Matrix Expression of Bayesian Game *** Supported partly by NNSF 62073315 of
China, and China Postdoctoral Science Foundation 2020TQ0184.

Daizhan Cheng†, Changxi Li‡

†Institute of Systems Science, Chinese Academy of Sciences, Beijing 100190,
P.R.China

E-mail<EMAIL_ADDRESS>
‡Shandong University

Abstract: A matrix-based framework for Bayesian games is presented, using
semi-tensor product of matrices. Static Bayesian games are considered first.
Matrix expression of Bayesian games is proposed. Three kinds of conversions,
which convert Bayesian games to complete information games are investigated,
certain properties are obtained, including two kinds of Bayesian-Nash
equilibriums. Finally, dynamic Bayesian games are considered. Markoven dynamic
equations are obtained for some strategy updating rules.

Keywords: static bayesian game, conversion, evolutionary bayesian game, semi-
tensor product (STP) of matrices.

##  1 Introduction

The idea of game theory has a very long history. But it is a common believe
that the symbol of modern game theory is the book of von Neumann and
Morgenstern [33]. Modern game theory and modern control theory are like a
twin. Not only because they were born in almost same time period, but also
they have similar purpose: to “manipulate” objects according to players’
purpose. Nowadays, more people try to use techniques in game theory to control
problems, which are sometimes called game-based control, or game theoretic
control [1, 3, 17, 32, 36, 38].

Bayesian game is also called the incomplete information game. Comparing with
complete information game, which is also called normal noncooperative game,
Bayesian has more practical applications, because games with incomplete
information, or uncertainty, are widely existing in real world, for instance
[18] gives some interesting examples such as Cournot competition under
asymmetric information, first-price, sealed-bid auction, etc. Its various
applications are investigated [26, 19, 13].

Recently, semi-tensor product (STP) of matrices has been used to study some
problems in game theory. For instance, there are STP-based investigations for
the evolution of dynamic games [20, 8], potential games [7, 27], vector space
structure of finite games [9, 21], and Boolean games [10], just to mention a
few.

Potential game was firstly proposed by [31], and then systematically developed
in [30]. It is of particular importance in game theoretic control [4]. Various
problems about Bayesian potential games have also been investigated [12, 14,
15, 16].

The purpose of this paper is to provide a framework for Bayesian game. Roughly
speaking, the major technique to deal with Bayesian game is to convert it into
a normal game, which is also called complete information game. This method was
firstly proposed by Harsanyi [22, 24, 25]. It is recently called the Harsanyi
transformation. Another commonly used transformation is Selten transformation
[39]. Using STP, a very recent work proposed a new transformation, called Ex-
ante agent transformation of Bayesian games, which keeps potential property
unchanged [34].

The paper is organized as follows: Section 2 consists of some necessary
preliminaries, including STP, normal noncooperative game, and potential game.
Section 3 considers the matrix expression of static Bayesian game. In Section
4, two types fo Bayesian-Nash equilibriums are proposed. In Section 5, three
kinds of conversions from Bayesian game to normal game. Section 6 considers
two kinds of Bayesian potential games. Their verifications and some properties
are presented. In Section 7, the dynamic Bayasian games are investigated.
Particular interest has been put on some stratify updating rules, which lead
to Markovin dynamic equations. Section 8 is a brief conclusion.

##  2 Preliminaries

###  2.1 STP of matrices

This subsection briefly reviews STP of matrices.

######  Definition 2.1.

[5, 6]: Let $M\in{\cal M}_{m\times n}$, $N\in{\cal M}_{p\times q}$, and
$t=\operatorname{lcm}\\{n,p\\}$ be the least common multiple of $n$ and $p$.
The semi-tensor product (STP) of $M$ and $N$, denoted by $M\ltimes N$, is
defined as

$\displaystyle\left(M\otimes I_{t/n}\right)\left(N\otimes
I_{t/p}\right)\in{\cal M}_{mt/n\times qt/p},$ (1)

where $\otimes$ is the Kronecker product.

Note that when $n=p$, $M\ltimes N=MN$. That is, the semi-tensor product is a
generalization of conventional matrix product. Moreover, it keeps all the
properties of conventional matrix product available [6]. Hence we can omit the
symbol $\ltimes$. Throughout this paper the matrix product is assumed to be
STP, and the symbol $\ltimes$ is mostly omitted.

The following are some basic properties:

######  Proposition 2.2.

  1. 1.

(Associative Law)

$\displaystyle(F\ltimes G)\ltimes H=F\ltimes(G\ltimes H).$ (2)

  2. 2.

(Distributive Law)

$\displaystyle\begin{cases}F\ltimes(aG\pm bH)=aF\ltimes G\pm bF\ltimes H,\\\
(aF\pm bG)\ltimes H=aF\ltimes H\pm bG\ltimes H,\quad
a,b\in{\mathbb{R}}.\end{cases}$ (3)

######  Proposition 2.3.

  1. 1.

Let $X\in{\mathbb{R}}^{m}$, $Y\in{\mathbb{R}}^{n}$ be two columns. Then

$\displaystyle X\ltimes Y=X\otimes Y.$ (4)

  2. 2.

Let $\omega\in{\mathbb{R}}^{m}$, $\sigma\in{\mathbb{R}}^{n}$ be two rows. Then

$\displaystyle\omega\ltimes\sigma=\sigma\otimes\omega.$ (5)

The following proposition shows that the STP satisfies the block-
multiplication law.

######  Proposition 2.4.

Let

$A=\begin{bmatrix}A^{11}&A^{12}&\cdots&A^{1q}\\\
A^{21}&A^{22}&\cdots&A^{2q}\\\ \vdots&~{}&~{}&~{}\\\
A^{p1}&A^{p2}&\cdots&A^{pq}\\\ \end{bmatrix},\quad
B=\begin{bmatrix}B^{11}&B^{12}&\cdots&B^{1r}\\\ B^{21}&B^{22}&\cdots&B^{2r}\\\
\vdots&~{}&~{}&~{}\\\ B^{q1}&B^{q2}&\cdots&B^{qr}\\\ \end{bmatrix},$

where

$\begin{array}[]{l}n_{c}(A^{ij})=n_{c}(A^{ik}),\quad 1\leq j,k\leq q,\;\forall
i,\\\ n_{r}(B^{ik})=n_{r}(B^{jk}),\quad 1\leq i,j\leq q,\;\forall
k.\end{array}$

($n_{c}(A)$ ($n_{r}(A)$) is the column (row) of $A$. Then

$A\ltimes B=\left(C^{i,j}\right),$

where

$C^{i,j}=\mathop{\sum}\limits_{k=1}^{q}A^{i,k}\ltimes B^{k,j},\quad
i=1,2,\cdots,p;\;j=1,2,\cdots,r.$

About the transpose, we have

######  Proposition 2.5.

$\displaystyle(A\ltimes B)^{\mathrm{T}}=B^{\mathrm{T}}\ltimes A^{\mathrm{T}}.$
(6)

About the inverse, we have

######  Proposition 2.6.

Assume $A$ and $B$ are invertible, then

$\displaystyle(A\ltimes B)^{-1}=B^{-1}\ltimes A^{-1}.$ (7)

The following property is for STP only.

######  Proposition 2.7.

Let $X\in{\mathbb{R}}^{m}$ be a column and $M$ a matrix. Then

$\displaystyle X\ltimes M=\left(I_{m}\otimes M\right)X.$ (8)

Let $m\geq 1$, $n\geq 1$, and $1\leq k\leq mn$. Define

$\displaystyle j_{k}:=k(mod~{}n);\quad i_{k}=(k-j)/n+1.$ (9)

######  Definition 2.8.

[5] A matrix $W_{[m,n]}\in{\cal M}_{mn\times mn}$, defined by

$\displaystyle\operatorname{Col}_{k}\left(W_{[m,n]}\right):=\delta_{mn}^{(j_{k}-1)m+i},\quad
k=1,\cdots,mn,$ (10)

is called the $(m,n)$-th dimensional swap matrix.

The basic function of the swap matrix is to “swap” two vectors. That is,

######  Proposition 2.9.

Let $X\in{\mathbb{R}}^{m}$ and $Y\in{\mathbb{R}}^{n}$ be two columns. Then

$\displaystyle W_{[m,n]}\ltimes X\ltimes Y=Y\ltimes X.$ (11)

Define a matrix $PR_{n}\in{\cal L}_{n^{2}\times n}$, called the order reducing
matrix, as

$\displaystyle\operatorname{Col}_{i}(PR_{n})=\delta_{n^{2}}^{(i-1)n+i},\quad
i=1,\cdots,n.$ (12)

######  Definition 2.10.

Let $A\in\mathcal{M}_{p\times n}$ and $B\in\mathcal{M}_{q\times n}$. Then the
Khatri-Rao Product of $A$ and $B$ is

$\displaystyle\begin{array}[]{ccl}A*B=[\operatorname{Col}_{1}(A)\ltimes\operatorname{Col}_{1}(B),\cdots,\operatorname{Col}_{n}(A)\ltimes\operatorname{Col}_{n}(B)]\in\mathcal{M}_{pq\times
n}.\end{array}$ (14)

###  2.2 Normal Non-cooperative Game

######  Definition 2.11.

A (finite non-cooperative) normal game is described as a triple $G=(N,S,C)$,
where

  * (i)

$N=\\{1,2,\cdots,n\\}$ is the set of players;

  * (ii)

$S=\prod_{i=1}^{n}S_{i}$ is called the profile, with

$\displaystyle S_{i}=\\{s^{i}_{1},s^{i}_{2},\cdots,s^{i}_{k_{i}}\\},\quad
i=1,\cdots,n,$ (15)

the set of strategies of player $i$, and $S_{-i}:=\prod_{j\neq i}^{n}S_{j}$;

  * (iii)

$C=(c_{1},\cdots,c_{n})\in{\mathbb{R}}^{n}$ with
$c_{i}:S\rightarrow{\mathbb{R}}$ defined as

$\displaystyle c_{i}:=c_{i}(s_{1},\cdots,s_{n}),\quad s_{j}\in
S_{j},\;j=1,\cdots,n,\quad i=1,\cdots,n,$ (16)

is called the payoff function of player $i$.

Denote the structure vector of $c_{i}$ by $V_{i}^{c}$, $i=1,\cdots,n$. Assume
$|N|=n$ and $k_{i}$, $i=1,\cdots,n$ are fixed, we denote the set of such games
by ${\cal G}_{[n;k_{1},\cdots,k_{n}]}$. Note that if $G\in{\cal
G}_{[n;k_{1},\cdots,k_{n}]}$, then $G$ is uniquely determined by
$\\{V_{i}^{c}\;\big{|}\;i=1,\cdots,n\\}$. Hence,

$\displaystyle{\cal G}_{[n;k_{1},\cdots,k_{n}]}\simeq{\mathbb{R}}^{n\kappa},$
(17)

where $\kappa=\prod_{i=1}^{n}k_{i}$.

######  Definition 2.12.

$G\in{\cal G}_{[n;k_{1},\cdots,k_{n}]}$ is expressed as
$V_{G}=(V_{1}^{c},\cdots,V_{n}^{c})\in{\mathbb{R}}^{n\kappa}$, where
$V_{i}^{c}$ is the structure vector of $c_{i}$ of $G$, $i=1,\cdots,n$.

######  Remark 2.13.

The decision-making (or strategy-selection of each player $j$) is based on the
difference of $c_{j}$ over two different strategy profiles

$c_{j}(s)-c_{j}(s^{\prime}),\quad j=1,\cdots,n;\;s,s^{\prime}\in S.$

For instance, the following concepts are depending on such differences:

  1. (i)

A strategy profile $s^{*}=(s_{1}^{*},\cdots,s_{n}^{*})$ is a Nash equilibrium
if

$\displaystyle c_{i}(s_{i}^{*},s_{-i}^{*})\geq
c_{i}(s_{i},s_{-i}^{*}),\quad\forall s_{i}\in S_{i},~{}s_{-i}^{*}\in
S_{-i},i=1,\cdots,n.$ (18)

It is easy to verify that Nash equilibrium depends on
$\\{c_{j}(s)-c_{j}(s^{\prime})\;\big{|}\;j=1,\cdots,n\\}$ only.

  2. (ii)

In the definition of potential games [30] it is assumed that there is a
pseudo-mixed-logical function $P:\prod_{i=1}^{n}{\cal
D}_{k_{i}}\rightarrow{\mathbb{R}}$ such that

$c_{i}(x_{i},s_{-i})-c_{i}(y_{i},s_{-i})=P(x_{i},s_{-i})-P(y_{i},s_{-i}),\quad
i=1,\cdots,n.$

It is clear that whether a game is potential depends on
$\\{c_{j}(s)-c_{j}(s^{\prime})\;\big{|}\;j=1,\cdots,n\\}$ only.

  3. (iii)

Consider an evolutionary game, two of the most commonly used strategy updating
rules (SURs) are the better and best response dynamics [11].

     * •

Better response SUR: updating its strategy to a strategy in
$\\{s_{i}\;\big{|}\;c_{i}(s_{i},s_{-i}(t))>c_{i}(s(t))\\}$, chosen uniformly
at random;

     * •

Best response SUR: updating its strategy to a strategy in
$\operatorname{argmax}_{s_{i}}c_{i}(s_{i},s_{-i}(t))$, chosen uniformly at
random;

One sees easily that they depend on
$\\{c_{j}(s)-c_{j}(s^{\prime})\;\big{|}\;j=1,\cdots,n\\}$ only.

The following proposition is obvious.

######  Proposition 2.14.

Two games $G$ and $G^{\prime}$ are equivalent (for decision making) if there
exists a row vector $v\in{\mathbb{R}}^{\kappa}$ such that

$\displaystyle V^{c}_{i}+v=V^{c^{\prime}}_{i},\quad i=1,\cdots,n.$ (19)

From Proposition 2.14 one sees easy that a game (precisely, an equivalent
class of games) can be expressed as

$\displaystyle
W_{G}:=[V^{c}_{2}-V^{c}_{1},\;V^{c}_{3}-V^{c}_{1},\;\cdots,\;V^{c}_{n}-V^{c}_{1}].$
(20)

We call (20) the equivalent vector form of $G$.

Using equivalent vector form of $G$, one sees that
$G_{[n;k_{1},\cdots,k_{n}]}$ is an $(n-1)\kappa$ dimensional vector space.
That is,

$\displaystyle G_{[n;k_{1},\cdots,k_{n}]}\sim{\mathbb{R}}^{(n-1)\kappa}.$ (21)

###  2.3 Potential Game

######  Definition 2.15.

[30] Consider a finite game $G\in{\cal G}_{[n;k_{1},\cdots,k_{n}]}$. If there
is a function $P:S\rightarrow{\mathbb{R}}$, such that for every $i\in N$ and
for every $s_{-i}\in S_{-i}$

$\displaystyle c_{i}(x,s_{-i})-c_{i}(y,s_{-i})=P(x,s_{-i})-P(y,s_{-i}),\quad
i=1,\cdots,n,$ (22)

then $G$ is called an exact potential game.

The followings are some fundamental properties of potential.

######  Theorem 2.16.

[30] If $G$ is a potential game, then the potential function $P$ is unique up
to a constant number. Precisely, if $P_{1}$ and $P_{2}$ are two potential
functions, then $P_{1}-P_{2}=c_{0}\in{\mathbb{R}}$.

######  Theorem 2.17.

[30] Let $P$ be a potential function for $G$. Then $s\in S$ is an equilibrium
point of $G$, if and only if

$\displaystyle P(s)\geq P(s_{-i},x),\quad\forall x\in S_{i},\;i=1,\cdots,n.$
(23)

Particularly, if $P$ admits a maximal value in $S$, then $G$ has at least one
pure Nash equilibrium.

