## Bayesian Games

Determine, with justification, wether the following ideals are prime, maximal, both, or neither.
(a) $\left\langle x^{3}-1\right\rangle$ in $\mathbb{Q}[x]$.
(b) $\left\langle x^{9}+7\right\rangle$ in $\mathbb{Q}[x]$
(c

A Bayesian game is a list $(N, A, \Theta, T, u, p)$

• $N$ : set of players
• $A=\left(A_{i}\right)_{i \in N}:$ set of action profiles
• $\Theta$ : set of payoff parameters
• $T_{i}:$ set of types for player $i ; T=\prod_{i \in N} T_{i}$
• $u_{i}: \Theta \times A \rightarrow \mathbb{R}:$ payoff function of player $i$

$p_{i}\left(\cdot \mid t_{i}\right) \in \Delta\left(\Theta \times T_{-i}\right):$ belief of type $t_{i}$
Each player $i$ knows his own type $t_{i}$ but does not necessarily know $\theta$ or other players’ types. .. belief $p_{i}\left(\cdot \mid t_{i}\right)$

The game has a common prior if there exists $\pi \in \Delta(\Theta \times T)$ such that
$$p_{i}\left(\cdot \mid t_{i}\right)=\pi\left(\cdot \mid t_{i}\right), \forall t_{i} \in T_{i}, \forall i \in N$$

## Infinite Hierarchies of Beliefs

When a researcher models incomplete information, there is often no ex-ante stage or explicit information structure in which players observe signals and make inferences. At the modeling stage, each player i has an infinite hierarchy of beliefs

• a first-order belief $\tau_{i}^{1} \in \Delta(\Theta)$ about payoffs (and other aspects of the world)
• a second-order belief $\tau_{i}^{2} \in \Delta\left(\Theta \times \Delta(\Theta)^{N \backslash{i}}\right)$ about $\theta$ and other players’ first-order beliefs $\tau_{-i}^{1}$
• a third-order belief $\tau_{i}^{3}$ about correlations in player i’s second-order uncertainty $\tau_{i}^{2}$ and other players’ second-order beliefs $\tau_{-i}^{2} \ldots$

## Formal Definition

For simplicity, consider two players.
Suppose that $\Theta$ is a Polish (complete separable metric) space.
Player $i$ has beliefs about $\theta$, about other’s beliefs about $\theta, \ldots$
\begin{aligned} X_{0} &=\Theta \ X_{1} &=X_{0} \times \Delta\left(X_{0}\right) \ & \vdots \ X_{n} &=X_{n-1} \times \Delta\left(X_{n-1}\right) \ \vdots & \end{aligned}
$\tau_{i}=\left(\tau_{i}^{1}, \tau_{i}^{2}, \ldots\right) \in \prod_{n=0}^{\infty} \Delta\left(X_{n}\right):$ belief hierarchy of player $i$
$H_{i}=\prod_{n=0}^{\infty} \Delta\left(X_{n}\right):$ set of $i$ ‘s hierarchies of beliefs
Every $X_{n}$ is Polish. Endow $X_{n}$ with the weak topology.

## Interpretation of Type Space

Harsanyi’s (1967) parsimonious formalization of incomplete information through a type space $(\Theta, T, p)$ naturally generates an infinite hierarchy of beliefs for each $t_{i} \in T_{i}$, which is consistent by construction:
first-order belief: $\quad h_{i}^{1}\left(\cdot \mid t_{i}\right)=\operatorname{marg}{\Theta} p\left(\cdot \mid t{i}\right)=\sum_{t_{-i}} p\left(\theta, t_{-i} \mid t_{i}\right)$
second-order belief: $\quad h_{i}^{2}\left(\theta, \hat{h}{-i}^{1} \mid t{i}\right)=\sum_{t_{-i} \mid h_{-i}^{1}\left(\cdot \mid t_{-i}\right)=\hat{h}{-i}^{1}} p\left(\theta, t{-i} \mid t_{i}\right) \ldots$
A type $t_{i} \in T_{i}$ in a space $(\Theta, T, p)$ models a belief hierarchy $\left(\tau_{i}^{1}, \tau_{i}^{2}, \ldots\right)$ if $h_{i}^{n}\left(\cdot \mid t_{i}\right)=\tau_{i}^{n}$ for each $n$

## Coherency

How expressive is Harsanyi’s language?
Is there $(T, p)$ s.t. $\left{h_{i}\left(\cdot \mid t_{i}\right) \mid t_{i} \in T_{i}\right}=H_{i} ?$
Hierarchies should be coherent:
$$\operatorname{marg}{X{n-2}} \tau_{i}^{n}=\tau_{i}^{n-1}$$
Different levels of beliefs should not contradict one another. $H_{i}^{0}$ : set of i’s coherent hierarchies.

