Games of Strategy - Ch 4, answers to unsolved PDF

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Warning: TT: undefined function: 32 Warning: TT: undefined function: 32Solutions to Chapter 4 ExercisesUNSOLVED EXERCISESU1. Find all Nash equilibria in pure strategies for the following games. First check for dominated strategies. If there are none, solve using iterated elimination of dominated str...

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Solutions to Chapter 4 Exercises UNSOLVED EXERCISES U1. Find all Nash equilibria in pure strategies for the following games. First check for dominated strategies. If there are none, solve using iterated elimination of dominated strategies. (a)

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For Colin, Right dominates Left, so Left is eliminated. Then for Rowena Up dominates Down, so Down is eliminated, giving the pure-strategy Nash equilibrium (Up,Right).

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Starting with Colin, Left dominates Right, so Right is eliminated. Then for Rowena Straight dominates both Up and Down, so both are eliminated. Then Left dominates Middle for Colin, so Middle is eliminated, leaving the pure-strategy Nash equilibrium of (Straight, Left).

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Beginning with Rowena, Straight dominates Up, so Up is eliminated. Then for Colin, Right dominates Middle is eliminated. Then for Rowena, Straight dominates Down, so it is eliminated, leaving only Straight. Right dominates Left, leaving only Right for Colin and giving the pure-strategy Nash equilibrium (Straight, Right).

(b)

(c)

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(d)

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The diversion is illuminated utilizing iterated strength. Colin has no iverwhelmed methodologies. For Rowena, Up overwhelms Down, so Down might be dispensed with. At that point

U2. For each of the four games in Exercise U1, identify whether the game is zero-sum or non-zero-sum. Explain your reasoning. (a) Not zero-sum. Payoffs sum to different totals in each cell (b) Not zero-sum. Payoffs sum to different totals in some cells. (c) Zero-sum (or constant sum). Payoffs sum to 8 in every cell. (d) Zero-sum (or constant sum). Payoffs sum to 10 in every cell. U3. As in Exercise S3 above, use the minimax method to find the Nash equilibria for the zero-sum games identified in Exercise U2. (a) Not zero-sum so minimax is not applicable. (b) Not zero-sum so minimax is not applicable. (c) The minima of Rowena’s strategies are 2 for Up, 3 for Straight, and 1 for Down, so Rowena chooses the maximum of 3, which comes from playing Straight. The minima of Colin’s strategies are 2 for Left, 2 for Middle, and 5 for Right, so Colin chooses the maximum of 5 which comes from playing Right the pure-strategy (minimax) Nash equilibrium (straight, Right). (d) The minima of Rowena’s strategies are 5 for Up, 3 for High, and 2 for Low, and 3 for Down, so Rowena chooses the maximum of 5, which comes from playing Up. The minima of Colin’s strategies are 2 for North, 3 for South, 5 for East, and 4 for West, so Colin chooses the maximum of 5 which comes from playing East, giving the pure-strategy (minimax) Nash equilibrium (Up, East). U4. Find all Nash equilibria in pure strategies in the following games. Describe the steps that you used in finding the equilibria. (a)

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Left dominates Right for Colin, so right is eliminated. With only left, down dominates up for Rowena, so the pure-strategy Nash equilibrium is (down, left)

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Neither player has any dominant strategies; best-response analysis shows that there are two pure-strategy Nash equilibria: (up, left) and (down, right).

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Down is dominant strategy for Rowena, so up may be eliminated. Then for Colin right dominates left, so left may be eliminated, leaving the pure-strategy Nash equilibrium of (down, right).

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Up dominates straight for Rowena, so straight is eliminated, but afterward, there are no more dominated strategies, so we cannot continue using iterated elimination of dominated strategies; that is, the game is not dominance solvable. We must analyse the remaining table (or even the full table) using best-response analysis to determine the pure-strategy Nash equilibria. They are (up, left) and (down, right).

(b)

(c)

(d)

U5. Use successive elimination of dominated strategies to solve the following game. Explain the steps you followed. Show that your solution is a Nash equilibrium.

Side 3 af 10

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Beginning with Colin, Middle dominates Right (remember that -2 < -1), so right is eliminated, leaving only left and middle for Colin. Then Down dominates Up for Rowena, so up is eliminated. Now Colin needs only to determine which action provides the large payoff against down, which is left, so the pure-strategy Nash equilibrium is (down, left)

U6. Find all of the pure-strategy Nash equilibria for the following game. Describe the process that you used to find the equilibria. Use this game to explain why it is important to describe an equilibrium by using the strategies employed by the players, not merely by the payoffs received in equilibrium.

U7. Consider the following game table:

(a) Complete the payoffs of the game table above so that Colin has a dominant strategy. State which strategy is dominant and explain why. (Note: There are many equally correct answers.)

