Dr Marek Jenöffy
email: mj@bueroamcarlsplatz.de

The Origin of Preferences, Economic Decision Theory, and Yaari

Keywords: Information, Uncertainty, Preferences, Communication, Preference Formation, Endogenous JEL: D81, D82, D83

Preferences" is a key concept in economic theory. However, economics has yet not achieved to explain the origin of preferences. Preferences originating means endogenous preferences. The Signal-Preference Model (SPM) provides a new algorithm for modeling endogenous preferences using the empirical result of communications research. This paper examines how the SPM performs in the framework of economic decision theory. The results of the SPM are compared with Expected Utility Theory, Bayesian learning and Prospect Theory. Surprisingly, the performance of the SPM in this field indicates that this holistic preference model seamlessly fits into microeconomics and decision theory.

Preferences and Economic Decision Theory

A main concept of economic theory is the differentiation between preferences and restrictions. Kirchgässner (2008, p.12) described this concept. "In a given situation the restrictions limit the individual's leeway of action.… Within this leeway, there are various alternatives of acting which are available and from which the individual can choose…According to these preferences, the individual assesses the various alternatives at his disposal, he weighs up the pros and cons, the costs and benefits of the alternatives against each other and finally chooses that (those) alternative(s) which come(s) closest to his preferences or which promise(s) to bring about the maximum net benefit."

However, the origin of the preferences is vague. Kirchgässner (2008, p.12) explained where preferences origin. "The preferences are derived from the intentions of the individual, they reflect the individual's ideas of value as they have been developed during the process of socialization, and they are principally independent of the actual possibilities of action." This thought raises questions: (1) How do preferences accrue during the socialization? (2) When does the socialization end? (3) Can preferences change after the process of socialization has ended? Or, (4) Is each child is born with an unchangeable grid of preferences?

There is little research in economics to clarify the conundrum of the origin of preferences. There is work by Krackhardt (1998) and [START_REF] Jenöffy-Lochau | Medien, Propaganda und Public Choice[END_REF], 2012, 2013[START_REF] Jenöffy-Lochau | The Origin of Preferences, Economic Decision Theory and Yaari[END_REF].

On the other hand, there is a huge literature in communications research how "opinions" form. There are countless empirical studies that examine how "signals" influence the "opinions" of "recipients." "Credibility" is an important variable. It is the opinion of the author that the different wording should scholars not prevent from looking and the results of the research.

Economic Decision Theory

Economic decision theory also links information to preferences. Von [START_REF] Von Neumann | The Theory of Games and Economic Behavior[END_REF] developed the concept of expected utilities. The basic model assumes that the preferences of individuals for alternatives depend on "states" and "probability of states." In the basic model, a state is a financial outcome that is assigned a probability. New information may alter this probability. The preferences of the individual are stable. The individual generally acts rationally and decides in favor of the alternative with the highest expected utility.

Bayes' Theorem [START_REF] Savage | The Foundation of Statistics[END_REF] is an endogenous way of depicting new information in an expected utility function. A signal can alter a "prior probability" before receiving the signal to a "posterior probability" after receiving the signal. The evaluation of the influence of a signal on the preferences of an individual needs exogenous given conditional probabilities of a signal given a state. Several subsequent studies found that individuals followed Baysian patterns (DeAngelo and Charness, 2009;[START_REF] Sloan | Deterring Re-arrests for Drinking and Driving[END_REF][START_REF] Sloan | Deterring Domestic Violence: Do Criminal Sanctions Reduce Repeat Offenses[END_REF]. Other studies focusing on ambiguity didn't find this pattern [START_REF] Poinas | Updating Beliefs with Imperfect Signals: Experimental Evidence[END_REF]Roussillon, 2012, andBaillon, Cabantous andWakker, 2012).

Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] developed Prospect Theory. This theory analyses "paradoxes" of Expected Utility Theory. Kahneman and Tversky work with an "editing phase" and an "evaluation phase." Individuals attach subjective "values" to the alternatives. These values determine the decision the individual makes, not the pure "consequence" of the alternative. There are several competing similar models by [START_REF] Chew | A Generalization of the Quasilinear Mean With Applications to the Measurement of Income Inequality and Decision Theory Resolving The Allais Paradox[END_REF], [START_REF] Machina | Expected Utility' Analysis Without the Independence Axiom[END_REF], [START_REF] Quiggin | A Theory of Anticipated Utility[END_REF] and Tversky and Kahneman (1992). Recently, [START_REF] Blavatskyy | Preference Reversals and Probablistic Decisions[END_REF] and [START_REF] Lindsay | The Arguments of Utility: Preference Reversals in Expected Utility of Income Models[END_REF] examined "preference reversals" in those models. Kahneman and Tversky measured a value function as being concave above a reference point and convex below. Several subsequent papers analyzed references points. [START_REF] Kőszegi | A Model of Reference-Dependent Preferences[END_REF] modeled endogenous reference points. [START_REF] Park | Loss Aversion and Consumption Plan with Stochastic Reference Points[END_REF] modeled the influence of reference points on consumption. Yaari (1987) developed Dual Theory. He transformed outcomes of decision problems into a [0,1] interval resulting in different axioms in comparison to Expected Utility Theory. [START_REF] Bikhchandani | A Theory of Fads, Fashion, Custom, and Cultural Change as Information Cascades[END_REF] and [START_REF] Banerjee | A Simple Model of Herd Behavior[END_REF] worked with signals in their models of information cascades and herd behavior. They are on the verge of disconnecting information that signals transfer from "states" and "probabilities of states." [START_REF] Viscusi | Risky Investment Decisions: How are Individuals Influenced by Their Groups[END_REF] measured that "groups" influence behavior even if people have full knowledge of probabilities and payoffs.

Empirical Results in Communications Research

As mentioned above, there are numerous empirical studies by communications researchers that analyzed the role of information in decision-making processes. [START_REF] Hovland | The Influence of Source Credibility on Communication Effectiveness[END_REF] concluded that credible information changed the decision of 23% of the individuals. Higher credible information is higher likely to change a decision than less credible infirmation. Whittaker (1964) examined that the "distance" between a new information and existing preferences determines whether a decision is reversed by an individual ("cognitive dissonance," "boomerang effect"). Freedman (1964) conducted experiments with "high-involvement" and "low-involvement" decisions. Under low involvement, he found greater changes in decisionmaking after the exposure to new information. [START_REF] Bochner | Communicator Discrepancy, Source Credibility, and Opinion Change[END_REF] conducted experiments on the credibility of new information. Highly credible information (a "Nobel prize winner"), changed decisions linearly. [START_REF] Laroche | A Model of Attitude Change in Groups Following A Persuasive Communication: An Attempt at Formalizing Research Findings[END_REF] structured the findings of various experiments and identified several variables, e.g., credibility. The concept of credibility goes back to Aristotle [START_REF] Rapp | Aristotle's Rhetoric[END_REF]. Several authors found an impact of food advertisements on the eating behavior of children (Viner and Cole, 2005;Beaudoin et al., 2007;[START_REF] Dixon | The Effects of Television Advertisements for Junk Food Versus Nutritious Food in Children's Food Attitudes and Preferences[END_REF][START_REF] Halford | Beyondbrand effect of Television (TV) food advertisements / commercials on calorie intake and food choice of 5-7-year-old children[END_REF]Zimmerman and Bell, 2010;[START_REF] Boyland | Television Advertising and Branding: Effects on Eating Behavior and Food Preferences in Children[END_REF][START_REF] Boyland | Television Advertising and Branding: Effects on Eating Behavior and Food Preferences in Children[END_REF]. The fact that information and credibility of information influences behavior is common knowledge in Communications research. "States" or "probabilities of states" are not used in this empirical research.

Useful Critics

In advocating for a holistic model of preference formation, Hausman (2012, p.118) identifies four sources of preferences. (1) Means-end reasoning, (2) attribute-based valuation, (3) emotional influence, and (4) physiological needs. He attributes the first two sources to economics and the latter two to psychology. According to Hausman, "means-end reasoning," which is basically economic decision theory as described above, is only one of four possible sources of preferences.

Insofar as it analyzes lotteries, Expected Utility Theory, Bayesian learning, Prospect Theory and Dual Theory deal with relatively rare cases that do not occur so frequently in "real life." This has triggered criticism, e.g., by Machina and Taleb. "Outside the gambling hall most economic decisions…are defined in terms of uncertain events rather than numerical probabilities" (Machina, 2003, p.25). [START_REF] Taleb | The Black Swan[END_REF] created the concept of the "Black Swan" as an allegory for unexpected events. How prevalent would be advertisement that contained information about lottery choices or probabilities of states? In business, individuals usually receive information that contains an information value. Credibility of information is a crucial variable.

The Signal-Preference Model of Endogenous Preferences Explaining the Origin of Preferences

These above critics, the intention of explaining the origin of preferences, and the countless empirical results of communications research lead to a radical new model of preferences.

The Signal-Preference Model (SPM) models preferences based on Hausman's four sources creating a broader model of preference formation. The SPM goes back to [START_REF] Jenöffy-Lochau | Medien, Propaganda und Public Choice[END_REF], 2012, 2013) and further develops the models of Bikhchandani et al. and Banerjee. There are two basic methodological differences. First, the model does not work with "states." Second, the model does not work with "probabilities of states." Instead, it uses the information value of a signal and the credibility of the signals. Preferences are the aggregated information received in past periods. I define the variables as it follows:

Definition 1: A "signal" is the transmission of exactly one information from a source of the signal to an individual.

Definition 2: All information is located between 0 and 1.

Definition 3: The "information value" of a signal describes the location of the signal an interval between 0 to 1. The information value of a signal is named ω.

Definition 4: The "credibility" of a signal describes whether the "information value" of the signal is believed to be true by the individual who receives the signal. The credibility of a signal is named γ.

