<p>In this analysis, we examine a public good contribution game involving three players. Each player can choose to either contribute or not contribute to fund a streetlamp in their town. The value of having the streetlamp is 3 for each player, while the value of not having it is 0. The game is structured such that if at least two players contribute, the lamp will be erected; otherwise, it will not be erected, and any contributing player will not get their money back.</p>
<p>Let <span class="math-container">$N = \{1, 2, 3\}$</span> denote the set of players, and let <span class="math-container">$\Sigma_i = \{0, 1\}$</span> represent each player <span class="math-container">$i$</span>'s strategy set, where 0 signifies not contributing and 1 signifies contributing. The payoff function <span class="math-container">$v_i(\sigma_1, \sigma_2, \sigma_3)$</span> determines the payoff of each player based on the chosen strategies of all players. It is defined as follows:
<br />
<span class="math-container">$v_i(\sigma_1, \sigma_2, \sigma_3)$</span>
<span class="math-container">$=$</span>
<span class="math-container">\begin{cases} 
0 &amp; \text{if } \sigma_i = 0 \text{ and } \sigma_j = 0 \text{ for some } j \neq i \\
3 &amp; \text{if } \sigma_i = 0 \text{ and } \sigma_j = 1 \text{ for both } j \neq i \\
-1 &amp; \text{if } \sigma_i = 1 \text{ and } \sigma_j = 0 \text{ for both } j \neq i \\
2 &amp; \text{if } \sigma_i = 1 \text{ and } \sigma_j = 1 \text{ for some } j \neq i 
\end{cases}</span></p>
<h2>Analysis</h2>
<h3>Part (a)</h3>
<p>We begin by examining each player's best-response correspondence:</p>
<ul>
<li>Player 1:
<ul>
<li>if <span class="math-container">$\sigma_2 = \sigma_3 = 1$</span>, Player 1 prefers not to contribute (<span class="math-container">$\sigma_1 = 0$</span>) to free-ride.</li>
<li>if <span class="math-container">$\sigma_2 = 1$</span> or <span class="math-container">$\sigma_3 = 1$</span>, Player 1 prefers to contribute (<span class="math-container">$\sigma_1 = 1$</span>) to receive the benefit.</li>
<li>if <span class="math-container">$\sigma_2 = \sigma_3 = 0$</span>, Player 1 prefers not to contribute (<span class="math-container">$\sigma_1 = 0$</span>) since the lamp won't be erected.</li>
</ul>
</li>
<li>Player 2 &amp; Player 3:
<ul>
<li>Symmetric reasoning applies to Player 2 and Player 3.</li>
</ul>
</li>
</ul>
<h3>Part (b)</h3>
<p>The best-response correspondence implies two kinds of Nash equilibria:</p>
<ol>
<li>No player contributes (0, 0, 0). In this case, no player can improve their payoff by deviating unilaterally.</li>
<li>Two players contribute while the third does not (e.g., 1, 1, 0). In this case, neither of the contributing players has an incentive to deviate.</li>
</ol>
<p>Credits:
<a href="https://press.princeton.edu/books/hardcover/9780691129082/game-theory" rel="nofollow noreferrer">https://press.princeton.edu/books/hardcover/9780691129082/game-theory</a></p>

CC BY-SA 4.0
2024-04-24T19:26:18.830Z
2024-04-24T19:26:18.830Z
GuyEternal
<p>Public Good Contribution: Three players live in a town, and each can choose
to contribute to fund a streetlamp. The value of having the streetlamp is 3 for
each player, and the value of not having it is 0. The mayor asks each player to
contribute either 1 or nothing. If at least two players contribute then the lamp
will be erected. If one player or no players contribute then the lamp will not
be erected, in which case any person who contributed will not get his money
back.</p>

<p>a. Write out or graph each player’s best-response correspondence.</p>

<p>b. What outcomes can be supported as pure-strategy Nash equilibria?</p>

<p>Consider player <span class="math-container">$i$</span> with beliefs about the choices of players <span class="math-container">$j$</span> and <span class="math-container">$k$</span>. If neither <span class="math-container">$j$</span> nor <span class="math-container">$k$</span> contribute then player <span class="math-container">$i$</span> does not want to contribute because the lamp would not be erected and he would lose his contribution. Similarly, if both <span class="math-container">$j$</span> and <span class="math-container">$k$</span> contribute then player <span class="math-container">$i$</span> does not want to contribute because the lamp would be erected without his contribution so he can free ride on their contributions. The remaining cases is where only one of the players <span class="math-container">$j$</span> and <span class="math-container">$k$</span> contribute, in which case by contributing <span class="math-container">$1$</span> player <span class="math-container">$i$</span> receives <span class="math-container">$2$</span>, while by not contributing he receives <span class="math-container">$0$</span>, and hence contributing is a best response. In summary,</p>

<p><span class="math-container">$BR_i(s_j,s_k)=\begin{cases} 0 , \text{ if $s_i=s_j$}\\1 ,\text{ if $s_k=1$ or $s_j=1$}\end{cases}$</span></p>

<p>Would this be correct for (a)?</p>

<p>for (b) :</p>

<p>The best response correspondence described in (a) above implies that there are two kinds of Nash equilibria: one kind (which is unique) is where no player contributes, and the other kind has two of the three players contributing and the third free riding. Hence, either the lamp being erected with two players contributing or the lamp not being erected with no player contributing can be supported as Nash equilibria.</p>

