# Choice Model by Structural Instability Theory

Considering a system whose behaviour is shown by the vector of variables X in the space of the states R^{n} and which is controlled by the vector of the variables Z in the control space R^{m}. If V: from R^{n} x R^{m} to R is a differential function, then for a point Z in the control space it is obtained the partial function from X to V_{z}(X), called potential function (Thom) or energy function (Zeeman). It results the map M of all the critical points of the potentials V_{z} of the family V, called a catastrophe variety which is represented by a behaviour surface, where X = (x_{1},….,x_{n}) in R^{n} and Z in R^{m }.

The stable local regimes of the system take place where the potential is minimum, according to Maxwell’s convention. It is noticed that the system must be of gradient type, which may be defined by means of the vector field F(X,Z) = – grad V(X,Z) so that when the system is moving the potential V is minimized. It is not needed the explicit knowledge of the function V, that may represent costs, energy, entropy, probability function, Lyapunov function of the equation system describing the system etc.

By minimizing the potential V_{z}, for an established command Z, the system shall evolve to a balance point X_{0}. It is proven that a critical point is structurally stable (non-sensitive to small perturbations) if the critical point is not degenerated.

Hereinafter, we show some exemplifications for the case n = 1, m = 2, defining the elementary catastrophe called cusp in which X is the state variable expressing the system behaviour (its output), and a and b are control variables (system input).

It results the equation of the behaviour surface, being the set of the critical points of the potential when the variables (a, b) move across the control surface: x^{3} + ax + b = 0.

The set S of singularities of the behaviour surface meets also the condition: 3x^{2} + a = 0.

The two branches of the singularities’ set are projected on control surface by the bifurcation set modified by the number and nature of the critical points of the potential. When passing through a point in the singularities’ set, a catastrophe occurs, i.e. a smooth variation in the control space generates a discontinuity, a jump in the behaviour space.

A point of demand/offer balance is actually a choice decision, practiced by the both sides generating the demand, and respectively the offer for products or services.

We shall try to define a model starting from a cusp catastrophe [Cuncev I., Theoretical concepts on transport mode choice, IM7Danube, Seminar 14 July 2006, Bucharest], such as the behaviour variable x becomes probability p_{2} for the selection of the alternative 2 considering the control variables b and a in which, b = c_{1} – c_{2} represent the difference of costs between the two alternatives in the conjuncture expressed by variable a (the oil price, the political situation, psychological factors etc). The behaviour variable may be expressed in different ways, through the probability that can have values from 0 to 1, as well as through absolute frequencies of the possible selections, and also through other expressing forms familiar to the decision-maker.

a. **Selecting a variant**

Let’s consider the following decision-making scenarios:

**Scenario a _{1}**– selection under permanent doubt, both variants are in competition, and the preference for the two variants is symmetrical.

**Scenario a _{2}**– the decisional inertia becomes significant by the decision-maker fidelity towards the current variant 1. Consequently, the probability of selection of alternative p

_{2}increments suddenly (jump) to reach the bearable limit of the tolerance of the cost difference b = c

_{1}– c

_{2}. In this case, the bearable limit is also a panic limit. The probability of selecting alternative p

_{2}is positive, although of a low value, even for certain negative values of the difference b, meaning that alternative 2 offers other utility value compensating somehow the cost higher than the current variant 1.

**Scenario a _{3}**– the decisional inertia or the fidelity towards the current variant is bigger, but the probability of selecting alternative p

_{2}jumps to even higher values.

**Scenario a _{4}**– inertia is excessive, the current variant is being maintained even for certain positive values b.

b. **Change of mind on the selected item**

Once selected the success alternative 2, the decision-maker may be bound to take into consideration the coming back to the previous variant 1. The reasons are included in the decreasing of b. It may be noticed a hysteresis in relation to the selection of the alternative: probability of giving-up, changing one’s mind or coming back to the previous variant will drop to lower values, for a value of b much lower than the value at which the probability jumped to higher values.

Some benefits of the model: it is considered the decisional inertia, the fidelity for current preference; the behaviour variable changes considering the parameter ‘a’ of circumstance: oil price, psychological factors etc.; the model distinguishes between the behaviour of the decision-maker selecting a variant or changing their mind (hysteresis).