One Person Games

Some problems from control theory, which derived from game theory, can be called one person games (note that this is not a specialistic term). They are grouped in many different categories of control theory. However we can often create an imaginary disinterested player who does not care about the outcome to transform those control problem to the language of game theory. We can call this player nature or chance.

A good example is a simple version of famous stopping problem which is often explained as choosing the best secretary. From a certain number of secretaries the best one must be chosen. Each lady comes ones to the office and choosing person must decide to employ her or to let her go and search further. The question is: when to stop? which secretary choose? Once a secretary is employed we stop looking for other persons in her place, so we never know if she was really the best one. But if we wait to long, it may happen that we learn we have missed better poeple for the post.

Our game: to put it simply instead of examining a secretary you have to choose the longest stick.

There are 5 straws. You have to select one of them and the payoff depends on the selected straw (varying from 1 to 5). The highest payoff is for the longest straw. You do not know in advance straws longitudes. The straws are displayed at random one at a time, one after another. Each straw is displayed only once. When you stop the game choosing the straw, you will obtain the payoff according to longitude of chosen straw. You will never see rest of straws, they are just rejected. Try it!

Solving the problem:

The only information about length of straws that player has is information about straws he already saw. That means during the selection process information about length increase after each new straw is shown. In our case of 5 straws, the optimal strategy is to simply note the length of the first two straws and then choose the first longer straw. If there are no straws longer than the first two, the player has to choose the last one.

Simplifying, the best strategy is to let first two pass and then choose the first longer than was we have seen.

Since the person has not seen all the straws till the end, the action is based on imperfect information. Examples of this are numerous in life and are easy to find.

1. Dauben, Joseph W "Game Theory." Microsoft Encarta. Microsoft Corp, 1999
2. McCain, Roger A. "Game Theory: An Introductory Sketch." Hypertext. 1999 (August 1999)
3. Morgenstern, Oscar. "The One-Person Game" Game Theory, A Nontechnical Introduction. New York: Basic Books, Inc., 1930, pp 3-7

 
This file was last modified on Wednesday, 22-Sep-2010 12:37:54 PDT