Fundamentals of Game Theory, Economics of Information, and Market Failure

Transcription

1 TOPIC XI: Fundamentals of Game Theory, Economics of Information, and Market Failure In this topic we begin to explore some of the core definitions and tools of what is commonly referred to as game theory. In summary, game theory is a method for thinking about strategies available to decision makers who may have competing goals. We will also turn out attention to several decision situations that highlight issues of economics of information. Broadly speaking, this area focuses on how incentives and outcomes can depend critically upon the information available to decision makers. Of course this is only an introduction into these two areas. We will only scratch the surface. However, you will begin to learn the terminology, to think like a game theorist and to think like an economist things about how information changes the strategic nature of decision making settings in real life. You will also have the opportunity to play some real games and learn how real players play these games in an experimental laboratory setting for cash payoffs. You can think of these areas of economics as attempts to model human behavior. Economists and other social scientists attempt to develop theories or models that help explain why individuals and groups make the decisions they make. They try to develop explanations for how decision makers will respond to changes in incentives. Something to think about when you are studying this section. Many real life decisions have a monetary benefit or cost associated with them. However, almost all decisions have impact on our lives that go beyond the pure monetary payoffs. As we focus on the issues in this class, you will begin to see how the monetary and non-monetary incentives don t always support or complement each other. This is a fact of life. However, it does create further complexities in modeling human behavior. XI 1

2 GAME THEORY The economic games we refer to in this chapter have four primary components: 1) the decision makers, 2) the strategies available to the decision makers, 3) the information of each decision maker, and 4) the possible outcomes of the game, which are dependent upon the strategies chosen. To get started, consider the following sequential game, played by two players - the first mover (FM) and the second mover (SM). Each player begins with an endowment of 10 tokens worth $1 dollar each. The first mover (FM) decides whether or not to send any of his/her tokens to the second mover (SM). Each token that the FM sends reduces the value of his/her token fund by $1, but increases the value of the token fund of the SM by $3. After the FM makes his/her decision, the SM makes his/her decision. The SM must decide how to divide the value of the token fund he/she holds between his/her self and the FM. That is, the SM decides how much of the fund to keep for his/her self and how much to send back to the FM. The game is played only once. Both players have full information about the endowments and the possible payoffs in the game. How do you think players would play this game? Do you think it would matter if they knew each other and would potentially meet again? Do you think it would matter if the game was repeated several times? To earn the most money in this game would require the FM to send his/her entire endowment to the SM, creating the largest possible surplus to be divided between the two ($40). Do you believe the FM would make this choice? Why or why not? XI 2

3 Some core definitions and terms in Game Theory Goals of Players - generally speaking, game theory assumes that individuals (players from here on) choose actions to achieve a maximum value or outcome in the game, based on the actions they expect other players to choose. Strategies - simply the possible choices or actions available to the players in the game. In the game above, the FM has 11 possible strategies - send 0, 1, 2,...10 tokens to the SM. The number of strategies available to the SM depends upon the strategy chosen by the FM. Outcomes - the payoffs to the players that result from the combination of strategies chosen by each player. In the games we discuss, we will talk about outcomes measured in dollars. In game theory, the outcomes are often measured in utilities or levels of satisfaction to each player from each combination of strategies. Hypothetically, we can describe a game in utilities. However, would we be able to actually measure utilities? Incentives at the margin - This is a term that is used widely by economists as a way of thinking about decision making. The focus is on the additional benefit or cost from taking an action. Marginal benefit would be the extra benefit from taking an action (incrementally). Marginal cost would be the extra cost of taking an action (incrementally). Best Response - the strategy that maximizes payoffs for a given players, contingent of the strategies chosen by the other players. Zero-sum games - games in which the total payoffs to players in the game are constant. That is, they do not vary if players use alternative strategies. Think of splitting a pie. The more one player gets, the less available to other players. Variable-sum games - games in which the total payoffs to the players depends on the strategies chosen by the players. These are the more interesting games. The total payoffs to the players in the game will vary according to the combination of choices made by the players. Nash Equilibrium - a combination of strategies chosen by each player, where no player would have an incentive to change his choice. Each player has made his best choice, given the choices of the other players. Social or Group Optimum - the outcome that maximizes the sum of the payoffs to all of the players in the group. Efficiency - earnings at a particular outcome as a percentage of the earnings at the social optimum. XI 3

