Artificial Intelligence A Modern Approach 1st Edition Stuart Russell, Peter Norvig - Solutions

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(b) Suppose instead that Nash equilibrium prices were given by:where 1 − a − b < 1. Would an increase in a result in an increase or softening of price competition? Would firm 2’s response be more or less aggressive than in part (4a)?
(a) Will firm 2 respond by increasing or decreasing its price? Does the change in firm 1’s location result in increased or softened price competition?
(c) What happens to market share and profits for each firm as a result of the increase in costs at firm 1?4 Firm 1 and firm 2 sell a similar but differentiated product, charging market price p1 and p2, respectively. Suppose fixed costs are equal to zero, and each firm has the same constant marginal
(b) What happens to each firm’s price as a result of the increase in costs at firm 1?
(a) What happens to firm 1’s reaction function? What happens to firm 2’s reaction function?
3 Using the information in the previous question, suppose that workers at firm 1 go on strike and demand a higher wage. To settle the dispute, managers at firm 1 increase wages paid to workers, so that the marginal cost of production at firm 1, c1 increases.
(e) Do firms earn positive profits in equilibrium? Does this result confirm or contradict the Bertrand paradox? Explain your answer.
(d) Use the equations of each firm’s reaction function from part (b) to solve for each firm’s equilibrium price pi, level of output qi, and profits .
(c) On a graph with p1 on the horizontal axis and p2 on the vertical axis, graph each firm’ reaction function.
(b) Solve for each firm’s first-order condition, and use it to solve for each firm’s reaction function.
(a) Write out each firm’s profit function, of the form
2 Suppose two firms sell products in a particular market, but consumers do not regard the output of each firm as perfectly identical. The two firms face the following demand curves:Each firm has the same constant marginal cost of production, so that c1 = c2= $15. Fixed costs are zero.
1 Can you use the model of product differentiation to explain the fact that all major brewers introduced a type of ice beer within a very short period of each other? Is this an example of maximal or minimal product differentiation?
5 Refer to Figure 6.3. What would happen if firm 2’s reaction function were steeper than firm 1’s reaction function? Can you relate your answer to stability of the Cournot equilibrium?
4 Suppose that the market demand function is given by:as in Section 6.1.1. Each firm has zero fixed costs and constant marginal cost of productionc. An increase in demand causes the demand function to shift upwards, reflected by an increase in the parameter a to ã >a. What effect does this have on
Using the same demand and cost functions, suppose that firms competed in prices instead of quantities. Solve for the Bertrand equilibrium, including market demand, market price, and profits earned by each firm.Contrast your answers to those in the previous questions.
2 Suppose instead that firm 1 was the Stackelberg leader in this market. Rewrite firm 1’s profit function, using firm 2’s reaction function, and solve for the Stackelberg equilibrium. Once you’ve solved for the equilibrium market price and total market demand, you should be able to tell
(d) Use the demand function to solve for the equilibrium market price, and then solve for the level of profits earned by each firm in equilibrium.
(c) Solve for the Cournot equilibrium
(b) Use the two first-order conditions to solve for each firm’s reaction function, and plot these two reaction functions.
(a) Write out each firm’s profit function, and solve for the two first-order profit-maximization conditions.
1 Consider a market where two firms are competing in quantities. Suppose that the market inverse demand function is given by:Each firm has zero fixed costs and constant marginal cost of production c.
(b) Is firm 2’s best response to ? Why or why not? If not, would firm 2 rather increase or decrease ?5 Repeat Question 4 when actions are strategic complements.6 Consider the reaction functions as displayed in Figure 5.8. What happens to each player’s Nash equilibrium strategy if player 1’s
(a) Suppose firm 2 adopts a strategy . Identify firm 1’s best response to and label it .
4 Use a diagram with reaction functions as in Figure 5.8 when actions are strategic substitutes. Notice that when firm 1’s strategy s1 is on the horizontal axis, its reaction function R1 is steeper than firm 2’s reaction function R2. Identify the Nash equilibrium and label it N, and label firm
(c) Is there a Nash equilibrium (in pure strategies) in this game? Is this game similar to any of the examples of Section 5.4?
(b) Find each player’s best-response strategy, given the strategy played by his/her opponent. Does either player have a dominant strategy?
