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Assignment Games

We have discussed various ways to order elements—ordering games, and various ways to group elements—grouping games. Now we will discuss various ways to assign characteristics to elements—these are assignment games. Assignment games will wind up our discussion of the three major types of games. They tend to be the hardest games, so it’s wise to save them for last.

ssignment games match a characteristic with an element of the game.
For example, you may be asked to assign a schedule: Bob works only Monday, Tuesday, or Friday. Or you may be told that a person is either a Democrat or a Republican.

Because the characteristics are typically assigned to groups of elements, assignment games can look very similar to grouping games. Additionally, in grouping games the groups are often identified by their characteristics. However, in assignment games you pair each element with one or more characteristics, whereas in grouping games you partition the elements into two or more groups.

It is important to identify the type of game you are dealing with because different methods are needed to solve each type. The following examples illustrate the distinction between these two kinds of games. The first, from the grouping games chapter, is the Olympic Dream Team game. The second is an assignment game.

Example: (Grouping game)

The starting line-up for the Olympic basketball “Dream Team” is chosen from the following two groups:
Group A Group B
Johnson, Drexler, Bryant, Ewing Laettner, Robinson, James, Malone, Pippen
The following requirements must be meet:
Two players are chosen from Group A, and three from Group B.
Jordan starts only if Bird starts.
Drexler and Bird do not both start.
If Jordan starts, then Malone does not.
Exactly 3 of the four fast-break specialists—Johnson, Bird, Jordan, Pippen—must be chosen.
In this game the goal is to select the starting line-up, thereby dividing the elements into two groups: those selected, and those not. Notice how the goal in the following game differs:

Example: (Assignment game)

There are eight players on a particular basketball team—A, B, C, D, F, G, H, I. Three are guards, three are forwards, and two are centers. Each player is either a free agent or not a free agent.
All of the guards are free agents.
A and C are forwards; F and H are not forwards.
Only one forward is a free agent.
In this game the team has already been selected; now the goal is to assign a position (characteristic) to each player and decide whether he is a free agent (characteristic). Notice that conditions, such as “All of the guards are free agents,” apply to groups of individuals; this makes the game at first glance appear to be a grouping game.

Many assignment games can be solved very efficiently by using a elimination grid. An example will illustrate this method of diagramming.

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