<p>Let $b_i\in\{0,B\}$ be $i$'s strategy. Then $i$'s payoff depends on the strategy profile $(b_i,b_{-i})$, where $b_{-i}=(b_j)_{j\ne i}$. </p>

<p>\begin{equation}
u_i(b_i,b_{-i})=
\begin{cases}
v_i&amp;\text{if $b_i=0$ and $b_j=B$ for some $j\ne i$}\\
0&amp; \text{if $b_i=0$ and $b_j=0$ for all $j\ne i$}\\
v_i-B&amp;\text{if $b_i=B$}
\end{cases}
\end{equation}</p>

<p>Thus, $b_i=B$ is a best response if 
\begin{equation}
u_i(B,b_{-i})\ge u_i(0,b_{-i})
\quad\Leftrightarrow\quad 
v_i-B\ge (1-\Pr(b_j=0,\;\forall j\ne i))v_i.\tag{1}
\end{equation}</p>

<p>Since the game is <em>ex ante</em> symmetric, we could further assume that every $i$ adopts a threshold strategy, i.e. 
\begin{equation}
b_i=\begin{cases}
0&amp;\text{if $v_i\le\overline v$}\\
B&amp;\text{if $v_i&gt;\overline v$}
\end{cases}\tag{2}
\end{equation}
where $\overline v$ is some common threshold value. Then, the probability in $(1)$ can be written as 
\begin{equation}
\Pr(b_j=0,\;\forall j\ne i)=\Pr(v_j\le \overline v,\;\forall j\ne i)=(\overline v)^{n-1},\tag{3}
\end{equation}
where the last equality is obtained from the assumption that $v_j$'s are i.i.d. and $v_j\sim U[0,1]$. </p>

<p>Consolidating $(1)$ to $(3)$, we can solve for the cutoff value $\overline v=B^{1/n}$. </p>