(a) Write out a payoff matrix which summarizes the strategies and payoffs for each player in this game.
3 Consider the following game describing the agency problem between a supervisor and a laborer. Each player has two strategies. The laborer can either work or shirk, and the supervisor can either monitor the laborer or not monitor. If the laborer works, he produces output valued by the supervisor
(d) Is this game similar to any of the examples of Section 5.4?
(c) Does either firm have a dominant strategy?
(b) Set up a payoff table like those in Section 5.4, and use it to solve for the Nash equilibrium.
(a) Describe the four elements of this game.
2 Suppose there are two firms, called A and B, which compete with each other in the market for a unique product. Each firm can charge one of two prices for this product: a high price or a low price . If both firms charge the high price, , they split total market profits of 1000. If they both charge
(b) Would this be most appropriately characterized as a non-cooperative game or a cooperative game?
(a) Imagine characterizing the relationship between OPEC members as a game. Identify the four elements of this game. You can find information to answer this question at http://www.opec.org/.
1 Representatives from the eleven member countries of the Organization of Petroleum Exporting Countries (OPEC) meet regularly to set production limits on the number of barrels of oil per day which each member country can produce.
(d) Calculate the total revenue, total cost, and the total profit of the monopoly.
(c) Suppose that the monopoly is able to observe who is the rich and who is the poor consumer. Determine the optimal purchase fee charged to the rich consumer (AR) and the optimal purchase fee charged to the poor consumer (AP).
(b) Determine the optimal price charged by the monopoly.
(a) Is this a short-run or long-run cost function? Explain.
(a) Is this a short-run or long-run cost function? Explain.
(a) Calculate equilibrium price and quantity.
26.7 Attempt to write definitions of the terms "intelligence," "thinking," and "consciousness."Suggest some possible objections to your definitions.
26.6 Under the correspondence theory, what kinds of propositions can be represented by a logical agent? A reflex (condition-action) agent?
26.5 Find and analyze an account in the popular media of one or more of the arguments to the effect that AI is impossible.
26.4 In the Brain Prosthesis argument, it is important to be able to restore the subject's brain to normal, such that its external behavior is as it would have been if the operation had not taken place. Can the sceptic reasonably object that this would require updating those neurophysiological
26.2 Does a refutation of the Chinese Room argument necessarily prove that appropriately programmed computers are conscious?
26.1 Go through Turing's list of alleged "disabilities" of machines, identifying which have been shown to be achievable, which are achievable in principle by a program, and which are still problematic because they require conscious mental states.
25.13 For the environment and motion uncertainty cone shown in Figure 25.28, find a navigation strategy that is guaranteed to reach the goal G.
25.12 We stated in our discussion of online algorithms that some special classes of environments are amenable to online algorithms with constant competitive ratios. An example is shown in Figure 25.27, which shows a robot on the bank of a river. We assume that the robot has an exact position
25.11 Explain how to approach the problem of online navigation for a cylindrical robot of significant diameter. Does the point-robot algorithm need to be modified? What happens when the robot is heading down a passageway that becomes too narrow to get through?
25.10 Implement a general environment in which to exercise the online navigation algorithm, such that arbitrary obstacles can be placed in the environment. Construct a specific environment corresponding to Figure 25.26. Then implement an agent that incorporates the online algorithm, and show that
25.9 For the environment in Figure 25.26, sketch the path taken by the robot in executing the online navigation strategy. This strategy always completely circumnavigates any obstacle it encounters, which is often unnecessary. Try to think of another strategy that will travel less distance around
25.8 Suppose a robot can see two point landmarks L\ and LI, and that it measures the angle between them as 17°. What is the shape of the recognizable set corresponding to this sensor reading? What is the shape of the recognizable set if the angle measurement has an uncertainty of ± 1 ° ? Sketch
25.6 Draw the cylindrical cell decomposition for the environment in Figure 25.25. With n boards in such an arrangement, how many regions would there be in the cell decomposition?
25.5 Consider the arm of a record player as a two-link robot. Let 0\ measure the elevation of the arm, and 6*2 measure its horizontal rotation. Sketch the configuration space of this robot with 6*1 and $2 axes. Include the configuration space obstacles for the platter and spindle.