Including the credibility of the signals, the SPM makes the following assumptions.

There

𝜔 𝛾 = .5 * ( 1 -𝛾 ) (1) Equation 2 defines the preference function PI of an individual with N ∈ N, ω ∈ [0,1], PI ∈ [0,1], |γ| ∈ [0,1], 𝛾̅ ∈ [0,1], 𝛾̅ ∈ [-1,0] and Nπ1 ∈ N, Nπ2 ∈ N. 𝑃 = 1 𝑁 . 5 * 1 -𝛾 = 𝑁 * .5 * (1 -𝛾̅ ) + 𝑁 * .5 * (1 -𝛾̅ ) 𝑁 (2)
It is ≤ 0; ≥ 0; ≤ 0 and ≤ 0. Ω1 is the number of new signals of the type π1 that causes a swing from Alternative 2 to Alternative 1.

𝛺 > 𝑁 -𝑁 𝛾̅ (3) 𝛺 > 𝑁 -𝑁 -𝛾̅ (4)
Ω2 is the number of new signals of the type π2 that causes a swing from Alternative 1 to Alternative 2.

𝑃 (𝛾̅ ) = 0. For Nπ2 > Nπ1 and |𝛾 | > |𝛾 | there is lim → 𝑃 (𝛾̅ ) = 1. If |𝛾 | > |𝛾 | there is lim ( )→ 𝑃 (𝑁 -𝑁 ) = 0. If |𝛾 | > |𝛾 | there is lim ( )→ 𝑃 (𝑁 -𝑁 ) = 1.

Transformation of the Outcomes into the Preference Interval

Lotteries generally work with outcomes Xj. There are normally few restrictions on these outcomes, so Xj ∈ R. The SPM, as developed above, works in the [0,1] interval of preferences (Equation 2). When outcomes are received that are outside [0,1], they are transformed into this interval. Dual Theory by Yaari (1987) worked with this method. Yaari transformed outcomes of lotteries so as to fit them into [0,1]. In order to scale the outcomes into the [0,1] interval, the outcomes Xj are transformed into ωj.

𝜔 = ⎩ ⎪ ⎨ ⎪ ⎧ .5 - 𝛾 * 𝑋 2 * max 𝑋 𝑖𝑓 𝑋 𝑋 ≥ 𝜉 .5 - 𝛾 2 𝑖𝑓 𝑋 𝑋 < 𝜉 (6)
2 Other 𝑓 𝑋 tested were

𝜔 = 0.5 + 𝛾 2 - 𝛾 * 𝑋 max 𝑋 (6𝑎) 𝜔 = ⎩ ⎪ ⎨ ⎪ ⎧ 0.5 - 𝛾 2 * 1 - 𝑋 max 𝑋 𝑖𝑓 𝑋 ≠ 0 0.5 - 𝛾 2 * 1 + 𝑋 max 𝑋 𝑖𝑓 𝑋 = 0 (6𝑏) 𝜔 = 0.5 + 𝛾 2 -𝛾 * l ( 𝑋 𝑋 )² (6𝑐)
In order to measure the performance of the SPM, I use the probabilities of states (pj) provided by the lotteries as credibility of the information value of the signals (γj). To do so, I use Equation 6with p1, p2 ∈ [0,1] and 𝛾 ∈ [0,1]. There is

𝛾 = 𝑝 𝑓𝑜𝑟 𝑠𝑖𝑔𝑛𝑎𝑙𝑠 𝑜𝑓 𝑡𝑦𝑝𝑒 𝜋 -𝑝 𝑓𝑜𝑟 𝑠𝑖𝑔𝑛𝑎𝑙𝑠 𝑜𝑓 𝑡𝑦𝑝𝑒 𝜋 (7)
Figure 1 depicts the ωj depending on γj depending on Equation 6.

Figure 1: ωj Depending on γj (Equation 6)

In Figure 1, ω(γj│Xj/Xmax = 1) = ω(γj│Xj/Xmax = 0) whereas for (Xj/Xmax) ≠ 0, there is

lim ( / )→ 𝜔 γ │ 𝑋 𝑋 = .5 (8)
A reference point is ξ. For Xj/Xmax ≥ ξ, the individual assumes the outcome is a "win." For Xj/Xmax < ξ, the outcome is regarded as a "loss." If there is no other description, there is ξ = 0. When the individual processes the signal, he or she determines ("framing") whether the outcome is a "win" or a "lose." The information value of the signal depends upon this. If the outcome is "win," the sign of γ is positive. If the outcome is "lose" the sign of γ is negative. This is in addition to the sign that γ has anyway, which is positive for π1 and negative for π2. Whether an outcome is "win" or "lose" depends on the other outcomes of the lottery. An outcome ".001" can be "lose" if all other outcomes are, e.g., "1." If ".001" is the maximum outcome, it may be "win."

The outcome may be "lose" if it is below a safe outcome for the other alternative. Signals are only processed if they deliver new information. 2) can be used The Expected Utility Theory leads to Equation 10. The SPM leads to Equation 10a. For symmetric choices and Xj > ξ Equation 10 equals Equation 10a. Equation 9 is a specific form of the SPM with the additional assumption that the credibility of the signals sums up to 1 (Equation 11).

0 1/4 1/2 3/4 1 -1 -1/2 0 1/2 1 ω j γ j ω(γj|Xj/
again with N ∈ N 0 , 𝜔 ∈ [0,1], PI ∈ [0,1] and Nπ1 + Nπ2 = N. Nπ1 ∈ N 0 . Nπ2 ∈ N 0 .

γ = 1 (11)

This implies that there are no contradictory signals. Thus, probabilities of states are special forms of the credibility of signals depending on an exogenous situation instead of a possible untruthfulness of the emitter. The difference between both approaches is the view of the world.

The Expected Utility Model assumes complete information and unchangeable preferences. The SPM does not require these assumptions, and in spite of this, delivers the same results.

The SPM and Subjective Probabilities

Bayes' Theorem is an endogenous way of depicting new information in an expected utility function ("subjective probabilities"). Bayesian learning differs between "prior probabilities" and "posterior probabilities." The "prior probability" is the probability of one or more particular states before the new signal is received. The "posterior probability" is the probability of the same states after receiving the signals (Hirshleifer and Riley 1992, pp.170ff). I modify the case above (Table 3) with new information. There is information about a state s3 = .10 concerning B with an outcome of 0. A remains unchanged. In the case above, the individual decides in favor of A if γ3 > (Equation 13). The SPM switches in the interval [ , ] to A whereas Bayes favors B. The SPM is more "conservative" when it comes to low outcome signals (Xj < ξ).

The SPM, Prospect Theory and Paradoxes

The results of the algorithm of the SPM are promising compared to Expected Utility Theory and Subjective Utility Theory. How does the SPM perform when it comes to the paradoxes that Prospect Theory analyzes like the isolation effect, the certainty effect, the reflection effect or Allais? What does the SPM provide for insight to coding and fanning out?

6.1. The SPM and Symmetric Choices with Zero Choices Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] presented several prospects in analyzing paradoxes. Some prospects are similar to symmetric choices but do not compare "states" of the same probability for all possible outcomes over all choices. Examples of this are Prospect C and D of the Allais Paradox. The prospects found by Kahneman and Tversky lead to additional rules when applying the SPM. In Allais C, the signal "win" 1,000k at .11 implicitly contains the information "lose" at .89. In this symmetric case, this is not new information, which leads us to Rule 1.

Rule 1: If a prospect offers two symmetric choices ("win"/ "lose"), this is regarded as one signal in the SPM.

Applying Rule 1, the SPM solves the prospects presented in Table 8, also K-T 2, K-T 4 and K-T 4' in Table 10 below. [START_REF] Machina | Choice Under Uncertainty: Problems Solved and Unsolved[END_REF] developed the prospects shown in Table 8 as a general pattern that he calls the "common ratio effect." This pattern is depicted in Table 9 for the SPM using the algorithm of Equations 2 and 6. There is p > q, 0 < X < Y and r D -4k .25 -2k -.25 0 -.5 30 -6k .42 .46 .75 .48 Table 11: The SPM Applied to Other "Isolation Effect" Prospects

∈ [0,1]. X1 γ1 X2 γ2 Xmax ω1 N Σωj, PI A X p 0 -(1-p) Y - 𝑝 * 𝑋 2 * 𝑌 + .5 2 1 + - * * B Y -q 0 1-q Y - -𝑞 * 𝑌 2 * 𝑌 + .5 2𝑌 + 𝑞𝑌 -𝑝𝑋 4 * 𝑌 C X rp 0 -(1-rp) Y - 𝑟𝑝 * 𝑋 2 * 𝑌 + .5 2 1 + * - * * D Y -rq 0 1-rq Y - -𝑟𝑞 * 𝑌 2 * 𝑌 + .5 2𝑌 + 𝑟𝑞𝑌 -𝑟𝑝𝑋 4 * 𝑌
The SPM also solves Lindsay, K-T 13 and K-T 13' (Table 11).

The SPM and the Certainty Effect

Kahnemann and Tversky found that people place greater weight on outcomes that are certain than outcomes that are merely probable. They called this pattern the certainty effect. Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] modified Allais using moderate instead of extremely large gains. Table 12 shows the findings of the SPM when applied to these prospects. The SPM predicts the same results that Kahneman and Tversky empirically measured in K-T 1, K-T 3', K-T 5, Jen1 and Yaari1. Altogether, the number of signals N seems to be a suitable variable that solves the certainty effect paradox. The SPM algorithm solves these problems because it weighs losses resulting in an outcome of zero higher than the Expected Utility Theory for a risk-neutral individual. Risk aversion is at a certain level inherent in the algorithm. [START_REF] Machina | Choice Under Uncertainty: Problems Solved and Unsolved[END_REF] pattern (Table 9) solves the cases above for p ≠ 1. Table 13 shows the general pattern for p = 1. For p = 1, Rule 1 does not apply, so the model uses all ωj.

