Why Credibility Theory?

Wang, Di

Supervisor: Zhou, Jian

23/4/18 School of management, Shanghai University

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Allais paradox

1 Statement of the Problem

2 Mathematical proof of inconsistencyExperiment 1Experiment 2

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Statement of the Problem

The Allais paradox arises when comparing participants' choices in two different experiments, each of which consists of a choice between two gambles, A and B. The payoffs for each gamble in each experiment are as follows:

Experiment 1 Experiment 2

Gamble 1A Gamble 1B Gamble 2A Gamble 2B

Winnings

Chance

Winnings

Chance

Winnings

Chance

Winnings

Chance

$1 million

100% $1 million

89%Nothing

89%Nothing

90%

Nothing

1%$1 million

11%

$5 million

10%$5 million

10%23/4/18 School of management, Shanghai University

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Mathematical proof of inconsistency Using the values above and a utility function U(W), where W is wealth, we can

demonstrate exactly how the paradox manifests. Because the typical individual prefers 1A to 1B and 2B to 2A, we can conclude that the expected utilities of the preferred is greater than the expected utilities of the second choices.

Experiment 1: 1.00U($1M) > 0.89U($1M) + 0.01U($0M) + 0.1U($ 5M)

Experiment 2: 0.89U($0M) + 0.11U($1M) < 0.9U($ 0M) + 0.1U($5M）

We can know from the experiment that:

Probability and Expect Utility Theory may not work in Some circumstances.

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Divakaran Liginlala,*, Terence T. Owb ， Modeling attitude to risk in human decision processes: Anapplication of fuzzy measures ， Fuzzy Sets and Systems 157 (2006) 3040 – 3054

Several models of the human decision process have been proposed, classical examples of which are utility theory and prospect theory.

In recent times, the theory of fuzzy measures and integrals has emerged as an alternative meriting further investigation.

For instance, of the nearly 196 articles on fuzzy measures published in the journal “Fuzzy Sets and Systems” since 1981, only 14 describe practical applications.

fuzzy measures and integrals represents a starting point of a nonadditive expected utility theory

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Table 1 provides a chronological listing of selected applications reported until the year 2005 along with the

methodology and the type of integral used in each case.

For instance, of the nearly 196 articles on fuzzy measures published in the journal “Fuzzy Sets and Systems”

since 1981, only 14 describe practical applications.

nonadditive expected utility theory

(i) " is additive if "(A & B) = "(A) + "(B).

(ii) " is superadditive if "(A & B)""(A) + "(B).

(iii) " is subadditive if "(A & B)!"(A) + "(B).

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Jae H. Min a, Young-Chan Lee b,*, A practical approach to credit scoring, A practical approach to credit scoring, Expert Systems with Applications 35 (2008) 1762–1770

So far, a variety of methods such as linear probability and multivariate conditional probability models, the recursive partitioning algorithm, artificial intelligence approaches

multi-criteria decision-making (MCDM),

mathematical programming approaches have been proposed to support the credit decision

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Arijit Laha, Building contextual classifiers by integrating fuzzy rule basedclassification technique and k-nn method for credit scoring, Advanced Engineering Informatics 21 (2007) 281–291

Credit-risk evaluation is a very challenging and important problem in the domain of financial analysis.

Statistical and neural network based approaches are among the most popular paradigms.

However, most of these methods produce so-called ‘‘hard’’ classifiers, those generate decisions without any accompanying confidence measure.

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application scoring: where the task is to classify credit applicants into ‘‘good’’ and ‘‘bad’’ risk groups.

1.financial information

2.demographic information

behavioral scoring

1.payment history information

the accuracy of the estimation techniques decreases with increased dimensions of the feature space

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especially those incorporating fuzzy set theoretic approach, termed as ‘‘soft’’ classifiers. These classifiers,along with the classification decision produce a confidence

other words, they have natural ability of handling uncertainty

Domain experts

The simplest way of tackling the above issues is to take the help of a domain expert and create the fuzzy rules to represent his/her domain knowledge.

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Gia Sirbiladze1, Irina Khutsishvili2, COMBINED DECISION PRECISING FUZZY TECHNOLOGY FOR CREDIT RISKEVALUATION OF BANK INVESTMENTS, The Third International Conference “Problems of Cybernetics and Informatics”, September 6-8,2010, Baku, Azerbaijan. Section #6 “Decision Making for Social-Economic Systems”

When a juridical person submits business-plan to the investment fund or the bank with the aim of receiving a credit, experts perform applicant's business analysis. In particular, they check up on certain factors that are essential to grant a credit.

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heuristic methods

the objective and expert data

possibilistic discrimination analysis

The classic version of the fuzzy discrimination

analysis uses so called frequency tabular-numeric knowledge base.

Psychometric

since one can hardly hope for the availability of statistical information data bases.

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Yue Jiaoa, Yu-Ru Syaub, E. Stanley Leec, Modelling credit rating by fuzzy adaptive network, Mathematical and Computer Modelling 45 (2007) 717–731

Various factors are used to determine the credit worth of a financial enterprise.

Vagueness may be caused by many different situations such as the difficult in defining precision, in deciding which variable is important, and how to define the variables and, sometimes, even how to define the problem.

Due to these problems, the modern computer cannot be used effectively and the judgment of human experts form an essential part of the overall evaluation.

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Essays in the theory of risk bearing

2.Exposition of the Theory of Choice Under Uncertainty

3.The Theory of Risk Aversion

8.Uncertainty and the Welfare Economics of Medical Care

11.Uncertainty and the Evaluation of Public Investments

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Uncertainty and the Welfare Economics of Medical Care

Patients are uncertain about the effectiveness of medical treatment

Another uncertainty may be quite different from that of his physician

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Uncertainty and the Evaluation of Public Investments

Discount with time and risk

Discount rate

Subjective probability

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Thank you!

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