25.4 Calculate the number of degrees of freedom of your hand, with the forearm fixed.
24.12 The Viterbi algorithm finds the most probable sequence of phones corresponding to the speech signal. Under the assumption that some words can be pronounced with more than one sequence of phones, explain why the Viterbi algorithm only computes an approximation to P(words\signal}.
24.11 Some sports announcers have been known to celebrate a score with the drawn out pronunciation [g ow ow ow ow ow ow el]. Draw a word HMM for "goal" such that the most probable path has a sequence of four [ow]s, but any number greater than 1 is possible.
24.10 Calculate the most probable path through the HMM in Figure 24.36 for the output sequence [C1,C2,C3,C4,C4,C6,C7]. Also give its probability.
24.9 Read this chapter from the beginning until you find ten examples of homophones. Does the status of a word as a homophone depend on the accent of the speaker?
24.8 We wish to use the alignment algorithm in an industrial situation where flat parts are moving along a conveyor belt and being photographed by a camera vertically above the conveyor belt. The pose of the part is specified by three variables, one for the rotation and two for the 2-D position.
24.7 In Figure 24.30, physically measure the cross ratio of the points ABCD as well as of the points A'B'C'D'. Are they equal?
24.6 A stereo system is being contemplated for terrain mapping. It will consist of two CCD cameras, each having 512x512 pixels on a 10 cm x 10 cm square sensor. The lenses to be used have focal length 16 cm, and focus is fixed at infinity. For corresponding points (u\, vi) in the left image and
24.5 Show that convolution with a given function/ commutes with differentiation, that is,(/*£)'=/*£'
24.4 Edges in an image can correspond to a variety of scene events. Consider the photograph on the cover of your book and assume that it is a picture of a real 3-D scene. Identify a set of ten different brightness edges in the image, and for each, decide whether it corresponds to a discontinuity in
24.3 Consider an infinitely long cylinder of radius r oriented with its axis along the y-axis. The cylinder has a Lambertian surface and is viewed by a camera along the positive z-axis. What will you expect.to see in the image of the cylinder if the cylinder is illuminated by a point source at
24.1 In the shadow of a tree with a dense, leafy canopy, one sees a number of light spots.Surprisingly, they all appear to be circular. Why? After all, the gaps between the leaves through which the sun shines through are not likely to be circular.
23.14 Implement a version of the chart-parsing algorithm that returns a packed tree for the longest leftmost edge, and then if that edge does not span the whole input, continues the parse from the end of that edge. Show why you will need to call PREDICT before continuing. The final result is a list
23.13 Implement a version of the chart-parsing algorithm that returns a packed tree of all edges that span the entire input.
23.12 We forgot to mention that the title of the text in Exercise 23.1 is "Washing Clothes." Go back and reread the text, and answer the questions in Exercise 23.3. Did you do better this time?Bransford and Johnson (1973) used this text in a better-controlled experiment and found that the title
23.11 One way to define the task of spelling correction is this: given a misspelled word and a dictionary of correctly spelled words, find the word(s) in the dictionary that can be transformed into the misspelled word in one insertion, deletion, substitution, or transposition. Given a dictionary of
23.10 Write grammar rules for the category Adjp, or adjective phrase, using reified categories.Show how to derive 3 g (g £ Fake(Guns) as the semantics of "a fake gun." An adjective phrase can be either a lone adjective (big), a conjunction (big and dumb), or an adjective phrase modified by an
23.9 We said that 3e, x e E Sleep(John, Past) A Name(x) = John was a plausible interpretation for "John slept." But it is not quite right, because it blurs the distinction between "John slept"and "Some person named John slept." The point is that the former sentence presupposes that speaker and
23.7 Collect some examples of time expressions, such as "two o'clock," "midnight," and"12:46." Also think up some examples that are ungrammatical, such as "thirteen o'clock" or"half past two fifteen." Write a grammar for the time language.
23.5 Open any book or magazine to a random page and copy down the first 10 sentences. How many of them are in £3 ? Show the parses of the sentences that are, and explain what went wrong for the sentences that are not.
23.4 Open any book or magazine to a random page and write down the first 20 nominal compounds you find. Characterize the semantic relations (e.g., made-of, used-for, etc.).
23.3 Without looking back at Exercise 23.1, answer the following questions:• What are the four steps that are mentioned?• What step is left out?• What is "the material" that is mentioned in the text?• What kind of mistake would be expensive?• Is it better to do too few or too many?• Why?
