econchap4-5 - u. __, .‘u Aaluuk IUUA 1U]. d _ - u...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: u. __, .‘u Aaluuk IUUA 1U]. d _ - u equinonum in mixed strategies, r the analysis of which is presented in Chapters 7 and 8. y: a Rs? KEY TERMS i assurance game (107) battle of the sexes (108) belief (89) best response (87) best-response analysis (99) cell-by-cell inspection (89) chicken (109) constant—sum game (85) convergence of expectations (107) coordination game (105) dominance solvable (96) dominant strategy (92) dominated strategy (92) enumeration (89) focal point (106) game matrix (84) example of a game that illustrates your an EXERCISES game table (84) iterated elimination of dominated strategies (96) maximin (100) minimax (100) minimax method (99) mixed strategy (84) Nash equilibrium (87) normal form (84) payoff table (84) prisoners' dilemma (90) pure coordination game (105) pure strategy (84) strategic form (84) successive elimination of dominated strategies (96) zero—sum game (85) SWEI’. (c) COLUMN 3 1 (d) 3. Find all Nash equilibria in pure strategies in the following non-zero—sum games. Describe the steps that you used in finding the equflibna. 5 [CH. 4] SlMULTANEOUS-MOVE GAMES WITH PURE STRATEGIES COLUMN (bl (c) m ‘nd all of the pure-strategy Nash equilibria for the following game. De- ribe the process that you used to find the equilibtia. Use this game to ex- EXERCISES 117 plain why it is important to describe an equilibrium by using the strategies employed by the players, not merely by the payoffs received in equilibrium. COLUMN 6. The game known as the battle of the Bismarck Sea (named for that part of the southwestern Pacific Ocean separating the Bismarck Archipelago from Papua—New Guinea) summarizes a well-known game actually played in a naval engagement between the United States and Japan in World War II. In 1943, a Japanese admiral was ordered to move a convoy of ships to New Guinea; he had to choose between a rainy northern route and a sunnier southern mute, both of which required 3 days sailing time. The Americans knew that the convoy would sail and wanted to send bombers after it. but they did not know which route it would take. The Americans had to send re- connaissance planes to scout for the convoy, but they had only enough re- connaissance planes to explore one route at a time. Both the Japanese and the Americans had to make their decisions with no knowledge of the plans being made by the other side. If the convoy was on the route explored by the Americans first. they could send bombers right away; if not, they lost a day of bombing. Poor weather on the northern route would also hamper bombing. 1f the Americans explored the northern route and found the Japanese right away. they could expect only 2 {of 3) good bombing days; if they explored the northern route and found that the Japanese had gone south, they could also expect 2 days of bombing. If the Americans chose to explore the southern route first, they could expect 3 full days of bombing if they found the Japanese right away but only 1 day of bombing if they found that the Japanese had gone north. (a) Illustrate this game in a game table. (b) Identify any dominant strategies in the game and solve for the Nash equilibrium. 7. An old lady is looking for help crossing the street. Only one person is needed to help her; more are okay but no better than one. You and I are the two people in the vicinity who can help; we have to choose simultaneously [CH. 4] SIMULTANEOUSrMOVE GAMES WITH PURE STRATEGIES ether to do so. Each of us will get pleasure worth a 3 from her success 0 matter who helps her). But each one who goes to help will bear a cost of this being the value of our time taken up in helping. Set this up as a me. Write the payoff table, and find all pure-strategy Nash equilibria. 0 players, lack and Iill, are put in separate rooms. Then each is told the les of the game. Each is to pick one of six letters; G, K, L, Q, R, or W. If the o happen to choose the same letter, both get prizes as follows: Letter G K L Q R W Ilack’s prize 3 2 5 3 4 5 Iill’s prize 6 5 4 3 2 1 they choose different letters, each gets 0. This whole schedule is re- aled to both players, and both are told that both know the schedules, (1 so on. ) Draw the table for this game. What are the Nash equilibria in pure strategies? ) Can one of the equilibria be a focal point? Which one? Why? ppose two players, A and B, select from three different numbers, 1, 2, and Both players get dollar prizes if their choices match, as indicated