######  Corollary 2.18.

Every finite potential game possesses a pure Nash equilibrium.

The following argument is a generalization of potential equation [7].

Assume $G\in{\cal G}_{[n;k_{1},\cdots,k_{n}]}$ is potential, define

$\displaystyle
d_{i}(s_{1},\cdots,s_{n}):=c_{i}(s_{1},\cdots,s_{n})-P(s_{1},\cdots,s_{n}),\quad
i=1,\cdots,n.$ (24)

Then it is easy to verify that $d_{i}$ is independent of $s_{i}$, denoted by
$d_{i}=d_{i}(s_{1},\cdots,\hat{s}_{i},\cdots,s_{n})$. In vector form (24)
becomes

$\displaystyle V^{d}_{i}\ltimes_{j\neq i}s_{j}=V^{c}_{i}s-V^{P}s,\quad
i=1,\cdots,n.$ (25)

where $s=\ltimes_{j=1}^{n}s_{j}$.

Define

$E_{i}:=I_{\alpha_{i}}\otimes{\bf 1}^{T}_{k_{i}}\otimes I_{\beta_{i}},\quad
i=1,\cdots,n,$

where

$\begin{array}[]{ll}\alpha_{1}=1,&\alpha_{i}=\prod_{j=1}^{i-1}k_{j},\;j\geq
2\\\ \beta_{n}=1,&\beta_{i}=\prod_{j=i+1}^{n}k_{j},\;j\leq n-1.\end{array}$

Then (25) can be expressed as

$\displaystyle V^{d}_{i}E_{i}=V^{c}_{i}-V^{p},\quad i=1,\cdots,n.$ (26)

Solving the first equation to get

$V^{P}=V^{c}_{1}-V^{d}_{1}E_{1},$

and plugging it into the other equations of (26) yield

$\displaystyle V^{d}_{i}E_{i}=V^{d}_{1}E_{1}=V^{c}_{i}-V^{c}_{1},\quad
i=2,3,\cdots,n.$ (27)

Denote

$\displaystyle\xi_{i}=(V^{d}_{i})^{T},\quad i=1,\cdots,n;\quad b:=W_{G}^{T}.$
(28)

Then (27) can be expressed as

$\displaystyle\Psi\xi=b,$ (29)

where $\xi=(\xi_{1}^{T},\cdots,\xi_{n}^{T})^{T}$ and

$\displaystyle\Psi=\begin{bmatrix}-E_{1}^{T}&E_{2}^{T}&0&\cdots&0\\\
-E_{1}^{T}&0&E_{3}^{T}&\cdots&0\\\ \vdots&~{}&~{}&~{}&~{}\\\
-E_{1}^{T}&0&0&\cdots&E_{n}^{T}.\end{bmatrix}$ (30)

Summarizing the above argument, we have

######  Theorem 2.19.

[9] Let $G\in{\cal G}_{[n,k_{1},\cdots,k_{n}]}$. $G$ is a potential game, if
and only if (29) has solution. Moreover, when a solution exists, then

$\displaystyle V^{P}=V^{c}_{1}-\xi_{1}^{T}E_{1}.$ (31)

##  3 Static Bayesian Game

######  Definition 3.1.

[18] A (finite) static Bayesian game (BG) $G=(N,T,A,c,Pr)$ consists of

  * (i)

Player set:

$\displaystyle N=\\{1,2,\cdots,n\\}.$ (32)

  * (ii)

Type set:

$\displaystyle T=T_{1}\times T_{2}\times\cdots\times T_{n},$ (33)

where $T_{i}$ is the type set of player $i$ with

$\displaystyle T_{i}=\\{t_{i}^{1},t_{i}^{2},\cdots,t_{i}^{\tau_{i}}\\},\quad
i=1,2,\cdots,n.$ (34)

  * (iii)

Profile and actions:

$\displaystyle A=A_{1}\times A_{2}\times\cdots\times A_{n},$ (35)

where $A_{i}$ is the action set of player $i$ with

$\displaystyle
A_{i}=\left\\{a_{i}^{1},a_{i}^{2},\cdots,a_{i}^{r_{i}}\right\\},\quad
i=1,2,\cdots,n,$ (36)

and $A_{i}(t_{i}^{j})$ is the admissible actions under type $t_{i}^{j}$ with

$\displaystyle A_{i}(t_{i}^{j})\subset A_{i},\quad
j=1,2,\cdots,\tau_{i},\;i=1,\cdots,n.$ (37)

  * (iv)

Payoff functions:

$\displaystyle c_{i}:A_{1}(t_{1}^{j_{1}})\times
A_{2}(t_{2}^{j_{2}})\times\cdots\times
A_{n}(t_{n}^{j_{n}})\rightarrow{\mathbb{R}},\quad i=1,2,\cdots,n.$ (38)

  * (v)

Beliefs:

$\displaystyle
p_{i}\left(t_{-i}\;|\;t_{i}\right)=\dfrac{Pr(t_{1},t_{2},\cdots,t_{n})}{Pr(t_{i})}=\dfrac{Pr(t_{i},t_{-i})}{\mathop{\sum}\limits_{t^{\prime}_{-i}}Pr(t_{i},t^{\prime}_{-i})},$
(39)

where $t_{-i}\in T_{-i}:=\prod_{j\neq i}T_{j}$.
$Pr(t_{1},t_{2},\cdots,t_{n})$, called the common prior, is a common
knowledge, and $t_{i}$ is a private knowledge for player $i$.

Identify $i\sim\delta_{l}^{i}\in D_{l}$, then $\delta_{l}^{i}$ is called the
vector representation of integer $i.$ By virtue of vector representation of
actions and types, then the action set $A_{i}$ and type set $T_{i}$ can be
expressed as

$A_{i}=\left\\{\delta_{r_{i}}^{j}\;|\;1\leq j\leq
r_{i}\right\\};T_{i}=\left\\{\delta_{\tau_{i}}^{j}\;|\;1\leq
j\leq\tau_{i}\right\\},\quad i=1,2,\cdots,n.$

Denote by $r=\prod_{i=1}^{n}r_{i}$, $\tau=\prod_{i=1}^{n}\tau_{i}$,
$R_{-\infty}=R\cup\\{-\infty\\}$, and

$\displaystyle\begin{array}[]{l}\bar{c}_{i}(a_{1},\cdots,a_{n};t_{1},\cdots,t_{n})=\begin{cases}c_{i}(a_{1},\cdots,a_{n};t_{1},\cdots,t_{n}),\quad
a_{j}\in A_{j}(t_{j}),\\\ -\infty,\quad\mbox{otherwise},\quad 1\leq i\leq
n.\end{cases}\end{array}$ (41)

Then we have

$\displaystyle\bar{c}_{i}:\Delta_{r\tau}\rightarrow{\mathbb{R}}_{-\infty},\quad
1\leq i\leq n.$ (42)

Using (41), it is obvious that for each $i$ there exists a unique row vector
$V_{i}\in{\mathbb{R}}_{-\infty}^{r\tau}$ such that

$\displaystyle\bar{c}_{i}(a_{1},\cdots,a_{n};t_{1},\cdots,t_{n})=V_{i}^{\bar{c}}ta,\quad
1\leq i\leq n,$ (43)

where $a=\ltimes_{i=1}^{n}a_{i}$, $t=\ltimes_{i=1}^{n}t_{i}$. Let
$V_{G}:=[V_{1}^{\bar{c}},\cdots,V_{n}^{\bar{c}}]$, which is called the
structure vector of Bayesian game $G$.

######  Remark 3.2.

  * (i)

Here we assume the purpose of each player is to maximize his payment. So using
$\bar{c}_{i}$ to optimize payment is equivalent to $c_{i}$. If the purpose of
each player is to minimize his cost, then in (41) $+\infty$ is used to replace
$-\infty$.

  * (ii)

In fact, (43) provides a natural vector space structure for
$c_{i}(a,t)\in{\mathbb{R}}^{r\tau}$, where $a=a_{1}a_{2}\cdots
a_{n}\in\Delta_{r}$, $t=t_{1}t_{2}\cdots t_{n}\in\Delta_{\tau}$.

######  Example 3.3.

Consider a finite BG

$\displaystyle G=(N,T,A,c,p),$ (44)

where

$\begin{array}[]{ccl}N&=&\\{1,2\\};\\\ T&=&\\{T_{1},T_{2}\\},\;\mbox{with}\\\
T_{1}&=&\\{t_{1}^{1},t_{1}^{2}\\},\\\ T_{2}&=&\\{t_{2}^{1},t_{2}^{2}\\};\\\
A&=&\\{A_{1},A_{2}\\},\;\mbox{with}\\\
A_{1}&=&\\{a_{1}^{1},a_{1}^{2},a_{1}^{3}\\},\\\
A_{2}&=&\\{a_{2}^{1},a_{2}^{2},a_{2}^{3}\\},\\\
A_{1}(t_{1}^{1})&=&\\{a_{1}^{1},a_{1}^{2}\\},\\\
A_{1}(t_{1}^{2})&=&\\{a_{1}^{2},a_{1}^{3}\\},\\\
A_{2}(t_{2}^{1})&=&\\{a_{2}^{1},a_{2}^{2},a_{2}^{3}\\},\\\
A_{2}(t_{2}^{2})&=&\\{a_{2}^{1},a_{2}^{3}\\}.\\\ \end{array}$

Table 1: Payment Function for $t_{1}^{1}-t_{2}^{1}$ $c_{1}\backslash c_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $2,~{}3$ | $1,~{}4$ | $1,~{}-2$
$a_{1}^{2}$ | $1,~{}-2$ | $2,~{}1$ | $0,-3$
Table 2: Payment Function for $t_{1}^{1}-t_{2}^{2}$ $c_{1}\backslash c_{2}$ | $a_{2}^{1}$ | $a_{2}^{3}$
---|---|---
$a_{1}^{1}$ | $-1,~{}2$ | $1,~{}3$
$a_{1}^{2}$ | $1,~{}-2$ | $-2,~{}0$
Table 3: Payment Function for $t_{1}^{2}-t_{2}^{1}$ $c_{1}\backslash c_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{2}$ | $3,~{}5$ | $2,~{}4$ | $2,~{}0$
$a_{1}^{3}$ | $2,~{}-2$ | $-2,~{}4$ | $3,~{}3$
Table 4: Payment Function for $t_{1}^{2}-t_{2}^{2}$ $c_{1}\backslash c_{2}$ | $a_{2}^{1}$ | $a_{2}^{3}$
---|---|---
$a_{1}^{2}$ | $2,~{}1$ | $-1,~{}-3$
$a_{1}^{3}$ | $2,~{}2$ | $-1,~{}-2$
Table 5: Distribution $t_{1}\backslash t_{2}$ | $t_{2}^{1}$ | $t_{2}^{2}$
---|---|---
$t_{1}^{1}$ | $0.3$ | $0.2$
$t_{1}^{2}$ | $0.1$ | $0.4$

Then

$\begin{array}[]{l}P_{r}(t^{1}_{2}\;|\;t_{1}^{1})=\frac{P_{r}(t_{1}^{1}\cup
t_{2}^{1})}{P_{r}(t^{1}_{1})}=\frac{P_{r}(t_{1}^{1}\cup
t_{2}^{1})}{P_{r}(t^{1}_{1}\cap t^{1}_{2})+P_{r}(t^{1}_{1}\cap
t^{2}_{2})}=0.75\\\ P_{r}(t^{2}_{2}\;|\;t_{1}^{1})=0.25;\quad
P_{r}(t^{1}_{2}\;|\;t_{1}^{2})=1/3;\quad P_{r}(t^{2}_{2}\;|\;t_{1}^{2})=2/3\\\
P_{r}(t^{1}_{1}\;|\;t_{2}^{1})=0.6;\quad
P_{r}(t^{2}_{1}\;|\;t_{2}^{1})=0.4;\quad P_{r}(t^{1}_{1}\;|\;t_{2}^{2})=0.2\\\
P_{r}(t^{2}_{1}\;|\;t_{2}^{2})=0.8\\\ \end{array}$

Hence the beliefs are:

$\displaystyle\begin{array}[]{l}p_{t_{1}^{1}}=(0.75,~{}0.25)^{T},\quad
p_{t_{1}^{2}}=(1/3,~{}2/3)^{T}\\\ p_{t_{2}^{1}}=(0.6,~{}0.4)^{T},\quad
p_{t_{2}^{2}}=(0.2,~{}0.8)^{T}.\end{array}$ (47)

Table 6: Payoffs $c\backslash(t,a)$ | $(\delta_{4}^{1},\delta_{9}^{1})$ | $(\delta_{4}^{1},\delta_{9}^{2})$ | $(\delta_{4}^{1},\delta_{9}^{3})$ | $(\delta_{4}^{1},\delta_{9}^{4})$
---|---|---|---|---
$\bar{c}_{1}$ | $2$ | $1$ | $1$ | $1$
$\bar{c}_{2}$ | $3$ | $4$ | $-2$ | $-2$
$c\backslash(t,a)$ | $(\delta_{4}^{1},\delta_{9}^{5})$ | $(\delta_{4}^{1},\delta_{9}^{6})$ | $(\delta_{4}^{1},\delta_{9}^{7})$ | $(\delta_{4}^{1},\delta_{9}^{8})$
$\bar{c}_{1}$ | $2$ | $0$ | $-\infty$ | $-\infty$
$\bar{c}_{2}$ | $1$ | $-3$ | $-\infty$ | $-\infty$
$c\backslash(t,a)$ | $(\delta_{4}^{1},\delta_{9}^{9})$ | $(\delta_{4}^{2},\delta_{9}^{1})$ | $(\delta_{4}^{2},\delta_{9}^{2})$ | $(\delta_{4}^{2},\delta_{9}^{3})$
$\bar{c}_{1}$ | $-\infty$ | $-1$ | $-\infty$ | $1$
$\bar{c}_{2}$ | $-\infty$ | $2$ | $-\infty$ | $3$
Table 7: Payoffs(Cont’d) $c\backslash(t,a)$ | $(\delta_{4}^{2},\delta_{9}^{4})$ | $(\delta_{4}^{2},\delta_{9}^{5})$ | $(\delta_{4}^{2},\delta_{9}^{6})$ | $(\delta_{4}^{2},\delta_{9}^{7})$
---|---|---|---|---
$\bar{c}_{1}$ | $1$ | $-\infty$ | $-2$ | $-\infty$
$\bar{c}_{2}$ | $-2$ | $-\infty$ | $0$ | $-\infty$
$c\backslash(t,a)$ | $(\delta_{4}^{2},\delta_{9}^{8})$ | $(\delta_{4}^{2},\delta_{9}^{9})$ | $(\delta_{4}^{3},\delta_{9}^{1})$ | $(\delta_{4}^{3},\delta_{9}^{2})$
$\bar{c}_{1}$ | $-\infty$ | $-\infty$ | $-\infty$ | $-\infty$
$\bar{c}_{2}$ | $-\infty$ | $-\infty$ | $-\infty$ | $-\infty$
$c\backslash(t,a)$ | $(\delta_{4}^{4},\delta_{9}^{3})$ | $(\delta_{4}^{3},\delta_{9}^{4})$ | $(\delta_{4}^{3},\delta_{9}^{5})$ | $(\delta_{4}^{3},\delta_{9}^{6})$
$\bar{c}_{1}$ | $-\infty$ | $3$ | $2$ | $2$
$\bar{c}_{2}$ | $-\infty$ | $5$ | $4$ | $0$
Table 8: Payoffs(Cont’d) $c\backslash(t,a)$ | $(\delta_{4}^{3},\delta_{9}^{7})$ | $(\delta_{4}^{3},\delta_{9}^{8})$ | $(\delta_{4}^{3},\delta_{9}^{9})$ | $(\delta_{4}^{4},\delta_{9}^{1})$
---|---|---|---|---
$\bar{c}_{1}$ | $2$ | $-2$ | $3$ | $-\infty$
$\bar{c}_{2}$ | $-2$ | $4$ | $3$ | $-\infty$
$c\backslash(t,a)$ | $(\delta_{4}^{4},\delta_{9}^{2})$ | $(\delta_{4}^{4},\delta_{9}^{3})$ | $(\delta_{4}^{4},\delta_{9}^{4})$ | $(\delta_{4}^{4},\delta_{9}^{5})$
$\bar{c}_{1}$ | $-\infty$ | $-\infty$ | $2$ | $-\infty$
$\bar{c}_{2}$ | $-\infty$ | $-\infty$ | $-1$ | $-\infty$
$c\backslash(t,a)$ | $(\delta_{4}^{4},\delta_{9}^{6})$ | $(\delta_{4}^{4},\delta_{9}^{7})$ | $(\delta_{4}^{4},\delta_{9}^{8})$ | $(\delta_{4}^{4},\delta_{9}^{9})$
$\bar{c}_{1}$ | $1$ | $2$ | $-\infty$ | $-1$
$\bar{c}_{2}$ | $-3$ | $2$ | $-\infty$ | $-2$