CC BY-SA 3.0
2018-02-27T22:25:05.777Z
Herr, thanks for the response. When you say ex-ante symmetric, how do you mean? And also, in general, when we use the notion of "symmetry" in solution concepts or games, like "symmetric game" or "symmetric Nash eq'm", what do they exactly mean?
Frank Swanton
CC BY-SA 3.0
2018-02-27T23:06:40.017Z
*Ex ante* symmetry is in the sense that every villager, before observing their private value, faces the same problem (same information, same prior, same strategy space, same payoff function, etc.). So $i$'s problem is the same as $j$'s problem, except the change of the name index. Therefore it's reasonable to assume that they will adopt the same strategy *ex ante* (as defined in $(2)$). If a profile of such strategies also happens to be mutual best responses, then we have a symmetric (Bayesian) Nash equilibrium. Read more [here](https://en.wikipedia.org/wiki/Symmetric_game).
Herr K.
CC BY-SA 3.0
2018-02-28T16:54:55.983Z
How did you get the IFF in (1), RHS probability?
Frank Swanton
CC BY-SA 3.0
2018-02-28T17:01:31.163Z
in your payoff, the third row seems odd to me. You would only submit to pay $B$ if $v_i>C$ and everyone else pays $0$.
Frank Swanton
CC BY-SA 3.0
2018-02-28T17:02:02.817Z
And you will submit $B$ if your valuation $v_i\leq c$ and everyone else pays 0.
Frank Swanton
CC BY-SA 3.0
2018-02-28T17:03:03.233Z
@FrankSwanton: The RHS in (1) is just the expected utility in the first two cases of $u_i(0,b_{-i})$. Or did you mean how I got (3)?
Herr K.
CC BY-SA 3.0
2018-02-28T17:07:22.547Z
@FrankSwanton: The utility $u_i(\cdot)$ describes the payoff to $i$ in every possible outcome, some of which is optimal. In writing down the payoff function, you should avoid bringing in strategic considerations, since these will likely cause confusion. After the payoff function is written, you can start analyzing the game and see which moves are strategically best for the player.
Herr K.
CC BY-SA 3.0
2018-02-28T17:11:41.803Z
Ok. Are you saying in my payoff function I shouldn't have considerations for where $v_i$ stands in relation to $B$? This is private information available to $i$, and this should taken into consideration in building up payoff function. The only difference I have with your payoff function is the last two part: I said you submit 0 if others pay 0 and your valuation is $\leq B$, and you submit $v_i-B$ if others pay $0$ and your valuation is higher than $B$.
Frank Swanton
CC BY-SA 3.0
2018-02-28T17:20:47.167Z
@FrankSwanton: The condition "if others pay 0 and your valuation is $≤B$" is not feasible: this is a simultaneous move game, and you don't get to observe what others have chosen. A payoff function *describes* the payoffs associated with *each possible outcome*, but what you're doing is to *prescribe* what *should be done* be a player if she were to behave in an optimal way, potentially ignoring outcomes that are suboptimal. This is precisely the confusion I was alluding to.
Herr K.
CC BY-SA 3.0
2018-02-28T23:56:59.487Z
Okay, very nice. Yes, static, incomplete information. Drawing a two-person, 2 by 2 matrix actually made sense along with your explanation. So your bottom payoff is essentially saying regardless of what others do, when I take the action of submitting the cost of the bridge, my payoff will be $v_i-B$ regardless.
Frank Swanton
CC BY-SA 3.0
2018-03-01T00:07:13.510Z
@FrankSwanton: Precisely. Glad that things are clarified.
Herr K.
CC BY-SA 3.0
2018-03-01T00:11:35.920Z
Yes, one thing what I realized is though : when you consider a best response of $i$ against the rest submitting $0$, $i$ gets $0$ if $b_i=0$. $i$ gets $b_i-B$ if she decides to pay for the bridge. So wouldn't the best response of $i$ depend on whether $v_i-B/geq 0$? In other words, your payoff function should reflect this conditional, no? I assume this condition statement, $vi_-B/geq 0$ gets factored into getting the CDF calculation?
Frank Swanton
CC BY-SA 3.0
2018-03-01T00:19:08.117Z
@FrankSwanton: Note that in (1) I allow the other players' strategies to be flexible, i.e. $(b_{-i})$ can be anything.
Herr K.
CC BY-SA 3.0
2018-03-01T00:36:22.810Z
This is great, Herr. Thanks. However, from ex ante symmetry, how did you go to the threshold rule? Did you use some proposition from Bayesian games? I remember something about this threshold value rule for continuous type space and discrete sets of actions in Bayesian games, but not exactly what that was
Frank Swanton
CC BY-SA 3.0
2018-03-01T00:38:30.660Z
Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/73862/discussion-between-herr-k-and-frank-swanton).
Herr K.
CC BY-SA 3.0
2018-03-01T19:46:42.217Z
Herr, can we chat?
Frank Swanton
CC BY-SA 3.0
2018-02-27T20:30:36.033Z
2018-02-27T20:35:42.417Z
2018-02-27T20:35:42.417Z
Herr K.
<p>Suppose a town is building a bridge, and it costs $B$.
There are $n$ villagers.</p>

<p>Each village's valuation of the bridge is private information, $v_i$.</p>

<p>It is common knowledge that this valuation is drawn from a uniform distribution $[0,1]$. $B\in[0,1]$.</p>

<p>Villager can only submit $0$ or $B$.</p>

<p>If one villager submits $B$, then the bridge is built and every other villager pays their submission.</p>

<p>If no bridge is built, everyone gets $0$.</p>

<p>How do I construct an expected payoff a village $i$?</p>

<p>What I got down to is having 2 scenarios: $v_i&gt;B$ and $v_i\leq B$.</p>

<p>But in each case, I have two possible payoffs. For the former case,</p>

<blockquote>
  <p>if every other player submits $0$, $i$ should submit B, because $v_i-B&gt;0$.
  <br/>if someone pays $B$, $i$ should submit 0, because she gets $v_i$.</p>
</blockquote>

<p>You get a similar type of for the other scenario.</p>

<p>But how would I incorporate this into expected payoff of $i$ and how should I go about constructing a social welfare function?</p>

<p>I feel like this is just a variation of all-payout auction with discrete action space for each $i$.</p>

