Ellsberg urn problems with multiple degrees of freedom

1 Introduction

The urn problem could trace back to the ancient Greek period [18] and biblical times [17]. In 1713, Bernoulli discussed the explicit urn problem of drawing “calculi” from urns. Afterwards, the urn problem refers to the model structured by a known number of urns containing balls of different colors. Why are people interested in urn problems? As explained by Balakrishnan and Kotz [9], an efficient method is given to model “random choice”, and urn model enables people to simulate the random process.

In 1923, Pólya and Eggenberger [2] proposed an urn model (called Pólya urn) containing b black balls and w white balls where b and w are known numbers. Pólya urn model was originally applied to issues dealing with the spread of contagious diseases. In order to deal with Pólya urn problems, probability theory successfully provides a mathematical tool [5]. Furthermore, we can employ stochastic process theory to discuss the problems about repeating the procedure that draw a ball at random and add some balls to the urn [1, 7]. In 1961, Ellsberg [4] presented another urn problems called Ellsberg urn problems: There is an urn containing one hundred balls and those balls are black or yellow but the numbers of black or white balls are unknown. The most remarkable feature of Ellsberg urn problems is that the number of balls in each color is unknown which seems to be connected to the belief degrees of human beings. Many scholars in management and economics studied these problems from a management perspective by the method of voting. For example, Ellsberg [4] showed that people prefer the option with no uncertainty when comparing options with uncertainty against options with no uncertainty. Eliaz and Ortoleva [3] studied Ellsberg urn problems with uncertainty in different dimensions and found that the majority of participants prefer no uncertainty when comparing options with no uncertainty against uncertainty in multiple dimensions.

Can Ellsberg urn problems be modeled by mathematical methods instead of voting? In 2019, Liu [14] provided a rigorous mathematical method to deal with Ellsberg urn problems by probability theory, uncertainty theory and chance theory; we will discuss this in Section 2. Liu discussed Ellsberg urn problem containing 30 red balls and 60 other balls (black or yellow balls) but the numbers of black or yellow balls are unknown. In this problem, there are two colors (black and yellow) and the number of balls in each of them is unknown. Following that, Lio [10] studied Ellsberg urn problems containing n balls which are either black, yellow or red. There are three colors and the number of balls in each of them is unknown. In this paper, we aim at studying Ellsberg urn problems with more than three colors and the number of balls of each color is also unknown. Then, we consider the question: what is the possibility of drawing a ball of each color?

The rest of the paper is organized as follows. Some basic concepts of uncertainty theory and chance theory are introduced in Section 2. In order to reasonably solve the Ellsberg urn problems with more than three colors, we apply probability theory, uncertainty theory and chance theory as rigorous mathematical tools in Section 3. Finally, a brief conclusion will be given in Section 4.

2 Preliminaries

Probability theory [8] is used for modeling frequency generated by samples while uncertainty theory is a mathematical tool for modeling belief degrees evaluated by domain experts. When we don’t have adequate historical samples (indispensable for probability theory), we can invite experts to provide their belief degrees. To model belief degrees, Liu [11] built uncertainty theory as an axiomatic system satisfying normality, duality and subadditivity. Later, the product axiom [12] was added into the axiomatic system.

In some situations, a complex system may contain uncertainty as well as randomness simultaneously. In order to model the phenomena, Liu [15] presented chance theory. Then Liu [15] introduced the fundamental concepts of chance measure and uncertain random variable. For uncertain random variables, there are some important contributions. For example, the operational law was proposed by Liu [16], the law of large numbers was proved by Yao and Gao [20], and the stronger law of large numbers was verified by Shen et al. [19].

In 1961, Ellsberg [4] proposed an urn problem: red and black balls filled in the urn are in unknown proportion. He then considered these problems from a management perspective by the method of voting. In this section, we study urn problems with multiple colors from a mathematical perspective and consider the fundamental question: what is the possibility of drawing a ball of each color?

In order to describe the general Ellsberg urn problems systematically, we first introduce the concept of degrees of freedom. The degrees of freedom of a system are the number of parameters of the system that may vary independently. Usually, this concept reflects the minimum number of parameters of the system that need to be identified in order to identify all the parameters. Suppose an Ellsberg urn is filled with n balls in m different colors where n and m are positive integers and each ball is marked in only one color. The number of balls of each color is unknown. In this system, the number of balls of each color is a parameter. Totally, there are m parameters. In order to determine all parameters, we need to determine at least (m - 1) parameters. Thus, the degrees of freedom of the system are (m - 1).