The cost functions are described as follows:

$\begin{cases}\bar{c}_{1}(a,t)=V^{\bar{c}}_{1}ta,\\\
\bar{c}_{2}(a,t)=V^{\bar{c}}_{2}ta,\end{cases}$

where

$\begin{array}[]{ccl}V^{\bar{c}}_{1}&=&[2,1,1,1,2,0,-\infty,-\infty,-\infty,-1,-\infty,1,1,\\\
{}\hfil&{}\hfil&-\infty,-2,-\infty,-\infty,-\infty,-\infty,-\infty,-\infty,3,2,2,\\\
{}\hfil&{}\hfil&2,-2,3,-\infty,-\infty,-\infty,2,-\infty,-1,2,-\infty,-1]\\\
V^{\bar{c}}_{2}&=&[3,4,-2,-2,1,-3,-\infty,-\infty,-\infty,2,-\infty,3,-2,\\\
{}\hfil&{}\hfil&-\infty,0,-\infty,-\infty,-\infty,-\infty,-\infty,-\infty,5,4,0,\\\
{}\hfil&{}\hfil&-2,4,3,-\infty,-\infty,-\infty,-1,-\infty,3,2,-\infty,-2]\\\
\end{array}$

##  4 Bayesian-Nash Equilibrium

This section proposes two kinds of types first.

######  Definition 4.1.

There are two kinds of types:

  * (i)

Types of Nature (TN):

The types are determined by pre-assigned distribution
$Pr(t_{1},t_{2},\cdots,t_{n})$, which is a common knowledge. Each player $i$
knows the type $t_{i}$ assigned to him. The type is assigned by Nature.

  * (ii)

Types of Human (TH):

Player $i$ has the right to choose $t_{i}$ for entering the game. Then $t_{i}$
becomes part of strategy for player $i$, $i=1,2,\cdots,n$.

The expected payoff functions under different kinds of types are defined as
follows.

######  Proposition 4.2.

  * (i)

TN:

The expected value of player $i$, using actions $a=(a_{1},a_{2},\cdots,a_{n})$
is

$\displaystyle\begin{array}[]{ccl}E^{N}_{i}(a)&:=&E_{i}(a(T))\\\
&=&\mathop{\sum}\limits_{t\in T}Pr(t)\bar{c}_{i}(a,t),\\\
&=&\mathop{\sum}\limits_{t\in T}Pr(t)V^{\bar{c}}_{i}(a,t),\quad
i=1,2,\cdots,n.\end{array}$ (51)

  * (ii)

TH:

The expected value of player $i$, using actions $a=(a_{1},a_{2},\cdots,a_{n})$
and type $t_{i}^{j}$ is

$\displaystyle\begin{array}[]{ccl}E^{H}_{i}(a,t_{i}^{j})&:=&E_{i}(a(T)|t_{i}=t_{i}^{j})\\\
{}\hfil&=&\mathop{\sum}\limits_{t_{-i}\in
T_{-i}}p_{i}(t_{-i}|t_{i}^{j})\bar{c}_{i}(a,t^{j}_{i},t_{-i}),\\\
{}\hfil&=&\mathop{\sum}\limits_{t_{-i}\in
T_{-i}}p_{i}(t_{-i}|t_{i}^{j})V^{\bar{c}}_{i}ta,\quad
i=1,2,\cdots,n.\end{array}$ (55)

Proof: It is easy to verify the results by virtue of (43) $\Box$

######  Definition 4.3.

Consider a (finite) static Bayesian game $G=(N,T,A,c,Pr)$.

  * (i)

TN:

A profile $\left(a_{1}^{*},a_{2}^{*},\cdots,a_{n}^{*}\right)$ is said to be a
pure Bayesian-Nash equilibrium, if for each $i$ and any $t_{i}^{j}\in T_{i}$
the following inequalities hold.

$\displaystyle\begin{array}[]{l}E^{N}_{i}(a^{*}(t)|t_{i})\geq
E^{N}_{i}\left(a_{1}^{*}(t_{1}),\cdots,a_{i-1}^{*}(t_{i-1}),a_{i},a_{i+1}^{*}(t_{i+1}),\cdots,a_{n}^{*}(t_{n})|t_{i}\right),\quad\forall
t_{i}\in T_{i};\;i=1,2,\cdots,n.\end{array}$ (57)

  * (ii)

TH:

A profile
$\left(a_{1}^{*}(t_{1}^{*}),a_{2}^{*}(t_{2}^{*}),\cdots,a_{n}^{*}(t_{n}^{*})\right)$
is said to be a pure Bayesian-Nash equilibrium, if for each $i$ and any $t\in
T$ the following inequalities hold.

$\displaystyle\begin{array}[]{l}E^{H}_{i}\left(a_{1}^{*}(t_{1}^{*}),a_{2}^{*}(t_{2}^{*}),\cdots,a_{n}^{*}(t_{n}^{*})\right)\geq
E^{H}_{i}\left(a_{1}^{*}(t_{1}^{*}),\cdots,a_{i-1}^{*}(t^{*}_{i-1}),\right.\\\
~{}~{}\left.a_{i}(t_{i}),a_{i+1}^{*}(t_{i+1}^{*}),\cdots,a_{n}^{*}(t_{n}^{*})\right),\forall
t_{i}\in T_{i},\;i=1,2,\cdots,n.\end{array}$ (60)

######  Definition 4.4.

Consider a (finite) static Bayesian game $G=(N,T,A,c,Pr)$. The belief vector
of player $i$ for $t_{i}^{j}$ is defined as

$\displaystyle\begin{array}[]{lllccc}p_{i}(t_{i}^{j}):&=&Pr(t_{-i}\in
T_{-i}|t_{i}=t_{i}^{j}){}{}{}{}{}{}{}{}{}\\\
&=&\begin{bmatrix}Pr\left(t_{-i}=(t_{1}^{1},\cdots,t_{i-1}^{1},t_{i+1}^{1}\cdots,t_{n}^{1})|t_{i}=t_{i}^{j}\right)\\\
Pr\left(t_{-i}=(t_{1}^{1},\cdots,t_{i-1}^{1},t_{i+1}^{1}\cdots,t_{n}^{2})|t_{i}=t_{i}^{j}\right)\\\
\cdots\\\
Pr\left(t_{-i}=(t_{1}^{\tau_{1}},\cdots,t_{i-1}^{\tau_{i-1}},t_{i+1}^{\tau_{i+1}}\cdots,t_{n}^{\tau_{n}})|t_{i}=t_{i}^{j}\right)\\\
\end{bmatrix},\\\ &{}{}&{}{}{}{}{}\\\
~{}~{}~{}~{}~{}p_{i}&=&[p_{i}(t_{i}^{1}),p_{i}(t_{i}^{2}),\cdots,p_{i}(t_{i}^{\tau_{i}})],~{}~{}~{}~{}i=1,2,\cdots,n.\end{array}$
(65)

######  Example 4.5.

Consider a Bayesian game, which has distribution as in Table 9.

Table 9: Distribution $t_{1}\backslash t_{2}$ | $t_{2}^{1}$ | $t_{2}^{2}$ | $t_{2}^{3}$
---|---|---|---
$t_{1}^{1}$ | $0.1$ | $0.2$ | $0.3$
$t_{1}^{2}$ | $0.15$ | $0.1$ | $0.15$

$p_{1}({t_{1}^{1}})=\begin{bmatrix}Pr(t_{2}^{1}|t_{1}^{1})\\\
Pr(t_{2}^{2}|t_{1}^{1})\\\ Pr(t_{2}^{3}|t_{1}^{1})\\\
\end{bmatrix}=\begin{bmatrix}\frac{Pr(t_{2}^{1},t_{1}^{1})}{Pr(t_{2}^{1},t_{1}^{1})Pr(t_{2}^{2},t_{1}^{1})Pr(t_{2}^{3},t_{1}^{1})}\\\
~{}~{}\\\
\frac{Pr(t_{2}^{2},t_{1}^{1})}{Pr(t_{2}^{1},t_{1}^{1})Pr(t_{2}^{2},t_{1}^{1})Pr(t_{2}^{3},t_{1}^{1})}\\\
~{}~{}\\\
\frac{Pr(t_{2}^{3},t_{1}^{1})}{Pr(t_{2}^{1},t_{1}^{1})Pr(t_{2}^{2},t_{1}^{1})Pr(t_{2}^{3},t_{1}^{1})}\\\
\end{bmatrix}=\begin{bmatrix}1/6\\\ 1/3\\\ 1/2\\\ \end{bmatrix}.$

Similarly,

$\begin{array}[]{ll}p_{1}({t_{1}^{2}})=(3/8,1/4,3/8)^{T},&p_{2}({t_{2}^{1}})=(0.4,0.6)^{T},\\\
p_{2}({t_{2}^{2}})=(2/3,1/3)^{T},&p_{2}({t_{2}^{3}})=(2/3,1/3)^{T}.\\\
\end{array}$

##  5 Conversions in Bayesian Games

The key idea to deal with Bayesian game is to convert it into a normal game,
which is also called complete information game. This method was firstly
proposed by Harsanyi [22, 24, 25], which is called Harsanyi conversion.
Another commonly used transformation is Selten conversion [39]. Besides
Harsanyi conversion and Selten conversion, we provide a new kind of
conversion, called Action-Type conversion.

######  Definition 5.1.

Consider a finite Bayesian game $G$ with prior probability distribution
$Pr(t_{1},t_{2},\cdots,t_{n})$. Three conversions are defined as follows:

  * (i)

Harsanyi Conversion (H-Conversion): Define

$\displaystyle c_{i}^{H}(a):=Ec_{i}(a),\quad i=1,2,\cdots,n.$ (66)

  * (ii)

Selten Conversion (S-Conversion): Player $i$ knows his type
$t_{i}=\bar{t}_{i}$. Define

$\displaystyle c_{i}^{S}(a):=E(c_{i}(a)|t_{i}=\bar{t}_{i}),\quad
i=1,2,\cdots,n.$ (67)

  * (iii)

Action-Type Conversion (AT-Conversion): Player $i$ is able to choose $t_{i}$.
Define

$\displaystyle\begin{array}[]{ccl}c_{i}^{AC}(a,t_{i}):=\left[E(c_{i}(a)|t_{i}=t_{i}^{1}),E(c_{i}(a)|t_{i}=t_{i}^{2}),\cdots,E(c_{i}(a)|t_{i}=t_{i}^{\tau_{i}})\right],\quad
i=1,2,\cdots,n.\end{array}$ (69)

The corresponding complete information games are called Harsanyi Bayesian
game, Selten Bayesian game, and Action-Type Bayesian game of Bayesian game
$G$, respectively.

######  Theorem 5.2.

The structure vectors of three conversions are as follows:

  * (i)

H-Conversion: Denote by

$p=\left[Pr(t_{1}^{1},\cdots,t_{n}^{1}),Pr(t_{1}^{1},\cdots,t_{n}^{2}),\cdots,Pr(t_{1}^{\tau_{1}},\cdots,t_{n}^{\tau_{n}})\right]^{T},$

then

$\displaystyle V_{i}^{H}=V^{\bar{c}}_{i}p,\quad i=1,2,\cdots,n,$ (70)

where $V_{i}^{H}$ is the structure vector of $c_{i}^{H}(a).$

  * (ii)

S-Conversion: For a given $\bar{t}=(\bar{t}_{1},\cdots,\bar{t}_{n})$

$\displaystyle
V_{i}^{S}=V_{i}^{\bar{c}}W_{[\tau_{i},\prod_{k=1}^{i-1}\tau_{k}]}\delta_{\tau_{i}}^{\bar{t}_{i}}p_{i}(\bar{t}_{i}),\quad
i=1,2,\cdots,n.$ (71)

where $V_{i}^{S}$ is the structure vector of $c_{i}^{S}(a).$

  * (iii)

AT-Conversion:

$\displaystyle\begin{array}[]{ccl}V_{i}^{AT}&=&V_{i}^{\bar{c}}W_{[\tau_{i},\prod_{k=1}^{i-1}\tau_{k}]}\left[I_{\tau_{i}}*p_{i}\right],\qquad
i=1,2,\cdots,n.\end{array}$ (73)

where $*$ is the Khatra-Rao product of matrices, and $V_{i}^{AT}$ is the
structure vector of $c_{i}^{AT}(a,t_{i}).$

Proof:

  * (i)

According to Harsanyi Conversion,

$\displaystyle\begin{array}[]{lllccc}c_{i}^{H}(a)&=&\sum_{t\in
T}Pr(t){\bar{c}}_{i}(a,t)\\\ &{}&\\\ &=&\sum_{t\in T}Pr(t)V^{\bar{c}}_{i}ta\\\
&{}&\\\ &=&V^{\bar{c}}_{i}pa:=V_{i}^{H}a.\end{array}$ (79)

  * (ii)

According to Selten Conversion, player $i$ knows his type $t_{i}=\bar{t}_{i}$

$\displaystyle\begin{array}[]{lllccc}c_{i}^{S}(a)&=&E(c_{i}(a)|t_{i}=\bar{t}_{i})\\\
&{}&\\\ &=&\sum_{t_{-i}\in
T_{-i}}p(t_{-i}|\bar{t}_{i}){\bar{c}}_{i}(a,\bar{t}_{i},t_{-i})\\\ &{}&\\\
&=&\sum_{t_{-i}\in
T_{-i}}p(t_{-i}|\bar{t}_{i})V^{\bar{c}}_{i}W_{[\tau_{i},\prod_{k=1}^{i-1}\tau_{k}]}\bar{t}_{i}t_{-i}a\\\
&{}&\\\
&=&V_{i}^{\bar{c}}W_{[\tau_{i},\prod_{k=1}^{i-1}\tau_{k}]}\delta_{\tau_{i}}^{\bar{t}_{i}}p_{i}(\bar{t}_{i})a\\\
&{}&\\\ &=&V_{i}^{S}a.\end{array}$ (89)

  * (iii)

According to Action-Type Conversion,

$\displaystyle\begin{array}[]{lllccc}c_{i}^{AT}(a,t_{i})&=&\left[E(c_{i}(a)|t_{i}=t_{i}^{1}),E(c_{i}(a)|t_{i}=t_{i}^{2}),\cdots,E(c_{i}(a)|t_{i}=t_{i}^{\tau_{i}})\right]\\\
&{}&\\\
&=&\left[V_{i}^{\bar{c}}W_{[\tau_{i},\prod_{k=1}^{i-1}\tau_{k}]}\delta_{\tau_{i}}^{t_{i}^{1}}p_{i}(t_{i}^{1})a,~{}\cdots,~{}V_{i}^{\bar{c}}W_{[\tau_{i},\prod_{k=1}^{i-1}\tau_{k}]}\delta_{\tau_{i}}^{t_{i}^{\tau_{i}}}p_{i}(t_{i}^{\tau_{i}})a\right]\\\
&{}&\\\
&=&V_{i}^{\bar{c}}W_{[\tau_{i},\prod_{k=1}^{i-1}\tau_{k}]}\left[\delta_{\tau_{i}}^{t_{i}^{1}}p_{i}(t_{i}^{1}),~{}\cdots,~{}\delta_{\tau_{i}}^{t_{i}^{\tau_{i}}}p_{i}(t_{i}^{\tau_{i}})\right]t_{i}a\\\
&{}&\\\
&=&V_{i}^{\bar{c}}W_{[\tau_{i},\prod_{k=1}^{i-1}\tau_{k}]}\left[I_{\tau_{i}}*p_{i}\right]t_{i}a\\\
&{}&\\\ &=&V_{i}^{AT}t_{i}a.\end{array}$ (99)

$\Box$

######  Definition 5.3.

Consider a (finite) static Bayesian game $G=(N,T,A,c,Pr)$.

