DELTA's view point of the problem

Basic characteristics

DELTA belongs to the area of DECISION TOOLS.

The name DELTA comes after its primary decision rule. This rule uses the concept of dominance in order to determine which set of consequences is more determinative in the final result.

The objective of DELTA is to present a method for evaluating choices under uncertainty. The emphases is on evaluation, and the representation used is that of standard probability theory. The DELTA method deals with decision situations modelled as a discrete number of alternatives with discrete outcomes.

DELTA handles decision problems in normal form.

Definition. A problem in a normal form means that a decision situation can be modelled in the following way : - First, the player (= the decision maker) selects one alternative from all strategies available to him. - Thereafter, nature selects one outcome from the available set of consequences, whose members might be dependent on the player's previous selection.

DELTA represents the information in the form of a decision tree. The tree consists of three types of nodes : decision nodes, chance nodes, and result nodes. DDT, the software that implements DELTA, supports only one level in the tree.

DELTA method selects the best way to act, given a decision situation and some estimates.

DELTA can handle imprecise information by the decision maker, using the concepts of probability base, value base and security levels, giving the possibility to specify ranges of values and probabilties instead of specific points.

In order for DELTA to help to take a decision, it is necessary to translate the problem into a decision frame.

The decision frame is composed by

- the structural information

- the probability base ( set of probability statements. ex. The event H1 is probable, )

- the value base (set of value statements. ex. The event H1 is desirable)

In a formal way.

Definition. Given a decision with m alternatives (A1,...,Am), each with mi consequences, and statements about the probabilities and values of these consequences. a decision frame is a structure <C,V,P>= <{{Cik}mi}m,P,V> containing the following information of the situation: - for each alternative Ai the corresponding consequence set {Cik}k Î Ki for Ki = {1,2,...,m}. - A probability base P containing all probability statements in the form of constrains and a core. - A value base V containing all value statements in the form of constrains and a core.

You can get much more details about DELTA and DDT at DDT's home page

Solving the problem

Delta method is the mathematical model, and DDT is the software that implements it.

DDT doesn't allow to specify more than one level in a decision tree. So I will select a particular situation.

Suppose the cells are addressed from the top to the bottom and from left to right as usual. Then, consider that the current location of the agent is l = (1,1), and the information of the field is as presented in the figure.

Figure 2 shows a decision situation for agent L. L is at the location 1,1 and the way out is at the location 0,3. The cost in time is represented by the number on the corner of each cell. The vector (D,Pa) represents the information of the mine, where D is the destructive level and Pa is the probabilistic function for A. The current time is 4, and the amount of damage of L is 15.

In order to simplify the problem, I will consider only the cells (0,0), (0,2), (1,1). It means that there are 3 alternatives L is facing.

If we name the ith alternative as Ai, where i is one of (0..5), we get the list :

A0 = L moves to (0,0),

A1 = L moves to (0,2),

A2 = L moves to (1,1), the agent decides to stay until the conditions around change.

For each alternative Ai we have two different consequences Ci0 and Ci1 with probabilities Pi0 and Pi1 respectivelly. We also have, for each Ci0 and Ci1, assigned values Vi0 and Vi1 respectivelly.

How to calculate the value for each consequence?

Every time L receives a damage, its condition C is updated in this way : C <- C-D. But is this enough?. What about the cost in time? Ok, we could consider the formula :

C <- C -D -PD, where PD = 5*(Tc+1),

PD is the potential damage at a certain location, and Tc is the cost in time (the little number at the corner). Why 5? I just compromise; the maximum damage is 10 and the minimum is zero (no damage at all). Why plus one? Because you can receive damage while you stay in a cell.

We then obtain the following table.

Table 1 shows the relation consequence-probability-value the agent L is facing. Ci0 represents mine active, and Ci1 represents mine not active.

Unfortunatelly, I could not use the tool DDT to get the result, because I don't have a license.

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