Games with incomplete information and uncertain payoff: from the perspective of uncertainty theory

Abstract

In a game with incomplete information, players do not possess full information about their opponents. For such a game, Bayesian approaches which are based on probability theory were employed to find the equilibria. Sometimes, players are lack of data for probabilistic reasoning in which case Bayesian methods cannot be used. In this paper, we adopt a new mathematical framework—uncertainty theory to solve such a game. The player’s type (incomplete information) is modeled as an uncertain variable. In addition to previous studies, the payoff for each player is uncertain in our research, which is a realistic assumption in practice. We first define the games with incomplete information with uncertain payoff (Iu-game). Then, we create a new game (U-game) as a method to solve Iu-game. A theorem is provided to show that the equilibria are the same for both games. We use an example to illustrate the application of the proposed theorem. Finally, we generalize the form of U-game and show the equilibria are independent of the incomplete information.

Appendix

The research framework is displayed in Fig. 2.

Fig. 2

The research framework

We introduce some fundamental concepts and properties in uncertainty theory including uncertain measure, uncertain variable and expected value.

Definition 7

(Liu 2007) Let $\mathcal {L}$ be a $\sigma$-algebra on a nonempty set $\varGamma$. A set function $\mathcal {M}: \mathcal {L}\rightarrow [0,1]$ is called an uncertain measure if it satisfies the following axioms,

Axiom 1.$\mathcal {M}\{\varGamma \}=1$for the universal set$\varGamma$;

Axiom 2.$\mathcal {M}\{\varLambda \}+\mathcal {M}\{\varLambda ^{c}\}=1$for any event$\varLambda$;

Axiom 3. For every countable sequence of events $\varLambda _1, \varLambda _2,$ $\ldots ,$ we have

$$\begin{aligned} \displaystyle \mathcal {M}\left\{ \bigcup _{i=1}^{\infty }\varLambda _i\right\} \le \sum _{i=1}^{\infty }\mathcal {M}\{\varLambda _i\}. \end{aligned}$$

In order to provide the operational law, Liu (2009a) defined the product uncertain measure on the product $\sigma$-algebra $\mathcal {L}$, called product axiom.

Axiom 4. Let$(\varGamma _k,\mathcal {L}_k,\mathcal {M}_k)$be uncertainty spaces for$k=1, 2, \ldots$The product uncertain measure$\mathcal {M}$is an uncertain measure satisfying

$$\begin{aligned} \displaystyle \mathcal {M}\left\{ \prod _{k=1}^\infty \varLambda _k\right\} =\bigwedge _{k=1}^\infty \mathcal {M}_k\{\varLambda _k\} \end{aligned}$$

where$\varLambda _k$are arbitrarily chosen events from$\mathcal {L}_k$for$k=1, 2, \ldots$, respectively.

Definition 8

(Liu 2007) An uncertain variable $\xi$ is a function from an uncertainty space $(\varGamma ,\mathcal {L},\mathcal {M})$ to the set of real numbers such that for any Borel set B of real numbers, the set

$$\begin{aligned} \{\xi \in B\} = \{\gamma \in \varGamma |\xi (\gamma )\in B\} \end{aligned}$$

is an event.

In order to describe uncertain variable in practice, uncertainty distribution $\varPhi :\mathfrak {R}\rightarrow [0,1]$ of an uncertain variable $\xi$ is defined as $\varPhi (x)=\mathcal {M}\left\{ \xi \le x\right\}$. An uncertainty distribution $\varPhi (x)$ is said to be regular if it is a continuous and strictly increasing function with respect to x at which $0<\varPhi (x)<1$, and

$$\begin{aligned} \lim _{x\rightarrow -\infty }\varPhi (x)=0,\lim _{x\rightarrow +\infty }\varPhi (x)=1. \end{aligned}$$

If $\xi$ is an uncertain variable with regular uncertainty distribution $\varPhi$, then we call the inverse function $\varPhi ^{-1}(\alpha )$ as the inverse uncertainty distribution of $\xi$.

An uncertain variable $\xi$ is called linear if it has a linear uncertainty distribution

$$\begin{aligned} \varPhi (x)=\left\{ \begin{array}{ll} 0, &{}\quad {x< a }\\ \displaystyle \frac{x-a}{b-a},&{} \quad {a \le x < b}\\ 1, &{} \quad {x \ge b} \end{array}\right. \end{aligned}$$

denoted by $\mathcal {L}(a,b)$, where a and b are real numbers with $a<b$. The inverse uncertainty distribution of linear uncertain variable $\mathcal {L}(a,b)$ is

$$\begin{aligned} \varPhi ^{-1}(\alpha )=(1-\alpha )a+\alpha b. \end{aligned}$$

Definition 9

(Liu 2009a) The uncertain variables $\xi _1,\xi _2,\ldots ,\xi _m$ are said to be independent if

$$\begin{aligned} \mathcal {M}\left\{ \bigcap \limits _{i=1}^m\left\{ \xi _i\in B_i\right\} \right\} =\bigwedge _{i=1}^m\mathcal {M}\left\{ \xi _i\in B_i\right\} \end{aligned}$$

for any Borel sets $B_1,B_2,\ldots ,B_m$ of real numbers.

The expected value operator of uncertain variable, proposed by Liu (2007), is the average value of uncertain variable in the sense of uncertain measure and represents the size of uncertain variable.

Definition 10

(Liu 2007) Let $\xi$ be an uncertain variable. Then the expected value of $\xi$ is defined as

$$\begin{aligned} {E}[\xi ]=\int _0^{+\infty }\mathcal {M}\{\xi \ge x\}\mathrm{d}x- \int _{-\infty }^0\mathcal {M}\{\xi \le x\}\mathrm{d}x \end{aligned}$$

provided that at least one of the two integrals is finite.

For an uncertain variable $\xi$ with uncertainty distribution $\varPhi (x)$, Liu (2010b) showed that its expected value can be obtained by

$$\begin{aligned} {E}[\xi ]=\displaystyle \int _{-\infty }^{+\infty }x\mathrm{d}\varPhi (x). \end{aligned}$$

(3)

Furthermore, if $\varPhi (x)$ is regular, then

$$\begin{aligned} {E}[\xi ]=\displaystyle \int _0^{1}\varPhi ^{-1}(\alpha )\mathrm{d}\alpha . \end{aligned}$$

(4)

Theorem 3

(Liu 2009a) Let $\xi$ and $\eta$ be independent uncertain variables with finite expected values. Then for any real numbers a and b, we have

$$\begin{aligned} {E}[a\xi +b\eta ]=a{E}[\xi ]+b{E}[\eta ]. \end{aligned}$$