  * (i)

Harsanyi Bayesian game:

$a^{*}\in A$ is called a Bayesian-Nash equilibrium for Harsanyi BG (H-BN-E),
if $a^{*}$ satisfies

$\displaystyle c_{i}^{H}(a_{i}^{*},a_{-i}^{*})\geq
c_{i}^{H}(a_{i},a_{-i}^{*}),\quad\forall a_{i}\in A_{i},\;i=1,\cdots,n.$ (100)

  * (ii)

Selten Bayesian game:

$a^{*}\in A$ is called a Bayesian-Nash equilibrium for Selten BG (S-BN-E) with
respect to pre-assigned $\bar{t}$, if $a^{*}$ satisfies

$\displaystyle c_{i}^{S}(a_{i}^{*},a_{-i}^{*})\geq
c_{i}^{S}(a_{i},a_{-i}^{*}),\quad\forall a_{i}\in A_{i},\;i=1,\cdots,n.$ (101)

  * (iii)

Action-Type Bayesian game:

$(t^{*},a^{*})\in T\times A$ is called a Bayesian-Nash equilibrium for Action-
Type BG (AT-BN-E), if $(t^{*},a^{*})$ satisfies

$\displaystyle\begin{array}[]{l}c_{i}^{AT}(a_{i}^{*},t_{i}^{*},a_{-i}^{*})\geq
c_{i}^{AT}(a_{i},t_{i},a_{-i}^{*}),\quad\forall t_{i}\in T_{i},\forall
a_{i}\in A_{i},\quad i=1,\cdots,n.\end{array}$ (103)

######  Theorem 5.4.

Let O-BN-E be the original B-N equilibrium from Definition 4.3. Compered with
original BN, we have the following results:

  * (i)

$a^{*}~{}\mbox{is an O-BN-E}\begin{array}[]{c}\Rightarrow\\\
\not\Leftarrow\end{array}a^{*}~{}\mbox{is an H-BN-E}.$

  * (ii)

$a^{*}~{}\mbox{is an O-BN-E}\begin{array}[]{c}\Rightarrow\\\
\Leftarrow\end{array}a^{*}~{}\mbox{is an S-BN-E for every}~{}\bar{t}\in T.$

  * (iii)

$a^{*}~{}\mbox{is an O-BN-E}\begin{array}[]{c}\not\Rightarrow\\\
\Leftarrow\end{array}(t^{*},a^{*})~{}\mbox{is an AT-BN-E}.$

######  Example 5.5.

Recall the Bayesian game in Example 3.3.

  * (i)

Harsanyi game:

It is easy to calculate that

$\begin{array}[]{l}V^{H}_{1}=[-\infty,-\infty,-\infty,1.6,-\infty,-0.6,-\infty,-\infty,-\infty],\\\
V^{H}_{2}=[-\infty,-\infty,-\infty,-0.1,-\infty,-2.1,-\infty,-\infty,-\infty],\\\
\end{array}$

Put them into bi-matrix form:

Table 10: Harsanyi Expected Payoff Bi-Matrix $a_{1}\backslash a_{2}$ | $\delta_{3}^{1}$ | $\delta_{3}^{2}$ | $\delta_{3}^{3}$
---|---|---|---
$\delta_{3}^{1}$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$
$\delta_{3}^{2}$ | $1.6,~{}-0.1$ | $-\infty,~{}-\infty$ | $-0.6,~{}-2.1$
$\delta_{3}^{3}$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$

Hence $(a_{1}^{2},a_{2}^{1})=(\delta_{3}^{2},\delta_{3}^{1})$ is an H-BN-E.

  * (ii)

Action-Type game:

We can also calculate that

$\begin{array}[]{l}\begin{array}[]{ccccccccccccc}V^{AT}_{1}&=&[1.3&-\infty&1&1&-\infty&-0.5&-\infty&-\infty&-\infty\\\
{}\hfil&{}{}\hfil&-\infty&-\infty&-\infty&2.3&-\infty&0&2&-\infty&0.3],\\\
\end{array}\\\ {}\\\
\begin{array}[]{ccccccccccccc}V^{AT}_{2}&=&[-\infty&-\infty&-\infty&0.8&2.2&-1.8&-\infty&-\infty&-\infty\\\
{}\hfil&{}{}\hfil&-\infty&-\infty&-\infty&0.4&-\infty&-2.4&-\infty&-\infty&-\infty].\end{array}\end{array}$

Put them into bi-matrix form (Table 11 and Table 12):

Table 11: TH: Expected Payoff Bi-Matrix $t_{1}a_{1}\backslash t_{2}a_{2}$ | $t_{2}^{1}a_{2}^{1}$ | $t_{2}^{2}a_{2}^{1}$ | $t_{2}^{1}a_{2}^{2}$
---|---|---|---
$t_{1}^{1}a_{1}^{1}$ | $1.3,~{}-\infty$ | $1.3.~{}-\infty$ | $-2.5,~{}-\infty$
$t_{1}^{2}a_{1}^{1}$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$
$t_{1}^{1}a_{1}^{2}$ | $1,~{}0.8$ | $1,~{}0.4$ | $-\infty,~{}2.2$
$t_{1}^{2}a_{1}^{2}$ | $2.3,~{}0.8$ | $2.3,~{}0.4$ | $-\infty,~{}2.2$
$t_{1}^{1}a_{1}^{3}$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$
$t_{1}^{2}a_{1}^{3}$ | $2,~{}-\infty$ | $2,~{}-\infty$ | $-\infty,~{}-\infty$
Table 12: A-T- Expected Payoff Bi-Matrix(cont’d) $t_{1}a_{1}\backslash t_{2}a_{2}$ | $t_{2}^{2}a_{2}^{2}$ | $t_{2}^{1}a_{2}^{3}$ | $t_{2}^{2}a_{2}^{3}$
---|---|---|---
$t_{1}^{1}a_{1}^{1}$ | $-2.5,~{}-\infty$ | $1,~{}-\infty$ | $1,~{}-\infty$
$t_{1}^{2}a_{1}^{1}$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$
$t_{1}^{1}a_{1}^{2}$ | $-\infty,~{}-\infty$ | $-0.5,~{}-1.8$ | $-0.5,~{}-2.4$
$t_{1}^{2}a_{1}^{2}$ | $-\infty,~{}-\infty$ | $0,~{}-1.8$ | $0,~{}-2.4$
$t_{1}^{1}a_{1}^{3}$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$ | $-\infty,~{}-\infty$
$t_{1}^{2}a_{1}^{3}$ | $-\infty,~{}-\infty$ | $0.3,~{}-\infty$ | $0.3,~{}-\infty$

It is easy to verify that there is no AT-BN-E.

##  6 Bayesian Potential Game

######  Definition 6.1.

[26] Consider a (finite) static Bayesian game $G=(N,T,A,c,Pr)$. It is called a
TN weighted potential game, if there exists a function $F:T\times
A\rightarrow{\mathbb{R}}$ such that for any $t\in T$

$\displaystyle\begin{array}[]{l}c_{i}(a_{i}^{\prime},a_{-i},t)-c_{i}(a_{i},a_{-i},t)=w_{i}\left(F(a_{i}^{\prime},a_{-i},t)-F(a_{i},a_{-i},t)\right),\\\
~{}~{}~{}~{}~{}a^{\prime}_{i},a_{i}\in A_{i},\;a_{-i}\in A_{-i},\;t\in
T,\end{array}$ (106)

where $w_{i}>0$. When $w_{i}=1$, it is a TN potential game, $\forall i$.

######  Definition 6.2.

Consider a (finite) static Bayesian game $G=(N,T,A,c,Pr)$. It is called a TH
weighted potential game, if there exists a function $F:T\times
A\rightarrow{\mathbb{R}}$ such that for any $t_{i},t_{i}^{\prime}\in
T_{i},~{}a_{i},a_{i}^{\prime}\in A_{i}$

$\displaystyle\begin{array}[]{l}c_{i}(a_{i}^{\prime},a_{-i},t_{i}^{\prime},t_{-i})-c_{i}(a_{i},a_{-i},t_{i},t_{-i})=w_{i}\left(F(a_{i}^{\prime},a_{-i},t_{i}^{\prime},t_{-i})-F(a_{i},a_{-i},t_{i},t_{-i})\right),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall\;a_{-i}\in
A_{-i},t_{-i}\in T_{-i}.\end{array}$ (109)

where $w_{i}>0$. When $w_{i}=1$, it is a TH potential game, $\forall i$.

The following proposition comes from definition immediately.

######  Proposition 6.3.

$\mbox{TH potential game}~{}\Rightarrow\mbox{TN potential game}.$

######  Remark 6.4.

[35] It is obvious that a Bayesian game $G=(N,T,A,c,Pr)$ is TN-potential, if
and only if, under each type it is potential. Hence, we can use potential
equation [6] to check whether a game is potential and to construct potential
function. Please refer to Theorem 2.19.

######  Example 6.5.

Consider a finite BG

$\displaystyle G=(N,T,A,c,p),$ (110)

where

$\begin{array}[]{ccl}N&=&\\{1,2\\};\\\ T&=&\\{T_{1},T_{2}\\},\;\mbox{with}\\\
T_{1}&=&\\{t_{1}^{1},t_{1}^{2},t_{1}^{3}\\},\\\
T_{2}&=&\\{t_{2}^{1},t_{2}^{2}\\};\\\ A&=&\\{A_{1},A_{2}\\},\;\mbox{with}\\\
A_{1}&=&\\{a_{1}^{1},a_{1}^{2}\\},\\\
A_{2}&=&\\{a_{2}^{1},a_{2}^{2},a_{2}^{3}\\},\\\ \end{array}$

$T$ is as follows:

Table 13: Distribution $t_{1}\backslash t_{2}$ | $t_{2}^{1}$ | $t_{2}^{2}$
---|---|---
$t_{1}^{1}$ | $0.1$ | $0.15$
$t_{1}^{2}$ | $0.15$ | $0.2$
$t_{1}^{3}$ | $0.3$ | $0.1$

Then

$p_{1}({t_{1}^{1}})=\begin{bmatrix}Pr(t_{2}^{1}|t_{1}^{1})\\\
Pr(t_{2}^{2}|t_{1}^{1})\\\ \end{bmatrix}=\begin{bmatrix}0.4\\\ 0.6\\\
\end{bmatrix}.$

Similarly, we have

$\begin{array}[]{ll}p_{1}({t_{1}^{2}})=(3/7,4/7)^{T},&p_{1}({t_{1}^{3}})=(0.75,0.25)^{T},\\\
p_{2}({t_{2}^{1}})=(2/11,3/11,6/11)^{T},&p_{2}({t_{2}^{2}})=(1/3,4/9,2/9)^{T}.\\\
\end{array}$

The payoff functions are as follows.

Table 14: Payment Function for $t_{1}^{1}-t_{2}^{1}$ $c_{1}\backslash c_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $5,~{}0$ | $2,~{}2$ | $0,~{}1$
$a_{1}^{2}$ | $2,~{}-1$ | $-1,~{}1$ | $1,~{}4$
Table 15: Payment Function for $t_{1}^{1}-t_{2}^{2}$ $c_{1}\backslash c_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $3,~{}0$ | $2,~{}3$ | $1,~{}1$
$a_{1}^{2}$ | $1,~{}-2$ | $0,~{}1$ | $2,~{}2$
Table 16: Payment Function for $t_{1}^{2}-t_{2}^{1}$ $c_{1}\backslash c_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $2,~{}4$ | $0,~{}-1$ | $5,~{}5$
$a_{1}^{2}$ | $1,~{}1$ | $3,~{}0$ | $1,~{}-1$
Table 17: Payment Function for $t_{1}^{2}-t_{2}^{2}$ $c_{1}\backslash c_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $1,~{}0$ | $-1,~{}2$ | $1,~{}3$
$a_{1}^{2}$ | $3,~{}-1$ | $0,~{}0$ | $2,~{}1$
Table 18: Payment Function for $t_{1}^{3}-t_{2}^{1}$ $c_{1}\backslash c_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $1,~{}1$ | $2,~{}2$ | $0,~{}3$
$a_{1}^{2}$ | $4,~{}2$ | $5,~{}3$ | $2,~{}3$
Table 19: Payment Function for $t_{1}^{3}-t_{2}^{2}$ $c_{1}\backslash c_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $-2,~{}1$ | $1,~{}2$ | $0,~{}0$
$a_{1}^{2}$ | $-4,~{}1$ | $-5,~{}-2$ | $-2,~{}0$

The potential function is obtained as

Table 20: Potential Function for $t_{1}^{1}-t_{2}^{1}$ $F$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $1$ | $3$ | $2$
$a_{1}^{2}$ | $-2$ | $0$ | $3$
Table 21: Potential Function for $t_{1}^{1}-t_{2}^{2}$ $F$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $1$ | $4$ | $2$
$a_{1}^{2}$ | $-1$ | $2$ | $3$
Table 22: Potential Function for $t_{1}^{2}-t_{2}^{1}$ $F$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $3$ | $-2$ | $4$
$a_{1}^{2}$ | $2$ | $1$ | $0$
Table 23: Potential Function for $t_{1}^{2}-t_{2}^{2}$ $F$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $-1$ | $1$ | $2$
$a_{1}^{2}$ | $1$ | $2$ | $3$
Table 24: Potential Function for $t_{1}^{3}-t_{2}^{1}$ $F$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $-1$ | $0$ | $1$
$a_{1}^{2}$ | $2$ | $3$ | $3$
Table 25: Potential Function for $t_{1}^{3}-t_{2}^{2}$ $F$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $4$ | $5$ | $3$
$a_{1}^{2}$ | $2$ | $-1$ | $1$

Finally, we have

$\begin{array}[]{ccl}V^{c}_{1}&=&[5,2,0,2,-1,1,3,2,1,1,0,2,2,0,5,1,3,1,\\\
{}\hfil&{}\hfil&1,-1,1,3,0,2,1,2,0,4,5,2,-2,1,0,-4,-5,-2],\\\
V^{c}_{2}&=&[0,2,1,-1,1,4,0,3,1,-2,1,2,4,-1,5,1,0,\\\
{}\hfil&{}\hfil&-1,0,2,3,-1,0,1,1,2,3,2,3,3,1,2,0,1,-2,0],\\\
V^{c}_{F}&=&[2,4,3,-1,1,4,0,3,1,-1,1,2,2,-3,3,1,0,-1,\\\
{}\hfil&{}\hfil&-3,-1,0,-1,0,1,-1,0,1,2,3,3,3,4,2,1,-2,0].\\\ \end{array}$

It is easy to verify that $F$ is a TN potential function.

