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What is game theory? • “Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent, rational decision-makers. Game theory provides general mathematical techniques for analyzing situations in which two ore more individuals make decisions that will influence one another’s welfare.” (-Myerson, Game Theory: Analysis of Conflict Why use game theory? • “Neo-classical” economic theory can be very limiting in the kinds of questions it can answer • Traditional theories of economic and social behavior are not generally useful for policy purposes • With so much data available today, we can analyze markets and strategic situations on very small scales, which would have been impossible before the invention of powerful computers and the internet, but what should the analysis look like? The perfectly competitive market: Firm behavior A representative firm receives a price p per unit of a good it sells, and it c costs C(q) = q 2 to produce q units. Then 2 • Total revenue of q units is pq c • Total costs are C(q) = q 2 2 • Profits are the difference between total revenue and total cost, π(q) = c pq − C(q) = pq − q 2 2 How do we find the profit-maximizing quantity, q ∗ , for which π(q ∗ ) ≥ π(q ′ ) for any other q ′ the firm could pick? The perfectly competitive market: Firm behavior Let’s start by thinking about “searching” for a maximum, starting from some arbitrary quantity q. Consider the change in profit from using some other quantity, q ′ : c c π(q) − π(q ′ ) = pq − q 2 − pq ′ − q ′2 2 2 c ′ ′ ′ = p(q − q ) − (q + q )(q − q ) 2 ′ Now, divide by q − q to get π(q) − π(q ′ ) c = p − (q + q ′ ) q − q′ 2 This represents the gain/loss to switching from q to q ′ . 1 The perfectly competitive market: Firm behavior We now take the limit as q ′ goes to q lim ′ q →q π(q) − π(q ′ ) c = lim p − (q + q ′ ) ′ ′ q →q q−q 2 or π ′ (q) = p − cq So the change in profit π(q) when evaluated at the optimal quantity q equals p − cq. If this is positive, it means that profits are increasing at q, so we should pick a larger value. If this is negative, it means that profits are decreasing at q, and we should pick a smaller value. Only when π ′ (q) = 0 do we know that there are no better options than q “nearby”. (Maximization in general) (Maximization in general) So for x∗ to solve max f (x), x ′ ∗ it is necessary that f (x ) = 0; we call the equation f ′ (x∗ ) = 0 the first-order necessary conditions. For some problems, this approach doesn’t work. Why not? (Maximizing quadratic functions) In this class, we’ll have to maximize functions like f (x) = a 2 x − bx + c 2 2 a lot, with a, b > 0. In this case, the derivative is f ′ (x) = ax − b where the equation f ′ (x∗ ) = 0 has the solution x∗ = b/a. When is this actually a maximizer? (Maximizing quadratic functions) (Common Derivatives and Differentiation Rules) • [log(x)]′ = √ 1 1 , [xb ]′ = bxb−1 , [ x]′ = √ , [ex ]′ = ex x 2 x • Addition/Scalar Multiplication: Dx [af (x) + bg(x)] = af ′ (x) + bg ′ (x) • Multiplication Rule: [f (x)g(x)]′ = f ′ (x)g(x) + f (x)g ′ (x) • Chain Rule: f ((g(x))′ = f ′ (g(x))g ′ (x) • Derivative Rule: f (x) g(x) ′ = f ′ (x)g(x) − f (x)g ′ (x) g(x)2 3 The perfectly competitive market: Firm behavior Now we know that a necessary condition for q ∗ to be a maximum of π(q) is that π ′ (q ∗ ) = p − cq ∗ = 0 or q∗ = p c In other words, the firm’s supply curve is q S = p/c. The perfectly competitive market: Firm behavior The perfectly competitive market: Consumer behavior There is a representative consumer who buy the good q along with another good, we’ll call m, the numeraire or “money”. The consumer has a utility function over bundles (q, m) of u(q, m) = b log(q) + m The consumer also faces a budget constraint, w = pq + m, so that total expenditures on the two goods cannot be more than w. What bundle should the consumer pick? The perfectly competitive market: Consumer behavior Rather than use Lagrange multipliers, we can rewrite the budget constraint as m = w − pq and substitute it into the utility function to get max b log(q) + w − pq q 4 which is unconstrained in q, making it a much easier problem to solve. Then our necessary condition is b b − p = 0 −→ q ∗ = ∗ q p giving us an demand curve, q D = b/p. The perfectly competitive market: Consumer behavior Market Equilibrium Definition 1. A price-quantity pair (p∗ , q ∗ ) is a price-taking equilibrium or perfectly competitive equilibrium if 1. Firms maximize profits, taking the price as given. 2. Consumers maximize utility, taking the price as given. 