Common Prior¶

Definition¶

In a game with types $\theta\in\Theta=\Theta_1\times\cdots\times\Theta_n$ $Θ_{n}$ , a common prior is a single probability distribution $p$ on $\Theta$ that is common knowledge among the players and from which every player's beliefs are derived. Player $i$ of type $\theta_i$ (with $p(\theta_i)>0$ $0$ ) holds the conditional belief

p(\theta_{-i}\mid\theta_i)=\frac{p(\theta_i,\theta_{-i})}{\sum_{\theta'_{-i}}p(\theta_i,\theta'_{-i})},

where $\theta_{-i}$ denotes the types of the other players. For finite type spaces, $p$ is a joint probability mass function of the type profile.

Interpretation¶

The common prior assumption says that players differ in what they know, not in how they interpret the world: any difference in beliefs comes from different private information. Each type's belief is the conditional probability of the others' types given her own, obtained from $p$ by Bayes' rule.

The 2-by-2 common prior of the correlated example with row a and column x outlined; each row is labelled with player 1's belief about x and each column with player 2's belief about a.

The figure shows the correlated example below as one table that both players read. Player 1 of type $a$ knows she is in the outlined row and rescales it by its total $0.5$ , giving $p(x\mid a)=0.8$ $0.8$ ; player 2 of type $x$ knows she is in the dashed column and rescales it by $0.6$ , giving $p(a\mid x)=\tfrac{2}{3}$ $\frac{2}{3}$ . Different types end up with different beliefs only because they condition on different parts of the same $p$ .

This is what lets a Bayesian game be written as one game with Nature moving first, the Harsanyi transformation: Nature draws $\theta$ from $p$ , and everyone knows that it does.

When types are correlated, a player's own type is informative about the others', so different types of the same player hold different beliefs; using the marginal distribution of $\theta_{-i}$ in that case is a frequent error. Only independent types give every type the same belief (Key Results).

Key Results¶

Beliefs consistent with a common prior satisfy $p(\theta_i)\,p(\theta_{-i}\mid\theta_i)=p(\theta)$ for all $i$ and $\theta$ .
With independent types, $p(\theta)=\prod_jp_j(\theta_j)$ and $p(\theta_{-i}\mid\theta_i)=p(\theta_{-i})$ for every type $\theta_i$ : all types of player $i$ hold the same belief, the product of the others' marginals.
Not every belief system comes from a common prior: two players without private information who assign different probabilities to the same event cannot share one. More generally (Aumann's agreement theorem), under a common prior, posteriors about an event that are common knowledge must be equal.
Under a common prior, a type's expected payoff is computed with $p(\cdot\mid\theta_i)$ $θ_{i})$ ; the ex ante expected payoff averages these with weights $p(\theta_i)$ .

Examples¶

Correlated types. Let $\Theta_1=\{a,b\}$ ${a, b}$ , $\Theta_2=\{x,y\}$ and

	$x$	$y$
$a$	$0.4$	$0.1$
$b$	$0.2$	$0.3$

The entries sum to 1. Type $a$ has $p(a)=0.5$ and believes $p(x\mid a)=0.4/0.5=0.8$ $0.8$ ; type $b$ believes $p(x\mid b)=0.2/0.5=0.4$ $0.4$ . Player 2 has $p(x)=0.6$ $0.6$ , $p(y)=0.4$ $0.4$ , so type $x$ believes $p(a\mid x)=0.4/0.6=\tfrac{2}{3}$ and type $y$ believes $p(a\mid y)=0.1/0.4=\tfrac{1}{4}$ $\frac{1}{4}$ .

Independent types. If each $\theta_i\in\{\tfrac{1}{4},2\}$ with probability $\tfrac{1}{2}$ independently, $p(\theta)=\tfrac{1}{4}$ for each of the four profiles and every type believes the other is low-cost with probability $\tfrac{1}{2}$ .

Prerequisites¶

Sources¶

Game Theory · Week 5

Aliases: common prior assumption, CPA, Harsanyi doctrine Keywords: prior distribution over types, p(theta), p(theta_{-i} | theta_i), posterior beliefs, consistent beliefs, agreeing to disagree Symbols: element of, product, sum, theta

Last updated: 17 October 2026 21:53