ECN 321 FINAL

(a) A Cooperative Solution is the combination of players' strategies and payoffs that the players would choose in a cooperative game (i.e., in a game where they could write enforceable contracts with one another). (b) A Social Dilemma is a non-cooperative game in which the equilibrium is not the cooperative solution ("Prisoners' Dilemma" is an example).

In game theory, (a) what is a Cooperative Solution, and (b) what is a Social Dilemma?

(a) Refinements are additional assumptions about a game or the situation in which a game occurs that will help predict the outcome of the game. (b) Refinements are typically used to help predict the outcome in games that have multiple equilibria. (c) Three common refinements are: (1) assume that players will play Randomized Strategies, in which they select actions at random, (2) assume that the game will be a Repeated Game or (3) assume that the game will be a Sequential Move Game.

In game theory, what are "refinements?" (b) When are refinements needed to solve a game? (c) List three types of refinements.

She has proved that the firm is a monopolist (which is the opposite of her original assumption). She has used the "Proof by Contradiction" method of proof.

Suppose an economist assumes that a firm is not a monopolist, and based on that assumption, she derives two conclusions: that the firm will set a very high price for its product, and, at the same time, that the firm will set a very low price for its product. What has she proved? What method of proof has she used in her analysis?

The economist has proved that the firms are not price takers. The economist has used proof by contradiction.

Suppose an economist is building a game theory model. The economist makes the assumption in the model that firms are price takers. Suppose the economist then derives two results from the model: (1) each firm will definitely shut down, and (2) each firm will definitely not shut down. - What, if anything, has the economist proved? - What is the name of the type of proof that the economist has used?

If a statement (for example, "will purchase a new car") and its exact opposite/negation /contradiction ("will not purchase a new car) are both derived from an assumption, then the assumption must be false. The economist has proved that the initial assumption is false by using the "proof by contradiction" method, also known as the "reductio ad absurdum" (reduce to absurdity) method.

Suppose an economist makes an assumption about consumer behavior and, based on that assumption, derives two results: (1) that the consumer will purchase a new car, and (2) that the consumer will not purchase a new car? What has the economist proved? What is the name of the type of proof that the economist has used?

A Hypothesis (plural Hypotheses) is a pattern in the data that is suspected to indicate a true relationship among variables. A Spurious Hypothesis is a hypothesis based on a pattern in the data that is discovered to have occurred by chance and that does not reflect a true pattern among variables.

What is a hypothesis? What is a spurious hypothesis?

Theorems are results/forecasts/predictions that are derived from theories/models; in other words, theorems are the implications of theories/models. Theorems are used (1) as forecasting/prediction tools, and (2) to test models/theories by comparing the theorem predictions against new data.

What is a theorem? How are theorems used?

A theory is a pattern believed to be observed in data. A model is a formal, precise statement of a theory. The three elements of a model are: variables, parameters and operators.

What is a theory? What is a model? What are the three elements of any model?

A valid argument is an argument where: - If the premise is true (we're not saying that it is), then the conclusion would be true. - A sound argument is (1) a valid argument, and (2) it is known that the premise is true.

What is a valid argument? What is a sound argument?

Sensitivity analysis determines the sensitivity of model results (the dependent variable) to changes in the values of model parameters.

What is sensitivity analysis?

The ceteris paribus assumption is the modeling assumption that everything in the world outside the model is assumed to remain constant. Each thing left outside the model is assumed to remain constant where it was when the data were collected for the model.

What is the ceteris paribus assumption?

Economics is the study of the rational allocation of resources under constraints to meet objectives.

What is the definition of economics as discussed in lecture and in the class handouts?

Ockham's Razor is the model building principle that states that unimportant details should be "cut away," or discarded, from a model. In practice, applying Ockham's Razor means that a model-builder should initially attempt to create a simple model that captures only the key variables in the situation under study.

What is the principle of Ockham's Razor?

Statistical Regression Analysis is typically used to determine the best values for the parameters in a model.

What type of analysis is used to determine the best values for the parameters in a model?

Linear relationships are important because they are common; however, it is almost always more important to correctly identify and analyze nonlinear (rather than linear) relationships in a model, because nonlinear relationships are most often the source of surprising model results, such as "exploding" relationships among variables.