######  Definition 6.6.

Consider a finite static Bayesian game $G=(N,T,A,c,Pr)$.

  * (i)

It is called a Harsanyi weighted potential game, if there exists a function
$Q^{H}:A\rightarrow{\mathbb{R}}$ such that for any $a_{i},a_{i}^{\prime}\in
A_{i}$

$\displaystyle\begin{array}[]{l}c_{i}^{H}(a_{i}^{\prime},a_{-i})-c_{i}^{H}(a_{i},a_{-i})=w_{i}\left(Q^{H}(a_{i}^{\prime},a_{-i})-Q^{H}(a_{i},a_{-i})\right),~{}~{}~{}\forall\;a_{-i}\in
A_{-i}.\end{array}$ (112)

where $w_{i}>0$. When $w_{i}=1$, $\forall i$, it is a Harsanyi potential game.

  * (ii)

It is called a Selten weighted potential game for the pre-assigned $t\in T$,
if there exists a function $Q^{S}:A\rightarrow{\mathbb{R}}$ such that for any
$a_{i},a_{i}^{\prime}\in A_{i}$

$\displaystyle\begin{array}[]{l}c_{i}^{S}(a_{i}^{\prime},a_{-i})-c_{i}^{S}(a_{i},a_{-i})=w_{i}\left(Q^{S}(a_{i}^{\prime},a_{-i})-Q^{S}(a_{i},a_{-i})\right),~{}~{}~{}\forall\;a_{-i}\in
A_{-i}.\end{array}$ (114)

where $w_{i}>0$. When $w_{i}=1$, $\forall i$, it is a Selten potential game.

  * (iii)

It is called a Action-Type weighted potential game, if there exists a function
$Q^{AT}:T\times A\rightarrow{\mathbb{R}}$ such that for any
$a_{i},a_{i}^{\prime}\in A_{i}$

$\displaystyle\begin{array}[]{l}c_{i}^{AT}(a_{i}^{\prime},t_{i}^{\prime},a_{-i})-c_{i}^{AT}(a_{i},t_{i},a_{-i})=w_{i}\left(Q^{AT}(a_{i}^{\prime},t_{i}^{\prime},a_{-i},t_{-i})-Q^{AT}(a_{i},t_{i},a_{-i},t_{-i})\right),~{}\forall\;a_{-i}\in
A_{-i}.\end{array}$ (116)

where $w_{i}>0$. When $w_{i}=1$, $\forall i$, it is a Action-Type potential
game.

The following proposition is obvious.

######  Proposition 6.7.

Consider a finite static Bayesian game $G=(N,T,A,c,Pr)$.

  * (i)

If $G$ is a Harsanyi potential game, then $G$ has at least one H-BN-E (the
potential maximizer).

  * (ii)

If $G$ is a Selten potential game, then $G$ has at least one S-BN-E (the
potential maximizer).

  * (iii)

If $G$ is a Action-Type potential game, then $G$ has at least one AT-BN-E (the
potential maximizer).

######  Proposition 6.8.

Consider a finite Bayesian game $G=(N,T,A,c,Pr)$. If $G$ is a TN or TH
potential game with potential function $F(a,t)$, then the corresponding
Harsanyi Bayesian game is a Harsanyi potential game with potential function

$\displaystyle\begin{array}[]{lllcc}Q^{H}(a)&=&\sum_{t\in T}Pr(t)F(a,t)\\\
&=&V_{F}^{c}pta\\\ &:=&V^{H}a,\\\ \end{array}$ (120)

where $V_{F}^{c}$ is the structure vector of $F(a,t).$

######  Example 6.9.

Consider Example 6.5 again.

  * (i)

Harsanli Bayesian game:

The payoff vectors of corresponding Harsanli BG are

$\begin{array}[]{ccl}V^{H}_{1}&=&V^{c}_{1}p=[1.55,1,1.1,1.9,1.35,1.35],\\\
V^{H}_{2}&=&V^{c}_{2}p=[1,1.7,2.5,0.25,0.95,1.65].\\\
V^{c}_{F}&=&[2,4,3,-1,1,4,0,3,1,-1,1,2,2,-3,3,1,0,-1,\\\
{}\hfil&{}\hfil&-3,-1,0,-1,0,1,-1,0,1,2,3,3,3,4,2,1,-2,0].\\\ \end{array}$

Back to matrix form:

Table 26: Harsanli-Bayesian  $c_{1}^{H}\backslash c_{2}^{H}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $1.55,~{}1$ | $1,~{}1.7$ | $1.1,~{}2.5$
$a_{1}^{2}$ | $1.9,~{}0.25$ | $1.35,~{}0.95$ | $\underline{1.35},~{}\underline{1.65}$

It is easy to verify that the Harsanli Bayesian game is potential with $V^{H}$
as its potential function, where

$V^{H}=V^{c}_{F}p=[-0.1,0.6,1.4,0.25,0.95,1.65].$

Back to matrix form:

Table 27: Harsanli Potential Function $V^{H}$ $Q_{H}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $-0.1$ | $0.6$ | $1.4$
$a_{1}^{2}$ | $0.25$ | $0.95$ | $1.65$
  * (ii)

Selten Bayesian game:

$V^{S}_{i}=V^{c}_{i}W_{[\tau_{i},\prod_{j=1}^{i-1}\tau_{j}]}\delta_{\tau_{i}}^{\theta}p_{i}({t_{i}^{\theta}}),\quad
1\leq\theta\leq\tau_{i},\;i=1,\cdots,n.$

Assume $t_{1}^{\theta}=t_{1}^{1}$, $t_{2}^{\theta}=t_{2}^{1}$. Then we have

$\begin{array}[]{ccl}V^{S}_{1}&=&[3.8,2,0.6,1.4,-0.4,1.6],\\\
V^{S}_{2}&=&[2.1818,0.6364,3.1818,-0.1818,0.4545,0.7273].\end{array}$

Back to matrix form

Table 28: Selten-Bayesian $P_{1}\backslash P_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $3.8,~{}2.1818$ | $2,~{}0.6364$ | $0.6,~{}3.1818$
$a_{1}^{2}$ | $1.4,-0.1818$ | $-0.4,~{}0.4545$ | $1.6,~{}0.7273$

Assume $t_{1}^{\theta}=t_{1}^{1}$, $t_{2}^{\theta}=t_{2}^{2}$. Then we have

$\begin{array}[]{ccl}V^{S}_{11}&=&[3.8,2,0.6,1.4,-0.4,1.6],\\\
V^{S}_{22}&=&[0.2222,2.3333,1.6667,-0.8889,-0.1111,1.1111].\end{array}$

Back to matrix form

Table 29: Selten-Bayesian $P_{1}\backslash P_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $3.8,~{}0.2222$ | $2,~{}2.3333$ | $0.6,~{}1.6667$
$a_{1}^{2}$ | $1.4,~{}-0.8889$ | $-0.4,~{}-0.1111$ | $1.6,~{}1.1111$

Assume $t_{1}^{\theta}=t_{1}^{2}$, $t_{2}^{\theta}=t_{2}^{1}$. Then we have

$\begin{array}[]{ccl}V^{S}_{12}&=&[1.8571,2,0.4286,2.7143,2.8571,2],\\\
V^{S}_{21}&=&[2.1818,0.6364,3.1818,-0.1818,0.4545,0.7273].\end{array}$

Back to matrix form

Table 30: Selten-Bayesian $P_{1}\backslash P_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $1.8571,~{}2.1818$ | $2,~{}0.6364$ | $0.4286,~{}3.1818$
$a_{1}^{2}$ | $2.7143,-0.1818$ | $2.8571,~{}0.4545$ | $2,~{}0.7273$
Table 31: Selten-Bayesian $P_{1}\backslash P_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $1.8571,~{}2$ | $2,~{}2.3333$ | $0.4286,~{}1.6667$
$a_{1}^{2}$ | $2.7143,~{}-0.8889$ | $2.8571,~{}-0.1111$ | $2,~{}1.1111$
Table 32: Selten-Bayesian $P_{1}\backslash P_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $1,~{}2.1818$ | $0.25,~{}0.6364$ | $3.75,~{}3.1818$
$a_{1}^{2}$ | $-0.25,-0.1818$ | $1,~{}0.4545$ | $0.25,~{}0.7273$

Assume $t_{1}^{\theta}=t_{1}^{3}$, $t_{2}^{\theta}=t_{2}^{2}$. Then we have

$\begin{array}[]{ccl}V^{S}_{13}&=&[1,0.25,3.75,-0.25,1,0.25],\\\
V^{S}_{22}&=&[0.2222,2.3333,1.6667,-0.8889,-0.1111,1.1111].\end{array}$

Back to matrix form

Table 33: Selten-Bayesian $P_{1}\backslash P_{2}$ | $a_{2}^{1}$ | $a_{2}^{2}$ | $a_{2}^{3}$
---|---|---|---
$a_{1}^{1}$ | $1,~{}2$ | $0.25,~{}2.3333$ | $3.75,~{}1.6667$
$a_{1}^{2}$ | $-0.25,~{}-0.8889$ | $1,~{}-0.1111$ | $0.25,~{}1.1111$

It is easy to verify that the Selten game is not potential for each type.

  * (iii)

Action-Type game:

$V^{AT}_{i}=\left[V^{S}_{i,1},V^{S}_{i,2},\cdots,V^{S}_{i,s_{i}}\right],\quad
i=1,2,\cdots,n.$

$\begin{array}[]{lllllll}V^{AT}_{1}=&{}&{}&{}&{}&{}&{}\\\
{}&[3.8&2&0.6&1.4&-0.4&1.6\\\ {}&1.8571&2&0.4286&2.7143&2.8571&2\\\
{}&1&0.25&3.75&-0.25&1&0.25],\\\ V^{AT}_{2}=&{}&{}&{}&{}&{}&{}\\\
{}&[2.1818&0.6364&3.1818&-0.1818&0.4545&0.7273\\\
{}&0.2222&2.3333&1.6667&-0.8889&-0.1111&1.1111].\end{array}$

Back to matrix form

Table 34: Action-Type-Bayesian $P_{1}\backslash P_{2}$ | $t_{2}^{1}a_{2}^{1}$ | $t_{2}^{2}a_{2}^{1}$ | $t_{2}^{1}a_{2}^{2}$
---|---|---|---
$t_{1}^{1}a_{1}^{1}$ | $3.8,~{}2.1818$ | $3.8,~{}0.6364$ | $2,~{}3.1818$
$t_{1}^{2}a_{1}^{1}$ | $1.4,~{}2.1818$ | $1.4,~{}0.6364$ | $-0.4,~{}3.1818$
$t_{1}^{3}a_{1}^{1}$ | $1.8571,2.1818$ | $1.8571,~{}0.6364$ | $2,~{}3.1818$
$t_{1}^{1}a_{1}^{2}$ | $2.7143,0.2222$ | $2.7143,~{}2.3333$ | $2.8571,~{}1.6667$
$t_{1}^{2}a_{1}^{2}$ | $1,0.2222$ | $1,~{}2.3333$ | $0.25,~{}1.6667$
$t_{1}^{3}a_{1}^{3}$ | $-0.25,~{}0.2222$ | $-0.25,~{}2.3333$ | $1,1.6667$
Table 35: Action-Type-Bayesian $P_{1}\backslash P_{2}$ | $t_{2}^{2}a_{2}^{2}$ | $t_{2}^{1}a_{2}^{3}$ | $t_{2}^{2}a_{2}^{3}$
---|---|---|---
$t_{1}^{1}a_{1}^{1}$ | $2,-0.1818$ | $0.6,0.4545$ | $0.6,0.7273$
$t_{1}^{2}a_{1}^{1}$ | $-0.4,~{}-0.1818$ | $1.6,0.4545$ | $1.6,0.7273$
$t_{1}^{3}a_{1}^{1}$ | $2,~{}-0.1818$ | $0.4286,0.4545$ | $0.4286,~{}0.7273$
$t_{1}^{1}a_{1}^{2}$ | $2.8571,~{}-0.8889$ | $2,~{}-0.1111$ | $2,~{}1.1111$
$t_{1}^{2}a_{1}^{2}$ | $0.25,-0.8889$ | $3.75,~{}-0.1111$ | $3.75,~{}1.1111$
$t_{1}^{3}a_{1}^{3}$ | $1,~{}-0.8889$ | $0.25,~{}-0.1111$ | $0.25,~{}1.1111$

The Action-Type game is not potential too.

According to the definition, a Bayesian game is a Harsanli potential game, if
and only if, the original game is potential for each type. Hence, we can use
potential equation (29) to check whether a Bayesian game is Harsanli potential
and to construct potential function. But the check for Selten potential game
and Action-Type potential game are not obvious.

In the following, we provide a method to verify whether a BG is Selten
potential or Action-Type potential. Denote by

$\phi_{i}=\otimes_{j=1}^{n}\gamma_{j},$

where

$\gamma_{j}=\begin{cases}I_{\tau_{i}},\quad j=i,\\\ {\bf
1}_{\tau_{j}}^{T},\quad j\neq i.\end{cases}$

And let $W_{G}^{S}$ and $W_{G}^{AT}$ be equivalent vectors of Selten Bayesian
game and Action-Type Bayesian game respectively

$\displaystyle\begin{array}[]{lcc}W_{G}^{S}&:=&[V_{2}^{S}-V_{1}^{S},V_{3}^{S}-V_{1}^{S},\cdots,V_{n}^{S}-V_{1}^{S}],{}{}{}{}{}{}{}\\\
{}&{}\hfil&{}\hfil\\\
W_{G}^{AT}&:=&[V_{2}^{AC}\phi_{2}-V_{1}^{AC}\phi_{1},\cdots,V_{n}^{AC}\phi_{n}-V_{1}^{AC}\phi_{1}].\end{array}$
(124)

It is easy to verify that

$W_{G}^{S}=V_{G}\Gamma^{S}_{\bar{t}},~{}~{}W_{G}^{AT}=V_{G}\Gamma^{AT},$

where
$V_{G}=[V_{1}^{\bar{c}},\cdots,V_{n}^{\bar{c}}],~{}\Gamma^{S}_{t_{i}}=W_{[\tau_{i},\prod_{k=1}^{i-1}\tau_{k}]}\delta_{\tau_{i}}^{\bar{t}_{i}}p_{i}(\bar{t}_{i}),~{}~{}~{}\Gamma^{AT}_{i}=W_{[\tau_{i},\prod_{k=1}^{i-1}\tau_{k}]}\left[I_{\tau_{i}}*p_{i}\right]\phi_{i},~{}\forall
i,$

$\displaystyle\Gamma^{S}_{\bar{t}}=\begin{bmatrix}-\Gamma^{S}_{t_{1}}&-\Gamma^{S}_{t_{1}}&\cdots&-\Gamma^{S}_{t_{1}}\\\
\Gamma^{S}_{t_{2}}&0&\cdots&0\\\ 0&\Gamma^{S}_{t_{3}}&\cdots&0\\\
\vdots&~{}&~{}&~{}\\\
0&0&\cdots&-\Gamma^{S}_{t_{n}}\end{bmatrix},~{}~{}\Gamma^{AT}=\begin{bmatrix}-\Gamma^{AT}_{1}&-\Gamma^{AT}_{1}&\cdots&-\Gamma^{AT}_{1}\\\
\Gamma^{AT}_{2}&0&\cdots&0\\\ 0&\Gamma^{AT}_{3}&\cdots&0\\\
\vdots&~{}&~{}&~{}\\\ 0&0&\cdots&-\Gamma^{AT}_{n}\end{bmatrix}.$ (125)

######  Theorem 6.10.

Consider a finite Bayesian game $G=(N,T,A,c,Pr)$.

