In a sideshow game. A player gets 3 balls to place into a Clown's mouth. Each ball is swallowed by the Clown and is then deposited into slots that can hold only 1 ball. Slots are numbered 1 to 9 and prizes are allocated to a player depending on the slot value of the three balls. If for example the first ball landed in slot 8, the second in slot 2 and the third in slot 1, the resulting prize number would be 821 as shown in Figure 3. 3rd 2nd DOL 1 2 3 4 5 6 6 1st JOL 7 7 8 9 Figure 3: A prize number of 821 is shown for a side show game. a) How many resulting prize numbers are possible in this game? b) How many resulting prize numbers are possible that end with the number 1?

In a sideshow game. A player gets 3 balls to place into a Clown's mouth. Each ball is swallowed by the Clown and is then deposited into slots that can hold only 1 ball. Slots are numbered 1 to 9 and prizes are allocated to a player depending on the slot value of the three balls. If for example the first ball landed in slot 8, the second in slot 2 and the third in slot 1, the resulting prize number would be 821 as shown in Figure 3. 3rd 2nd DOL 1 2 3 4 5 6 6 1st JOL 7 7 8 9 Figure 3: A prize number of 821 is shown for a side show game. a) How many resulting prize numbers are possible in this game? b) How many resulting prize numbers are possible that end with the number 1?

Principles of Microeconomics

7th Edition

ISBN:9781305156050

Author:N. Gregory Mankiw

Publisher:N. Gregory Mankiw

Chapter22: Frontiers Of Microeconomics

Section: Chapter Questions

Problem 6PA

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Cameron and Luke are playing a game called ”Race to 10”. Cameron goes first, and the players take turns choosing either 1 or 2. In each turn, they add the new number to a running total. The player who brings the total to exactly 10 wins the game. a) If both Cameron and Luke play optimally, who will win the game? Does the game have a first-mover advantage or a second-mover advantage? b) Suppose the game is modified to ”Race to 11” (i.e, the player who reaches 11 first wins). Who will win the game if both players play their optimal strategies? What if the game is ”Race to 12”? Does the result change? c) Consider the general version of the game called ”Race to n,” where n is a positive integer greater than 0. What are the conditions on n such that the game has a first mover advantage? What are the conditions on n such that the game has a second mover advantage?
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