Which type of relationship, linear or nonlinear, is more important to correctly identify and analyze when building a model? Explain why in one or two sentences.

Economists often use "Natural Experiments" instead of controlled experiments. Such experiments are called "natural" because they occur by chance; they are not "set up" by scientists. As an example, consider a situation in which all factors that affect the purchasing behavior of two groups of consumers are the same, except for one factor, say, price. This situation is a natural experiment. Differences in behavior between the two groups (differences in quantities consumed, for example) may be explained by differences in the one factor that differs between the groups (in this example, price), because all other factors are the same for the two groups.

Because economists study human behavior, it is often difficult to conduct controlled experiments to test economic theories. How do economists overcome this problem?

A Nash Equilibrium is an equilibrium in which each player has and chooses to play a Nash Strategy, that is, each play plays a strategy that maximizes a player's payoff given the strategies actually chosen by all other players.

Briefly define/describe the term "Nash Equilibrium."

Statistical hypothesis testing is used in two ways: (1) to test the reliability of the parameter estimates in the model, and (2) to determine whether "real world" observations are so far from the theorems (predictions) derived from a model that the model should be revised. [In (2), Statistical hypothesis testing methods are used to calculate Confidence Interval numbers for theorems/model predictions. If real-world observations are inside the Confidence Interval numbers, then we continue to have confidence in our model. If real-world observations are outside the Confidence Interval numbers, then we lose confidence in our model, and it's "Back to the drawing board!" to revise the model.]

How is statistical hypothesis testing used in the model building process?

a) "smartphone producers are paid a higher price for smartphones" b) "the quantity of smartphones produced will increase (ceteris paribus)" c) Yes, the Law of Supply states that if the price of an item increases, producers will supply more of the item (ceteris paribus). So, if it were true that the price of smartphones increased, then it would be true that producers would produce more smartphones, so if the premise were true, then the conclusion would be true, so it is a valid argument. d) Uncertain, because we don't know whether the premise "smartphone producers are paid a higher price for smartphones" is actually true. If we knew for sure that the premise were true, then we would know that the argument is sound in addition to being valid. Once we knew the argument was sound, we would be comfortable concluding that the quantity of smartphones produced will increase. e) The converse argument is "If the quantity of smartphones produced increases, then smartphone producers are paid a higher price." This converse argument is not necessarily valid, because, for example, producers may decide to produce more phones if the cost of production decreases, even if the price they are paid for the phones remains the same (or decreases a bit). The point is that converse arguments are not always valid; you must check them. f) The inverse argument is "If smartphone producers are paid a lower price for smartphones, then the quantity of smartphones produced will decrease (ceteris paribus)." The inverse statement in this example is valid under ceteris paribus conditions; however, in general, inverse statements are not always valid. For example, consider the statement "If a firm is a monopoly, then the firm will seek to maximize profits." If this statement is valid, does it follow that the inverse is valid "If a firm is not a monopoly, then the firm will not seek to maximize profits?" Of course not, all firms seek to maximize profits, not just monopolies. The point is that inverse arguments are not always valid; you must check them. g) The contrapositive argument is "If the quantity of smartphones produced does not increase (ceteris paribus), then smartphone producers are not being paid higher prices." Under ceteris paribus conditions, this is always true, so the contrapositive argument is valid. In general, if an argument is valid, then the contrapositive argument is always valid as well.

Consider the following argument: "If smartphone producers are paid a higher price for smartphones, then the quantity of smartphones supplied will increase (ceteris paribus)." a) What is the premise in the argument above? b) What is the conclusion in the argument above? c) Assuming the Law of Supply holds, is the argument above valid? Briefly, why? d) Is the argument above sound? Yes, no, or uncertain? Briefly, why? e) What is the converse of the argument above? Is this converse argument necessarily valid? If not, give a counterexample to refute the converse argument. f) What is the inverse of the argument above? Is this inverse argument necessarily valid? If not, give a counterexample to refute the inverse argument. g) What is the contrapositive of the argument above? Is this contrapositive argument necessarily valid? If not, give a counterexample to refute the contrapositive argument.