  * (i)

$G$ is a Selten potential game, if and only if, the following linear equation
has a solution

$\displaystyle\Psi\xi=b^{S},$ (126)

where $\xi=(\xi_{1}^{T},\cdots,\xi_{n}^{T})^{T},~{}b^{S}=(W_{G}^{S})^{T}$, and
$\Psi$ is defined by (30). Moreover, when a solution exists, the structure
vector of Selten potential function $Q^{S}(a)$ is

$\displaystyle\begin{array}[]{llcc}V^{S}_{Q}&=&V^{S}_{1}-\xi_{1}^{T}E_{1}{}{}{}{}{}{}{}{}{}{}{}{}\\\
&=&V^{c}_{1}\delta_{\tau_{1}}^{\bar{t}_{1}}p_{1}(\bar{t}_{1})-\xi_{1}^{T}E_{1}.\end{array}$
(129)

  * (ii)

$G$ is a Action-Type potential game, if and only if, the following linear
equation has a solution

$\displaystyle\Psi^{AT}\xi=b^{AT},$ (130)

where
$\xi=(\xi_{1}^{T},\cdots,\xi_{n}^{T})^{T},~{}b^{AT}=(W_{G}^{AT})^{T},~{}\varphi_{i}=I_{\theta_{i}}\otimes{\bf
1}^{T}_{\tau_{i}}\otimes I_{\vartheta_{i}}\otimes{\bf
1}^{T}_{r_{i}},\vartheta_{i}=\prod_{j=i+1}^{n}\tau_{i}\prod_{l=1}^{i-1}r_{l}$,

$\displaystyle\theta_{i}=\prod_{j=1}^{i-1}\tau_{i},~{}~{}\Psi^{AT}=\begin{bmatrix}-\varphi_{1}^{T}&\varphi_{2}^{T}&0&\cdots&0\\\
-\varphi_{1}^{T}&0&\varphi_{3}^{T}&\cdots&0\\\ \vdots&~{}&~{}&\ddots&~{}\\\
-\varphi_{1}^{T}&0&0&\cdots&\varphi_{n}^{T}\end{bmatrix}.$ (131)

Moreover, when a solution exists, the structure vector of Action-Type
potential function $Q^{AT}(a,t)$ is

$\displaystyle\begin{array}[]{llcc}V^{AT}_{Q}&=&V^{AT}_{1}-\xi_{1}^{T}\varphi_{1}{}{}{}{}{}{}{}{}{}{}\\\
&=&V^{c}_{1}\left[I_{\tau_{i}}*p_{i}\right]\phi_{1}-\xi_{1}^{T}\varphi_{1}.\end{array}$
(134)

Proof: The proof of (i) is similar with Theorem 2.19, so we only prove (ii).
According to (116), there exist functions
$d_{i}(a_{-i},t_{-i},\hat{a}_{i},\hat{t}_{i})$ such that

$\displaystyle
c_{i}^{AT}(a,t_{i})=Q^{AT}(a,t)+d_{i}(a_{-i},t_{-i},\hat{a}_{i},\hat{t}_{i}),~{}\forall
i,$ (135)

where the symbol “hat” means that
$d_{i}(a_{-i},t_{-i},\hat{a}_{i},\hat{t}_{i})$ is independent of $a_{i}$ and
$t_{i}.$ Taking its vector form, (135) becomes

$\displaystyle\begin{array}[]{llcc}~{}~{}~{}~{}V_{i}^{AT}t_{i}a&=&V_{Q}^{AT}ta+V^{d}_{i}t_{-i}a_{-i}\\\
\Leftrightarrow
V_{i}^{AT}\phi_{i}ta&=&V_{Q}^{AT}ta+V^{d}_{i}\varphi_{i}ta.\end{array}$ (138)

Equation (138) is equivalent to

$\displaystyle V_{Q}^{AT}=V_{i}^{AT}\phi_{i}-V^{d}_{i}\varphi_{i}.$ (139)

Then

$\displaystyle
V_{i}^{AT}\phi_{i}-V_{1}^{AT}\phi_{1}=V^{d}_{i}\varphi_{i}-V^{d}_{1}\varphi_{1},~{}i=2,3,\cdots,n.$
(140)

Equation (140) is equivalent to the following linear equations

$\Psi^{AT}\xi=b^{AT},$

where $\xi_{i}=(V_{i}^{d})^{T},~{}i=1,2,\cdots,n.$ $\Box$

######  Example 6.11.

Consider a finite Bayesian game $G=(N,T,A,c,Pr)$, where

$\begin{array}[]{ccl}N&=&\\{1,2\\};\\\ T&=&\\{T_{1},T_{2}\\},\;\mbox{with}\\\
T_{1}&=&\\{t_{1}^{1},t_{1}^{2}\\},\\\ T_{2}&=&\\{t_{2}^{1},t_{2}^{2}\\};\\\
A&=&\\{A_{1},A_{2}\\},\;\mbox{with}\\\ A_{1}&=&\\{a_{1}^{1},a_{1}^{2}\\},\\\
A_{2}&=&\\{a_{2}^{1},a_{2}^{2}\\},\\\ \end{array}$

The distribution $Pr$ is as follows:

Table 36: Distribution $t_{1}\backslash t_{2}$ | $t_{2}^{1}$ | $t_{2}^{2}$
---|---|---
$t_{1}^{1}$ | $0.2$ | $0.3$
$t_{1}^{2}$ | $0.4$ | $0.1$

Then

$p_{1}=\begin{bmatrix}Pr(t_{2}^{1}|t_{1}^{1}),Pr(t_{2}^{1}|t_{1}^{2})\\\
Pr(t_{2}^{2}|t_{1}^{1}),Pr(t_{2}^{2}|t_{1}^{2})\\\
\end{bmatrix}=\begin{bmatrix}2/5,4/5\\\ 3/5,1/5\\\ \end{bmatrix}.$
$p_{2}=\begin{bmatrix}Pr(t_{1}^{1}|t_{2}^{1}),Pr(t_{1}^{1}|t_{2}^{2})\\\
Pr(t_{1}^{2}|t_{2}^{1}),Pr(t_{1}^{2}|t_{2}^{2})\\\
\end{bmatrix}=\begin{bmatrix}1/3,3/4\\\ 2/3,1/4\\\ \end{bmatrix}.$

The payoff vectors are

$\begin{array}[]{ccl}V_{1}^{c}&=&[a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},b_{3},b_{4},c_{1},c_{2},c_{3},c_{4},d_{1},d_{2},d_{3},d_{4}],\\\
V_{2}^{c}&=&[e_{1},e_{2},e_{3},e_{4},f_{1},f_{2},f_{3},f_{4},g_{1},g_{2},g_{3},g_{4},h_{1},h_{2},h_{3},h_{4}].\end{array}$

We can calculate the payoff vectors of Action-Type Bayesian game.

$\begin{array}[]{ccl}V_{1}^{AT}&=&\frac{1}{5}[2a_{1}+3b_{1},2a_{2}+3b_{2},2a_{3}+3b_{3},2a_{4}+3b_{4},4c_{1}+d_{1},4c_{2}+d_{2},4c_{3}+d_{3},4c_{4}+d_{4}]\\\
&:=&[\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6},\alpha_{7},\alpha_{8}],\\\
V_{2}^{AT}&=&\frac{1}{12}[4e_{1}+8f_{1},4e_{2}+8f_{2},4e_{3}+8f_{3},4e_{4}+8f_{4},9g_{1}+3h_{1},9g_{2}+3h_{2},9g_{3}+3h_{3},9g_{4}+3h_{4}]\\\
&:=&[\beta_{1},\beta_{2},\beta_{3},\beta_{4},\beta_{5},\beta_{6},\beta_{7},\beta_{8}].\\\
\end{array}$

According to Theorem 6.10

$\displaystyle\Psi^{AT}=\begin{bmatrix}-1&0&0&0&1&0&0&0\\\ 0&-1&0&0&1&0&0&0\\\
-1&0&0&0&0&1&0&0\\\ 0&-1&0&0&0&1&0&0\\\ 0&0&-1&0&1&0&0&0\\\
0&0&0&-1&1&0&0&0\\\ 0&0&-1&0&0&1&0&0\\\ 0&0&0&-1&0&1&0&0\\\
-1&0&0&0&0&0&1&0\\\ 0&-1&0&0&0&0&1&0\\\ -1&0&0&0&0&0&0&1\\\
0&-1&0&0&0&0&0&1\\\ 0&0&-1&0&0&0&1&0\\\ 0&0&0&-1&0&0&1&0\\\
0&0&-1&0&0&0&0&1\\\ 0&0&0&-1&0&0&0&1\\\
\end{bmatrix},~{}~{}b^{AT}=\begin{bmatrix}\alpha_{1}-\beta_{1}\\\
\alpha_{2}-\beta_{2}\\\ \alpha_{3}-\beta_{3}\\\ \alpha_{4}-\beta_{4}\\\
\alpha_{1}-\beta_{5}\\\ \alpha_{2}-\beta_{6}\\\ \alpha_{3}-\beta_{7}\\\
\alpha_{4}-\beta_{8}\\\ \alpha_{5}-\beta_{1}\\\ \alpha_{6}-\beta_{2}\\\
\alpha_{7}-\beta_{3}\\\ \alpha_{8}-\beta_{4}\\\ \alpha_{5}-\beta_{5}\\\
\alpha_{6}-\beta_{6}\\\ \alpha_{7}-\beta_{7}\\\ \alpha_{8}-\beta_{8}\\\
\end{bmatrix}.$

Equation (130) has a solution if and only if

$\operatorname{rank}(\Psi^{AT})=\operatorname{rank}([\Psi^{AT},b^{AT}]),$

which implies that

$\displaystyle\begin{cases}\alpha_{1}-\alpha_{2}-\alpha_{3}+\alpha_{4}-\beta_{1}+\beta_{2}+\beta_{3}-\beta_{4}=0\\\
\beta_{2}-\beta_{4}-\beta_{6}+\beta_{8}=0\\\
\alpha_{1}-\alpha_{2}-\alpha_{5}+\alpha_{6}=0\\\
\beta_{1}-\beta_{3}-\beta_{5}+\beta_{7}=0\\\ \end{cases}.$

Particularly, if

$\begin{array}[]{cccccccccccl}\alpha_{1}=\alpha_{2},&\alpha_{3}=\alpha_{4},&\alpha_{5}=\alpha_{6},&\alpha_{7}=\alpha_{8};\\\
\beta_{1}=\beta_{3},&\beta_{2}=\beta_{4},&\beta_{5}=\beta_{7},&\beta_{6}=\beta_{8}.\\\
\end{array}$

The Bayesian game is an Action-Type potential game with AT potential function

$Q^{AT}(a,t)=c^{AT}_{1}(a,t_{1})+c^{AT}_{2}(a,t_{2}).$

##  7 Dynamic Bayesian Games

###  7.1 Dynamics of Selten Bayesian Games

For a repeated Bayesian game, its dynamics depends on two facts: (1) the
strategy updating rule (SUR); (2) the conversion. Cosider a Bayesian game, a
fundamental assumption in this section is, for a given conversion, the player
updates his action or type according to his SUR to optimize the corresponding
conversion payoff.

######  Definition 7.1.

  * (i)

The dynamics is called asynchronous if at each time there is only one player
who is allowed to update his action or type.

  * (ii)

The dynamics is called synchronous if at each time all players can update
actions or types synchronously.

Consider a Bayesian game $G=(N,T,A,c,Pr)$. Assume the conversion is
S-Conversion. As the dynamics depends on the information acquired, we assume
player $i$ knows his type $\bar{t}_{i}\in T_{i}$ and other players’ actions
$a_{-i}(k)$ at time $k+1$. We introduce the following asynchronous dynamics
for Selten Bayesian games.

  * (i)

If the SUR is asynchronous myopic best response adjustment (MBRA), then at
time $k$ there is only one player (say player $i$) to update his action as
follows

$\displaystyle a_{i}(k+1)=\operatorname{argmax}_{a_{i}\in
A_{i}}E(c_{i}(a_{i},a_{-i}(k))|t_{i}=\bar{t}_{i}),\quad i=1,2,\cdots,n.$ (141)

where $\bar{t}=(\bar{t}_{1},\bar{t}_{2},\cdots,\bar{t}_{n})$ the pre-assigned
type profile.

  * (ii)

If the SUR is asynchronous logit response (LR), then at time $k$ there is only
one player (say player $i$) to select $a_{i}$ according to the following
probability

$\displaystyle
Pr(a_{i}(k+1)=a_{i}|a(k))=\dfrac{e^{\frac{1}{T}E(c_{i}(a_{i},a_{-i}(k))|t_{i}=\bar{t}_{i})}}{\sum_{a^{\prime}_{i}\in
A_{i}}e^{\frac{1}{T}E(c_{i}(a^{\prime}_{i},a_{-i}(k))|t_{i}=\bar{t}_{i})}},\quad
i=1,2,\cdots,n.$ (142)

We give an example to show how to get the dynamic equation for repeated
Bayesian game under S-Conversion.

######  Example 7.2.

Consider a Bayesian game $G=(N,A,T,c,Pr)$, where $|N|=2$,
$A_{1}=\\{a_{1}^{1},a_{1}^{2}\\}$, $A_{2}=\\{a_{2}^{1},a_{2}^{2}\\}$,
$T_{1}=\\{t_{1}^{1},t_{1}^{2}\\}$, $T_{2}=\\{t_{2}^{1},t_{2}^{2}\\}$. The
payoff vectors $V_{1}^{c},V_{2}^{c}$ are

$\begin{array}[]{l}V^{c}_{1}=[2,1,0,1,-1,1,3,-2,2,3,2,-2,3,3,-2,1],\\\
V^{c}_{2}=[1,3,2,-1,2,2,1,-2,-1,0,-2,2,2,3,-1,0].\\\ \end{array}$

The distribution shown in Table 37.

Table 37: Distribution $t_{1}\backslash t_{2}$ | $t_{2}^{1}$ | $t_{2}^{2}$
---|---|---
$t_{1}^{1}$ | $0.1$ | $0.3$
$t_{1}^{2}$ | $0.4$ | $0.2$

Assume $t_{1}=t_{1}^{1}$, $t_{2}=t_{2}^{2}$, then we have

$p_{1}({t_{1}^{1}})=(0.25,0.75)^{T},\quad p_{2}({t_{2}^{2}})=(0.6,0.4)^{T}.$

According to (71), we can obtain the S-Conversion payoff vector

$\begin{array}[]{ccl}V^{S}_{1}&=&V^{c}_{1}\delta_{2}^{1}p_{1}({t_{1}^{1}})\\\
{}\hfil&=&[-0.25,1,2.25,-1.25],\\\
V^{S}_{2}&=&V^{c}_{2}W_{[2,2]}\delta_{2}^{2}p_{2}({t_{2}^{2}})\\\
{}\hfil&=&[2,2.4,0.2,-1.2].\end{array}$

Assume the SUR is synchronous myopic best response adjustment, then it is easy
to get the dynamic equation as

$\begin{array}[]{l}a_{1}(k+1)=\delta_{2}[2,1,2,1]a_{1}(k)a_{2}(k),\\\
a_{2}(k+1)=\delta_{2}[2,2,1,1]a_{1}(k)a_{2}(k).\end{array}$

Finally, we have

$a(k+1)=\delta_{4}[4,2,3,1]a(k).$

######  Theorem 7.3.

Consider a Bayesian game $G=(N,A,T,c,Pr)$. Suppose $G$ is a Selten potential
game with potential function $Q^{S}(a)$, if each player updates its action
according to asynchronous MBAR, then the dynamics converges to an S-BN-E.