X1 γ1 X2 γ2 Xmax ω1 ω2 N Σωj, PI A Y q 0 -1+q Y - 𝑞 * 𝑌 2 * 𝑌 + .5 (1 -𝑞) 2 + .5 2 -𝑞 + 𝑋 2 * 𝑌 B X -1 Y 𝑋 2 * 𝑌 + .5 3 2 3 - 𝑞 3 + 𝑋 6 * 𝑌

Table: 13: A General Pattern for p = 1 Applying the SPM

There is q ∈ (0,1) and 0 < X < Y. Using Table 13, the individual prefers B to A if

2 -𝑞 + 𝑋 2 * 𝑌 > 1.5 𝑞 < .5 + 𝑋 2 * 𝑌 (15)
The SPM predicts that there will be a decision in favor of A if q > .5. Otherwise, the individual decides in favor of B for all Y (X, Y > .5). The certainty effect is inherent in the algorithm. Critics may argue that the individual would prefer B even if Y = 30 and X = 1 (Prospect Jen1). In Jen1, the individual decides in favor of A. In comparison to "30," many individuals obviously regard "1" as a "lose." Xj/Xmax does not seem to lie beyond the reference point ξ where the individual starts to regard the outcome as a "win." So the signal (1,.1) is regarded as a signal of type π1 with γ1 > 0. As long as "1" is regarded a "loss," the individual chooses A for all Y.

Altogether the SPM is helpful in explaining several certainty effect prospects.

The SPM and the Reflection Effect

Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] found strong empirical evidence for the "reflection effect." The individual is risk-averse as to prospects involving gains (K-T 3, Yaari1, K-T 11) and risk-loving regarding prospects involving losses (K-T 3',K-T 4' and K-T 12). Table 14 shows several such prospects and the outcomes using the SPM. Summed up, the SPM seems to depict the reflection effect. The SPM predicts the results that Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] empirically measured. I apply Equation 15 to K-T 3' for Y < X < 0. The individual prefers B as long as Y < -7,500. Equation 15 works for Y < X with Y, X ∈ R.

X1 γ1 X2 γ2 D Xmax ω1 ω2 N Σω, PI SPM K-T 3 A 4,
Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] and [START_REF] Lindsay | The Arguments of Utility: Preference Reversals in Expected Utility of Income Models[END_REF] further examine the case in which an individual receives money and is asked to make a choice afterwards. In K-T 11, the individual receives 1,000 prior to the decision. With a probability of .5, he or she loses everything or keeps 1,000 (A). This is compared to B, a sure win of 500. In K-T 12, the individual receives 2,000 prior to the decision. Both choice problems are identical. The SPM solves these problems for gains and losses without the need to reflect the previous receipt of the money. The first stage of the choice seems to be disconnected from the second stage when applying the SPM algorithm.

6.5. The SPM, Allais and Coding Allais (1953) compared the choices of alternatives and inconsistent decisions. Table 14 shows the four possible alternatives A, B, C and D. According to [START_REF] Allais | Le comportement de l'homme rationnel devant le risque: critique des postulates et axiomes de l'école Américaine[END_REF], Kahneman and Tversky (1979) and [START_REF] Varian | Microeconomic Analysis[END_REF] most people prefer A to B and D to C.

Even Savage chose A and corrected himself later, as Machina mentioned (Machina, 1987, p.128f). A and B can be transformed into D and C. This behavior violates the substitution axiom of the Expected Utility Theory. [START_REF] Machina | Choice Under Uncertainty: Problems Solved and Unsolved[END_REF] showed that this is part of a general empirical pattern called the common consequences effect. Indifference curves may fan out. This fanning out can be explained by difficulties of the individual in processing small probabilities.

Alternative Choice X1 X2 X3 A 1,000k at 1 B 5,000k at .10 1,000k at .89 0 at .01 C 1,000k at .11 0 at .89 D 5,000k at .10 0 at .90 Table 14: The Allais Paradox according to [START_REF] Varian | Microeconomic Analysis[END_REF] The pure SPM does not depict Allais. Equations 2 and 6 predict B if γj(B) is (.10,.83,.07). The Expected Utility Theory predicts B for (.10,.50,.40). The phenomenon in which indifference curves fan out is better described by the SPM. The SPM chooses B as long as (Equation 16)

𝑌 > 𝑌 * 𝑞 -𝑋 1 -2𝑞 -𝑞 (16)
For Allais, the individual decides in favor of B if Ymax > 1,222k. The Expected Utility Theory predicts B if (Equation 17)

𝑌 > 𝑋 -𝑞 * 𝑌 𝑞 ( 17
)
The Expected Utility Theory predicts B if Ymax > 1,100k.

Before solving this case, it seems necessary to discuss what the overall information value of a signal is. Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] call these considerations "coding." People perceive outcomes as gains and losses. Whether an outcome is a gain or a loss depends on the position of the individual.

The maximum "win" in Table 14 is $5,000k. This is for most individuals a huge win, meaning most of the individuals will not have to work again. This was in particular true in the 1950s when Allais developed his paradox. I differentiate three information values of signals.

(1) lifetime income, (2) monthly income, and (3) a small sum. These information values exist as gains and losses. 15 Comparing A and B in this way, the individual decides in favor of A as Allais measured. Obviously, the general information value of signals differs between countries and between affluent, middle class or poor individuals. Lifetime income differs between Albania and Park Avenue in New York. Five thousand dollars might be a small sum for a billionaire or a sufficient "monthly income" for a middle class individual.

The SPM and Reference Point Topics

Whether an outcome is a gain or loss is defined with respect to some reference point. The location of the reference point affects whether the outcomes are coded as gains or losses. Table 17 shows other prospects that involve reference point topics and applies the SPM algorithm. In Jen2, the individual has clear preferences for B. The outcome of A is 0. [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF], also Discussed by [START_REF] Katsikopoulos | One-reason decision-making: Modeling violations of expected utility theory[END_REF] Apparently, the individual isn't capable of processing the credibility ".001." If the individual processes ".1" instead of ".001," the individual is not indifferent anymore. The same thing applies to the case with the negative outcome (K-T 14'). Or, the individual compares the case "monthly income = 1" with the case of a "small sum" that he or she neglects. From this, follows Rule 3. Kahnemann and Tversky described a similar rule as "simplification."

Rule 3: If a prospect contains signals with very large or very small differences in credibility, individuals are not capable of processing them. Instead, they differentiate between credibility at a level they can process.

It is useful to examine at which outcome the switch towards F in K-T 14 occurs. Prospect 8 and 8' also indicate that the individual has difficulties processing small differences in credibility. The individual doesn't seem to differentiate between ".001" and ".002." It appears that he or she assesses the credibility of both signals similarly. A related phenomenon is depicted in K-T 7 and K-T 7' ( In K-T 7, the individual decides in favor of B, whereas in K-T 7' most of the individuals choose A. In this case, the SPM works. However, Rule 1 does not apply. The individual does not seem to process this case as a symmetric choice. The number ".90" seems to signal to the individual a sure win. So, he or she decides in favor of Prospect B. Conversely, A is chosen in Problem 7' to avoid the high probability of a loss. It appears that the fanning out of indifference curves also occurs between .9 and 1.

The SPM and Contradictory Signals

Lotteries imply that there are no contradictory cases. In everyday business decisions, this is not always the case. Frequently, one expert predicts an outcome that is contrary to the prediction of another expert. Both experts may have the same credibility and that may not necessarily sum up to 1. Cases like this are not part of Expect Utility Models. Its probabilities add up to 1. Table 20 shows two cases.

signals directly influence the preferences of the individual. It works in Yaari's [0,1] space. When testing the model on lotteries, several interesting outcomes can be seen. First, the SPM delivers the same results that the Expected Utility Theory for symmetric choices. Second, the SPM is similar to Bayesian learning but is more conservative when it comes to low outcome signals.

Third, the SPM predicts several anomalies of Prospect Theory, like the "reflection effect," the "certainty effect" and the "isolation effect." Fourth, information processing of signals gives insight into coding and the fanning out of indifference curves. Fifth, the SPM can also process contradictory signals or the mixture of signals, which of course are everyday cases in business life. Altogether, the model is an astonishingly powerful tool to explain the formation of preferences. It provides a method to better understand how the processing of information provided by signals affects decisional outcomes and it seamlessly fits into economic decision theory. It also offers a great opportunity to reduce doctrinal friction between economics and the other social sciences.

1Figure 3 :

 3 Figure 3: Stability of Preferences Depending on Credibility γ of New Signals π2

  Differences between Economic Decision ModelsThis paper analyzes how the new algorithm provided by the SPM performs as a decision model in comparison to Expected Utility Theory, Bayesian learning and Prospect Theory. It shows additionally how to work with contradictory signals and "signal mixture." The paper shows that a holistic model seamlessly performs in the framework of economic decision theory.