(b) Complete the payoffs of the game table above so that neither player has a dominant strategy, but also so that each player does have a dominated strategy. State which strategies are dominated and explain why. (Again, there are many equally correct answers.) Side 4 af 10

U8. The Battle of the Bismarck Sea (named for that part of the southwestern Pacifc Ocean separating the Bismarck Archipelago from Papua New Guinea) was a naval engagement played between the United States and Japan during 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 route, both of which required three 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 reconnaissance planes to scout for the convoy, but they had only enough reconnaissance 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 that the Americans explored 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. If the Americans explored the northern route and found the Japanese right away, they could expect only two (of three) good bombing days; if they explored the northern route and found that the Japanese had gone south, they could also expect two days of bombing. If the Americans chose to explore the southern route frst, they could expect three full days of bombing if they found the Japanese right away but only one day of bombing if they found that the Japanese had gone north. (a) Illustrate this game in a game table.

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(b) Identify any dominant strategies in the game and solve for the Nash equilibrium.

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The Japanese Navy has a weakly dominant strategy to go North. The Americans will also go North. Equilibrium is (North, North) with payoffs of (2,-2).

U9. Two players, Jack and Jill, are put in separate rooms. Then each is told the rules of the game. Each is to pick one of six letters: G, K, L, Q, R, or W. If the two happen to choose the same letter, both get prizes as follows:

If they choose different letters, each gets 0. This whole schedule is revealed to both players, and both are told that both know the schedules, and so on. (a) Draw the table for this game. What are the Nash equilibria in pure strategies?

(b) Can one of the equilibria be a focal point? Which one? Why?

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U10. Three friends (Julie, Kristin, and Larissa) independently go shopping for dresses for their high-school prom. On reaching the store, each girl sees only three dresses worth considering: one black, one lavender, and one yellow. Each girl furthermore can tell that her two friends would consider the same set of three dresses, because all three have somewhat similar tastes. Each girl would prefer to have a unique dress, so a girl’s utility is 0 if she ends up purchasing the same dress as at least one of her friends. All three know that Julie strongly prefers black to both lavender and yellow, so she would get a utility of 3 if she were the only one wearing the black dress, and a utility of 1 if she were either the only one wearing the lavender dress or the only one wearing the yellow dress. Similarly, all know that Kristin prefers lavender and secondarily prefers yellow, so her utility would be 3 for uniquely wearing lavender, 2 for uniquely wearing yellow, and 1 for uniquely wearing black. Finally, all know that Larissa prefers yellow and secondarily prefers black, so she would get 3 for uniquely wearing yellow, 2 for uniquely wearing black, and 1 for uniquely wearing lavender. (a) Provide the game table for this three-player game. Make Julie the row player, Kristin the column player, and Larissa the page player.

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(b) Identify any dominated strategies in this game, or explain why there are none. - There are no dominated strategies. No player always does best choosing one colour. (c) What are the pure-strategy Nash equilibria in this game? - There are six pure-strategy Nash equilibria: (Yellow, Lavender, Black), (lavender, Yellow, Black), (Black, Yellow, Lavender), (yellow, black, Lavender), (Lavender, Black, Yellow) and (Black, Lavender, Yellow). U11. Bruce, Colleen, and David are all getting together at Bruce’s house on Friday evening to play their favourite game, Monopoly. They all love to eat sushi while they play. They all know from previous experience that two orders of sushi are just the right amount to satisfy their hunger. If they wind up with less than two orders, they all end up going hungry and don’t enjoy the evening. More than two orders would be a waste, because they can’t manage to eat a third order and the extra sushi just goes bad. Their favourite restaurant, Fishes in the Raw, packages its sushi in such large containers that each individual person can feasibly purchase at most one order of sushi. Fishes in the Raw offers takeout, but unfortunately doesn’t deliver. Suppose that each player enjoys $20 worth of utility from having enough sushi to eat on Friday evening, and $0 from not having enough to eat. The cost to each player of picking up an order of sushi is $10. Unfortunately, the players have forgotten to communicate about who should be buying sushi this Friday, and none of the players has a cell phone, so they must each make independent decisions of whether to buy (B) or not buy (N) an order of sushi. (a) Write down this game in strategic form.

(b) Find all the Nash equilibria in pure strategies.

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(c) Which equilibrium would you consider to be a focal point? Explain your reasoning.

U12. Roxanne, Sara, and Ted all love to eat cookies, but there’s only one left in the package. No one wants to split the cookie, so Sara proposes the following extension of “Evens or Odds” (see Exercise S12) to determine who gets to eat it. On the count of three, each of them will show one or two fingers, they’ll add them up, and then divide the sum by 3. If the remainder is 0, Roxanne gets the cookie, if the remainder is 1, Sara gets it, and if it is 2, Ted gets it. Each of them receives a payoff of 1 for winning (and eating the cookie) and 0 otherwise. (a) Represent this three-player game in normal form, with Roxanne as the row player, Sara as the column player, and Ted as the page player. - Game theory is the study of conflict and co-operation between intelligent rational decision makers. - Given person R, Person S and person T loves to eat cookies. As there is only one left no one wants to split the cookies. Below strategic game theory explains the splitting up of the cookie.

(b) Find all the pure-strategy Nash equilibria of this game. Is this game a fair mechanism for allocating cookies? Explain why or why not.

U13. (Optional) Construct the payoff matrix for your own two-player game that satisfies the following requirements. First, each player should have three strategies. Second, the game should not have any dominant strategies. Third, the game should not be solvable using minimax. Fourth, the game should have exactly two pure-strategy Nash equilibria. Provide your game matrix, and then demonstrate that all of the above conditions are true. Side 9 af 10

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