Proof: Suppose the action profile at time $k$ is $a(k)$. Suppose the updating
player is $i$, then

$\begin{array}[]{cccl}&{}\hfil&Q^{S}(a(k+1))-Q^{S}(a(k)){}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
&=&E(c_{i}(a(k+1))|t_{i}=\bar{t}_{i})-E(c_{i}(a(k))|t_{i}=\bar{t}_{i}){}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
&=&E(c_{i}(a_{i}(k+1),a_{-i}(k))|t_{i}=\bar{t}_{i})-E(c_{i}(a_{i}(k),a_{-i}(k))|t_{i}=\bar{t}_{i})\geq
0.\end{array}$

Since the action profile is finite, then after finite steps, the asynchronous
MBAR will converge to the potential maximizer, which is an S-BN-E of $G$.
$\Box$

######  Theorem 7.4.

Consider a Selten Bayesian potential game $G=(N,A,T,c,Pr)$. Suppose each
player updates its action according to asynchronous LR. Then

  * (i)

The invariant distribution, denote by $\mu_{T}$, of asynchronous LR is

$\displaystyle\mu_{T}(a)=\dfrac{e^{\frac{1}{T}Q^{S}(a)}}{\sum_{a^{\prime}\in
A}e^{\frac{1}{T}Q^{S}(a^{\prime})}}.$ (143)

  * (ii)

Let $\mu_{0}(a)=\lim_{T\rightarrow 0}\mu_{T}(a).$ Then the support of
$\mu_{0}$ is equal to the set of maximizers of $Q^{S}(a).$

Proof:

  * (i)

The asynchronous LR defines an irreducible and aperiodic Markov chain on the
state space $A$. Denote by $P(a;a^{\prime})$ the probability transferring from
profile $a$ to $a^{\prime}$ under the dynamics of asynchronous LR. To prove
$\mu_{T}(a)$ is an invariant distribution, we only need to prove that the
following detailed balance condition is satisfied

$\mu_{T}(a)P(a;a^{\prime})=\mu_{T}(a^{\prime})P(a^{\prime};a),~{}~{}\forall
a,a^{\prime}\in A.$

If $a=a^{\prime}$, the detailed balance condition is clearly fulfilled. If $a$
and $a^{\prime}$ differ in more than one player, then
$P(a;a^{\prime})=P(a^{\prime};a)=0.$ Hence we only consider that $a$ and
$a^{\prime}$ differ exactly in one player $i$. That is, $a_{i}\neq
a^{\prime}_{i},~{}a_{-i}=a^{\prime}_{-i}$. Then

$\begin{array}[]{cccl}\mu_{T}(a)P(a;a^{\prime})&=&\dfrac{e^{\frac{1}{T}Q^{S}(a)}}{\sum_{\bar{a}\in
A}e^{\frac{1}{T}Q^{S}(\bar{a})}}\cdot\dfrac{e^{\frac{1}{T}c_{i}^{S}(a^{\prime}_{i},a_{-i})}}{\sum_{\tilde{a}_{i}\in
A_{i}}e^{\frac{1}{T}c_{i}^{S}(\tilde{a}_{i},a_{-i})}}\\\
&=&\dfrac{e^{\frac{1}{T}Q^{S}(a)}}{\sum_{\bar{a}\in
A}e^{\frac{1}{T}Q^{S}(\bar{a})}}\cdot\dfrac{e^{\frac{1}{T}Q^{S}(a^{\prime}_{i},a_{-i})}}{\sum_{\tilde{a}_{i}\in
A_{i}}e^{\frac{1}{T}Q^{S}(\tilde{a}_{i},a_{-i})}}\\\
&=&\dfrac{e^{\frac{1}{T}Q^{S}(a^{\prime}_{i},a_{-i})}}{\sum_{\bar{a}\in
A}e^{\frac{1}{T}Q^{S}(\bar{a})}}\cdot\dfrac{e^{\frac{1}{T}Q^{S}(a)}}{\sum_{\tilde{a}_{i}\in
A_{i}}e^{\frac{1}{T}Q^{S}(\tilde{a}_{i},a_{-i})}}\\\
&=&\mu_{T}(a^{\prime})P(a^{\prime};a).{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\end{array}$

  * (ii)

As $T\rightarrow 0$, the asynchronous LR converges to the myopic best reply
rule. Therefore, the support of $\mu_{0}$ is equal to the set of maximizers of
$Q^{S}(a).$

$\Box$

######  Example 7.5.

Recall Example 7.2. Assume the SUR is synchronous LR, the conversion is
S-Conversion. Assume the type assigned by nature is $t_{1}=t_{1}^{1}$,
$t_{2}=t_{2}^{2}$, then the expected payoff is

$\begin{array}[]{ccccccl}V^{S}_{1}&=&[-0.25,&1,&2.25,&-1.25],\\\
V^{S}_{2}&=&[2.000,&2.4,&0.20,&-1.20].\end{array}$

By virtue of (142), we have

$\displaystyle
Pr(a_{1}(k+1)=a_{1}|a_{2}(k))=\begin{cases}\dfrac{e^{\frac{1}{T}c_{1}^{S}(a_{1}=a_{1}^{1},a_{2}(k)=a_{2}^{1})}}{e^{\frac{1}{T}c_{1}^{S}(a_{1}=a_{1}^{1},a_{2}(k)=a_{2}^{1})}+e^{\frac{1}{T}c_{1}^{S}(a_{1}^{2},a_{2}(k)=a_{2}^{1})}}:=\alpha_{1},\\\
\dfrac{e^{\frac{1}{T}c_{1}^{S}(a_{1}=a_{1}^{1},a_{2}(k)=a_{2}^{2})}}{e^{\frac{1}{T}c_{1}^{S}(a_{1}=a_{1}^{1},a_{2}(k)=a_{2}^{2})}+e^{\frac{1}{T}c_{1}^{S}(a_{1}^{2},a_{2}(k)=a_{2}^{2})}}:=\beta_{1}.\\\
\end{cases}$

Similarly, we have

$\displaystyle
Pr(a_{2}(k+1)=a_{2}|a_{1}(k))=\begin{cases}\dfrac{e^{\frac{1}{T}c_{1}^{S}(a_{2}=a_{2}^{1},a_{1}(k)=a_{1}^{1})}}{e^{\frac{1}{T}c_{1}^{S}(a_{2}=a_{2}^{1},a_{1}(k)=a_{1}^{1})}+e^{\frac{1}{T}c_{1}^{S}(a_{2}^{2},a_{1}(k)=a_{1}^{1})}}:=\alpha_{2},\\\
\dfrac{e^{\frac{1}{T}c_{1}^{S}(a_{2}=a_{2}^{1},a_{1}(k)=a_{1}^{2})}}{e^{\frac{1}{T}c_{1}^{S}(a_{2}=a_{2}^{1},a_{1}(k)=a_{1}^{2})}+e^{\frac{1}{T}c_{1}^{S}(a_{2}^{2},a_{1}(k)=a_{1}^{2})}}:=\beta_{2}.\\\
\end{cases}$

Then

$\displaystyle a_{1}(k+1)=M_{1}a(k),~{}~{}a_{2}(k+1)=M_{2}a(k),$

where $a(k)=a_{1}(k)a_{2}(k)$,

$\displaystyle
M_{1}=\begin{bmatrix}\alpha_{1}&\beta_{1}&\alpha_{1}&\beta_{1}\\\
1-\alpha_{1}&1-\beta_{1}&1-\alpha_{1}&1-\beta_{1}\\\ \end{bmatrix}.$
$\displaystyle
M_{2}=\begin{bmatrix}\alpha_{2}&\beta_{2}&\alpha_{2}&\beta_{2}\\\
1-\alpha_{2}&1-\beta_{2}&1-\alpha_{2}&1-\beta_{2}\\\ \end{bmatrix}.$

Finally, we have

$\displaystyle a(k+1)=Ma(k)=(M_{1}*M_{2})a(k),$

whre $*$ is the Khatra-Rao product.

Let $T=2$, then

$\displaystyle M_{1}=\begin{bmatrix}0.0067&0.9890&0.0067&0.9890\\\
0.9933&0.0110&0.9933&0.0110\\\ \end{bmatrix}.$ $\displaystyle
M_{2}=\begin{bmatrix}0.2315&0.2315&0.9975&0.9975\\\
0.7685&0.7685&0.0025&0.0025\\\ \end{bmatrix}.$ $\displaystyle
M=\begin{bmatrix}0.0015&0.2289&0.0067&0.9866\\\ 0.0051&0.7601&0.0000&0.0024\\\
0.2299&0.0025&0.9909&0.0110\\\ 0.7634&0.0084&0.0025&0.0000\\\ \end{bmatrix}$

Therefore the steady action profile is

$a^{*}(t_{1}^{1},t_{2}^{1})=[0.0357,0.0011,0.9336,0.0296]^{T}.$

Player $1$’s steady strategy is

$a_{1}^{*}(t_{1}^{1},t_{2}^{1})=(I_{2}\otimes
J_{2}^{T})a^{*}(t_{1}^{1},t_{2}^{1})=[0.0369,0.9631]^{T}.$

Player $2$’s steady strategy is

$a_{2}^{*}(t_{1}^{1},t_{2}^{1})=(J_{2}^{T}\otimes
I_{2})a^{*}(t_{1}^{1},t_{2}^{1})=[0.9693,0.0307]^{T}.$

Similarly, we can conclude that

  * •

If $t_{1}^{\theta}(t)=t_{1}^{1}$ $t_{2}^{\theta}(t)=t_{2}^{2}$, the steady
action profile is

$a^{*}(t_{1}^{1},t_{2}^{2})=[0.1932,0.0680,0.5466,0.1922]^{T}.$

Player $1$’s steady strategy is

$a_{1}^{*}(t_{1}^{1},t_{2}^{2})=(I_{2}\otimes
J_{2}^{T})a^{*}(t_{1}^{1},t_{2}^{2})=[0.2612,0.7388]^{T}.$

Player $2$’s steady strategy is

$a_{2}^{*}(t_{1}^{1},t_{2}^{2})=(J_{2}^{T}\otimes
I_{2})a^{*}(t_{1}^{1},t_{2}^{2})=[0.7398,0.2602]^{T}.$

  * •

If $t_{1}^{\theta}(t)=t_{1}^{2}$ $t_{2}^{\theta}(t)=t_{2}^{1}$, the steady
action profile is

$a^{*}(t_{1}^{2},t_{2}^{1})=[0.2359,0.7556,0.0020,0.0064]^{T}.$

Player $1$’s steady strategy is

$a_{1}^{*}(t_{1}^{2},t_{2}^{1})=(I_{2}\otimes
J_{2}^{T})a^{*}(t_{1}^{2},t_{2}^{1})=[0.9915,0.0085]^{T}.$

Player $2$’s steady strategy is

$a_{2}^{*}(t_{1}^{2},t_{2}^{1})=(J_{2}^{T}\otimes
I_{2})a^{*}(t_{1}^{2},t_{2}^{1})=[0.2379,0.7621]^{T}.$

  * •

If $t_{1}^{\theta}(t)=t_{1}^{2}$ $t_{2}^{\theta}(t)=t_{2}^{2}$, the steady
action profile is

$a^{*}(t_{1}^{2},t_{2}^{2})=[0.0858,0.9109,0.0003,0.0030]^{T}.$

Player $1$’s steady strategy is

$a_{1}^{*}(t_{1}^{2},t_{2}^{2})=(I_{2}\otimes
J_{2}^{T})a^{*}(t_{1}^{2},t_{2}^{2})=[0.9967,0.0033]^{T}.$

Player $2$’s steady strategy is

$a_{2}^{*}(t_{1}^{2},t_{2}^{2})=(J_{2}^{T}\otimes
I_{2})a^{*}(t_{1}^{2},t_{2}^{2})=[0.0861,0.9139]^{T}.$

###  7.2 Dynamics of Action-Type Bayesian Games

Consider a Bayesian game $G=(N,T,A,c,Pr)$. Assume the conversion is AT-
Conversion. Unlike Selten Bayesian game, the SUR of AT Bayesian game consists
of action updating rule (AUR) and type updating rule (TUR). If the AUR of
player $i$ is $f_{i}$, and the TUR of player $i$ is $g_{i}$, then the SUR is

$\displaystyle\begin{array}[]{l}\begin{cases}a_{i}(k+1)=f_{i}(a_{1}(k),\cdots,a_{n}(k),t(k)),\\\
t_{i}(k+1)=g_{i}(a_{1}(k),\cdots,a_{n}(k),t(k)).\end{cases}\end{array}$ (145)

The player can update its action and type, concurrently or separately. Assume
in the decision-making process player $i$ knows his type $t_{i}(k)$ and other
player’s actions $a_{-i}(k)$ at time $k+1$, but he doesn’t know other player’s
type $t_{-i}(k)$.

Firstly, we design different asynchronous MBARs for Action-Type Bayesian
games, which are shown as follows:

  * (i)

The SUR is called asynchronous concurrent MBRA (C-MBRA), if at time $k$ the
updating player $i$ chooses his action and type concurrently, which is shown
as follows

$\displaystyle a_{i}(k+1)\ltimes
t_{i}(k+1)=\operatorname{argmax}_{(a_{i},t_{i})\in A_{i}\times
T_{i}}c_{i}^{AT}(a_{i},t_{i},a_{-i}(k)),\quad i=1,2,\cdots,n.$ (146)

  * (ii)

The SUR is called asynchronous separate MBRA (S-MBRA), if at time $k$ the
updating player $i$ either chooses his action as follows

$\displaystyle a_{i}(k+1)=\operatorname{argmax}_{a_{i}\in
A_{i}}c_{i}^{AT}(a_{i},t_{i}(k),a_{-i}(k)),\quad i=1,2,\cdots,n.$ (147)

or selects type as follows

$\displaystyle t_{i}(k+1)=\operatorname{argmax}_{t_{i}\in
T_{i}}c_{i}^{AT}(t_{i},a_{i}(k),a_{-i}(k)),\quad i=1,2,\cdots,n.$ (148)

######  Theorem 7.6.

Consider an Action-Type Bayesian potential game $G=(N,A,T,c,Pr)$. If each
player updates its action according to asynchronous C-MBAR or S-MBAR, then the
dynamics converges to AT-BN-E.

Proof: Suppose the action-type profile at time $k$ is $(a(k),t(k))$. Suppose
the updating player is $i$ at time $k$ using asynchronous C-MBAR, then

$\begin{array}[]{cccl}&{}\hfil&Q^{AT}(a(k+1),t(k+1))-Q^{AT}(a(k),t(k)){}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
&=&c^{AT}_{i}(a(k+1),t_{i}(k+1))-c^{AT}_{i}(a(k),t_{i}(k)){}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
&=&c^{AT}_{i}(a_{i}(k+1),a_{-i}(k),t_{i}(k+1))-c^{AT}_{i}(a_{i}(k),a_{-i}(k),t_{i}(k))\geq
0,\\\ \end{array}$

where $Q^{AT}$ is the potential function of the Action-Type Bayesian potential
game $G$. Since the action profile is finite, after finite steps, the
asynchronous MBAR will converge to the potential maximizer, which is an AT-
BN-E of $G$. The proof to asynchronous S-MBAR is the same, so we omit the
details.