  𝑓 𝑋 . Equation 6 is a function 𝑓 𝑋 that does the transformation with 𝜔 ∈ [0,1], 𝑋 , 𝑋 ∈ R, 𝛾 ∈ [0,1]. I preferred Equation 6 over other functions because it depicts best the information processing of a signal. 2

  Figure2: ωj Depending on Xj/Xmax (Equation6)Figure2depicts ωj depending on (Xj/Xmax). It shows a kink in the function of ω(Xj/Xmax) occurring at a reference point ξ. ξ is depicted as .05 in Figure2. If Xj/Xmax ∈ [0,ξ] the outcome of an alternative is regarded a "lose." ξ depends on the outcome of the other alternatives. After the outcomes Xj are transformed, the initial preference function of the SPM (Equation2) can be used

Table 1 :

 1 Advantages of the SPM Compared to Other Economic Decision ModelsEquation 1 defines the information value ω of a signal.

	are two alternatives and two types of signals. A signal of the type π1 provides the
	information: "Alternative 1 is better." The information value parameter is ω1. 𝜔 ∈ [0,.5]. A
	signal of the type π2 provides the information: "Alternative 2 is better." The information value
	parameter is ω2. 𝜔 ∈ [.5,1]. Perfectly credible signals of the type π1 have γ1 = 1. Perfectly
	credible signals of the type π2 have γ2 = -1. Entirely incredible signals have an information value
	ω close to .5. For π1: 0 ≤ γ ≤ 1. For π2: -1 ≤ γ ≤ 0. The less credible a signal is the closer γπ is
	to 0.
	Table 1 sums up the advantages of the SPM in comparison to traditional decision models.
		Advantages of the SPM; The model…
	(1)	…explains origin of preferences
	(2)	…predicts stability or instability of preferences
	(3)	…reduces complexity
	(4)	…uses credibility an important variable
	(5)	…is compatible with other social sciences; Marketing
	(6)	…can include "emotions", "needs"

Table 2

 2

		Expected Utility	Bayes	Prospect Theory	Information Cascades	SPM
	Does new
	information	No	Yes	Yes	Yes	Yes
	exist?
	New		New	New
	information	---	probability	probability of	Signal	Signal
	transmitted by		of a state	a state
	New information influences	---	Probability of states	Probability of states	Probability of states	Preferences
				Constant;
	Preferences	Constant	Constant	analyzes	Constant	Endogenous
				paradoxes

shows the key differences between the SPM and other economic decision models. 1 For Nπ1 > Nπ2 and |𝛾 | > |𝛾 | there is lim →

Table 3 :

 3 I apply Equation 9 to a simple case in order to compare the Expected Utility Theory and the SPM. In Table3, EUA is 210 and EUB is 220. The rational individual decides in favor of B. Given the assumptions above, the individual has preferences for B. EU of Two Alternatives I now apply the holistic SPM to this case. There are two alternatives. Alternative 1 is "A" Alternative 2 is "B." There are two types of signals. A signal of type π1 provides the information: "Alternative 1 is better." The information value parameter is ω1. The credibility γ1 is assigned to ω1. A signal of type π2 provides the information: "Alternative 2 is better." The information value parameter is ω2 with γ2 assigned to ω2. There is ω ∈ [0,.5], ω ∈ [.5,1], 0 ≤ γ ≤ 1 and -1 ≤ γ ≤ 0. Perfectly credible signals have |γ| = 1. The less credible a signal, the closer |γ| is to 0. Table4shows the signals in the case above.

	𝐸𝑈 =	𝑝 𝑢(𝑥 )	(9)

Nπ1 is the number of signals π1 an individual has received. Nπ2 is the number of signals π2.

The functions 6a and 6c do not have a kink. However, they may depict "win" outcomes as "loses."

4. The SPM, Symmetric Choices and Von Neumann Morgenstern Utility

The concept of expected utilities links information and preferences. Preferences of individuals for alternatives depend on uncertain samples from the future. The outcome of an individual depends on possible states in the future. States are assigned probabilities and have probabilities for all alternatives. Equation 9 shows an expected utility of a risk-neutral individual.

Table 4 :

 4 Solving Table 3 Applying the SPM Table 4 shows four signals. Two signals are of type π1. They deliver information in favor of Alternative 1 that is A. Two signals are of type π2. They deliver information favoring B. Applying Equation 6 delivers PI = .50417. The individual decides in favor of B. The results are the same as when using Equation 9. The individual decides in favor of B as long as

	𝑢 > 𝑢 +	𝑝 * (𝑢 -𝑢 ) 𝑝	(10)
	𝑋 > 𝑋 +	𝛾 * (𝑋 -𝑋 ) 𝛾	(10𝑎)

  Table 5 shows the new probabilities of B.

	State	p(s)	A	pnew(s3)	B
		γ	uAj	γnew	uBj
	I	.60	150	.54	200
	II	.40	300	.36	250
	III	0		.10	0
		EUA	210	EUBnew	198

Table 5 :

 5 Probabilities of States Influenced by a Signal This leads to EUBnew = 198. The rational individual now prefers A with EUA = 210. The individual has preferences for A. New information has caused behavioral change.I model this case using the SPM. Table6shows the signals and the information value parameters. The fifth signal provides 0 with .10. It is a signal of type π1 because it favors Alternative 1. So γ > 0. It provides ω = .45 shifting the preferences towards A. The new signal does not necessarily change the credibility of the old signals.

	Signal	Signal Type	Xj	𝛾	ωj
	1	π1	150	.60	.35
	2	π1	300	.40	.30
	3	π2	200	-.60	.70
	4	π2	250	-.40	.67
	5	π1	0	.10	.45
	Σ				2.47
	N			5	.493

Table 6 :

 6 ωj, PI and the Decision of the Individual I compare the SPM to Bayes with modifying Prospect B. In B', the probability of X3 is .04 and respectively .02 in B''. For B', the individual still decides in favor of Prospect A using the SPM. Using Bayes, the individual decides in favor of Prospect B' and B''(Table7).

		X1	γ1	X2	γ2 X3 γ3 Bayes Xmax ω1 ω2 ω3 N Σω, PI SPM
	A	150 .60 300 .40		300 .35 .30	5 2.467
	B	200 -.60 250 -.40 0 .10	A	300 .70 .67 .45	.493	A
	B' 200 -.60 250 -.40 0 .04	B'	300 .70 .67 .48	.499	A
	B'' 200 -.60 250 -.40 0 .02	B''	300 .70 .67 .49	.501	B''

Table 7 :

 7 Decisions Comparing the Signal-Preference Model and Bayes

	Using Bayes, the individual decides in favor of A, if
		𝑝 (𝑠 ) >	*	*	*	*	*	*	(12)
	In the case above the individual decides in favor of A if pnew(s3) > (Equation 12). The SPM
	chooses A if
	𝛾 >	-𝑋 * 𝛾 -𝑋 * 𝛾 -𝑋 * 𝛾 -𝑋 * 𝛾 𝑋	(13)

Table 8 :

 8 Table 8 shows such prospects and the empirically measured decisions. Applying the SPM to Symmetric Choices Including Zero Choices

		X1	γ1 X2 γ2 X3 γ3 D	Xmax	ω1 ω2 N Σω, PI SPM
	Allais C 1,000k .11	0	-.89		5,000k .49	2	1.039
	D 5,000k -.10	0	.90	X 5,000k .55		.520	X
	K-T 6 C	3	.05	0	-.95	67	3	.48	2	.992	X
	D	1	-.1	0	.9	33	3	.52		.496
	Yaari2 C	.4	.2	0	-.8	65	.4	.40	2	.994	X
	D	.3	-.25	0	.75	35	.4	.59		.497

Table 9 :

 9 The SPM and Machina's "Common Ratio Effect" Different choice problems can be decomposed in different ways. This can lead to inconsistent preferences. Kahnemann and Tversky call this phenomenon the isolation effect. Kahneman and[START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] modify their prospect K-T 4 by adding a second stage (K-T 10). The individual receives money in a first stage and then must make a second decision in which the individual keeps the money or loses it (Table10).

	Applying the SPM, the individual prefers B to A if
	2 * 𝑌 + 𝑞 * 𝑌 -𝑝 * 𝑋 4 * 𝑌	> .5
	2 * 𝑌 + 𝑞 * 𝑌 -𝑝 * 𝑋 > 2 * 𝑌
	𝑌 >	𝑝 𝑞	* 𝑋	(14)
	Accordingly, the individual prefers D to C if 𝑟 * 𝑞 * 𝑌 > 𝑟 * 𝑝 * 𝑋. In Allais C/D (Table 8), the
	SPM predicts that the individual will prefer C as long as X1D >	. .	* X1C.
	6.2. The SPM and the Isolation Effect

Table 10 :

 10 The SPM Applied to Isolation Effect Prospects Applying the SPM, this second stage produces a fifth signal and a third signal to be processed (Rule 1) so that the individual decides in favor of B in K-T 10. It appears that the algorithm of the SPM helps to explain decisions when the isolation effect occurs. This is because the SPM analyzes the perception of signals by the individuals. The additional signal swings the preferences.

	X1 γ1 X2 γ2 X3 γ3 D Xmax ω1 ω2 ω3 N Σω, PI SPM
	Lindsay A 49	1				X 200 .78	3	1,13	X
	B 200 -.25	0	.75			200 .63 .13	.38
	K-T 13 A 6k .25	0	-.25 0 -.5 18	6k	.38 .63 .75 6	3.13
	B 4k -.25 2k -.25 0	.5 82	6k	.58 .54 .25	.52	X
	K-T 13' C -6k -.25 0	.25	0	.5 70 -6k .63 .38 .25 6	2.88	X

Table 12 :

 12 The SPM Applied to Certainty Effect Prospects

		X1	γ1	X2	γ2 X3 γ3	D	Xmax	ω1 ω2 ω3 N Σω, PI SPM
	K-T 1 A 2,500 .33 2,400 .66 0 -.01 18 2,500 .34 .18 .51 4 2.003
	B 2,400 -1			82 2,500 .98	.501	X
	K-T 3' A -4,000 -.8	0	.2	92 -4,000 .90 .40	3 1.425	X
	B -3,000 -1			8 4,000 .13	.475
	K-T 5 A	3	.5	0	-.5	22	3	.25 .75	3 1.667
	B	1	-1			78	3	.67	.556	X
	Jen1 A	30	.5	0	-.5	X	30	.25 .75	3 1.483	X
	B	1	1				30	.48	.494
	Yaari1 A	.4	.8	0	-.2	20	.4	.10 .60	3 1.575
	B	.3	-1			80	.4	.88	.525	X

Table 14 :

 14 The SPM Applied to Reflection Effect Prospects

			000	.8	0 -.2 20 4,000	.100 .600 3	1.575
		B 3,000	-1		80 4,000	.875	.525	X
	Yaari1	A	.4	.8	0 -.2 20	.4	.100 .600 3	1.575
		B	.3	-1		80	.4	.875	.525	X
	K-T 3'	A -4,000 -.8 0	.2 92 -4,000 .900 .400 3	1.425	X
		B -3,000 -1		8	4,000	.125	.475
	K-T 11	A 1,000	.5	0 -.5 16 1,000	.250 .750 3	1.750
		B	500	-1		84 1,000	.750	.583	X
	K-T 12	C -1,000 -.5	0	.5 69 -1,000 .750 .250 3	1.250	X
		D	-500	1		31 -1,000 .250	.417

Table 15 :

 15 Table 15 shows six possible general information values of signals. Possible General Information Value of Signals From these considerations, I derive Rule 2, leading to Table 16.Rule 2: If a prospect involves a lifetime income, individuals tend not to differ between it and an n-fold of it with n ∈ N.