in the llowing table. What are the Nash equilibria of this game? Which, if any, is likely to emerge as the (focal) outcome? Explain. ) Consider a slightly changed game in which the Choices are again just numbers but the two cells with (15, 15) in the table become (25, 25). What is the expected (average) payoff to each player if each flips a coin to decide whether to play 2 or 3? Is this better than focusing on both choosing 1 as a focal equilibrium? How should you account for the risk that A might do one thing while B does the other? . _....._'_,.._... _ ._ ___.. ._.. ._... n- .t...,..... .. _.. .. EXERCISES 119 10. In Chapter 3, the three gardeners, Emily, Nina, and Talia, play a sequential 11. 12. version of the street-garden game in which there are four distinguishable outcomes (rather than the six different outcomes specified in the example of Section 4.6). For each player, the four outcomes are: (1) player does not contribute, both of the others do [pleasant garden, saves cost of own contribution) (ii) player contributes, and one or both of the others do (pleasant garden, incurs cost of contribution) (iii) player does not contribute, only one or neither of the others does (sparse garden, saves cost of own contribution) (iv) player contributes, but neither of the others does (sparse garden, incurs cost of own contribution) Of them, outcome i is the best (payoff 4) and outcome iv is the worst (payoff 1). if each player regards a pleasant garden more highly than her own contribution, then outcome ii gets payoff 3 and outcome iii gets payoff 2. (a) Suppose that the gardeners play this game simultaneously, deciding whether to contribute to the street garden without knowing what choices the others will make. Draw the three-player game table for this version of the game. (13) Find all of the Nash equilibria in this game. (c) How might this simultaneous version of the street-garden game be played out in reality? Consider a game in which there is a prize worth $30. There are three contes— tants, A, B, and C. Each can buy a ticket worth $15 or $30 or not buy a ticket at all. They make these choices simultaneously and independently. Then, knowing the ticket-purchase decisions, the game organizer awards the prize. If no one has bought a ticket, the prize is not awarded. Otherwise, the prize is awarded to the buyer of the highest-cost ticket if there is only one such player or is split equally between two or three if there are ties among the highest-cost ticket buyers. Show this game in strategic form. Find all pure-strategy Nash equilibria. in the film A Beautiful Mind, Iohn Nash and three of his graduate school colleagues find themselves faced with a dilemma while at a bar. There are four brunettes and a single blonde available for them to approach. Each young man wants to approach and win the attention of one of the young women. The payoff to each of winning the blonde is 10: the payoff of win- ning a brunette is 5; the payoff from ending up with no girl is 0. The catch is that, if two or more young men go for the blonde, she rejects all of them and then the brunettes also reject the men because they don’t want to be second .. .n__..___ EXERClSES 151 SUMMARY When players in a simultaneous-move game have a continuous range of actions to choose, best-response analysis yields mathematical best—response rules that can be solved simultaneously to obtain Nash equilibrium strategy choices. The best»response rules can be shown on a diagram in which the intersection of the two curves represents the Nash equilibrium. Firms choosing price or quantity from a large range of possible values and political parties choosing campaign advertising expenditure levels are examples of games with continuous strategies. The results of laboratory tests of the Nash equilibrium concept show that a common cultural background is essential for coordinating in games with multiple equilibria. Repeated play of some games shows that players can learn from expe- rience and begin to choose strategies that approach Nash equilibrium choices. Further, predicted equilibria are accurate only when the experimenters’ assump- tions match the true preferences of players. Real-world applications of game the- ory have helped economists and political scientists, in particular, to understand important consumer, firm, voter, legislature, and government behaviors. Theoretical criticisms of thteash equilibrium concept have argued that the concept does not adequately accOunt for risk, that it is of limited use because many games have