4 Dominant Strategy - a particular type of Nash equilibrium, where a players best strategy does not depend on the choice of the other players. Pure-strategy equilibrium - an equilibrium where the strategies are played with a definite plan. Mixed strategy equilibrium - an equilibrium where different strategies are played according to different probabilities. One-shot game - a game that is played only once (and the players know ahead of time that it will be played only once). Finitely repeated game - a game is played more than once and the players know ahead of time how many times it will be played. Infinitely repeated game - a game is played more than once and will be repeated indefinitely with some know probability, and the players know ahead of time the probability of repeating. Backward induction - viewing the game from the perspective of looking backwards from the end of the game to the first round. More specifically, consider play in the final round (T), then play in round T-1, based on expected play in round T, then play in round T-2, based on expected play in round T-1,...etc. until... play in round 1, based on play expected play in round 2. Simultaneous play games - players make their choices simultaneously. Sequential play games - at least one player moves after the other players have moved. Presentation of the game The game described at the beginning of this chapter, was simply described using words and the rules of grammar. This was made possible, partially because the game was relatively simple. There is no right way to describe a game. Other than, the presentation should make clear the possible actions, information, and outcomes in the game. There are two standard ways, however, that are often used to describe games: normal form and extensive form. normal form games - presentation of the game using a payoff table (often called a payoff matrix). The cells of the table display strategy options and the outcomes of combinations of strategies. XI 4

5 Here is an Example of a 2-person, 2 strategy, normal form game Player 2 Player 1 strategy 1 strategy 2 strategy 1 P1,1; P2,1 P1,1; P2,2 strategy 2 P1,2; P2,1 P1,2; P2,2 Read the game in the following way. Player 1 and Player 2 both have two strategies (call them strategy 1 and strategy 2). The payoffs of the combinations of strategies are show in italics. For example, P1,1; P2,1 would represent the payoffs to player 1 when he/she chooses strategy 1 and to player 2 when he/she chooses strategy 1. P1,2; P2,1 would represent the payoffs to player 1 when he/she chooses strategy 2 and to player 2 when he/she chooses strategy 2. P1,1; P2,2 would represent the payoffs to player 1 when he/she chooses strategy 1 and to player 2 when he/she chooses strategy 2. P1,2; P2,2 would represent the payoffs to player 1 when he/she chooses strategy 2 and to player 2 when he/she chooses strategy 2. Now consider an example of the game above, with real numbers. It s easier!! The strategies or C & D. Player 2 Player 1 C C $15; $15 $7.5; $17.50 D D $17.50; $7.50 $10; $10 You would read the payoffs of this game as above. If both players choose C, they both earn $15. If both players choose D, they both earn $10. If Player 1 chooses C and Player 2 chooses D, Player 1 receives $7.50, and Player 2 receives $ If Player 1 chooses D and Player 2 chooses C, Player 1 receives $17.50, and Player 2 receives $7.50. XI 5

6 Suppose this game was played as a one-shot game, and the players chose their strategies simultaneously, what would be the Nash equilibrium? If both players chose strategy C, they would both earn $15. Together they would earn $30. This total is greater than the total of any of the other payoff cells. Why isn t the combination where both players choose C a Nash equilibrium? Note: A common way to represent choices by two players is in pairs. For example (C,C) would mean that player 1 chose strategy C and player 2 chose strategy C. Similarly, (D,C) would mean that player 1 chose strategy D and player 2 chose strategy C, and so on... See if you can determine why (D,D) is the only Nash equilibrium of the game and it is a dominant strategy. Note: This game is a good illustration of how individuals pursuing their own self-interest can lead to a result that is not optimal for the group. In the social science literature, this situation is known as a social dilemma. Now we consider the same game shown in extensive form. Extensive form games - presentation of the game using a tree diagram. In this form, the parts of the tree are made up of nodes for each possible action, and nodes are connected by branches. The final notes show the outcomes associated with particular combinations of strategies. Consider the game tree shown below, which is the extensive form of the game presented above. NOTE: The Dash line in the game implies Player 2 does NOT know the choice of Player 1 prior to his/her own choice. Player 2 plays the game above without knowing the choice of Player 1. XI 6