		Gains		Losses
		Small Sum		Small Sum
		Monthly Income		Monthly Income
		Lifetime Income		Lifetime Income
	Signal	Signal Type	Sum	Xj	γ	Ω
	1	π1	1,000k	1 ("lifetime income")	1	0
	2	π1	0	0	.01	.495
	3	π2	5,000k	1 ("lifetime income")	-.10	.55
	4	π2	1,000k	1 ("lifetime income")	-.89	.945
	Σ					1.99
	N					>.497

Table 16 :

 16 Applying the SPM to Allais Using Table

Table 17 :

 17 Applying the SPM Determining Whether an Outcome is a Gain or a Loss In Jen3, there are five outcomes. Four outcomes can be regarded as "losses." Only one outcome is a "win." There are three signals of the type π2 and two signals of the type π1. The outcome ".001" is positive but does not lie beyond ξ in this prospect. However, solving the prospect with the SPM without introducing the reference point would result in C. The individual "codes" the prospects with ξ. In doing so, results in preferences for D (Jen3ξ). The references point determines the algebraic sign of γ of a signal in the algorithm. Prospect Jen1 (Table12) worked in the same way.

		X1	γ1 X2 γ2	X3		γ3 D Xmax	ω1	ω2 ω3 N Σω, PI SPM
	Jen2 A	0	-.99 0 -.01				5,000k .995 .505	5	2.90
	B	0	.09	0 .51 5,000k -.4 X 5,000k .455 .245 .70		.58	X
	Jen3 C .001 -.99 0 -.01				5,000k .500 .505	5	2.45	X
	D .001 .09	0 .51 5,000k -.4 X 5,000k .500 .245 .70		.49
	Jen3ξ C .001 -.99 0 -.01				5,000k .995 .505	5	2.90
	D .001 .09	0 .51 5,000k -.4 X 5,000k .455 .245 .70		.58	X
	6.7. The SPM, High Gains, Small Credibility and Fanning Out
	K-T 14 and K-T 14' also deal with the fanning out of indifference curves based on small
	probabilities. Table 18 shows some of these prospects.
			X1	γ1	X2	γ2	D	Xmax	ω1	ω2 N Σω. PI SPM
	K-T 8	C	6,000	.001	0	-.999 73	6,000	.4995	2	.4995	X
		D	3,000	-.002	0	.998	27 6,000			.2498
	K-T 8'	C -6,000 -.001	0	.999	30 -6,000 .5005	2	.5005
			-3,000	.002	0 -.998 70 -6,000			.2503	X
	K-T 14 E 5,000	.001	0	-.999 72	5,000	.4995	1	.4995	X
		F	5	-1			28	5,000			.4995
	K-T 14' G -5,000 -.001 0	.999	17 -5,000 .5005	1	.5005
		H	-5	1			83 -5,000			.5005	X

Table 18 :

 18 The SPM and Prospects by Kahneman and

Table 19 :

 19 Table 19). The SPM and the Fanning Out of Indifference Curves

	X1	γ1	X2	γ2	D	Xmax	ω1	ω2	N Σω. PI SPM
	K-T 7 A 6,000	.45	0 -.55 14 6,000 .275 .775 4 2.225
	B 3,000	-.9	0	.1	86 6,000 .725 .450	.556	X
	K-T 7' A -6,000 -.45 0	.55 92 -6,000 .725 .225 4 1.775	X
	B -3,000	.9	0	-.1	8 -6,000 .275 .550	.444

No "logical" reason is given. Channing's opinion is a signal that carries an ω. Although this signal is emotional, it can be processed by the SPM, as Table 21 shows. Due to the ω3 = 1 of Channing's signal, the decision swings from A to B. The SPM algorithm enables economics to process cases like this that are very close to reality and to integrate them into economic decision theory.

Conclusion

The Signal-Preference Model (SPM) is a holistic model designed to explain the origin of preferences. The SPM is an algorithm that puts the information an individual has received in the past ("signals") and the preferences of an individual into context without the use of "states." New Yaari, Menachem. (1987). The Dual Theory of Choice Under Risk. Econometrica 55(1), 95-115.


Marek Jenőffy

Büro Am Carlsplatz

Information, Credibility, and Endogenous Preferences

Keywords: Information, Credibility, and Endogenous Preferences

published or not. The documents may come

Preferences: Concept and Interpretation

There have been several attempts to further develop preference theory. Multi-attribute utility theory [START_REF] Lancaster | Consumer Demand: A New Approach[END_REF]) opined that individuals have preferences not for goods but for characteristics of goods. [START_REF] Stigler | De Gustibus Non Est Disputandum[END_REF] developed a theory of "commodities," which assumes that preferences do not exist for specific consumer goods but for specific psychological states that they call "commodities." [START_REF] Becker | Habits, Addictions, and Traditions[END_REF] formulated a concept of stable "meta-preferences" (Becker 1992, p.340) that include "past choices and choices by others as arguments in a person's current utility function." [START_REF] Sen | Rational Fools: A Critique of the Behavioral Foundations of Economic Theory[END_REF] developed another concept of "moral rankings," which are "rankings of rankings." [START_REF] Kirchgässner | Homo Oeconomicus[END_REF] outlined the differentiation between preferences and restrictions. He stated that economic theory does not explain the origin of preferences. Preferences accrue "over the process of socialization" (p.12).

Going back to von [START_REF] Von Neumann | The Theory of Games and Economic Behavior[END_REF], the concept of expected utilities links information and preferences. New information may alter the probabilities of the states. Due to changed expected utilities, the rational individual may decide in favor of a different alternative. The preferences of the individual remain unchanged. In Bayes' Theorem [START_REF] Savage | The Foundation of Statistics[END_REF], a signal can alter a "prior probability" before receiving a signal to a "posterior probability" after receiving a signal. This requires exogenous given conditional probabilities of a signal given state s.

Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] analyzed paradoxes of the Expected Utility Theory. Their Prospect Theory defines an "editing phase," in which the individual reformulates the available options. In an "evaluation phase," "values" are attached to the outcomes reflecting the subjective value of the outcome for the individual. The value function is concave above a reference point and convex and steeper below. A "decision weight" is attached to the probability. There are other models by [START_REF] Chew | A Generalization of the Quasilinear Mean With Applications to the Measurement of Income Inequality and Decision Theory Resolving The Allais Paradox[END_REF], [START_REF] Machina | Expected Utility' Analysis Without the Independence Axiom[END_REF], [START_REF] Quiggin | A Theory of Anticipated Utility[END_REF] and Tversky and Kahneman (1992, "rank dependent expected utility"). [START_REF] Kőszegi | A Model of Reference-Dependent Preferences[END_REF] modeled preferences determined by endogenous reference points. [START_REF] Yaari | The Dual Theory of Choice Under Risk[END_REF] Dual Theory transforms decision problems into a [0,1] interval. [START_REF] Bikhchandani | A Theory of Fads, Fashion, Custom, and Cultural Change as Information Cascades[END_REF] developed a model of "information cascades." Individuals observe the behavior of others about the quality of a good ("signal"). A signal influences the behavior of the next individual. His or her behavior produces the next signal, thereby starting an information cascade. [START_REF] Banerjee | A Simple Model of Herd Behavior[END_REF] similarly modeled "herd behavior." [START_REF] Viscusi | Risky Investment Decisions: How are Individuals Influenced by Their Groups[END_REF] measured that "groups" influence behavior even if people have full knowledge over probabilities and payoffs. [START_REF] Krackhardt | Endogenous Preferences: A Structural Approach[END_REF] modeled "adaptive preferences." These preferences depend on past preferences and a weight variable that depicts interaction with other individuals. Endogenous time preferences including future consumption levels are modeled by [START_REF] Becker | The Endogenous Determination of Time Preferences[END_REF]. [START_REF] Brocas | Information Processing and Decision-making: Evidence from the brain Sciences and Inplications for Economics[END_REF] worked on transferring findings of neuroscience research into economics. Palacios-Huerta and Santos ( 2004) and [START_REF] Anderson | Are Risk Preferences Stable? Comparing an Experimental Measure with a Validated Survey-based Measure[END_REF] found empirically evidence for changed risk preferences. Palacios-Huerta and Santos (2004, p.625) concluded: "Although much progress has been made in economics by taking preferences as constant and assuming that they are stable, many important matters cannot be explained without addressing the fundamental endogeneity of tastes."