CC BY-SA 3.0
2018-02-27T17:52:10.970Z
Is $v_i\le c$ a typo for $v_i\le B$? Also, is the value of $B$ common knowledge?
Herr K.
CC BY-SA 3.0
2018-02-27T17:52:50.420Z
Writing down expected payoff is one thing, constructing a social welfare function is an entirely different matter...
Herr K.
CC BY-SA 3.0
2018-02-27T18:08:53.903Z
Yes, I am asking both lol. The cost of bridge is common knowledge.
Frank Swanton
CC BY-SA 3.0
2018-02-27T20:34:38.237Z
See my answer for the expected payoff, and a symmetric BNE. A lot more assumptions are needed for the construction of a social welfare function (see [Wikipedia](https://en.wikipedia.org/wiki/Social_welfare_function) for a detailed discussion).
Herr K.
<game-theory><public-goods>
Free rider problem in game theory
Frank Swanton
# Free rider problem in game theory

(Asked by: username_0 on 2018-02-27)

Suppose a town is building a bridge, and it costs $B$.
There are $n$ villagers.

Each village's valuation of the bridge is private information, $v\_i$.

It is common knowledge that this valuation is drawn from a uniform distribution $[0,1]$. $B\in[0,1]$.

Villager can only submit $0$ or $B$.

If one villager submits $B$, then the bridge is built and every other villager pays their submission.

If no bridge is built, everyone gets $0$.

How do I construct an expected payoff a village $i$?

What I got down to is having 2 scenarios: $v\_i>B$ and $v\_i\leq B$.

But in each case, I have two possible payoffs. For the former case,

> if every other player submits $0$, $i$ should submit B, because $v\_i-B>0$.
>   
> if someone pays $B$, $i$ should submit 0, because she gets $v\_i$.

You get a similar type of for the other scenario.

But how would I incorporate this into expected payoff of $i$ and how should I go about constructing a social welfare function?

I feel like this is just a variation of all-payout auction with discrete action space for each $i$.

**Tags:** game-theory, public-goods
> **Comments:**
> **username_1**: Is $v_i\le c$ a typo for $v_i\le B$? Also, is the value of $B$ common knowledge?  
> **username_1**: Writing down expected payoff is one thing, constructing a social welfare function is an entirely different matter...  
> **username_0**: Yes, I am asking both lol. The cost of bridge is common knowledge.  
> **username_1**: See my answer for the expected payoff, and a symmetric BNE. A lot more assumptions are needed for the construction of a social welfare function (see [Wikipedia](https://en.wikipedia.org/wiki/Social_welfare_function) for a detailed discussion).  

---

## 1 answers

### Answer by username_1 on 2018-02-27. Score: 2 | Accepted answer

Let $b\_i\in\{0,B\}$ be $i$'s strategy. Then $i$'s payoff depends on the strategy profile $(b\_i,b\_{-i})$, where $b\_{-i}=(b\_j)\_{j\ne i}$.

\begin{equation}
u\_i(b\_i,b\_{-i})=
\begin{cases}
v\_i&\text{if $b\_i=0$ and $b\_j=B$ for some $j\ne i$}\\
0& \text{if $b\_i=0$ and $b\_j=0$ for all $j\ne i$}\\
v\_i-B&\text{if $b\_i=B$}
\end{cases}
\end{equation}

Thus, $b\_i=B$ is a best response if
\begin{equation}
u\_i(B,b\_{-i})\ge u\_i(0,b\_{-i})
\quad\Leftrightarrow\quad
v\_i-B\ge (1-\Pr(b\_j=0,\;\forall j\ne i))v\_i.\tag{1}
\end{equation}

Since the game is *ex ante* symmetric, we could further assume that every $i$ adopts a threshold strategy, i.e.
\begin{equation}
b\_i=\begin{cases}
0&\text{if $v\_i\le\overline v$}\\
B&\text{if $v\_i>\overline v$}
\end{cases}\tag{2}
\end{equation}
where $\overline v$ is some common threshold value. Then, the probability in $(1)$ can be written as
\begin{equation}
\Pr(b\_j=0,\;\forall j\ne i)=\Pr(v\_j\le \overline v,\;\forall j\ne i)=(\overline v)^{n-1},\tag{3}
\end{equation}
where the last equality is obtained from the assumption that $v\_j$'s are i.i.d. and $v\_j\sim U[0,1]$.