7 Notice that we can make the game above a sequential game, simply by removing the dashed line. Does this change the game s Nash equilibrium? Do you think it would change behavior? What do you think of presenting a game in words verus normal form versus extensive form? Many would argue that it is simpler to present simultaneous games in normal form and sequential games in extensive form. Would you agree? Generally speaking, choosing the form of the presentation is somewhat subjective. Some games are simpler to explain in one form and others in another form. It can also be simply a question of one s own preferences about the style one like s best. The games we discussed above are relatively simple, partially because they have a single Nash equilibrium and that equilibrium is a dominant strategy. As games become more complex, the number of equilibrium may increase and these equilibrium may not be dominant strategies. As we play various games over the course of the next few weeks, we will discuss how strategies and equilibrium change with changes in the structure and incentives of the game. ECONOMICS OF INFORMATION Generally speaking, strategic situations vary with the rules or structure of the game, the incentives of the various players, and information players have about themselves and other players in the game. The economics of information is an area of focus in economics that examines how situations and behavior can be effected by varying the information available to decision makers. In this section, we explore a few of the commonly discussed issue in this area. First, some terminology. Concepts as they relate to the degree of information Perfect Information - a situation in which a player knows all relevant information when a decision is made. Think about the games described above, both players have perfect information. Imperfect Information - a situation in which a player does not know all relevant information when a decision is made. Think about a round of 5-card draw poker, players have imperfect information about the cards drawn by others. Concepts as they relate to differing information across players Symmetric Information - a situation in which all players have the same information. The games described above have both perfect information and symmetric information. Asymmetric Information - a situation in which players differ in the information available to them. Each player in 5-card draw poker has different information. XI 7

8 Principal-Agent Problems - An important class of economic decision problems are what is referred to as principal-agent problems. Think of two individuals, where one contracts with the other to fulfill a particular action. Examples include, an entertainment or sports agent, your financial advisor, etc. In these cases, the principal would like for the agent to do his/her best to make the principal as well off as possible. However, in most cases, this will require and investment of time or money on the part of the agent. In other words, what is best for the agent is not necessarily best for the principal. How do principals and agents try to solve these problems? They create contracts that are an attempt to better realign incentives. But, this can be very complicated. A major complication is that in most situations there is imperfect and asymmetric information on the part of the principal about the actions of the agent. The principal may not be able to observe if the agent is doing his/her best to fulfill the contract. Bargaining Problems - In many situations in life, decisions come down to a bargaining problem between two individuals or two groups of individuals. In many cases, these are said to be zero-sum games. Think of the bargaining as determining who gets the larger share of the pie. What if you are trying to buy a new automobile from a local dealership. We can think of the pie as the difference between your maximum willingness to pay for the automobile and the seller s minimum willingness to accept for he automobile. In most cases, these bargaining problems are games of imperfect and asymmetric information. In some cases, a trade/agreement does not occur, even when the maximum willingness to pay is greater than the minimum willingness to sell. This means there is a loss in possible gains or surplus to the buyer and the seller. How has the internet changed the bargaining problem in the buying and selling of a car? Moral-Hazard Problems - Generally speaking, a moral hazard problem occurs when a contract or agreement leads an individual to take risks that they would not have taken if the contract or agreement had not been implemented. Examples include the way in which insurance may cause the insured to take greater risks. This idea can also be applied to some laws. Is it possible that requiring individuals to wear seat belts or wear helmets could actually lead them to be more careless while driving or riding a vehicle? Moral hazard problems can also be viewed as situations in which, after the contract or agreement, one party to the contract has an incentive to change his/her behavior in a way that is beneficial to him/her and costly to the other party in the contract. Think of fraudulent behavior in an insurance market. The Lemons-Problem - In many situations, buyers are shopping for goods for which they do not have perfect information about the quality of the good. Often there is asymmetric information between the buyer and the seller. Think of used cars. Almost always, the seller will have better information about the used car compared to the potential buyer. How would this asymmetric information affect the market for used cars? If sellers are not able to make buyers fully aware of the quality of their car, the sellers with high quality used cars will not be able to signal the superb quality to potential buyers. This means that in a relative sense, high quality cars would sell at a lower price than with perfect information and low quality cars would sell at a higher price than with perfect information. But, how would sellers respond to this incentive? The argument goes that many sellers with good cars would decide not to sell their high quality cars (they aren t able to get enough for them). But this means that low quality cars are driving high quality cars out of the market, due to the asymmetric information. Further, this means that the XI 8

