<ol>
<li>Set of players, I guess this is quite clear. As an example, take player <span class="math-container">$A$</span> and <span class="math-container">$B$</span></li>
<li>Actions. This just tells you what the two players can do. For example <span class="math-container">$A$</span> can choose <span class="math-container">$U$</span>(p) or <span class="math-container">$D$</span>(own) and <span class="math-container">$B$</span> can choose <span class="math-container">$L$</span>(eft) or <span class="math-container">$R$</span>(ight).</li>
<li>Types: A type is attached to a player. For example, player <span class="math-container">$A$</span> can be of two different types, say <span class="math-container">$A_1$</span> or <span class="math-container">$A_2$</span>. The type that player <span class="math-container">$A$</span> has is usually known by the player herself, but usually not known by the other players (why we need a probability distribution). Similarly assume that player 2 can have two types <span class="math-container">$B_1$</span> or <span class="math-container">$B_2$</span>. Again, one normally assume that player 2 knows his own type but <span class="math-container">$A$</span> does not know the type of player <span class="math-container">$B$</span>.</li>
<li>A belief over the type combinations, which in our case gives probabilities that nature draws the particular types for <span class="math-container">$A$</span> and <span class="math-container">$B$</span>. In our example, this gives 4 numbers that add up to 1. For example <span class="math-container">$\pi(A_1, B_1) = 1/8, \pi(A_2, B_1) = 2/8, \pi(A_1, B_2) = 3/8, \pi(A_2, B_2)= 4/8$</span>. From these beliefs we can determine the posterior beliefs using Bayes theorem. Assume, for instance, that player <span class="math-container">$A$</span> knows she is of type <span class="math-container">$A_1$</span> then the probability that she beliefs player 2 to be of type <span class="math-container">$B_1$</span> is given by:
<span class="math-container">$$
\pi(B_1|A_1) = \frac{\pi(A_1, B_1)}{\pi(A_1, B_1) + \pi(A_1, B_2)} = \frac{1/8}{4/8} = \frac{1}{4}.
$$</span></li>
<li>A payoff function <span class="math-container">$u$</span> that specifies for every type and all possible combination of actions and combinations of types the payoff the particular type will receive. In our case, this gives 16 numbers. For example, the number <span class="math-container">$u_{A_1}(U,L, B_2)$</span> gives the payoff of player <span class="math-container">$A$</span> when she is of type <span class="math-container">$A_1$</span>, when she chooses the action <span class="math-container">$U$</span>,  when she faces a type <span class="math-container">$B_2$</span> adversary and when this adversary chooses <span class="math-container">$L$</span>.</li>
</ol>
<p>A strategy of a certain player specifies for each type an action to play. For example, one strategy for player <span class="math-container">$B$</span> could be to choose <span class="math-container">$L$</span> if he is of type <span class="math-container">$B_1$</span> and to choose <span class="math-container">$R$</span> when he is of type <span class="math-container">$B_2$</span>. Player <span class="math-container">$B$</span> has therefore 4 possible strategies <span class="math-container">$(L, L)$</span>, <span class="math-container">$(L,R)$</span>, <span class="math-container">$(R,L)$</span> and <span class="math-container">$(R,R)$</span> where each tuple specifies to do when he is of type <span class="math-container">$B_1$</span> and what to do if he is of type <span class="math-container">$B_2$</span>.</p>
<p>Player <span class="math-container">$A$</span> also has 4 strategies <span class="math-container">$(U,U)$</span>, <span class="math-container">$(U,D)$</span>, <span class="math-container">$(D,U)$</span>, <span class="math-container">$(D,D)$</span>. For example, the strategy <span class="math-container">$(U,D)$</span> says to choose <span class="math-container">$U$</span> if she is of type <span class="math-container">$A_1$</span> and to choose <span class="math-container">$D$</span> if she is of type <span class="math-container">$A_2$</span> also.</p>
<p>In total there are 16 possible strategy profiles (4 for <span class="math-container">$A$</span> times 4 for <span class="math-container">$B$</span>). We could summarize this profile by quadruples, for example <span class="math-container">$((L, R), (U, D))$</span></p>
<p>Each such strategy profile gives an expected payoff for each player type. For example, the expected payoff of <span class="math-container">$((L,R), (U,D))$</span> for player type <span class="math-container">$A_1$</span> is then given by:
<span class="math-container">$$
V_{A_1}((L,R),(U,D)) = \pi(B_1|A_1) u_{A_1}(U, L, B_1) + \pi(B_2|A_1) u_{A_1}(U,R,B_2).
$$</span>
The first term gives the probability of facing a type <span class="math-container">$B_1$</span> adversary times the payoff this generates (i.e. <span class="math-container">$A_1$</span> plays <span class="math-container">$U$</span> and <span class="math-container">$B_1$</span> plays <span class="math-container">$L$</span>). The second term gives the probability of facing a type <span class="math-container">$B_2$</span> adversary and receiving the payoff this generates (i.e. when <span class="math-container">$A_1$</span> plays <span class="math-container">$U$</span> and <span class="math-container">$B_2$</span> plays <span class="math-container">$R$</span>).</p>
<p>These payoffs, in term, determine the best responses (as the strategies of a particular player type that maximize expected payoff given the strategies of the other player), and this, in turn, gives the definition of a (Bayesian) Nash equilibrium as the strategy profiles which are mutual best responses.</p>

