<p>The interpretation of <span class="math-container">$\sigma_j(s_j)$</span> is that it's the probability with which player <span class="math-container">$j$</span> uses his pure strategy <span class="math-container">$s_j$</span>. In your image example, that would be the probability of the row player choosing a particular row or the column player choosing a particular column. Given that each player decides independently, the product <span class="math-container">$\Pi_{j=1}^I\sigma_j(s_j)$</span> represents the joint probability that outcome <span class="math-container">$s$</span> occurs when each player <span class="math-container">$j$</span> uses mixed strategy <span class="math-container">$\sigma_j$</span>.</p>
<p>In contrast, <span class="math-container">$\sigma_j(s)$</span> would signify the probability that player <span class="math-container">$j$</span> assigns to the pure <em>strategy profile</em> <span class="math-container">$s$</span>. In other words, <span class="math-container">$\sigma_j(s)$</span> is the probability of a particular <em>cell</em> in the matrix game. But <span class="math-container">$j$</span> cannot &quot;choose&quot; a particular outcome of the game, as the players are assumed to be non-cooperative in this context. It also doesn't make sense to multiply the probability that each player assigns to an outcome <span class="math-container">$s$</span>, i.e. <span class="math-container">$\Pi_{j=1}^I\sigma_j(s)$</span>.</p>
<p>In the textbook, the last paragraph page 5 has a detailed example of how the payoff function is applied to the game in Figure 1.1. That could help with understanding.</p>

CC BY-SA 4.0
2024-06-13T01:49:45.407Z
2024-06-13T01:49:45.407Z
Herr K.
<p>In the beginning chapter of the book <em>Game Theory</em> by Fudenberg a game is shown as an example (<a href="https://i.sstatic.net/uzJlf.png" rel="nofollow noreferrer">https://i.sstatic.net/uzJlf.png</a>) and a function is suggested to calculate the payoff. Player <span class="math-container">$i$</span>'s payoff to profile <span class="math-container">$\sigma$</span> is given by
<span class="math-container">$$
\sum_{s\in S}\left(\Pi^I_{j=1}\sigma_j(s_j)\right)u_i(s)
$$</span></p>
<p>What keeps bugging me is that I think, strictly speaking, <span class="math-container">$s$</span> should not have the subscript <span class="math-container">$j$</span> so that
<span class="math-container">$$
\sum_{s\in S}\left(\Pi^I_{j=1}\sigma_j(s)\right)u_i(s)
$$</span></p>
<p>If you could point out and correct where I'm getting confused, it would be really appreciated.</p>

CC BY-SA 4.0
2024-06-12T23:05:58.607Z
My gues is that (1) is more restrictive than (2). In (1) $s$ is a vector of length $I$ but $\sigma_j$ only depends on $s_j$ the $j$'th entry of $s$. In (2) $\sigma_j$ could depend on all entries of $s=(s_1,...,s_I)$.  But I have not read the book so I'm only basing this on the question. And of course this does not answer whether such a restriction is valid or not in the given context.
Jesper Hybel
<game-theory>
Please help understanding the equation for player $i$ 's payoff to mixed strategy profile $\sigma$
Howard Dwight
# Please help understanding the equation for player $i$ 's payoff to mixed strategy profile $\sigma$

(Asked by: username_0 on 2024-06-12)

In the beginning chapter of the book *Game Theory* by Fudenberg a game is shown as an example (<https://i.sstatic.net/uzJlf.png>) and a function is suggested to calculate the payoff. Player $i$'s payoff to profile $\sigma$ is given by
$$
\sum\_{s\in S}\left(\Pi^I\_{j=1}\sigma\_j(s\_j)\right)u\_i(s)
$$

What keeps bugging me is that I think, strictly speaking, $s$ should not have the subscript $j$ so that
$$
\sum\_{s\in S}\left(\Pi^I\_{j=1}\sigma\_j(s)\right)u\_i(s)
$$

If you could point out and correct where I'm getting confused, it would be really appreciated.

**Tags:** game-theory
> **Comments:**
> **username_1**: My gues is that (1) is more restrictive than (2). In (1) $s$ is a vector of length $I$ but $\sigma_j$ only depends on $s_j$ the $j$'th entry of $s$. In (2) $\sigma_j$ could depend on all entries of $s=(s_1,...,s_I)$.  But I have not read the book so I'm only basing this on the question. And of course this does not answer whether such a restriction is valid or not in the given context.  

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## 1 answers

### Answer by username_2 on 2024-06-13. Score: 4

The interpretation of $\sigma\_j(s\_j)$ is that it's the probability with which player $j$ uses his pure strategy $s\_j$. In your image example, that would be the probability of the row player choosing a particular row or the column player choosing a particular column. Given that each player decides independently, the product $\Pi\_{j=1}^I\sigma\_j(s\_j)$ represents the joint probability that outcome $s$ occurs when each player $j$ uses mixed strategy $\sigma\_j$.

In contrast, $\sigma\_j(s)$ would signify the probability that player $j$ assigns to the pure *strategy profile* $s$. In other words, $\sigma\_j(s)$ is the probability of a particular *cell* in the matrix game. But $j$ cannot "choose" a particular outcome of the game, as the players are assumed to be non-cooperative in this context. It also doesn't make sense to multiply the probability that each player assigns to an outcome $s$, i.e. $\Pi\_{j=1}^I\sigma\_j(s)$.

In the textbook, the last paragraph page 5 has a detailed example of how the payoff function is applied to the game in Figure 1.1. That could help with understanding.


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