoJ.Bobonis

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