9 overall level of used cars would decline on average? This is an example of what is called adverse selection. How do buyers and sellers try to solve this problem? Car-fax? Extended Warranties? Complete records of service? Profiling Based on Statistical Grouping - Because of imperfect information, there are many markets in which sellers must choose a price based on statistical properties of a group of potential clients, instead of individual information about each client. A very good example of this is automobile insurance. Young drivers pay significantly more than older drives. Young men pay significantly more than young women. Why is this the case. Are you a worse driver than your sister? Or you a worse driver than your 91 year old neighbor? A DISCUSSION OF MARKET FAILURE - MARKET STRUCTURE AND THE CHARACTERISTICS OF GOODS For markets to allocate resources to their highest valued uses, we need a situation (an economy) where the following exist. 1. Firms can move resources from one market to another and there exists competition between firms (the issue of COMPETITION versus MONOPOLY POWER). 2. Firms need to "know" that if they invest in activities that move resources to alternative uses, they will receive the "profits" of doing so. This is an important issue of stable PROPERTY RIGHTS which involves a very important role of government. 3. When consumers or producers make decisions about what to buy (demand) or the cost of supplying (supply), their decisions take into account all relevant (from society s point of view) benefits to consumption or costs of production. This relates to issues of EXTERNALITIES, PUBLIC GOODS, AND COMMON POOL RESOURCES -- leading to a discussion of two other roles of government: (1) the provision of "public goods" and the "regulation" of some consumption and/or production activities. 4. Finally there is the issue of WELFARE. Markets create gains from trade - but they are not necessarily "sensitive" to the allocation of those gains across individuals. We can end up with societies with significant differences in the welfare (incomes, opportunities) of different individuals. This leads to another argument for governments, the REDISTRIBUTION of wealth. XI 9

10 THE MARKET AS AN ALLOCATOR In the following summaries, we reflect further on some of the issues raised above. We begin with the case of perfect competition where we focus on situations where: (a) demand represents the true value of the good to society and (b) supply represents the true cost to society of producing the good. As we will see, this means that there are no externalites in consuming the good and no externalities in producing the good. Following this analysis we then: a) move from perfect competition to the case of monopoly. b) consider what happens if we have perfect competition, but with externalities. PERFECT COMPETITION with no externalities: THE MARKET AS AN ALLOCATOR Are perfectly competitive industries efficient allocators? The answer is a qualified yes. Assuming all costs and benefits are being taken into account, perfectly competitive firms will provide goods to those individual who are willing to pay a price that at least covers the MC of providing the good. This can be seen by referring to the figure below. All individuals willing to pay a price of P E or higher will be able to purchase the good. Those willing to pay a maximum price below P E will not purchase the good. But, does P E represent the MC of producing the good? The answer is yes. Refer to the decision by the individual firm. The firm will produce output as long as MC of additional units of output is less than P E. This means that firms are providing the goods to anyone who is willing to pay at least the MC of having the good produced. XI 10

