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Ray_App1_ElemGameTheory - 756 Chapter 18 Mlultflateml...

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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 real-life 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 self-explanator y: Elli-lit? 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 “7-1123 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; Final-13:5 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 cross-border 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’ Dilemma-like 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 Prisoner-3’ 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 real-life 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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