The Signal-Preference Model of Endogenous Preferences

Whenever preferences are obviously "changing," there is new information at the same time. At the stock exchange, it is common knowledge that stock prices are influenced by new information. Laws ban the use of "insider information" to gain profits. High-frequency trading at the stock exchange aims to react some milliseconds earlier to new information than other market participants. [START_REF] Kirchgässner | Homo Oeconomicus[END_REF] uses the term "state of information" in order to express changes in information. He does not posit a change in the preferences if "changes in the restrictions and/or in the information of individuals are sufficient to explain changes in their behavior" (Kirchgässner 2008, p.37). The "state of information" seems to be a third, hybrid element. In Bikhchandani's et al. and Banerjee's models, the individual receives a signal. However, what is a "signal"? How do we know that signals and not something else determines people's preferences? "Signals" transmit information. Hausman (2012, p.118) mentions four sources of preferences: "(1) means-end reasoning, (2) attribute-based valuation, (3) emotional influence, and (4) physiological needs."

He attributes the first two sources to economics. "Means-end reasoning…derives preferences over alternative actions from preferences over their possible consequences and beliefs about the probabilities of those consequences." Multi-attribute utility derives preferences among actions from "attitudes toward their attributes."

Hausman attributes the latter two sources of preferences to the field of psychology and consumer research. "Emotions" are "a source of information." They provide "a common currency to compare alternatives." Emotions "focus a decision-maker on new information" and "motivate information processing" [START_REF] Hausman | Preference, Value, Choice, and Welfare[END_REF], p.124, Peters et al. 2006, pp.80-81, Peters 2006 pp.454-455).

"Needs" of individuals are discussed by [START_REF] Maslow | A Theory of Human Motivation[END_REF] who defines a need hierarchy of an individual: (1) physiological needs, such as food, sleep and healthcare; (2) safety needs; (3) belongingness and love needs; (4) esteem needs; and (5) the self-actualization needs. If the individual has satisfied his basic needs (such as the need for food), he or she will, basically, seek to fulfill needs that are higher in the hierarchy (Maslow 1943, pp.385f). There are counterexamples that information may lead to neglect basic needs.

In order to process all sources of preference formation, Hausman (2012, p.118f) requests that "economist need to accommodate within their models the findings of psychologists concerning the flaws that deform decision making and the heuristics that guide and sometimes distort people's reasoning," so going beyond economic decision theory ("means-end reasoning"). "Genes" is a possible fifth source of preferences.

Hausman's interdisciplinary considerations require that an interdisciplinary model of preference formation can depict all influences mentioned above. In either case, there is information that provides an information value and this information value finally accrues the preferences. The difference between the sources is the "envelope" of the information. Bikhchandani's et al. "signal" is capable of depicting these influences. None of the sources presented above contradicts the model of signals shaping preferences. A "signal" can be a snarling stomach (a "need"), a lottery outcome ("means and reasoning"), a color of a car ("attribute-based valuation") or a beautiful face ("emotional influence").

The key element of the Expected Utility Theory and Bayesian learning is the definition of "states," which are assigned "probabilities of states." An economic model seems to be capable of depicting preferences in a manageable way provided the relevant "state" consists of one element. If there are more elements ("state dependent utility") the models become very complex [START_REF] Hirshleifer | The Analytics of Uncertainty and Information[END_REF]. Machina (2003, p.25) remarks: "Outside of a gambling hall, most economic decisions and transactions involving uncertainty -investment decisions, search decisions, insurance contracts, financial instruments -are defined in terms of uncertain events rather than numerical probabilities." The use of numerical probabilities without clarifying the origin is heavily criticized by Taleb (2010, p.279) with the background of a Wall Street broker. The SPM addresses these critics when working with the information value of signals instead of "states." The SPM makes the following assumptions.

There are alternatives. Signals πi contain information about the alternatives delivering information value parameters (ωi). An information value parameter influences the preferences of the individual. An individual receives N signals.

To begin with, I simplify the model making the following additional assumptions. Preferences are generated from the information an individual has gathered in the past. Alternative 1 is preferred to Alternative 2 if PI < 0.5. Alternative 2 is preferred to Alternative 1, if PI > 0.5. The individual is indifferent between the alternatives if PI = 0.5.

The Source of Signals and the Credibility of Signals

Individuals generally receive signals from two different sources: Personal experiences and input from others. "Others" means signals obtained by observing the behavior of others or signals that are transmitted by the media. Orr (1987, p.91) remarks: "Market advertising and production can continue only so long as the advertised good is accepted by purchasers." After a one-time purchase, an individual will purchase the advertised product again as long as the product provides characteristics that are at least as good as the characteristics of a competing good. Signals from advertising can influence the individual only if the signal is not voided by the personal experiences of the individual. The individual would normally value his or her own experiences higher than signals from advertisements.

Credibility is one of the core variables of experimental communications research. Several studies are from Yale, leading to the label the "Yale Attitude Change Approach." The significance of the variable credibility is basic knowledge in communication research. It has been examined in various studies, e.g. [START_REF] Hovland | The Influence of Source Credibility on Communication Effectiveness[END_REF], [START_REF] Bergin | The Effect of Dissonant Persuasive Communications Upon Changes in a Selfreferring Attitude[END_REF], [START_REF] Aronson | Communicator Credibility and Communication Discrepancy as Determinants of Opinion Change[END_REF] and [START_REF] Laroche | A Model of Attitude Change in Groups Following A Persuasive Communication: An Attempt at Formalizing Research Findings[END_REF].

Altogether, the studies show that a less credible signal has less impact on the preferences of an individual. Signals that individuals receive beyond their personal experiences differ further regarding perceptibility and salience. I integrate the credibility of a signal in the SPM. I state the assumptions upon which the model is based.

All regarded signals are perfectly perceptible and salient. A perfectly credible signal π1 delivers ω 1 = 0 whereas a perfectly credible signal π2 delivers ω = 1. Signals with an imperfect credibility are reduced in their information value ω by the credibility factor γ. For π1: 0 ≤ γ ≤ 1. For π2: -1 ≤ γ ≤ 0.

Perfectly credible signals of the type π1 have γ1 = 1. Perfectly credible signals of the type π2 have γ2 = -1. Entirely incredible signals have an information value ω close to 0.5. The less credible a signal is, the closer γπ is to 0. The information value ω of a signal is defined according to Equation 2.

𝜔 (𝛾 ) = 0.5 * ( 1 -𝛾 ) (2)
0.5 can be regarded as a reference point. A possible endogeneity of this reference point is not discussed in this paper. Equation 2 leads to a preference function PI adjusted by the credibility parameter (Equation 3)

with N ∈ N, ω ∈ [0, 1], PI ∈ [0, 1], |γ| ∈ [0, 1], 𝛾 ∈ [0,1], 𝛾 ∈ [-1,0], Nπ1 ∈ N and Nπ2 ∈ N. 𝑃 = 1 𝑁 0.5 * 1 -𝛾 = 𝑁 * 0.5 * (1 -𝛾 ) + 𝑁 * 0.5 * (1 -𝛾 ) 𝑁 (3)
The case of a perfectly credible signal complies with the model in the previous chapter. Using credibility instead of probabilities reflects Machina's (2003, p.25) remark quoted above.

Ω1 is the number of new signals of the type π1 with existing preferences based on Nπ1 and Nπ2 that were received prior to Ω1 that causes a swing from Alternative 2 to Alternative 1 when applying Equation 3. Equation 6 shows why the assumption of constant preferences makes sense in many economic models. If there is a great number of signals the individual has received in the past (𝑁 → ∞), preferences tend to be constant (Equation 7). This is the case that mainstream economics implicitly assumes.

lim → 𝑃 (𝑁, 𝜔) = 𝑃 (7)
Thus, the omission of the origin of preferences does not void the models in these cases. Ongoing consuming processes rest upon a greater number of signals received in the past. Consequently, it makes sense to treat preferences as constant and exogenous in order to reduce the complexity of the models ("complete information").

Figure 1 shows the stability of preferences in several cases. All signals are completely credible. In this case, the preferences can't swing from one alternative to another. The relevance of preference forming considerations requires at least one credible new signal πj.

Neuroscience

The SPM goes back to Jenőffy-Lochau (1997). [START_REF] Brocas | Information Processing and Decision-making: Evidence from the brain Sciences and Inplications for Economics[END_REF] developed a similar threshold structure using empirical results from neurosciences. [START_REF] Shadlen | A Computational Analysis of the Relationship between Neuronal and Behavioral Responses to Visual Motion[END_REF] and [START_REF] Mazurek | A Role for Neural Integrators in Perceptual Decision Making[END_REF] exposed monkeys to different standardized stimuli and measured the behavior of the monkeys afterwards. [START_REF] Mazurek | A Role for Neural Integrators in Perceptual Decision Making[END_REF]) built the decision-making process in three stages: (1) the reception of sensory evidence;

(2) the accumulation of sensory evidence; and (3) the categorization of evidence leading to a choice. [START_REF] Mazurek | A Role for Neural Integrators in Perceptual Decision Making[END_REF]Mazurek et al. ( , p.1257) ) concluded: "The model instantiates the hypothesis that neurons in sensorimotor association areas compute the time integral of sensory signals from the visual cortex, construed as evidence for or against a proposition, and that the decision is made when the integrated evidence reaches a threshold."

Brocas (2012) concluded: "Decision-making is consistent with a threshold model in which evidence is pooled into a measure Δ that is compared to a decision-making threshold Δ*. Then Δ > Δ* is evidence of A and a is triggered, while Δ < Δ* is evidence of B and b is triggered." This structure is similar to the SPM. [START_REF] Brocas | Information Processing and Decision-making: Evidence from the brain Sciences and Inplications for Economics[END_REF] adds: "Decision-making is consistent with a noisy process that accumulates information Δt over time to one or two boundaries x and y, and which terminates when a boundary is reached. Reaching x is evidence of A and triggers a while reaching y is evidence of B and triggers b."

