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Sequential Games

Unlike spatial and hybrid games, sequential games do not order elements in space. Sequential games can be classified according to the criteria used to order the elements:
  • Chronological (before, after, etc.)
  • Quantifiable (size, height, etc.)
  • Ranking (first, second, etc.)
Chronological Games
Chronological games order elements in a time-sequence.
 
For example, James was born before George who was born before Kim who was born before Sara.

In the line-up games that we studied earlier, the elements were ordered spatially. In chronological ordering games, the elements are ordered sequentially. This is true of many of the games that we will study in this chapter. Because these games are sequential in nature, their diagrams can be quite different from those used to solve spatial and hybrid games.

One of the most common and efficient types of diagrams is the flow chart. In these diagrams the elements are connected by arrows.*

Now that we have a second way to diagram linear ordering games, we need, of course, some means of deciding which method to use.
 
In general, a game with no fixed elements should be solved using a flow chart.

In constructing a flow chart, follow these guidelines:
  1. Look for a condition that starts the “flow”.
  2. Build the chart around the element that occurs in the greatest number of conditions.
  3. Keep the chart flexible; it will probably have to evolve with the changing conditions.
An example will illustrate the flow chart method of diagramming:

Chronological Game

Eight people—S, T, U, V, W, X, Y, Z—were each born in a different year, 1971 through 1978. The following is known about their ages.
 
W is older than V.
 
S is younger than both Y and V.
 
T is not younger than Y.
 
Z is younger than Y, but older than U.

This game does not have any fixed elements (such as T was born in 1972), so the flow chart method is indicated. We’ll use an arrow to indicate that one person is older than another. The condition “W is older than V” is naturally symbolized as W—>V. The other conditions are symbolized in like manner, giving the following schematic: 
  1. W—>V
  2. Y—>S
  3. V—>S
  4. T—>Y
  5. Y—>Z
  6. Z—>U
Now we construct the diagram. Following the guidelines on page 93, look for the element that occurs in the greatest number of conditions. It is Y, so we build the chart around Y. Start with condition 4:
T—>Y

Adding conditions 2 and 5 gives


Adding conditions 1 and 3 gives


Finally, adding condition 6 gives

 
There are no conditions on the element X, so it can not be placed in the diagram. However, we note it below the diagram as follows:

 

Two properties of the diagram should be noted before turning to the questions. First, if two elements are in different rows and no sequence of arrows connects them, then the diagram does not tell us which one is older.
 
For example, since W and Y are in different rows and are not connected by a sequence of arrows, the diagram does not tell us who is older. However, the diagram does tell us that T is older than U, because the arrows “flow” from T to Y to Z to U. Second, the diagram tells us that only T, W, or X can be the oldest, and likewise that only U, S, or X can be the youngest.
 
Example-1
Which one of the following is a possible sequence of births from first to last?
  1. W V T Y Z U X S
  2. W V T U Y Z X S
  3. U T Y W V S Z X
  4. T W Y S Z U X V
  5. T Y W V S U X Z
Solution
This is a straightforward elimination problem. (B) and (E) are not possible sequences because the diagram shows that Z must be older than U. (C) is not a possible sequence because the diagram shows that T must be older than U. The arrows “flow” from T to Y to Z to U. Finally, (D) is not a possible sequence because the diagram shows that V must be older than S.
 
Hence, by process of elimination, the answer is (A).
 
 
Example-2
If S was born in 1975, which one of the following must be false?
  1. V was born in 1973.
  2. V was born in 1972.
  3. Z was born in 1977.
  4. Y was born in 1974.
  5. Z was born in 1974.
Solution
The diagram shows that T, Y, W, and V—not necessarily in that order—were all born before S. So they must have been born in the years ‘71 through ‘74. This gives the following possible diagram.
(Because one of the births, S, is fixed, it is now more convenient to use a line-up diagram.)
 
71 72 73 74 75 76 77 78
T Y W V S      
 
Clearly, this diagram shows that Z must have been born after ‘75. Choice (E), therefore, makes the necessarily false statement.
The answer is (E).
 

 

Example-3
If S was born in 1976 and X was born in 1973, then the year of birth of exactly how many other people can be determined?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
Solution
Since two births—S and X—are fixed, we again revert to a line-up diagram:
 
71 72 73 74 75 76 77 78
    X     S    
 
The original diagram shows that W, V, T, and Y were all born before S, so they must be placed to the left of S on the new diagram. However, we cannot uniquely determine their positions: W and V could have been born in ‘71 and ‘72, respectively, or T and Y could have been. So one possible diagram is
 
71 72 73 74 75 76 77 78
W V X T Y S    
 
Clearly this diagram forces Z and U to have been born in the years ‘77 and ‘78, respectively. Hence, only two other births can be determined.
 
The answer is (C).
 
 
Example-4
If S was born before X, then which one of the following could be true?
  1. Z was born before T.
  2. V was born before W.
  3. U was born before S.
  4. W was born after S.
  5. W was born in 1976.
Solution
Add the condition “S was born before X” to the diagram:


In this diagram U and S are in different rows and are not connected by a sequence of arrows, so U could have been born before S.
 
The answer, therefore, is (C).
 
 
Example-5
If the condition “S is younger than both Y and V” is dropped, then the year of birth of exactly how many people could be determined?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
Solution
The condition “S is younger than both Y and V” anchored the diagram. Without it we get the following diagram:
 
T—>Y—>Z—>U
 
W—>V
 
(X and S “wild”)

Because the two parts of this diagram are independent (they are not connected by a sequence of arrows), W and V could have been born before T or after U. Hence, the year of birth cannot be determined for any of the people.
 
The answer is (A).
 

Points to Remember

  1. The three types of sequential games are
    • Chronological (before, after, etc.)
    • Quantifiable (size, height, etc.)
    • Ranking (first, second, etc.)
  2. Most sequential games can be solved most efficiently with a flow chart.
  3. In general, a game with no fixed elements should be solved using a flow chart.
  4. When constructing a flow chart, follow these guidelines.
    • Look for a condition that starts the “flow”.
    • Build the chart around the element that occurs in the greatest number of conditions.
    • Keep the chart flexible; it will probably have to evolve with the changing conditions




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