In 2019, Liu [14] discussed Ellsberg urn problems with one degree of freedom using probability theory, uncertainty theory, and chance theory. Lio [10] studied the case with two degrees of freedom. In this section, we consider Ellsberg urn problems with more than two degrees of freedom. Suppose an Ellsberg urn is filled with n balls in m different colors where n and m are positive integers and the number of balls of each color is unknown. The first step is numbering. Imagine that all balls are numbered from 1 to n in order of color 1, color 2, ⋯, color m. Take an uncertainty space (Γ, ℒ , ℳ) to be Γ = { ( γ 1 , γ 2 , ⋯ , γ m ) | γ 1 + γ 2 + ⋯ + γ m = n , γ k = 0 , 1 , ⋯ , n , k = 1 , ⋯ , m }(2) with power set and uncertain measure

M { Λ } = | Λ | | Γ | = | Λ | C n + m - 1 m - 1(3) where |Λ| represents the cardinality of the event Λ and C n + m - 1 m - 1 = ( n + m - 1 ) ! n ! ( m - 1 ) ! .(4)

Example 3.1. Let Γ be the set made up of all possible combinations of differently colored balls in the previously stipulated order. Each element (γ₁, γ₂, ⋯ , γ _m ) ∈ Γ means that there are γ _k balls in color k, k = 1, 2, ⋯ , m, respectively.

For example, if m = 1 and n = 1, the urn has one ball in one color resulting in only one case which is in accord with | Γ | = C 1 0 = 1 in Equation 4. If m = 2 and n = 2, we have | Γ | = C 3 1 = 3 which indicates that the urn with two balls of two colors includes three cases: both balls marked in color 1 or both balls marked in color 2 or one ball marked in color 1 and the other ball marked in color 2.

Since the number of balls of color i is completely unknown for each i with 1 ≤ i ≤ m and can be any integer between 0 and n, Liu [14] suggests treating it as an uncertain variable ξ i ( γ 1 , γ 2 , ⋯ , γ m ) = γ i .(5) Thus, ξ₁ + ξ₂ + ⋯ + ξ _m  = n. In addition, the balls of color i are numbered from 1 + ∑ j = 1 i - 1 γ j to ∑ j = 1 i γ j for i = 1, 2, ⋯ , m, respectively.

Theorem 3.1. Let n and m be positive integers, Γ be a set defined in Equation (2) and ξ₁, ξ₂, ⋯ , ξ _n be uncertain variables defined in Equation (5). For each element (k₁, k₂, ⋯ , k _m ) ∈ Γ, we have M { ξ 1 = k 1 , ξ 2 = k 2 , ⋯ , ξ m = k m } = 1 C n + m - 1 m - 1 .

Proof: It follows from Equation (3) that we have M { ξ 1 = k 1 , ξ 2 = k 2 , ⋯ , ξ m = k m } = M { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | γ 1 = k 1 , γ 2 = k 2 , ⋯ , γ m = k m } = | { ( k 1 , k 2 , ⋯ , k m ) } | | Γ | = 1 C n + m - 1 m - 1 . The theorem is proved.□

Remark 3.1. Theorem 3.1 suggests that all elements in Γ have the same uncertain measure.

To illustrate the meaning of the representations for the unknown numbers of balls with different colors, we focus attention on the uncertainty distribution of ξ _i for each i with 1 ≤ i ≤ m.

Theorem 3.2. Let n and m be positive integers. For any uncertain variables ξ₁, ξ₂, ⋯ , ξ _m defined in Equation (5), we have M { ξ i = k } = C n + m - k - 2 m - 2 C n + m - 1 m - 1 for k = 0, 1, ⋯ , n, respectively.

Proof: It follows from Equation (3) that M { ξ i = k } = M { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | ξ i ( γ 1 , γ 2 , ⋯ , γ m ) = k } = M { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | γ i = k } = | { γ i = k } | | Γ | = C n + m - k - 2 m - 2 C n + m - 1 m - 1 for i = 1, 2, ⋯ , m and k = 0, 1, ⋯ , n, respectively. The theorem is proved.□

Theorem 3.3. Let n and m be positive integers. The uncertainty distribution of ξ _i defined in Equation 5 is Φ i ( x ) = { 0 , ifx < 0 1 - C n + m - k - 2 m - 1 C n + m - 1 m - 1 , ifk ≤ x < k + 1 , k = 0 , 1 , ⋯ , n - 1 1 , ifx ≥ n for each i with 1 ≤ i ≤ m.