Eating Behavior

Boyland and Halford (2013) examine the impact of food advertisements on the eating behavior of children. I model this case.

I assume that there are two alternatives. Alternative 1 is "high fat, high sugar food is good." Alternative 2 is "high fat, high sugar food is bad." There are two signals. Signal π1 provides the information "Alternative 1 is better." The information value parameter ω is 0. Signal π2 provides the information "Alternative 2 is better." The information value parameter ω is 1.

This leads to Hypothesis H0 of the model: "If a child is exposed to signals of type π1, it develops preferences, for Alternative 1." [START_REF] Halford | Beyondbrand effect of Television (TV) food advertisements / commercials on calorie intake and food choice of 5-7-year-old children[END_REF] exposed children from five to seven years to different signals. The children were divided in two groups. One group was exposed to food related advertisements. The other group was exposed to non-food related advertisements. Applying the SPM, one group received received signals π1, the other group didn't. Afterwards, the eating behavior of both groups was measured (grapes, jellies, chocolate, snack-a-jacks and crisps). [START_REF] Halford | Beyondbrand effect of Television (TV) food advertisements / commercials on calorie intake and food choice of 5-7-year-old children[END_REF] found that the amount of food in kcal eaten in the session with no π1 is 545 kcal. The other session with π1 shows an amount of 657 kcal (Table 1). Both numbers are measured for normal weight children. For overweight and obese children, the numbers are 594 kcal (no π1) and 684 kcal with π1. These findings support H0. [START_REF] Halford | Beyondbrand effect of Television (TV) food advertisements / commercials on calorie intake and food choice of 5-7-year-old children[END_REF] show an increased kcal intake in the session with π1. The change in behavior also works in the opposite direction. Hypothesis H1 posits: "If a child is exposed to signals of type π2, it develops preferences for Alternative 2." [START_REF] Dixon | The Effects of Television Advertisements for Junk Food Versus Nutritious Food in Children's Food Attitudes and Preferences[END_REF] found "the strongest evidence supporting an effect of food advertising occurred in relation to the healthy food ads. Several attitudes toward healthy foods improved in response to exposure to healthy food ad" [START_REF] Dixon | The Effects of Television Advertisements for Junk Food Versus Nutritious Food in Children's Food Attitudes and Preferences[END_REF](Dixon et al. , p.1320)). [START_REF] Beaudoin | Promoting Healthy Eating and Physical Activity[END_REF] also provide evidence for H1. They examined a campaign to promote fruit and vegetable consumption in a low-income urban population in New Orleans. They showed that "attitudes toward fruit and vegetable consumption improved over time, in association with an increase in fruit and vegetable message recall…These findings suggest a media effects model in which the dissemination of media campaign messages led to improvements in people's attitudes related to healthy diet…" (Beaudoin et al. 2007, p.220f). Boyland and Halford (2013, p.240) conclude that the impact of media exposure on "eating behavior (total intake and food choice) has been robustly shown in a number of recent studies with a variety of experimental paradigms."

Signal

Drinking and Driving

There is more evidence on the influence of information on behavior. Sloan, Platt and Chepke (2011) examine re-arrests for drinking and driving in North Carolina. I apply the SPM to this case.

There are two alternatives. Alternative 1 is "Drinking and driving is good." Alternative 2 is "Drinking and driving is bad." There are two signals. Signal π1 contains the information: "Alternative 1 is better." Signal π2 contains the information: "Alternative 2 is better." Signal π1 delivers ω1= 0 whereas signal π2 delivers ω2 = 1.

Signal π1 is all information that drinking and driving isn't a problem. Signal π2 is all information that drinking and driving is a problem like a conviction, a fine or jail. Hypothesis H1 of the model is the following: If individuals receive signals of the type π2 then they develop preferences for Alternative 2. A possible signal π2 is prosecution following an index arrest for DWI (driving while intoxicated). Another stronger signal π2 is a conviction as a consequence of the prosecution.

Sloan, Platt and Chepke (2011) examine 305,310 index DWI arrests during the period from 2001 to 2006 using data from North Carolina Administrative Office of the Courts (AOC). They found that "persons who are prosecuted following the index arrest for DWI are less likely to be re-arrested for DWI during the next two years" (Sloan, Platt and Chepke 2011, p.19). The reduction of the rearrest rate ranges from 8.1% to 9.7% relative to the observational mean of the re-arrest rate. Among those who were convicted, the reduction ranged from 14.2% to 45.2%.

The signals of type π2 resulted in more preferences for Alternative 2 as Hypothesis H1 predicted. DeAngelo and Charness ( 2009) found similar results for speeding. Sloan et al. (2013, p.75) conclude: "People presumably learn about the working of the criminal justice system from direct experience with it."

Experience Goods

Another case in which to apply the SPM is the model of experience goods. [START_REF] Nelson | Information and Consumer Behavior[END_REF] assumes that there are "search costs" before purchasing a good for the first time. The individual invests in the "search" until the marginal costs of the search equals its marginal return. The object of the search is obviously information. Nelson's model implicitly assumes that the final preferences of the individual accrue after information has been received. The more information needed to accrue the preferences, the more stable the preferences become. I model this case with a very simple experience good. A child eats a guava for the first time in his or her live. The model makes the following assumptions.

There are two alternatives. Alternative 1 is "A guava is good." Alternative 2 is "A guava is bad." There are two standardized signals. Signal π1 contains the information: "Alternative 1 is better." Signal π2 contains the information: "Alternative 2 is better." Signal π1 delivers ω1= 0 whereas signal π2 delivers ω2 = 1.

When the child tastes a guava for the first time in his or her life, he or she receives a signal. This signal is either π1 or π2. If the child likes the guava, the signal is π1. If the child doesn't like the guava, the signal is π2. So the first tasting of the guava created preferences. These preferences are stable until the individual either forgets the signal or he or she receives signals that outweigh the first signal.

5.5.The Allais Paradox

Microeconomic textbooks mention the Allais Paradox as an example that individuals act inconsistently. [START_REF] Allais | Le comportement de l'homme rationnel devant le risque: critique des postulates et axiomes de l'école Américaine[END_REF] compares the choices of alternatives and inconsistent decisions. Table 2 shows four possible alternatives A, B, C and D.

Alternative Choice A 1,000k at 1 B 5,000k at .10 1,000k at .89 0 at 1 C 1,000k at .11 0 at .89 D 5,000 at .10 0 at .90 Table 2: The Allais Paradox according to [START_REF] Varian | Microeconomic Analysis[END_REF] According to [START_REF] Allais | Le comportement de l'homme rationnel devant le risque: critique des postulates et axiomes de l'école Américaine[END_REF], Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] and [START_REF] Varian | Microeconomic Analysis[END_REF] most people prefer A to B and D to C. However, A and B can be transformed into D and C. This behavior violates the substitution axiom of the Expected Utility Theory. [START_REF] Machina | Choice Under Uncertainty: Problems Solved and Unsolved[END_REF] showed that this is part of a general empirical pattern called the "common consequences effect." Indifference curves may "fan out." A probability p** stochastically dominates p*, if the more preferred outcome has a higher probability, and the less preferred outcome has a lower probability, in p** than in p*. There have been several modulations of expected utility theory to explain this and similar behavior (e.g. Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF][START_REF] Tversky | Loss Aversion in Riskless Choice: A Reference Dependent Model[END_REF]Tversky and Kahneman 1992;[START_REF] Yaari | The Dual Theory of Choice Under Risk[END_REF][START_REF] Loewenstein | Risk as Feelings[END_REF][START_REF] Katsikopoulos | One-reason decision-making: Modeling violations of expected utility theory[END_REF]Lindsey 2013). This case can be explained using the Signal Preference Model. In regard to C and D, the individual receives two signals. Signal π1 provides the information "win", signal π2 provides the information "loose." The individual regards the information value of both signals similarly. Therefore he or she chooses D, which would also deliver the highest expected utility for the risk neutral individual. Comparing A and B, the individual receives only one signal π1 for A. For B, the individual also receives a second signal π2. The reception of this signal makes the individual prefer A to B in order to avoid the signal π2. This consideration is similar to what Kahneman and [START_REF] Kahnemann | Prospect Theory: An Analysis of Decision under Risk[END_REF] called the "certainty effect." People overweight outcomes that are certain. "Coding" makes people perceive outcomes as gains and losses rather than final states or wealth or welfare. The SPM matches such effects of "coding." The individual does not "rationally" assess the probabilities of the states but reacts to the signals. The number of received types of signals appears to be a variable in this game.

Car Colors

The Federal Motor Transport Authority of Germany publishes every year the numbers of new cars sold in Germany by color. There have been several changes in the colors of new cars during the last ten years. Table 3 I assume that there are two alternatives. Alternative 1 is "Kirch is creditworthy." Alternative 2 is "Kirch is not creditworthy." There are two signals. Signal π1 provides the information "Alternative 1 is better." The information value parameter ω is 0. Signal π2 provides the information "Alternative 2 is better." The information value parameter ω is 1.

Breuer transferred in the interview a signal of the type π2 ("Kirch is not creditworthy"). This signal was highly credible (γ2 = -1). Breuer had insider information about the Kirch Media Group because the Kirch Media Group was a customer of Deutsche Bank. Therefore, the interview was capable of shifting the preferences towards Alternative 2, triggering Kirch's bankruptcy.