multiple equilibria, and that it cannot be justified on the basis of rationality alone. In many cases, a better description of the game and its pay- off structure or a refinement of the Nash equilibrium concept can lead to better predictions or fewer potential equilibria. The concept of rationaliznbility relies on the elimination of strategies that are never a best response to obtain a set of rationalizable outcomes. When a game has a Nash equilibrium, that outcome will be rationalizable; but rationalizability also allows one to predict equilibrium outcomes in games that have no Nash equilibria. KEY TER M S .; best-response curves (127) rationalizability (145) best-response rule [124) rationalizable (145) continuous strategy (124) refinement [143) never a best response (145) EXERCISES ? l. in the political campaign advertising game in Section LB, party L chooses an advertising budget, x (millions of dollars}, and party R similarly chooses a budget, y (millions of dollars]. We showed there that the best-response rules n that game are y = VE— x, for party R, and x = W — y for party L. USE hese best-response rules to verify that the Nash equilibrium advertising Judgets are x = y : 1/4, or $250,000. the bistro game of Figure 5.1 defines demand functions for Xavier’s (0,.) 1nd Yvonne’s (le as Qx = 44 — 2 P, + Py, and Q, = 44 e 2 P}, + PX. Profits for :ach firm depend, in addition, on their costs of serving each customer. Sup— Jose, here, that Yvonne’s is able to reduce its costs to $6 per customer by re- listributing the serving tasks and laying off several servers; Xavier’s :ontinues to incur a cost of $8 per customer. a) Recalculate the best-response rules and the Nash equilibrium prices for the two firms, given the change in the cost conditions. b) Graph the two best-response curves and describe the differences be- tween your graph and Figure 5.1. In particular, which curve has moved and by how much? Can you account for the changes in the diagram? Tuppietown has two food stores, La Boulangerie, which sells bread, and La 2romagerie, which sells cheese. It costs $1 to make a loaf of bread and $2 to nake a pound of cheese. If La Boulangerie’s price is P1 dollars per loaf of need and La Fromagerie’s price is P; dollars per pound of cheese, their re- .pective weekly sales, Q1 thousand loaves of bread and Q2 thousand pounds If cheese, are given by the following equations: QIZIO—Pl—0‘5P2! 02:1270.5P1_P2. a) Find the two stores’ best-response rules. Show the best—response curves, and find the Nash equilibrium prices in this game. b) Suppose that the two stores collude and set prices jointly to maximize the sum of their profits. Find the joint profit-maximizing prices for the stores. c) Provide a short intuitive explanation for the differences between the Nash equilibrium prices and those that maximize joint profit. Why is joint profit maximization not a Nash equilibrium? d) In this problem, bread and cheese are mutual complements. They are often consumed together; that is why a drop in the price of one in- creases the sales of the other. The products in our bistro example in Section 1.A are substitutes for each other. How does this difference ex- plain the differences between your findings for the best-reSponse rules, the Nash equilibrium prices, and the joint profit-maximizing prices in this question, and the corresponding entities in the bistro example in the text? ‘wo carts selling coconut milk (from the coconut) are located at 0 and 1. 1 rule apart on the beach in Rio de Ianeiro. (They are the only two coconut milk carts on the beach.) The carts—Cart 0 and Can I—charge prices p0 and p1, respectively, for each coconut. Their customers are the beach goers uni- formly distributed along the beach between 0 and 1. Each beach goer will purchase one coconut milk in the course of her day at the beach and, in ad- dition to the price, each will incur a transport cost of 0.5 times d2, where dis the distance (in miles) from her beach blanket to the coconut cart. In this system, Cart 0 sells to all of the beach goers located between 0 and x, and Cart 1 sells to all of the beach goers located between it and 1, where x is the location of the beach goer who pays the same total price if she goes to 0 or 1. Location x is then defined by the expression pa + 0.5x2 = p, + 0.5[1 — x)? The two carts will set their prices to maximize their bottom-line profit fig- ures, B; profits are determined by revenue (the cart’s price times its number of customers) and cost (the carts each incur a cost of $0.25 per