Proposition 1 (Brandenburger and Dekel 1993)

There exists a homeomorphism $f_{i}: H_{i}^{0} \rightarrow \Delta\left(\Theta \times H_{-i}\right)$ s.t.
$$\operatorname{marg}{X{n-1}} f_{i}\left(\cdot \mid \tau_{i}\right)=\tau_{i}^{n}, \forall n \geq 1$$

## Common Knowledge of Coherency

Is there $(T, p)$ s.t. $\left{h_{i}\left(\cdot \mid t_{i}\right) \mid t_{i} \in T_{i}\right}=H_{i}^{0} ?$
We need to restrict attention to hierarchies of beliefs under which there is common knowledge of coherency:
\begin{aligned} \text { – } & H_{i}^{1}=\left{\tau_{i} \in H_{i}^{0} \mid f_{i}\left(H_{-i}^{0} \mid \tau_{i}\right)=1\right} \ & H_{i}^{2}=\left{\tau_{i} \in H_{i}^{1} \mid f_{i}\left(H_{-i}^{1} \mid \tau_{i}\right)=1\right} \ & \ldots \ \text { – } & H_{i}^{*}=\bigcap_{k \geq 0} H_{i}^{k} \end{aligned}

## The Interim Game

For any Bayesian game $\mathcal{B}=(N, A, \Theta, T, u, p)$, define the interim game $I G(\mathcal{B})=(\hat{N}, \hat{S}, U)$
\begin{aligned} \hat{N} &=U_{i \in N} T_{i} \ \hat{S}{t{i}} &=A_{i} \ \hat{U}{t{i}}(\hat{s}) &=E_{p_{i}\left(\cdot \mid t_{i}\right)}\left[u_{i}(\theta, \hat{s})\right] \equiv \sum_{\left(\theta, t_{i}\right)} p_{i}\left(\theta, t_{-i} \mid t_{i}\right) u_{i}\left(\theta, \hat{s}{t{i}}, \hat{s}{t{-i}}\right), \forall t_{i} \in \hat{N} \end{aligned}
where $\hat{s}=\left(\hat{s}{t{i}}\right){t{i} \in \hat{N}}$
Assume finite $\Theta \times T$ to avoid measurability issues.

## Th Ex Ante Game

For a Bayesian game $\mathcal{B}=(N, A, \Theta, T, u, \pi)$ with a common prior $\pi$, the ex-ante game $G(\mathcal{B})=(N, S, U)$ is given by
$$\begin{gathered} S_{i}=A_{i}^{T_{i}} \ni s_{i}: T_{i} \rightarrow A_{i} \ U_{i}(s)=E_{\pi}\left[u_{i}(\theta, s(t))\right] \end{gathered}$$

## Bayesian Nash Equilibrium

Strategies of player $i$ in $\mathcal{B}$ are mappings $s_{i}: T_{i} \rightarrow A_{i}$ (measurable when $T_{i}$ is uncountable).
Definition 1
In a Bayesian game $\mathcal{B}=(N, A, \Theta, T, u, p)$, a strategy profile $s: T \rightarrow A$ is a Bayesian Nash equilibrium (BNE) if it corresponds to a Nash equilibrium of $I G(\mathcal{B})$, i.e., for every $i \in N, t_{i} \in T_{i}$
$$E_{p_{i}\left(\cdot \mid t_{i}\right)}\left[u_{i}\left(\theta, s_{i}\left(t_{i}\right), s_{-i}\left(t_{-i}\right)\right)\right] \geq E_{p_{i}\left(\cdot \mid t_{i}\right)}\left[u_{i}\left(\theta, a_{i}, s_{-i}\left(t_{-i}\right)\right)\right], \forall a_{i} \in A_{i}$$
Interim rather than ex ante definition preferred since in models with a continuum of types the ex ante game has many spurious equilibria that differ on probability zero sets of types.

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