- A proposition is an assertion that can be only true or false. A proposition is also sometimes called a statement or sentence; when it is, we mean the special kind of sentence that can only be true or false, but not both, nor neither. - An argument is a set of statements such that the truth of one of the statements implies the truth of the other. The premise is the statement that is known to be true, and the conclusion is the statement that is true based on the knowledge that the premise is true. An argument is also known as a conditional statement (because the truth of the conclusion in the argument is conditional on (that is, depends on) the truth of the premise).

In deductive reasoning, what is a proposition? What is an argument?

Situations between perfect competition and monopoly/monopsony.

In economics, which types of situations are typically modeled using game theory?

The operators in a model are determined by using Nested Modeling (which is a "sneaky" way of using Statistical Regression Analysis). Although Statistical Regression Analysis is primarily used to find the best values for the parameters in a model, it also helps you find the best operators for a model, when used in the special way called Nested Modeling.

How are the operators in a model determined? Suppose you are trying to decide which of two possible equations best fits the patterns in the data. Which type of analysis should you use? (Hint: It's the answer to question 7 above, but used in a "sneaky" way.) (Notice that determining which equation best fits the data also tells you which operators should be used in the model and where they should be located in the model equation.)

-Inductive Reasoning is the process of finding a pattern/theory/model from data. -Deductive reasoning is the process of finding the implications/ forecasts/ predictions/ theorems that come from a model.

How does Inductive Reasoning differ from Deductive Reasoning?

A Dominant Strategy is a strategy that maximizes a player's payoff for all possible strategies of all other players. A Nash Strategy is a strategy that maximizes a player's payoff given the strategies actually chosen by all other players.

How does a Dominant Strategy differ from a Nash Strategy?

(a) "Unraveling" describes the process by which a game that is a prisoners' dilemma on the last turn will, by backward induction, also be a prisoners' dilemma on every other turn, including the first turn. NOTE: The problem with unraveling, from the players' perspective, is that they end up stuck in a prisoners' dilemma on every turn of the game. (b) Unraveling occurs in finitely-repeated games in which the last turn is a prisoners' dilemma. (c) If the game is repeated an infinite number of times, or the game has an uncertain last turn, then the unraveling problem can be overcome if the interest rate is low enough and the probability of the game continuing one more turn is high enough.

In game theory: (a) What is the meaning of the term "Unraveling?" (b) When does unraveling occur? (c) What can be done to overcome the unraveling problem?

- In the valid argument A ==> B, A is called the sufficient statement and B is called the necessary statement. - A is called the sufficient statement because, given that A ==> B is valid, we know that if A is true, then B must be true. However, other things besides A could also cause B to be true. This is why the truth of B is not sufficient for the truth of A: if B is true, A may or not be true, because other things besides the truth of A could be causing the truth of B. - B is called the necessary statement because, given that A ==> B is valid, if B is not true then A cannot be true. In other words, the truth of B is necessary for the truth of A. However, the truth of A is not necessary for the truth of B because other things besides A could cause B to be true.

In the following valid argument, A ==> B, which statement is the necessary statement, and which is the sufficient statement?

These statements are not true. The methodology of infinitely repeated games can be used to model situations that potentially last forever, and the method of finite repeated games with an uncertain last turn can be used to model situations that are uncertain in that you do not know when the situation will end.

Suppose a business person says, "It is impossible to model situations that potentially last forever. It is also impossible to model situations that are uncertain, that is, situations in which you do not know when the situation will end." Are these statements true? If so, briefly explain why. If not, name the method(s) that could be used to model the situations.

- The converse argument is: "If we need to change banking regulations, then we need to give $700 billion in loans to banks." The converse is not necessarily valid. - The inverse argument is: "If we do NOT need to give $700 billion in loans to banks, then we do NOT need to change banking regulations." The inverse is not necessarily valid. - The contrapositive argument is: "If we do NOT need to change banking regulations, then we do NOT need to give $700 billion in loans to banks." If the original argument is valid, then the contrapositive argument is necessarily valid. - If the original argument is valid, then the argument would also be sound if it were true that "we need to give $700 billion in loans to banks."