coconut times the number of coconuts sold). (a) Determine the expression for the number of customers served at each cart. (Recall that Cart 0 gets the customers between 0 and x, or just x. while Cart 1 gets the customers between xand 1, or 1 — x.) (b) Write out profit functions for the two carts and find the two best- response rules for their prices. (c) Graph the best—response rules, and then calculate (and show on your graph) the Nash equilibrium price level for coconuts on the beach. . The game illustrated in Figure 5.3 has a unique Nash equilibrium in pure strategies. However, all nine outcomes in that game are rationalizable. Con- firm this assertion, explaining your reasoning for each outcome. . The game illustrated in Figure 5.4 has a unique Nash equilibrium in pure strategies. Find that Nash equilibrium, and then show that it is also the unique rationalizable outcome in that game. . Section 4.B describes a fishing game played in a small coastal town. When the response rules for the two boats have been derived, rationalizability can be used to justify the Nash equilibrium in the game. In the description in the text, we take the process of narrowing down strategies that can never be best responses through three rounds. By the third round. we know that X (the number of barrels of fish brought home by boat 1) must be at least 9, and that Y (the number of barrels of fish brought home by boat 2) must be at least 4.5. The narrowing process in that round restricted X to the range be- tween 9 and 13.75 while restricting Yto the range between 4.5 and 7.5. Take this process of narrowing through one additional (fourth) round and show the reduced ranges ()f X and Ythat are obtained at the end of the round. — 154 [CH. 5] SIMULTANEOUSeMOVE GAMES WiTH PURE STRATEGIES 9. Nash equilibrium through rationalizability can be achieved in games with up- ward-sloping best-response curves if the rounds of eliminating never-best- response strategies begin with the smallest possible values. Consider the pricing game between Xavier's Tapas Bar and Yvonne’s Bistro that is illus« trated in Figure 5.1. Use Figure 5.1 and the best-response rules from which it is derived to begin rationalizing the Nash equilibrium in that game. Start with the lowest possible prices for the two firms and describe [at least] two rounds of narrowing the set of rationalizable prices toward the Nash equilibrium. 10. Optional, requires calculus Recall the political campaign advertising exam» pie from Section LB concerning parties L and R. In that example, when i L spends $x million on advertising and R spends $y million, L gets a share x/ (x + y) of the votes and R gets y/ (x + y). We also mentioned that two types of asymmetries can arise between the parties in that model. One panywsay, R—may be able to advertise at a lower cost or R’s advertising dollars may be more effective in generating votes than L’s. To allow for both possibilities, we can write the payoff funchons of the two parties as ky _ 3C 7 __ _ V]. - x x and VR — k cy. These payoff functions show that R has an advantage in the relative effec- tiveness of its ads when k is high and that R has an advantage in the cost of its ads when 6 is low. (a) Use the payoff functions to derive the best-response functions for R (which chooses y] and L (which chooses x). (b) Use your calculator or your computer to graph these best-response functions when k = 1 and c : 1. Compare the graph with the one for the case in which k : 1 and c : 0.8. What is the effect of having an advan- tage in the cost of advertising? (c) Compare the graph from part b, when k : 1 and c : 1 with the one for the case in which k 2 2 and c = 1. What is the effect of having an advan- tage in the effectiveness of advertising dollars? ((1) Solve the best-response functions that you found in part a, jointly for x and y, to show that the campaign advertising expenditures in Nash equilibrium are ck k = d : r. x [c+k)2 an y (Ht)Z (e) Let k : 1 in the equilibrium spending level equations and show how the two equilibrium spending levels vary with changes in c. The let c : 1 and show how the two equilibrium spending levels vary with changes in k. Do your answers support the effects that you observed in parts b and c of this exercise? ...
View Full Document

This note was uploaded on 04/14/2010 for the course ECON 398 taught by Professor Emre during the Spring '07 term at University of Michigan.

Page1 / 6

econchap4-5 - u. __, .‘u Aaluuk IUUA 1U]. d _ - u...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online