Consolidating $(1)$ to $(3)$, we can solve for the cutoff value $\overline v=B^{1/n}$.

> **Comments:**
> **username_0**: Herr, thanks for the response. When you say ex-ante symmetric, how do you mean? And also, in general, when we use the notion of "symmetry" in solution concepts or games, like "symmetric game" or "symmetric Nash eq'm", what do they exactly mean?  
> **username_1**: *Ex ante* symmetry is in the sense that every villager, before observing their private value, faces the same problem (same information, same prior, same strategy space, same payoff function, etc.). So $i$'s problem is the same as $j$'s problem, except the change of the name index. Therefore it's reasonable to assume that they will adopt the same strategy *ex ante* (as defined in $(2)$). If a profile of such strategies also happens to be mutual best responses, then we have a symmetric (Bayesian) Nash equilibrium. Read more [here](https://en.wikipedia.org/wiki/Symmetric_game).  
> **username_0**: How did you get the IFF in (1), RHS probability?  
> **username_0**: in your payoff, the third row seems odd to me. You would only submit to pay $B$ if $v_i>C$ and everyone else pays $0$.  
> **username_0**: And you will submit $B$ if your valuation $v_i\leq c$ and everyone else pays 0.  
> **username_1**: @FrankSwanton: The RHS in (1) is just the expected utility in the first two cases of $u_i(0,b_{-i})$. Or did you mean how I got (3)?  
> **username_1**: @FrankSwanton: The utility $u_i(\cdot)$ describes the payoff to $i$ in every possible outcome, some of which is optimal. In writing down the payoff function, you should avoid bringing in strategic considerations, since these will likely cause confusion. After the payoff function is written, you can start analyzing the game and see which moves are strategically best for the player.  
> **username_0**: Ok. Are you saying in my payoff function I shouldn't have considerations for where $v_i$ stands in relation to $B$? This is private information available to $i$, and this should taken into consideration in building up payoff function. The only difference I have with your payoff function is the last two part: I said you submit 0 if others pay 0 and your valuation is $\leq B$, and you submit $v_i-B$ if others pay $0$ and your valuation is higher than $B$.  
> **username_1**: @FrankSwanton: The condition "if others pay 0 and your valuation is $≤B$" is not feasible: this is a simultaneous move game, and you don't get to observe what others have chosen. A payoff function *describes* the payoffs associated with *each possible outcome*, but what you're doing is to *prescribe* what *should be done* be a player if she were to behave in an optimal way, potentially ignoring outcomes that are suboptimal. This is precisely the confusion I was alluding to.  
> **username_0**: Okay, very nice. Yes, static, incomplete information. Drawing a two-person, 2 by 2 matrix actually made sense along with your explanation. So your bottom payoff is essentially saying regardless of what others do, when I take the action of submitting the cost of the bridge, my payoff will be $v_i-B$ regardless.  
> **username_1**: @FrankSwanton: Precisely. Glad that things are clarified.  
> **username_0**: Yes, one thing what I realized is though : when you consider a best response of $i$ against the rest submitting $0$, $i$ gets $0$ if $b_i=0$. $i$ gets $b_i-B$ if she decides to pay for the bridge. So wouldn't the best response of $i$ depend on whether $v_i-B/geq 0$? In other words, your payoff function should reflect this conditional, no? I assume this condition statement, $vi_-B/geq 0$ gets factored into getting the CDF calculation?  
> **username_1**: @FrankSwanton: Note that in (1) I allow the other players' strategies to be flexible, i.e. $(b_{-i})$ can be anything.  
> **username_0**: This is great, Herr. Thanks. However, from ex ante symmetry, how did you go to the threshold rule? Did you use some proposition from Bayesian games? I remember something about this threshold value rule for continuous type space and discrete sets of actions in Bayesian games, but not exactly what that was  
> **username_1**: Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/73862/discussion-between-herr-k-and-frank-swanton).  
> **username_0**: Herr, can we chat?  

---