Proof: It follows from Theorem 3.2 that we have Φ i ( x ) = M { ξ i ≤ x } = M { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | ξ i ( γ 1 , γ 2 , ⋯ , γ m ) ≤ x } = M { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | γ i ≤ x } = | { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | γ i ≤ x } | | Γ | = { 0 , ifx < 0 ∑ j = 0 k C n + m - j - 2 m - 2 C n + m - 1 m - 1 , ifk ≤ x < k + 1 , k = 0 , 1 , ⋯ , n - 1 1 , ifx ≥ n = { 0 , ifx < 0 1 - C n + m - k - 2 m - 1 C n + m - 1 m - 1 , ifk ≤ x < k + 1 , k = 0 , 1 , ⋯ , n - 1 1 , ifx ≥ n for each i with 1 ≤ i ≤ m. The theorem is proved.□

Remark 3.2. It follows from Theorem 3.3 that ξ₁, ξ₂, ⋯, ξ _m are identically distributed.

Theorems 3.1 and 3.3 suggest that it is fair for the balls’ drawing in terms of the definitions of numbering all balls from 1 to n in order of color 1, color 2, ⋯, color m.

The second step is drawing a ball from the Ellsberg urn. Take a probability space (Ω, ℱ, Pr) where Ω = {1, 2, ⋯ , n}, ℱ is the power set of Ω, and probability measure is Pr { Λ } = | Λ | | Ω | = | Λ | n . Since the ball is drawn randomly and the unknown number of balls is uncertain, drawing a ball contains uncertainty and randomness. It should be represented by an event in the chance space ( Γ , ℒ , M ) × ( Ω , ℱ , Pr ) .

The ball of color 1 is drawn if and only if ω ≤ γ₁ and the ball of color m is drawn if and only if ω > ∑ j = 1 m - 1 γ j . Thus, drawing a ball of color 1 is represented by the event “ color 1 ″ = { ( γ 1 , γ 2 , ⋯ , γ m , ω ) ∈ Γ × Ω | ω ≤ γ 1 } and drawing a ball of color m is the event " color m " = { ( γ 1 , γ 2 , ⋯ , γ m , ω ) ∈ Γ × Ω | ω > ∑ j = 1 m − 1 γ j } .

Furthermore, drawing a ball of color i is represented by the event, " color i " = { ( γ 1 , γ 2 , ⋯ , γ m , ω ) ∈ Γ × Ω | ∑ j = 1 i − 1 γ j < ω ≤ ∑ j = 1 i γ j }(6) for each i with 1 ≤ i ≤ m.

Theorem 3.4. Let n and m be positive integers. For each i with 1 ≤ i ≤ m, the chance measure of the event “color i” defined in Equation 6 is Ch { " color i " } = 1 m .

