FROM THE EDITOR

K. ROSCOE DAVIS, Editor University of Georgia

Periodically I receive short articles from authors who provide insights into different issues in the decision sciences. One of these issues with which we all struggle is that "more information will lead to better decisions." Professor Ernest H. Forman (George Washington University) argues that this is not always true. Following is Professor Forman's discussion regarding this issue.

HOW ADDITIONAL INFORMATION CAN LEAD TO INFERIOR DECISIONS -- A PARADOX

by Ernest H. Forman, George Washington University

Abstract: More information does not necessarily result in better decisions. A paradox showing how more information can result in a decreased expected return is presented and explained.

An implicit assumption in many deciison-making situations is that more information will lead to a better decision. This, however, is not always true. Glazer, Steckel, and Winer [1] found "the rather ironic result that decision makers would be better off in some situations had the information not been available in the first place." This can be due to several factors, including information overload, the tendency to focus on objectives for which information is readily available, or the improper processing of information. An example of how easy it is to process information incorrectly is given in the following paradox which arose from a problem discussed at length by Morgan, Chaganty, Dahiya and Doviak in "Let's Make a Deal: The Player's Dilemma"[2].

Before examining the paradox, let us first look at the context for the decision. The decision problem appeared in a column titled "Ask Marilyn," in Parade Magazine [4]:

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, "do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

While most people intuitively decide to stay with their original choice, the average payoff from a decision to switch is twice as great as from a decision to stay with the original choice. There are several explanations of why it is better to switch ([1] contains explanations of several fallacious arguments that not switching is just as good). The most academic argument involves Bayes theorem. However, because Bayes theorem is not intuitive to many people, I decided to develop a simple Analytic Hierarchy Process [3] (AHP) model (see Figure 1) that I think is more intuitive. The purpose of this article is not to present an alternative solution, however, but to examine a paradox that arose when performing a sensitivity analysis on the AHP solution: if the decision maker is given more information (e.g., from being able to sniff at each of the doors before making the first choice), it appears that the expected return decreases!

The basic AHP model shows two possible states below the goal node: (1) that the contestant has chosen a door with a car, or (2) that the contestant has chosen a door with a goat. The decision alternatives--whether or not to switch after the MC reveals a door with a goat--appear below each of these "nodes."

The `L' on the diagram stands for "local" priority and designates a priority relative to a node's parent (node above). The `G' stands for "global" priority and represents the priority relative to the goal node, which in this case is to decide whether or not to switch doors. Local and global priorities for nodes directly below the goal are identical, and, in this decision, represent the likelihood that either a car door is chosen (.333) or a door with a goat (.667).

Looking at the nodes below the CAR DOOR, the diagram shows that if the car door is chosen, a decision to switch doors returns nothing (local priority of 0 for SWITCH). The reason is clear--if the car door is chosen and the decision is to switch, the decision maker will lose the car. Conversely, a decision to stay with the chosen door yields 1 car (local priority of 1). If these two local priorities are multiplied by the global priority of the parent node (CARDOOR = .333), the global priorities of 0 and .333 are obtained for the SWITCH and NOSWITCH alternatives under the CARDOOR node.

Now looking at the elements below the GOAT DOOR, we see that if a goat door is chosen, a decision to switch doors yields a car (local priority of 1). This is because a switch is being made from a door with a goat and the MC has eliminated the other door with a goat so the only door left must have the car. Conversely, not switching yields zero. The global priorities of the nodes below GOATDOOR are again the product of the local priorities and the global priority of the parent: .67 and 0 for the SWITCH and NOSWITCH alternative, respectively.

The synthesized results shown in Figure 2 indicate that the expected return from a decision to switch is .667 cars, twice the expected return of the decision not to switch.

The Paradox of Additional Information

Now suppose that the contestant is allowed to "sniff" each door before making the original choice. Suppose that, even with sniffing, the contestant could not tell for certain which doors have the goats, but the additional information gained from sniffing increases the probability of being able to choose the car door from 33 1/3% to something like 45%. What would this do to the expected return?

This can be seen with a "sensitivity analysis" graph (see Figure 3). The vertical axis represents the expected value from playing the game and the horizontal axis is the probability of initially choosing the door with the car. The vertical line at .333 on the horizontal axis represents the likelihood of initially choosing the car door. The heights of the intersection of this line with the two lines for the decision alternatives, SWITCH and NOSWITCH, show the expected returns for the two decision alternatives. Looking at these intersections we see that it is better to SWITCH since the expected return (height) is about .667 as opposed to .333 for the NOSWITCH decision.

If, however, the contestant can sniff at each door before making the initial choice, and, by doing so, can increase the likelihood of choosing the car from 33 1/3% to say, 45%, the vertical line would move to the right and the expected return (the height of the intersection with the SWITCH alternative) decreases. Thus we have the paradox that more information leads to a diminished expected return. What does this imply to the decision maker? Should he or she ignore the additional information?

Note: We suggest that the reader develop his or her own explanation before reading our explanation below.

Explanation of Paradox

Simply having more information does not ensure a better decision. Care must be taken to use the information properly. If the additional information obtained from sniffing is used to increase the probability of choosing the car from .333 to, say, 45, the expected return does indeed go down. However, the information gained from sniffing can be used instead to increase the likelihood of picking a door with a goat, i.e., avoiding the choice of the door with the car. If this is done, the probability of choosing the car door would decrease. Thus, the vertical line in the graph would move to the left and the expected return would increase.