Empirical Evidence for the Impact of Credibility

The impact of the credibility of signals is examined in the field of communications research for over 60 years. There are various studies that prove the impact of credibility in decision-making processes. I work with old, basic studies. [START_REF] Hovland | The Influence of Source Credibility on Communication Effectiveness[END_REF] worked with credible and less credible signals on four topics. These topics were "anti-histamine," "atomic submarines," "steel shortage," and the "future of movies." Table 5 shows the changes on "opinions" depending on the credibility of the "source" (Hovland and Weiss, 1951, p.643). [START_REF] Hovland | The Influence of Source Credibility on Communication Effectiveness[END_REF] concluded, in economic wording, that high credibility signals altered the preferences of 23.0% of the individuals on average in comparison to a shift of 6.6% that was triggered by low credibility signals. [START_REF] Hovland | The Influence of Source Credibility on Communication Effectiveness[END_REF] Similar findings were obtained by [START_REF] Bochner | Communicator Discrepancy, Source Credibility, and Opinion Change[END_REF]. They conducted experiments on preference change depending on the credibility of a signal and worked with written signals advocating a specific amount of sleep at night. The two sources of the signals were "Sir John Eccles, Nobel prize winning physiologist" and "Mr. Harry J. Olsen, director of the Fort Worth YMCA." [START_REF] Bochner | Communicator Discrepancy, Source Credibility, and Opinion Change[END_REF] Table 6 shows the result of their findings. They found preference change for a highly credible source (a "Nobel prize winner") to be bigger and linearly related to the signal discrepancy (from 7.5 hours to 6 hours). For a signal having medium credibility, preference change was found from 7.5 hours to 6.5 hours.

Individuals have personal experiences regarding their appropriate amount of sleep. Therefore, it seems surprising that the signal had an impact on the expressed preferences of the individuals at all. As long as the signals seemed reasonable, they did influence the preferences of the individual. [START_REF] Bochner | Communicator Discrepancy, Source Credibility, and Opinion Change[END_REF] didn't find significant differences between the two sources regarding their impact. There was no difference in γ. Beyond a threshold the signals lost their impact on the individuals because the signal was given less credibility. The threshold was 2 hours for the medium credibility source and 0 hours for the high credibility source. The medium credibility source reached the threshold earlier.

In the SPM, signals of no credibility (γ = 0) provide an information value ω = 0.5. Such a signal does not swing existing preferences. However, it may weaken those preferences. If an individual has instable preferences and he or she receives many signals providing ω = 0.5 the preferences of the individual go to PI = 0.5. Hovland and Weiss (1951, p.645) found out that individuals tend to disassociate a signal and the source with the result that the original skepticism fades and an "untrustworthy" signal is accepted later. They call this the "Sleeper Effect." The structure of the model reflects a weak "Sleeper Effect."

Conclusion

The Signal-Preference Model (SPM) puts into context the information an individual has received in the past ("signals") and the preferences of an individual without the use of "states." New signals directly influence the preferences of the individual. Received signals differ regarding their credibility. The impact of the credibility of the signal is integrated into the SPM. The SPM explains how preferences form. The SPM has several methodological advantages in comparison to Bayesian

  There are two alternatives. There are only two signals. Signal π1 contains the information: "Alternative 1 is better." Signal π2 contains the information: "Alternative 2 is better." The signals are standardized and contain only one element. Signal π1 delivers ω1= 0 whereas signal π2 delivers ω2 = 1. The preference function of an individual is depicted as shown in Equation 1 with N ∈ N 0 , 𝜔 ∈ [0, 1] and PI ∈ [0, 1]. I use the Yaari interval. Nπ1 + Nπ2 = N. Nπ1 ∈ N 0 . Nπ2 ∈ N 0 . Nπ1 is the number of signals π1 an individual has received. Nπ2 the number of signals π2.

  number of new signals of the type π2 with existing preferences based on Nπ1 and Nπ2 that were received prior to Ω2. that causes a swing from Alternative 1 to Alternative 2 when applying Equation 3. 𝑁 + 𝛺 * 0.5 * (1 -𝛾 ) 𝑁 + 𝛺 > 0.5 𝛺 * 0.5 * (1 -𝛾 ) > 0.5 * 𝑁 -𝑁 + 0.5 * 𝛺 0.5 * 𝛺 -0.5 * 𝛾 * 𝛺 -0.5 * 𝛺 > 0.5 * 𝑁 -𝑁 𝛺 > 0.5 * 𝑁 -𝑁 -0.5 * 𝛾 = 0.5 * 𝑁 + 0.5 * 𝑁 -𝑁 -0the model posits: "If an individual is exposed to signals of type π1, his or her preferences move towards Alternative 1." Hypothesis H1 of the model states: "If an individual is exposed to signals of type π2, his or her preferences move towards to Alternative 2." The impact of a new signal πj on the formation of preferences depends on the strength of the existing preferences and the number N of signals on which the preferences rest on prior to receiving the new signal πj. The influence of a new signal πj is depicted in Equation 6.

Figure 1 :

 1 Figure 1: Stability of Preferences Depending on N Received Prior to New Signals.

Figure 2

 2 Figure2shows the stability of preferences in several cases depending on the credibility of a received signal.

Figure 2 :

 2 Figure 2: Stability of Preferences Depending on γ of New Signals π2

  shows the shares of blue and black cars in Germany between 2003 and 2012.

	Year	Blue Cars	Black Cars
	2003	18.65%	22.63%
	2004	16.91%	24.17%
	2005	16.37%	25.45%
	2006	16.30%	27.83%
	2007	15.10%	30.61%
	2008	13.03%	31.29%
	2009	13.64%	27.58%
	2010	10.48%	30.37%
	2011	9.07%	31.31%
	2012	8.24%	29.48%

Table 3 :

 3 Car Colors in Germany between 2003 and 2012 according to the Federal Motor Transport Authority of GermanyTable3shows that the number of blue cars decreased during the last ten years while the customers purchased a higher percentage of black cars in 2012 in comparison to 2003. I apply the SPM to this case.I assume that there are two alternatives. Alternative 1 is "Blue car." Alternative 2 is "Black car." There are two signals. Signal π1 provides the information "Alternative 1 is better." The information value parameter ω is 0. Signal π2 provides the information "Alternative 2 is better." The information value parameter ω is 1.The SPM explains the behavior of the individuals as follows. The individuals act rationally. During the last ten years, the individuals received more signals of the type π2. Possible signals π2 could be that their neighbors bought a black car and they liked it. They received the signals from advertisements that showed more black cars. They received signals from opinion leaders who were driving black cars. Thus, one may infer when Channing Tatum is filmed with a black car this may produce a signal π2. The presence of more signals of the type π2 triggered a shift of the preferences of the individuals towards Alternative 2.5.7. The 2012 Presidential DebateThe opinion polls of the 2012 United States presidential election show President Obama in the lead during early 2012. After the first debate on October 3, Mitt Romney was able to reduce the lead of the president in the polls. Table4shows the poll numbers and the electors with no toss-ups.

	Date	Electors Obama	Electors Romney	National Average
	4/12	227	170	Obama +4.3
	5/15	243	170	Obama +1.8
	6/14	221	170	Obama +0.8
	7/18	221	181	Obama +1.4
	8/11	237	191	Obama +4.6
	9/12	237	191	Obama +3.5
	10/1	269	191	Obama +4.0
	10/2	269	181	Obama +3.3
	10/5	251	181	Obama +2.6
	10/10	217	181	Romney +1.5
	10/13	201	191	Romney +1.3
	10/18	201	206	Romney +1.0

Table 4 :

 4 Opinion Polls of the 2012 United States Presidential Election according to Realclearpolitics.com I apply the SPM to this case. The model makes the following assumptions.I assume that there are two alternatives. Alternative 1 is "Obama/Biden." Alternative 2 is "Romney/Ryan." There are two signals. Signal π1 provides the information "Alternative 1 is better." The information value parameter ω is 0. Signal π2 provides the information "Alternative 2 is better." The information value parameter ω is 1.The debate on October 3 transferred signals. There was a consensus among commentators that President Obama performed poorly in the debate. The president himself stated: "I had a bad day." Commentators lauded the performance of Mitt Romney. So, this debate transferred signals of the type π2. These signals moved the preferences of the individuals towards Alternative 2 as the numbers in Table5show. President Obama performed better in the subsequent debates on October 16 and October 22, as did Vice-president Biden. There they were able to transfer signals of the type π1, which outweighed the signals π2 from October 3. As a result, Obama eventually gained reelection.5.8. The Deutsche Bank -Kirch LawsuitIn an Bloomberg TV interview in February 2002, the CEO of Deutsche Bank, Rolf E. Breuer said "everything that you can read and hear" shows that "the financial sector isn't prepared to provide further" loans or equity to Leo Kirch. Within months, the Kirch Media Group filed Germany's biggest bankruptcy since World War II.Kirch blamed Breuer of having triggered his bankruptcy through the Bloomberg TV interview. He sued Deutsche Bank about 1.5 billion Euros in damages. In December 2012, the Munich Appeals Court held Deutsche Bank liable for the Kirch Media bankruptcy. Deutsche Bank appealed the verdict at the Federal Court of Justice of Germany. Many pundits now say the phrase Breuer used in the interview was the most expensive statement ever made in an interview. I model this case applying the SPM.

Table 5 :

 5 Net Changes of Opinion according to

	Topic	High Credibility Source	Low Credibility Source
		Changed opinion in direction	Changed opinion in direction
		of communication	of communication
	Anti-Histamines	.23	.13
	Atomic Submarines	.36	.00
	Steel Shortage	.23	-.04
	Future of Movies	.13	.17
	Average	.23	.07
	Difference: 16.4%; P=<0.01

Table 6 :

 6 Credibility Effects on Opinion Change according to

	Promoted Sleep	Preferences of Individuals	Preferences of the Individuals
		Source "Sir John Eccles"	Source "Harry J. Olsen"
	8	7.71	7.50
	7	7.14	7.57
	6	7.29	7.14
	5	7.00	6.50
	4	7.21	7.14
	3	6.64	6.50
	2	5.86	6.93
	1	5.71	6.86
	0	6.79	7.29