3. Markets clear, so that supply equals demand, or q S = q D . We’ve already solved for supply and demand above, so all that’s left to do is equate them: p b qS = = = qD c p √ √ √ so that p∗ = bc and q ∗ = b/ c. 5 Market Equilibrium (Wait, why models?) In any economic model, we have a number of variables that are determined within the model, (p, q), and other variables that are taken as given, and the model is not meant to explain them (b, c). • A variable is endogenous if its value is determined within the model. (p, q) • A variable is exogenous if its value is given, and we use it to solve for the endogenous variables. (c) • A model is good if the story it tells about how exogenous variables determine endogenous variables is convincing. By using variation in what we observe, we can measure the magnitude and significance of different “forces” in the economic environment. Here, an economic model is a positive explanation of how the world is: It is an attempt to describe how the world really works. Perfectly competitive markets in general • A representative consumer chooses a bundle of goods (q, m) to maximize its utility, v(q) + m, subject to the constraint pq + m = w, taking prices as given. • A representative firm chooses how much of the good q to supply to maximize its profits, π(q) = pq − C(q) where C(q) represents its total costs, taking prices as given. 6 • The quantity demanded by the consumer equals the quantity supplied by the firm, so that markets clear. What are reasonable assumptions to make about what v(q) and C(q) look like? The perfectly competitive market: Firm behavior in general With the idea of a derivative, we know that max pq − C(q) q has first-order necessary condition p − C ′ (q ∗ ) = 0 or “price equals marginal cost”. The perfectly competitive market: Consumer behavior in general Similarly, the consumer solves max v(q) − pq + w q which has first-order necessary condition v ′ (q ∗ ) = p or “marginal benefit equals price”. The perfectly competitive market: Equilibrium in general The combined conditions for the consumer and firm imply C ′ (q ∗ ) = p∗ = v ′ (q ∗ ) so that price equals the marginal cost of providing the good to the firm, and the marginal benefit of the last unit to the consumer. The Welfare Theorem Rather than letting the market determine the outcome, what if a “benevolent social planner” decided q? How would the outcome be different? Definition 2. The social planner’s problem is found by summing the payoffs of all the agents, and maximizing the resulting social welfare function. The outcome that maximizes the social welfare function is efficient. Here, the social welfare function is the sum of profits and utility, giving us W (q) = (pq − C(q)) + (v(q) + w − pq) = v(q) + w − C(q) 7 The Welfare Theorem Now, maximizing over q gives W ′ (q ∗ ) = v ′ (q ∗ ) − C ′ (q ∗ ) = 0 or “marginal benefit of the good equals the marginal cost of supplying it”, exactly as in the price-taking equilibrium. Theorem 3. In a perfectly competitive market, the outcome is efficient. The Welfare Theorem When does the welfare theorem fail? 1. Agents have price power 2. There are externalities: Agents’ actions have direct consequences for others that are not reflected through the price system 3. The government is intervening in ways that give agents strange incentives 4. Agents have different information about the value or cost of goods and services 5. Agents have trouble finding one another quickly, leading to “search frictions” 6. There are “missing markets” (Wait, why models?) • Now we have another motivation for using a model: It provides us with an argument about why perfectly competitive markets are desirable. In other words, it is a normative explanation of how the world should be. • In addition, while you may disagree with the conclusion that perfectly competitive markets are desirable, at least you can argue with the assumptions of the model or provide an alternative model to illustrate your point. Price power: Monopoly • Suppose there is a single firm in the market that knows it controls quantity and price. How does it behave? Suppose the inverse demand curve is p(q). Then a monopoly solves max p(q)q − C(q) q with FONC p′ (q ∗ )q ∗ + p(q ∗ ) − C ′ (q ∗ ) = 0 {z } | {z } | <0 Price-MC Is this efficient? This seems to solve the issue of studying market power. What happens when there are, say, seven firms? 8 Some Puzzles What is missing in a perfectly competitive model of a market? 1. Why do firms offer quantity discounting? 2. Why are insurance or mortgages available to some people, but not others, even when the others are willing to pay a higher price? (i.e., markets fail to clear) 3. Why are fastfood restaurants usually located right next to each other, rather than uniformly distributed around town? 4. Why do we need lawyers and contracts? 