Suppose a government official says: "If the government needs to give a $700 billion bailout loan to banks, then we need to change banking regulations." This argument may or may not be valid, but let's suppose that it is valid. What is the converse of this argument, and is the converse necessarily valid? What is the inverse of this argument, and is the inverse necessarily valid? What is the contrapositive of the argument, and is the contrapositive necessarily valid? Suppose the original argument is valid; what would make the argument a sound argument for concluding that we need to change banking regulations?

The word "data" is plural. Data are measured, recorded observations.

What are data? Is the word "data" singular or plural?

- The truth of "Business investment falls" is necessary for the truth of "if interest rates rise." That is, assuming the argument is valid, if business investment is not falling, then interest rates cannot be rising. - The truth of "Interest rates rising" is sufficient for the truth of "Business investment falls." That is, if interest rates are rising, then business investment must be falling. Other things could also cause business investment to fall, but we know that if interest rates rise, that is one of the (possibly many) things that will certainly cause business investment to fall (assuming the argument is valid, of course).

Suppose it is a valid argument that "If interest rates rise, then business investment falls." The truth of which statement in the argument is necessary for the truth of the other? The truth of which statement in the argument is sufficient for the truth of the other?

Alternative finding "(c)" would confirm her original argument, because it is the contrapositive argument to the original argument. If the economist finds that alternative finding (c) is valid, then "Proof by Contrapositive" confirms the validity of her original argument.

Suppose that an economist believes that the following argument is true "if consumers have lower incomes, then consumers spend less money on Paris vacations." Suppose also, however, that it is not (legally) possible to take income away from consumers to directly test this argument. Which of the following alternative findings would confirm her original argument: (a) if consumers have higher incomes, then consumers spend more money on Paris vacations, (b) if consumers spend less money on Paris vacations, then consumers have lower incomes, or (c) if consumers spend more money on Paris vacations, then consumers have higher incomes. Using one of the following terms: direct proof, proof by induction, proof by contrapositive, proof by contradiction, or proof by counterexample, briefly explain why your chosen alternative finding supports the economist's original belief.

We can conclude that ALL fast food restaurants on the street lower their prices, because we know that at least one firm (Firm A) lowered its price, and we know that if any one firm lowers its price, then the firms on either side will lower their prices. Once the firms on either side of Firm A lower their prices, then BY INDUCTION the firms on either side of those firms lower their prices, and so on, until ALL firms on the street have lowered their prices. We have used the "proof by induction" method to derive the conclusion that ALL firms will lower priced based on knowledge that just one firm lowered price. For this method of proof to work, it is critical that the "inductive assumption" applies to ANY one of the objects (in this example, firms) under consideration. In the inductive assumption in this example: "if any one firm lowers the price of its product, then the firms on either side of it will lower their prices as well," the word "any" in is critical, because it justifies the assumption that the price-reduction behavior will continue to "chain" its way from firm to firm down the street, continuing beyond the firms immediately adjacent to Firm A.

Suppose there are many fast food restaurants located along a major street in a large city. Suppose it is true that if any one firm lowers the price of its product, then the firms on either side of it will lower their prices as well. Suppose Firm A lowers its price. What can we conclude will happen to the prices set by ALL fast food restaurants on the street? What method of proof has been used to derive the conclusion?

We can conclude that all financial trading firms in the global markets will eventually use short selling practices. The "Proof by Induction" method of proof has been used to derive this conclusion.

Suppose there are many financial trading firms interacting with one another in global financial markets. Suppose an economist proves that if any one of these firms uses "short selling" practices, then its trading partners also will be forced to use "short selling" practices. Suppose we find out that Firm A is now using short selling practices. What can we conclude about the eventual trading behavior of all financial trading firms in the global markets? What method of proof has been used to derive this conclusion?

Converse: If NSE, then DSE. No, not necessarily true. Contrapositive: If not NSE, then not DSE. Yes, necessarily true. Inverse: If not DSE, then not NSE. No, not necessarily true.

The following conditional statement is valid: If the equilibrium of a game is a Dominant Strategy Equilibrium (DSE), then it is a Nash Strategy Equilibrium (NSE). - What is the converse of the preceding statement; is the converse necessarily valid? - What is the contrapositive of the preceding statement; is the contrapositive necessarily valid?- - What is the inverse of the preceding statement; is the inverse necessarily valid?