Proof: When i = 1, it follows from Equation (1) chance measure that the chance measure of drawing a ball of color 1 is Ch { " color 1 " } = ∫ 0 1 Pr ⁡ { ω ∈ Ω | ℳ { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | ( γ 1 , γ 2 , ⋯ , γ m , ω ) ∈ " color 1 " } ≥ x } dx = ∫ 0 1 Pr ⁡ { ω ∈ Ω | M { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | ω ≤ γ 1 } ≥ x } dx = ∫ 0 1 Pr ⁡ { ω ∈ Ω | C n + m − ω − 1 m − 1 C n + m − 1 m − 1 ≥ x } dx = ∑ k=2 n+1 ∫ C n + m − k − 1 m − 1 / C n + m − 1 m − 1 C n + m − k m − 1 / C n + m − 1 m − 1 Pr ⁡ { ω ∈ Ω | ω ≤ k − 1 } dx = ∑ k=2 n+1 k − 1 n ⋅ C n + m − k m − 1 − C n + m − k − 1 m − 1 C n + m − 1 m − 1 = ∑ k = 1 n C n + m - k - 1 m - 1 n C n + m - 1 m - 1 = 1 m ∑ k = 1 n C n + m - k - 1 m - 1 C n + m - 1 m = 1 m . When i = m, the chance measure of drawing a ball of color m is Ch { " color m " } = ∫ 0 1 Pr ⁡ { ω ∈ Ω | M { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | ( γ 1 , γ 2 , ⋯ , γ m , ω ) ∈ " color m " } ≥ x } dx = ∫ 0 1 Pr ⁡ { ω ∈ Ω | M { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | ω > ∑ j = 0 m − 1 γ 1 } ≥ x } dx = ∫ 0 1 Pr ⁡ { ω ∈ Ω | C m + ω − 2 m − 1 C n + m − 1 m − 1 ≥ x } dx = ∑ k=1 n ∫ C m + k − 3 m − 1 / C n + m − 1 m − 1 C m + k − 2 m − 1 / C n + m − 1 m − 1 Pr ⁡ { ω ∈ Ω | ω ≥ k } dx = ∑ k=1 n n − k + 1 n ⋅ C m + k − 2 m − 1 − C m + k − 3 m − 1 C n + m − 1 m − 1 = ∑ k = 1 n C m + k − 2 m − 1 n C m + k − 1 m − 1 = 1 m ∑ k = 1 n C m + k − 2 m − 1 C n + m − 1 m = 1 m When 2 ≤ i ≤ m - 1, we have Ch { " color i " } = ∫ 0 1 Pr ⁡ { ω ∈ Ω | M { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | ( γ 1 , γ 2 , ⋯ , γ m , ω ) ∈ " color i " } ≥ x } dx = ∫ 0 1 Pr { ω ∈ Ω | M { ( γ 1 , γ 2 , ⋯ , γ m ) ∈ Γ | ∑ j = 1 i - 1 γ j < ω ≤ ∑ j = 1 i γ j } ≥ x } x = ∫ 0 1 Pr { ω ∈ Ω | C ω + i - 2 i - 1 · C n + m - ω - i m - i C n + m - 1 m - 1 ≥ x } x = ∫ 0 1 Pr { ω ∈ { 1 , 2 , ⋯ , ω * } | C ω + i - 2 i - 1 · C n + m - ω - i m - i C n + m - 1 m - 1 ≥ x } x + ∫ 0 1 Pr { ω ∈ { ω * + 1 , ω * + 2 , ⋯ , n } | C ω + i - 2 i - 1 · C n + m - ω - i m - i C n + m - 1 m - 1 ≥ x } x = ∑ k = 1 ω * ∫ C k + i - 3 i - 1 · C n + m - k - i + 1 m - i / C n + m - 1 m - 1 C k + i - 2 i - 1 · C n + m - k - i m - i / C n + m - 1 m - 1 Pr { ω ∈ Ω | k ≤ ω ≤ ω * } x + ∑ k = ω * + 2 n + 1 ∫ C k + i - 2 i - 1 · C n + m - k - i m - i / C n + m - 1 m - 1 C k + i - 3 i - 1 · C n + m - k - i + 1 m - i / C n + m - 1 m - 1 Pr { ω ∈ Ω | ω * + 1 ≤ ω ≤ k - 1 } x = ∑ k = 1 ω * ω * - k + 1 n ( C k + i - 2 i - 1 · C n + m - k - i m - i C n + m - 1 m - 1 - C k + i - 3 i - 1 · C n + m - k - i + 1 m - i C n + m - 1 m - 1 ) + ∑ k = ω * + 2 n + 1 k - ω * - 1 n ( C k + i - 3 i - 1 · C n + m - k - i + 1 m - i C n + m - 1 m - 1 - C k + i - 2 i - 1 · C n + m - k - i m - i C n + m - 1 m - 1 ) = ∑ k = 1 n C k + i - 2 i - 1 · C n + m - k - i m - i n C n + m - 1 m - 1 = 1 m ∑ k = 1 n C k + i - 2 i - 1 · C n + m - k - i m - i C n + m - 1 m = 1 m where ω * = ⌈ ( i - 1 ) n k - 1 ⌉ denotes the greatest integer not exceeding ( i - 1 ) n k - 1 . The theorem thus proved.□

In summary, suppose that an Ellsberg urn contains n balls of m different colors and the number of balls of each color is unknown. The chance measure of drawing a ball of color i is 1/m for each i with 1 ≤ i ≤ m. That means, all events: drawing a ball of color 1, drawing a ball of color 2, ⋯, and drawing a ball of color m have a same chance measure.

This paper proposed the general Ellsberg urn problem that contains n balls of m different colors and the number of balls of each color is unknown. The degrees of freedom of this problem are (m - 1). Liu [14] has discussed Ellsberg urn problems with one degree of freedom, and Lio [10] has studied problems with two degrees of freedom. Both of them concluded that the chance measure of drawing a ball of one color is equal to the chance measure of drawing a ball of another color. This paper studied the general cases of more than two degrees of freedom with the help of probability theory, uncertainty theory, and chance theory. It is concluded that drawing a ball of one color has the same chance measure to that drawing a ball in another color. As possible applications, some issues about incomes can be studied in the future.