5. The most recent financial crisis largely concerned whether particular sheets of paper were valuable or not. How is that possible? 6. What is going on when people “haggle”? 7. Suppose college has no impact on your productivity as a worker (for many students I’ve seen, this is certainly true). Why do so many people go? 8. Why do firms exist? Game Theory Game theory is the study of strategic behavior through mathematical models. Its goal is to provide a comprehensive approach to explaining and predicting the behavior of agents who act deliberately. Game theory also gives us substantially more ability to study how endogenous variables are determined by allowing us to look at the actual “mechanics” of markets: Contracts, bidding, bargaining, etc. Generally, we assume • Agents are self-regarding, in that their preferences over outcomes do not depend on the preferences of the other agents per se, or “directly” • Agents are rational, meaning that they have complete, transitive preferences over all outcomes (this is the real definition of “rationality” in economics). Game Theory A game is composed of • Players : Those agents who take actions • Actions : The choices which players can select • Payoffs : The numerical value that players associate with different outcomes of the game, which are allowed to depend on each player’s action • Timing : A description of which players take actions when • Information : A description of what players know, and when they know it 9 Game Theory • We begin with simultaneous-move games of complete information, in which all players make their decisions at the same time and know everything about the game and about each other. • We then add timing, giving us dynamic games of complete information, where players make decisions in sequence, and cannot revisit their earlier choices. • Finally, we add incomplete or imperfect information to get Bayesian games (think poker). Prisoners’ Dilemma • There are two burglars, who have been captured in the process of committing a crime. They have been very careful, and actually do not even know each other’s real name. The district attorney tells them: “If you both remain silent, I have enough evidence to send each of you to jail for two years. However, if one of you confesses and the other tells the truth, I will give the confessor a lighter sentence, sending him to jail for only one year, while I prosecute the other aggressively and send him to jail for five years. If both of you confess, there won’t be a trial, and you both get three years.” Prisoners’ Dilemma Row Silent Confess Column Silent -2,-2 -1,-5 Confess -5,-1 -3,-3 What is the best outcome for both burglars (what would a social planner pick)? What do you think the burglars do, and why? What other economic situations have similar incentives? Battle of the Sexes • Two people have decided to go on a date. The two options are a Football game, and the Ballet. The male prefers football, while the female prefers ballet. They discuss which option they will pick, but both happen to forget which they decided on. Worse, they both forgot their smart phones at work, and the two events are about to begin. Both prefer to be together rather than apart. 10 Battle of the Sexes Male Football Ballet Female Football 2,1 0,0 Ballet 0,0 1,2 What is the best outcome for the couple (what would a social planner pick)? What do you think they do, and why? What if we made the payoffs to Ballet (10, 20)? What other economic situations have similar incentives? Matching Pennies • You are waiting for a plane with a friend. Both of you have plenty of pocket change, so you propose the following game: You both secretly pick Heads or Tails. If both coins are heads, you get both coins. If both coins are tails, your friend gets both coins. Matching Pennies You Heads Tails Friend Heads 1,-1 -1,1 Tails -1,1 1,-1 What is the best outcome for the couple (what would a social planner pick)? What do you think they do, and why? What happens if we made the payoff to (Tails, Tails) equal to (10, −1)? The Strategic Form The matrix of players/actions/payoffs that we’ve been using to describe games is very helpful, since it summarizes all of the relevant information from a game theory perspective. We call it the strategic form. Row Player U D Column Player L urow (U, L), ucolumn (U, L) urow (D, L), ucolumn (D, L) R urow (U, R), ucolumn (U, R) urow (D, L), ucolumn (D, L) So you can think of game theory as a generalization of a regular economics model where consumers have preferences over bundles of goods or firms have preferences over quantities produced, to a setting where agents have preferences over how the other agents act. 11