$\Box$

Similarly, we can design asynchronous concurrent LR (C-LR) and separate LR
(S-LR) for Action-Type Bayesian games, which are shown as follows:

  * (i)

The SUR is called asynchronous C-LR, if at time $k$ the updating player $i$
chooses his action and type concurrently according to the following
probability

$\displaystyle
Pr\big{(}a_{i}(k+1)=a_{i},t_{i}(k+1)=t_{i}|a(k),t_{i}(k)\big{)}=\dfrac{e^{\frac{1}{T}c_{i}^{AT}(a_{i},t_{i},a_{-i}(k))}}{\sum_{(a^{\prime}_{i},t^{\prime}_{i})\in
A_{i}\times
T_{i}}e^{\frac{1}{T}c_{i}^{AT}(a^{\prime}_{i},t^{\prime}_{i},a_{-i}(k))}},\quad
i=1,2,\cdots,n.$ (149)

  * (ii)

The SUR is called asynchronous S-LR, if at time $k$ the updating player $i$
either chooses his action according to the following probability

$\displaystyle
Pr\big{(}a_{i}(k+1)=a_{i}|a(k),t_{i}(k)\big{)}=\dfrac{e^{\frac{1}{T}c_{i}^{AT}(a_{i},t_{i}(k),a_{-i}(k))}}{\sum_{a^{\prime}_{i}\in
A_{i}}e^{\frac{1}{T}c_{i}^{AT}(a^{\prime}_{i},t_{i}(k),a_{-i}(k))}},\quad
i=1,2,\cdots,n.$ (150)

or chooses his type according to the following probability

$\displaystyle
Pr\big{(}t_{i}(k+1)=t_{i}|a(k),t_{i}(k)\big{)}=\dfrac{e^{\frac{1}{T}c_{i}^{AT}(t_{i},a_{i}(k),a_{-i}(k))}}{\sum_{t^{\prime}_{i}\in
T_{i}}e^{\frac{1}{T}c_{i}^{AT}(t^{\prime}_{i},a_{i}(k),a_{-i}(k))}},\quad
i=1,2,\cdots,n.$ (151)

######  Theorem 7.7.

Consider an Action-Type Bayesian potential game $G=(N,A,T,c,Pr)$. Suppose each
player updates his action according to asynchronous C-LR or S-LR. Then

  * (i)

The invariant distribution, denote by $\nu_{T}$, of asynchronous LR (either
C-LR or S-LR) is

$\displaystyle\nu_{T}(a,t)=\dfrac{e^{\frac{1}{T}Q^{AT}(a,t)}}{\sum_{(a^{\prime},t^{\prime})\in
A\times T}e^{\frac{1}{T}Q^{AT}(a^{\prime},t^{\prime})}}.$ (152)

  * (ii)

Let $\nu_{0}(a,t)=\lim_{T\rightarrow 0}\nu_{T}(a,t).$ Then the support of
$\nu_{0}$ is equal to the set of maximizers of potential function
$Q^{AT}(a,t).$

Proof: We only prove (i), because the proof of (ii) is similar to the proof of
Theorem 7.4.

The asynchronous C-LR defines an irreducible and aperiodic Markov chain on the
state space $A\times T$. Denote by $P(a,t;a^{\prime},t^{\prime})$ the
probability transferring from profile $(a,t)$ to $(a^{\prime},t^{\prime})$
under the dynamics of asynchronous C-LR. To prove $\nu_{T}(a,t)$ is an
invariant distribution, we only need to prove that the following detailed
balance condition is satisfied

$\nu_{T}(a,t)P(a,t;a^{\prime},t^{\prime})=\nu_{T}(a^{\prime})P(a^{\prime},t^{\prime};a,t),~{}~{}\forall
a,a^{\prime}\in A,~{}t,t^{\prime}\in T.$

If $a=a^{\prime},t=t^{\prime}$, the detailed balance condition is clearly
fulfilled. If $a,a^{\prime}$ or $t,t^{\prime}$ differ in more than one player,
then $P(a,t;a^{\prime},t^{\prime})=P(a^{\prime},t^{\prime};a,t)=0.$ Hence we
only consider that $a,a^{\prime}$ or $t,t^{\prime}$ differ exactly in one
player $i$. That is, $a_{i}\neq
a^{\prime}_{i},~{}a_{-i}=a^{\prime}_{-i};~{}t_{i}\neq
t^{\prime}_{i},~{}t_{-i}=t^{\prime}_{-i}$. Then,

$\begin{array}[]{cccl}\nu_{T}(a,t)P(a,t;a^{\prime},t^{\prime})&=&\dfrac{e^{\frac{1}{T}Q^{AT}(a,t)}}{\sum_{(\bar{a},\bar{t})\in
A\times
T}e^{\frac{1}{T}Q^{AT}(\bar{a},\bar{t})}}\cdot\dfrac{e^{\frac{1}{T}c_{i}^{AT}(a^{\prime}_{i},a_{-i},t^{\prime}_{i})}}{\sum_{(\tilde{a}_{i},\tilde{t}_{i})\in
A_{i}\times
T_{i}}e^{\frac{1}{T}c_{i}^{AT}(\tilde{a}_{i},\tilde{t}_{i},a_{-i})}}\\\
&=&\dfrac{e^{\frac{1}{T}Q^{AT}(a,t)}}{\sum_{(\bar{a},\bar{t})\in A\times
T}e^{\frac{1}{T}Q^{AT}(\bar{a},\bar{t})}}\cdot\dfrac{e^{\frac{1}{T}Q^{AT}(a^{\prime}_{i},a_{-i},t^{\prime}_{i},t_{-i})}}{\sum_{(\tilde{a}_{i},\tilde{t}_{i})\in
A_{i}\times
T_{i}}e^{\frac{1}{T}Q^{AT}(\tilde{a}_{i},\tilde{t}_{i},a_{-i},t_{-i})}}\\\
&=&\dfrac{e^{\frac{1}{T}Q^{AT}(a^{\prime}_{i},a_{-i},t^{\prime}_{i},t_{-i})}}{\sum_{(\bar{a},\bar{t})\in
A\times
T}e^{\frac{1}{T}Q^{AT}(\bar{a},\bar{t})}}\cdot\dfrac{e^{\frac{1}{T}Q^{AT}(a,t)}}{\sum_{(\tilde{a}_{i},\tilde{t}_{i})\in
A_{i}\times
T_{i}}e^{\frac{1}{T}Q^{AT}(\tilde{a}_{i},\tilde{t}_{i},a_{-i},t_{-i})}}\\\
&=&\nu_{T}(a^{\prime},t^{\prime})P(a^{\prime},t^{\prime};a,t).{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\end{array}$

The proof for asynchronous S-LR is the same, so it is omitted.

$\Box$

We give an example to show how to get the dynamic equation for repeated
Bayesian game under AT-Conversion.

######  Example 7.8.

Recall the Bayesian game $G=(N,A,T,c,Pr)$ in Example 6.11. Suppose

$\begin{array}[]{cccccccccccl}\alpha_{1}=-1,&\alpha_{2}=3,&\alpha_{3}=-4,&\alpha_{4}=2,&\alpha_{5}=-1,&\alpha_{6}=5,&\alpha_{7}=1,&\alpha_{8}=2;\\\
\beta_{1}=7,&\beta_{2}=5,&\beta_{3}=6,&\beta_{4}=4,&\beta_{5}=-2,&\beta_{6}=3,&\beta_{7}=1,&\beta_{8}=2.\\\
\end{array}$

According to Example 6.11, $G$ is Action-Type potential.

Assume the SUR is asynchronous C-MBRA, player $1$ updates his action and type
first and then player $2$ updates his action and type. It is easy to get the
dynamic equation as

$\begin{array}[]{l}\begin{cases}t_{1}(2k+1)a_{1}(2k+1)=\delta_{4}[4,3,4,3,4,3,4,3]t_{1}(2k)a(2k),\\\
t_{2}(2k+1)a_{2}(2k+1)=t_{2}(2k)a(2k),~{}~{}~{}k=1,2,\cdots,\end{cases}\end{array}$

and

$\begin{array}[]{l}\begin{cases}t_{2}(2k)a_{2}(2k)=\delta_{4}[1,1,1,1,1,1,1,1]t_{2}(2k-1)a(2k-1),\\\
t_{1}(2k)a_{1}(2k)=t_{1}(2k-1)a(2k-1),~{}~{}~{}k=1,2,\cdots.\end{cases}\end{array}$

Denote by $x(k):=t(k)\ltimes a(k)$. It is obvious that the fixed points of
asynchronous C-MBRA dynamics are

$x_{1}^{f}=(t_{1}^{2},t_{2}^{1},a_{1}^{1},a_{2}^{1}),~{}~{}~{}x_{2}^{f}=(t_{1}^{2},t_{2}^{1},a_{1}^{2},a_{2}^{1}).$

We can verify that both $x_{1}^{f}$ and $x_{2}^{f}$ are the AT-BN-E of $G$.

##  8 Conclusion

The main purpose of this paper is to provide a fundamental framework for
finite Bayesian games, using STP expression of finite games.

Three conversions from incomplete information to complete information are
discussed, which are Harsanyi, Selten, and AT conversions respectively.
Formulas are obtained for three conventions respectively. In addition,
Bayesian potential game is also investigated. Finally, the evolutive equations
for dynamic Bayesian games are also provided according to SURs and
conversions.

There are many problems remain for further investigation. For instance, the
following are some challenging topics:

  * (i)

A generalization: State-based game. (As time-varying types.)

  * (ii)

Deal with all history based knowledge SUR. (Truncation)

  * (iii)

Learning-based optimization.

  * (iv)

Calculating Bayesian Nash equilibrium.

  * (v)

Dynamic Bayesian Nash equilibrium.

  * (vi)

Application of Potential Bayesian game.

## References

  * [1] M. Aurangzeb, F.L., Lewis, Internal structure of coalitions in competitive and altruistic graphical coalitional games, Automatica, Vol. 50, 335-348, 2014.
  * [2] R. Gibbons, A Primer in Game Theory, Bell & Bain Ltd., Glasgow, 1992.
  * [3] T. Basar, G.J. Olsder, Dynamic Noncooperative Game Theory, SIAM, 1998.
  * [4] O. Candogan, I. Menache, A. Ozdaglar, P.A. Parrilo, Flows and decompositions of games: Harmonic and potential games, Mathematcs of Operations Research, Vol. 36, No. 3, 474-503, 2011.
  * [5] D. Cheng, H. Qi, Z. Li, Analysis and Control of Boolean Networks - A Semi-tensor Product Approach, Springer, London, 2011.
  * [6] D. Cheng, H. Qi, Y. Zhao, An Introduction to Semi-tensor Product of Matrices and Its Applications, World Scientific, Singapore, 2012.
  * [7] D. Cheng, On finite potential games, Automatica, Vol. 50, 1793-1801, 2014.
  * [8] D. Cheng, F. He, H. Qi, et al.,Modeling, analysis and control of networked evolutionary games, IEEE Trans. Aut. Contr., Vol. 60, No. 9, 2401-2415, 2015.
  * [9] D. Cheng, T. Liu, K. Zhang, et al., On decomposition subspaces of finite games, IEEE Trans. Aut. Contr., Vol. 61, No. 11, 3651-3656, 2016.
  * [10] D. Cheng, T. Liu, From Boolean game to potential game, Automatica, Vol. 96, 51-60, 2018
  * [11] O. Candogan, A. Ozdaglar, and P. A. Parrilo, Dynamics in near-potential games, Games and Econ. Behav., vol. 82, pp. 66-90, 2013.
  * [12] E. Einy, O. Haimanko, Equilibrium existence in games with a concase Bayesian potential, Games and Economic Behavior, Vol. 123, 288-294, 2020.
  * [13] E.C. Eksin, A. Ribeiro, Distributed fictitious play in potential games of incomplete information, Proc. 54th CDC, 5190-5196, 2015.
  * [14] C. Eksin, B. Swenson, S. Kar, Learning pure-strategy Nash equilibria in networked multi-agent systems with uncertainty, IEEE 55th Conference on Decision and Control, pp. 5292-5297, 2016.
  * [15] C. Eksin, A. Ribeiro, Distributed Fictitious Play for Multi-Agent Systems in Uncertain Environments, IEEE Trans. Aut. Contr., 2017.
  * [16] G. Facchini, F.V. Megen, P. Borm, Tijs, congestion models and weighted bayesian potential games, Theory and Decision, Vol. 42, No. 2, 193-206, 1997.
  * [17] P. Frihauf, M. Krstic, T. Basar, Nash equilibrium seeking in noncooprative games, IEEE Trans. Aut. Contr., Vol. 57, 1192-1207, 2012.
  * [18] R. Gibbons, A Primer in Game Theory, Prentice, London, 1992.
  * [19] L. Giupponi, C. Ibars, Bayesian potential games to model cooperation for cognitive radios with incomplete information, Proc. 2009 Int. Conf. Commun., DOI:10.1109.5198780, 2009.
  * [20] R. Gopalakrishnan, J. R. Marden, A. Wierman, An architectural view of game theoretic control, Performance Evaluation Review, Vol. 38, No. 3, 31-36, 2011.
  * [21] Y. Hao, D. Cheng, On skew-symmetric games, J. Franklin Institute, Vol. 355, 3196-3220, 2018.
  * [22] J.C. Harsanyi, Games with incomplete information played by Bayisian players, Part I, the basic model, Management Science, Vol. 14, 159-182, 1967.
  * [23] A. E. Carlos, N. Netzer, The logit-response dynamics, Games and Economic Behavior, Vol. 68, No. 2, 413-427, 2010.
  * [24] J.C. Harsanyi, Games with incomplete information played by Bayisian players, Part II, bayesian equilibrium points, Management Science, Vol. 14, 320-334, 1968.
  * [25] J.C. Harsanyi, Games with incomplete information played by Bayisian players, Part III, the basic probability distribution of the game, Management Science, Vol. 14, 486-502, 1968.
  * [26] R.V. Heumen, B. Peleg, S. Tijs, P. Borm, Axiomatic characterizations of solutions for Bayesian games, Theory and Decision, Vol. 40, 103-129, 1996.
  * [27] X. Liu, J. Zhu, On potential equations of finite games, Automatica, Vol. 68, 245-253, 2016.
  * [28] J.R. Marden, State based potential games, Automatica, Vol. 48, 3075-3088, 2012.
  * [29] V.G.L. Mejia, Y. Wan, F.L. Lewis, Bayesian graphical games for synchronization in networks of dynamical systems, IEEE Trans. Cont. Network Sys., doi:10.1109/tcns.2019.2927201.
  * [30] D. Monderer, L.S. Shapley, Potential Games Games and Economic Behavior, Vol. 14, 124-143, 1996.
  * [31] R.W. Rosenthal, A class of games possessing pure-strategy Nash equilibria, Int. J. Game Theory, Vol. 2, 65-67, 1073.
  * [32] H. Tembine, E. Altman, R. El-Azouzi, Y. Hayel, Evolutionary games in wireless networks, IEEE Trans. Sys. Man, Cyb.-Part B: Cyb., Vol. 40, No. 3, 634-646, 2010.
  * [33] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944.
  * [34] Y. Wu, S. Le, K. Zhang, et al., Ex-ante agent transformation of Bayesian games, (preprint).
  * [35] G. Facchini, F. V. Megen, P. Borm, et al., Congestion models and weighted Bayesian potential games. Theory and Decision, Vol. 42, No. 2, 193-206, 1997.
  * [36] A.Y. Yazicioglu, M. Egerstedt, J.S. Shamma, A game theoretic approach to distributed coverage of graphs by heterogeneous mobile agents, Est. Contr. Netwk. Sys., Vol. 4, 309-315, 2013.
  * [37] Suten Yue, Equilibrium Study of Weighted Congestion games and Bayesian Games via the Semi-tensor Product Method, Master dethes, Dalian Univ. of Technology, 2018.
  * [38] R. Zhang, L. Guo, Controllability of Nash equilibrium in game-based control systems, IEEE Trans. Aut. Contr., Vol. 64, No. 10, 4180-4187, 2019.
  * [39] S. Zamir, Bayesian gams: Games with incomplete information, in Computational Compleity: Theory Techniques and Applications, pp 238-253, Springer, New York, 2012.