CC BY-SA 4.0
2020-05-08T14:16:01.707Z
The upper line of the rhs of your best response function contains a typo, it should be $s_k=s_j$. The lower line should be an exclusive "or". You could write the rhs in a single line as $|s_k-s_j|$. However, I think they are actually asking for $i$'s best response correspondence on $[0,1]^2$, not just $\{0,1\}^2$, i.e. including mixed strategies.
VARulle
CC BY-SA 4.0
2020-05-08T14:17:56.557Z
... and pls correct the first word of the title! ;)
VARulle
<game-theory>
public good contribution game theory
Slim Shady
# public good contribution game theory

(Asked by: username_0 on 2020-05-08)

Public Good Contribution: Three players live in a town, and each can choose
to contribute to fund a streetlamp. The value of having the streetlamp is 3 for
each player, and the value of not having it is 0. The mayor asks each player to
contribute either 1 or nothing. If at least two players contribute then the lamp
will be erected. If one player or no players contribute then the lamp will not
be erected, in which case any person who contributed will not get his money
back.

a. Write out or graph each player’s best-response correspondence.

b. What outcomes can be supported as pure-strategy Nash equilibria?

Consider player $i$ with beliefs about the choices of players $j$ and $k$. If neither $j$ nor $k$ contribute then player $i$ does not want to contribute because the lamp would not be erected and he would lose his contribution. Similarly, if both $j$ and $k$ contribute then player $i$ does not want to contribute because the lamp would be erected without his contribution so he can free ride on their contributions. The remaining cases is where only one of the players $j$ and $k$ contribute, in which case by contributing $1$ player $i$ receives $2$, while by not contributing he receives $0$, and hence contributing is a best response. In summary,

$BR\_i(s\_j,s\_k)=\begin{cases} 0 , \text{ if $s\_i=s\_j$}\\1 ,\text{ if $s\_k=1$ or $s\_j=1$}\end{cases}$

Would this be correct for (a)?

for (b) :

The best response correspondence described in (a) above implies that there are two kinds of Nash equilibria: one kind (which is unique) is where no player contributes, and the other kind has two of the three players contributing and the third free riding. Hence, either the lamp being erected with two players contributing or the lamp not being erected with no player contributing can be supported as Nash equilibria.

**Tags:** game-theory
> **Comments:**
> **username_1**: The upper line of the rhs of your best response function contains a typo, it should be $s_k=s_j$. The lower line should be an exclusive "or". You could write the rhs in a single line as $|s_k-s_j|$. However, I think they are actually asking for $i$'s best response correspondence on $[0,1]^2$, not just $\{0,1\}^2$, i.e. including mixed strategies.  
> **username_1**: ... and pls correct the first word of the title! ;)  

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## 1 answers

### Answer by username_2 on 2024-04-24. Score: 1

In this analysis, we examine a public good contribution game involving three players. Each player can choose to either contribute or not contribute to fund a streetlamp in their town. The value of having the streetlamp is 3 for each player, while the value of not having it is 0. The game is structured such that if at least two players contribute, the lamp will be erected; otherwise, it will not be erected, and any contributing player will not get their money back.

Let $N = \{1, 2, 3\}$ denote the set of players, and let $\Sigma\_i = \{0, 1\}$ represent each player $i$'s strategy set, where 0 signifies not contributing and 1 signifies contributing. The payoff function $v\_i(\sigma\_1, \sigma\_2, \sigma\_3)$ determines the payoff of each player based on the chosen strategies of all players. It is defined as follows:
  
$v\_i(\sigma\_1, \sigma\_2, \sigma\_3)$
$=$
\begin{cases}
0 & \text{if } \sigma\_i = 0 \text{ and } \sigma\_j = 0 \text{ for some } j \neq i \\
3 & \text{if } \sigma\_i = 0 \text{ and } \sigma\_j = 1 \text{ for both } j \neq i \\
-1 & \text{if } \sigma\_i = 1 \text{ and } \sigma\_j = 0 \text{ for both } j \neq i \\
2 & \text{if } \sigma\_i = 1 \text{ and } \sigma\_j = 1 \text{ for some } j \neq i
\end{cases}

## Analysis

### Part (a)

We begin by examining each player's best-response correspondence:

* Player 1:
  + if $\sigma\_2 = \sigma\_3 = 1$, Player 1 prefers not to contribute ($\sigma\_1 = 0$) to free-ride.
  + if $\sigma\_2 = 1$ or $\sigma\_3 = 1$, Player 1 prefers to contribute ($\sigma\_1 = 1$) to receive the benefit.
  + if $\sigma\_2 = \sigma\_3 = 0$, Player 1 prefers not to contribute ($\sigma\_1 = 0$) since the lamp won't be erected.
* Player 2 & Player 3:
  + Symmetric reasoning applies to Player 2 and Player 3.

### Part (b)

The best-response correspondence implies two kinds of Nash equilibria:

1. No player contributes (0, 0, 0). In this case, no player can improve their payoff by deviating unilaterally.
2. Two players contribute while the third does not (e.g., 1, 1, 0). In this case, neither of the contributing players has an incentive to deviate.

Credits:
<https://press.princeton.edu/books/hardcover/9780691129082/game-theory>


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