11 An alternative way of looking at this issue is in terms of what economists call consumer surplus and producer surplus from trading with each other. Consumer surplus is the difference between the maximum consumers would pay for a given good and what they actually pay. You can think of consumer surplus as the gains consumers receive from trading (buying) a particular good. In the graph above, consumer surplus is the area below the demand curve and above the equilibrium price for all units bought. Producer surplus is the difference between the minimum sellers would be willing to receive for a given good and what they actually receive. You can think of producer surplus as the gains producers receive from trading (selling) a particular good. In the graph above, producer surplus is the area above the supply curve and below the equilibrium price for all units sold. Can you tell from the graph why perfect competition leads to the maximization of consumer and producer surplus? Why did we answer a qualified yes to the question Are perfectly competitive industries efficient allocators? The qualification deals with whether the costs to the firm represents all costs to society. We will turn to this subject (and others) in the topics which follow. MONOPOLY: THE MARKET AS AN ALLOCATOR Are monopolistic industries efficient allocators? Generally, the answer is no, at least for a monopolist that charges the same price to every customer. From an efficiency point of view, single price monopolists charge too high of a price and produce too little. This may sound strange. Why would a monopolist produce too little? Consider the figure below. XI 11

12 The single price monopolist would charge P M and produce Q M. Notice, however, at this output level the price the monopolist is charging to all of its customers is greater than the MC of producing additional units of output. That is, there are units of output up to Q SO where individuals would pay a price greater than the MC of producing those units of output. Thus, from society s point of view, the monopolist should produce Q and charge a price of P. SO SO What solutions exist for remedying this market failure created by monopoly power? We discuss two very different solutions. 1) If the monopolist can price discriminate, he/she will maximize profits by producing where P= MC. Refer to the figure shown below for the perfect price discriminating monopolist. This price discriminating monopolist, producing where MR= MC, produces the socially optimal level of output. That is, by being able to charge each individual their limit price the monopolist will maximize profits by producing up to the point where MC= MR. That output level, however, is also the output level that maximizes social welfare. 2) As an alternative solution, the government can theoretically regulate the monopolist to provide the optimal level of output, requiring them to charge a lower price and produce a greater output level. At the end of this section, we summarize some of the issues involved in regulation. XI 12

13 EXTERNALITIES, PUBLIC GOODS, AND COMMON POOL RESOURCES One way of thinking about the issue of efficiency in allocating resources is to focus on the implications of individuals following their own self interest. That is, what if individuals make decisions that do not account for the impact of those decisions on others? Unlike the profit seeker in perfect competition, not taking into account the implications of our actions on others can lead to outcomes that are not beneficial (efficient) for society as a whole. We investigate this issue of self interest by considering three types of situations where individuals following their narrowly defined self interests imply inefficiencies with the allocation of scarce resources. Externalities Externalities are byproducts of a consumption or production action that affect individuals other than the individual consuming or producing. Examples include: immunizations, smoke, education, CO 2 emissions, etc. In each of these activities, the party conducting the activity creates a byproduct on others. In some cases these byproducts have positive effects and in others negative effects. Why do externalities matter? Basically, they imply that either the benefits to society or the costs to society of an individual taking an action are not being fully accounted for when that person decides to take the action or decides at what level to take the action. Some examples help. 1. Society benefits by you getting a better education. Do you take into account the impact on society when you decide where to go to school and for how long? 2. Society is harmed by your smoking. Do you take into account the impact on society when you decide whether to smoke or how much to smoke? 3. Society is harmed by a the air pollution from a coal burning steam plant. Does the firm take into account the impact on society when they make their decisions regarding how to produce steam or how much? Externalities and Perfect Competition We can investigate the impact of externalities on production with perfect competition by considering how they affect supply or demand. Consider the following two examples: XI 13

14 Case 1: Negative externalities created by producers. In this case the supply curve of producers underestimates the true cost of producing the good. That is if all costs to society were taken into account, costs would be higher, and the supply curve would lie to the left of the supply curve generated by considering only private costs. See the figure below. In the case of negative externalites, the output produced by the perfectly competitive industry is greater than the output that would be efficient (optimal) from society s point of view. One possible solution would be to have the government tax the producers of this good, increasing their private costs to be more in line with true social costs. Case 2: Positive externalities created by producers. In this case the supply curve of producers overestimates the true cost of producing the good. That is if all costs to society were taken into account, costs would be lower, and the supply curve would lie to the right of the supply curve generated by considering only private costs. Draw this situation for yourself and demonstrate graphically how positive externalties produced by perfectly competitive firms lead to underproduction from societies point of view. XI 14