CC BY-SA 4.0
2023-08-02T12:48:06.317Z
2023-08-02T12:48:06.317Z
tdm
<p>I am having trouble understanding the definition of a Bayesian game based on the following definition from class. I would appreciate it if you could explain the notations and overall meaning for point 2, 3, 4, and 5. I am sure it is pretty simple once you know it, but they make no sense to me whatsoever. Thank you, and let me know if you need anything else.</p>
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CC BY-SA 4.0
2023-07-28T12:59:40.070Z
Was an example not provided after (or before) the definition? Usually examples are the easiest way to illustrate and understand a definition.
smcc
<utility><nash-equilibrium><bayesian-game><probability><dominant-strategy>
Understanding the notations in Bayesian game definition
coderDcoder
# Understanding the notations in Bayesian game definition

(Asked by: username_0 on 2023-07-27)

I am having trouble understanding the definition of a Bayesian game based on the following definition from class. I would appreciate it if you could explain the notations and overall meaning for point 2, 3, 4, and 5. I am sure it is pretty simple once you know it, but they make no sense to me whatsoever. Thank you, and let me know if you need anything else.

[![enter image description here](https://i.sstatic.net/5XrDt.png)](https://i.sstatic.net/5XrDt.png)

**Tags:** utility, nash-equilibrium, bayesian-game, probability, dominant-strategy

> **Comments:**
> **username_1**: Was an example not provided after (or before) the definition? Usually examples are the easiest way to illustrate and understand a definition.  

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## 1 answers

### Answer by username_2 on 2023-08-02. Score: 2 | Accepted answer

1. Set of players, I guess this is quite clear. As an example, take player $A$ and $B$
2. Actions. This just tells you what the two players can do. For example $A$ can choose $U$(p) or $D$(own) and $B$ can choose $L$(eft) or $R$(ight).
3. Types: A type is attached to a player. For example, player $A$ can be of two different types, say $A\_1$ or $A\_2$. The type that player $A$ has is usually known by the player herself, but usually not known by the other players (why we need a probability distribution). Similarly assume that player 2 can have two types $B\_1$ or $B\_2$. Again, one normally assume that player 2 knows his own type but $A$ does not know the type of player $B$.
4. A belief over the type combinations, which in our case gives probabilities that nature draws the particular types for $A$ and $B$. In our example, this gives 4 numbers that add up to 1. For example $\pi(A\_1, B\_1) = 1/8, \pi(A\_2, B\_1) = 2/8, \pi(A\_1, B\_2) = 3/8, \pi(A\_2, B\_2)= 4/8$. From these beliefs we can determine the posterior beliefs using Bayes theorem. Assume, for instance, that player $A$ knows she is of type $A\_1$ then the probability that she beliefs player 2 to be of type $B\_1$ is given by:
   $$
   \pi(B\_1|A\_1) = \frac{\pi(A\_1, B\_1)}{\pi(A\_1, B\_1) + \pi(A\_1, B\_2)} = \frac{1/8}{4/8} = \frac{1}{4}.
   $$
5. A payoff function $u$ that specifies for every type and all possible combination of actions and combinations of types the payoff the particular type will receive. In our case, this gives 16 numbers. For example, the number $u\_{A\_1}(U,L, B\_2)$ gives the payoff of player $A$ when she is of type $A\_1$, when she chooses the action $U$, when she faces a type $B\_2$ adversary and when this adversary chooses $L$.

A strategy of a certain player specifies for each type an action to play. For example, one strategy for player $B$ could be to choose $L$ if he is of type $B\_1$ and to choose $R$ when he is of type $B\_2$. Player $B$ has therefore 4 possible strategies $(L, L)$, $(L,R)$, $(R,L)$ and $(R,R)$ where each tuple specifies to do when he is of type $B\_1$ and what to do if he is of type $B\_2$.

Player $A$ also has 4 strategies $(U,U)$, $(U,D)$, $(D,U)$, $(D,D)$. For example, the strategy $(U,D)$ says to choose $U$ if she is of type $A\_1$ and to choose $D$ if she is of type $A\_2$ also.

In total there are 16 possible strategy profiles (4 for $A$ times 4 for $B$). We could summarize this profile by quadruples, for example $((L, R), (U, D))$

Each such strategy profile gives an expected payoff for each player type. For example, the expected payoff of $((L,R), (U,D))$ for player type $A\_1$ is then given by:
$$
V\_{A\_1}((L,R),(U,D)) = \pi(B\_1|A\_1) u\_{A\_1}(U, L, B\_1) + \pi(B\_2|A\_1) u\_{A\_1}(U,R,B\_2).
$$
The first term gives the probability of facing a type $B\_1$ adversary times the payoff this generates (i.e. $A\_1$ plays $U$ and $B\_1$ plays $L$). The second term gives the probability of facing a type $B\_2$ adversary and receiving the payoff this generates (i.e. when $A\_1$ plays $U$ and $B\_2$ plays $R$).

These payoffs, in term, determine the best responses (as the strategies of a particular player type that maximize expected payoff given the strategies of the other player), and this, in turn, gives the definition of a (Bayesian) Nash equilibrium as the strategy profiles which are mutual best responses.


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