23.2 Describe how a simple pseudo-natural-language (template-based) explanation facility can be built for a vanilla, backward-chaining, logical reasoning system. The explanation facility should be written as a program WHY that generates an explanation after ASK has answered a question from the
22.5 This exercise concerns grammars for very simple languages.a. Write a context-free grammar for the language a"b".b. Write a context-free grammar for the palindrome language: the set of all strings whose second half is the reverse of the first half.c. Write a context-sensitive grammar for the
22.4 Augment the grammar from this chapter so that it handles the following:a. Pronoun case.b. Subject/verb agreement.c. Article/noun agreement: "agents" is an NP but "agent" is not. In general, only plural nouns can appear without an article.
22.3 Determine what semantic interpretation would be given to the following sentences by the grammar in this chapter:a. It is a wumpus.b. The wumpus is dead.c. The wumpus is in 2,2.
22.1 Outline the major differences between Pascal (or any other computer language with which you are familiar) and English, and the "understanding" problem in each case. Think about such things as grammar, syntax, semantics, pragmatics, compositionality, context-dependence, lexical ambiguity,
21.7 Using the data from the family tree in Figure 21.5, or a subset thereof, apply the FOIL algorithm to learn a definition for the Ancestor predicate.
21.4 Fill in the missing values for the clauses C\ and/or €2 in the following sets of clauses, given that C is the resolvent of C\ and €2-a. C= True => P(A,B}, C\ = P(x,y) =^ Q(x,y), C2 =??.b. C = True => P(A,B), C, =11, C2 =??.c. C = P(x,y) => P(x,f(y)), C{ =11, C2 =11.If there is more than
21.3 Would a probabilistic version of determinations be useful? Suggest a definition.
21.1 Show, by translating into conjunctive normal form and applying resolution, that the conclusion drawn on page 633 concerning Brazilians is sound.
20.11 (Discussion topic.) Is reinforcement learning an appropriate abstract model for human learning? For evolution?
20.10 Extend the standard game-playing environment (Chapter 5) to incorporate a reward signal. Put two reinforcement learning agents into the environment (they may of course share the agent program) and have them play against each other. Apply the generalized TD update rule(Equation (20.8)) to
20.9 Write down the update equation for Q-learning with a parameterized implicit representation.That is, write the counterpart to Equation (20.8).
20.8 Adapt the vacuum world (Chapter 2) for reinforcement learning by including rewards for picking up each piece of dirt and for getting home and switching off. Make the world accessible by providing suitable percepts. Now experiment with different reinforcement learning agents. Is input
20.7 How can the value determination algorithm be used to calculate the expected loss experienced by an agent using a given set of utility estimates U and an estimated model M, compared to an agent using correct values?
20.6 Prove formally that Equations (20.1) and (20.3) are consistent with the definition of utility as the expected reward-to-go of a state.
20.5 The description of reinforcement learning agents in Section 20.1 uses distinguished terminal states to indicate the end of a training sequence. Explain how this additional complication could be eliminated by modelling the "reset" as a transition like any other. How will this affect the
20.4 The environments used in the chapter all assume that training sequences are finite. In environments with no clear termination point, the unlimited accumulation of rewards can lead to problems with infinite utilities. To avoid this, a discount factor 7 is often used, where 7 < 1.A reward k
20.2 Implement a passive learning agent in a simple environment, such as that shown in Figure 20.1. For the case of an initially unknown environment model, compare the learning performance of the LMS, TD, and ADP algorithms.
20.1 Show that the estimates developed by the LMS-UPDATE algorithm do indeed minimize the mean square error on the training data.
19.7 The network in Figure 19.13 has four hidden nodes. This number was chosen somewhat arbitrarily. Run systematic experiments to measure the learning curves for networks with different numbers of hidden nodes. What is the optimal number? Would it be possible to use a crossvalidation method to
19.6 Suppose that a training set contains only a single example, repeated 100 times. In 80 of the 100 cases, the single output value is 1; in the other 20, it is 0. What will a back-propagation network predict for this example, assuming that it has been trained and reaches a global optimum?(Hint:

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Artificial Intelligence A Modern Approach 1st Edition Stuart Russell, Peter Norvig - Solutions

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