15 Case 3: Positive externalities created by consumers. In this case the demand curve of consumers underestimates the true benefit of consuming the good. That is if all benefits to society were taken into account, benefits would be higher, and the demand curve would lie to the right of the demand curve generated by considering only private benefits. See the figure below. In this case, the output produced by the perfectly competitive industry is less than the output that would be efficient (optimal) from society s point of view. One possible solution would be to have the government subsidize the buyers of this good, increasing their private valuation of the good to be more in line with true social values. Case 4: Negative externalities created by consumers. In this case the demand curve of consumers overestimates the true benefit of consuming the good. That is if all benefits to society were taken into account, benefits would be lower, and the demand curve would lie to the left of the demand curve generated by considering only private benefits. Draw this situation for yourself and demonstrate graphically how negative externalties produced by consumers lead to over production from societies point of view. XI 15

16 Public Goods and Common Pool Resources In this section, we consider the two attributes of some goods that create resource allocation problems. These two attributes are: 1. rivalness (subtractability) - the extent to which one person s use prohibits another person s use. 2. excludability - the ability to exclude users. Most goods we think of that are produced by markets are rival in nature and are excludable. But if a good tends to be either nonrival in character or property rights are such that it is hard to exclude users, markets tend to inefficiently produce those goods. In particular, we consider two types of goods. Public goods are goods with low excludability and low rivalness: defense (security, policing, etc.), knowledge, charities, churches, etc. Common pool resources are low on excludability and high on rivalness: ocean fisheries, atmosphere, ground water, Internet, etc. Why does a low degree of excludability or a low degree of rivalness create special problems for society? Consider individual incentives (the self interest assumption). If there exists a low degree of excludability, in the case of public goods, individuals can consume the product without helping to provide the product (the free rider problem). If all individuals act this way, the good will be under produced from society s perspective. If there exists a low degree of excludability, in the case of common pool resources, individuals can over appropriate from the resource (the free rider problem). If all individuals act this way, the resource will be overused from society s perspective and possibly destroyed. If there exists a low degree of rivalness, any market price will exclude some users (users with a reservation price below the market price). But, those users should not be left out. The MC of providing them the good is essentially zero. What role can "government" play in relieving these problems? What are some of the problems in relying on governments? XI 16

17 GOVERNMENT Government Solutions Governments create a variety of policies that have both beneficial and adverse effects. Policy tools at the hand of government can be summarized under two basic areas -- laws related to restricting activities and laws relating to taxing. Following our discussion above, it is theoretically possible to have government policies that help to remedy each of the problems of market failure. In particular, governments can create legislation that: 1) strengthens and protects property rights, 2) regulates monopolies, 3) taxes producers of negative Externalities and subsidizes producers of positive externalities, 4) calls for the provision or subsidization of public goods, 5) restricts the use of common pool resources, and 6) provides for programs of income redistribution (welfare) in many different forms. Some Basic Problems with Government Solutions The solutions summarized above sound pretty good. The problems rest in implementation - what might be called government failures. Below is a list of concerns one might have in relying on government solutions. They fall into two major areas: information and incentives. 1) Information is costly to acquire. How could government be expected to have the correct information necessary to devise solutions? 2) What is good for some may be bad for others. How does a government pass legislation that is good for society as a whole? 3) Government can be thought of as a firm, producing or enforcing various forms of legislation. What are the incentives of the managers and workers within this firm? If they follow their own self interest, how do we get legislation and enforcement that is correct from society s point of view? 4) Legislation creates new incentives within the economy. In some cases those new incentives can adversely affect the productivity of individuals in the economy. We close on a few last principles of political economy that represent food for thought when considering the ability of government to remedy market failures. XI 17