<p>These are the errors in the original diagram:</p>
<ol>
<li>Your game tree does not show the opportunistic-type tax payer's income-reporting choices. They choose between reporting an income of 0 or 1 (you need two extra tax-payer decision nodes and four extra branches at the bottom of the tree). You appear to have assumed that high-income opportunistic tax payers always report low income. It may be they want to report truthfully. If the probability of audit is sufficiently high, then they will report truthfully. Otherwise they will under-report.</li>
<li>Also, you appear to have assumed low-income opportunistic tax payers report low income (optimally they will, but the decision should be shown in the tree) and are always taxed <span class="math-container">$t$</span> (the same amount as for high income) and fined if audited. There should be no fine for under-reporting and there should be no tax for a low-income opportunistic tax payer (who reports truthfully).</li>
<li>You should have dashed lines between all tax-collector decision nodes where the same income is reported (rather than between all nodes where the tax payer has the same income).</li>
</ol>
<p>The top part of the tree (for the honest tax payer) appears to be correct.</p>
<p>Below is the corrected tree. Note I have not drawn the decision of the opportunistic taxpayer with low income as they optimally choose not to report regardless of the tax collector's choices. Note also that for nature you could have just one decision node with four branches.</p>
<p><a href="https://i.sstatic.net/g8lVn.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/g8lVn.jpg" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-10-18T11:56:40.950Z
Dear @smcc thank you for your help. I see how to draw the tree. I tried to solve it, but I could not do. I am preparing an exam, can you please help me to solve for the part b and c. I will be really glad. Many thanks:)
studentp
CC BY-SA 4.0
2022-10-18T16:19:33.213Z
It needs to be done from the top to the bottom, otherwise it has no sense. Where are the stages of the game, here ?
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-18T16:37:38.700Z
In the tree, two natures are used. What you mean? That’s I need to start with assign beliefs in the nodes $(\mu_1, \mu_2, 1-\mu_1-\mu_2)$ and $(\beta, 1-\beta)$ and then I will calculate the ecpected payoffs of Tax collector(TC) @MatteoBulgarelli
studentp
CC BY-SA 4.0
2022-10-18T18:33:16.583Z
You can remove the branch where the poor are dishonest.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-18T19:12:14.833Z
Additionally, when the collector discovers the dishonest, she earns $t + f - c$, not just $f -c$.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-18T19:23:58.977Z
Yes, it should be $t+f-c$. 

In response to the previous comments: 

Yes, the game tree could have been labelled better (the first node is the move of nature "N" between the words "Honest " and "Opportunistic"). I could also have consolidated the two moves of nature into one.

The branch where the low income opportunistic tax payer is dishonest has not been drawn (i.e. has already been removed as suggested).
smcc
CC BY-SA 4.0
2022-10-19T18:04:49.300Z
I solved this part -b according to the answers and comments. What do you think? @smcc
studentp
CC BY-SA 4.0
2022-08-26T10:16:40.147Z
2022-08-31T14:55:48.440Z
smcc
<p>a) Without loss of generality, assume that every poor is honest. Indeed, form the perspective of the collector, the expected utility of Auditing given that someone is poor and honest or poor and dishonest is the same.</p>
<p>The tree takes the following form:</p>
<p><a href="https://i.sstatic.net/ZsQ0E.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ZsQ0E.png" alt="enter image description here" /></a></p>
<p>b) First, It's clear that everytime a rich reports 1, the tax collector won't audit. If the rich reports 0, the Bayesian belief of the tax collector that he has high income given that he reported 0 is:</p>
<p><span class="math-container">$\sigma(p,q) = Pr(H|0) = \frac{Pr(0|H)Pr(H)}{Pr(0|H)Pr(H) + Pr(0|\bar H)Pr(\bar H)} = \frac{p(1-q)}{p(1-q) + 1-p}$</span></p>
<p>Hence the collector audits if and only if</p>
<p><span class="math-container">$E(U(A|0)) &gt; E(U(T|0)) \iff (t+f-c)\sigma(p,q) -c(1-\sigma(p,q)) \geq 0$</span></p>
<p>Thus iff <span class="math-container">$ \sigma(p,q) \geq c/(t+f)$</span>, she audits when she receives a 0 report.</p>
<p>The PBEa are thus <span class="math-container">$[0,-c],[1-t,t-c],[1-t,t-c]$</span> if <span class="math-container">$\sigma(p,q) \geq c/(t+f)$</span> and  <span class="math-container">$[0,0],[1-t,t],[1,0]$</span> if <span class="math-container">$\sigma(p,q) &lt; c/(t+f)$</span></p>
<p>c) <span class="math-container">$t+f = 1$</span> maximizes the tax revenue (has no economic sense, but I can't find out any mixed strategy that solves this paradox...)</p>

CC BY-SA 4.0
2022-10-19T11:23:37.080Z
The TC's payoff in the rightmost outcome is $t$, not $0$. Also, the information sets are missing.
VARulle
CC BY-SA 4.0
2022-10-19T12:30:43.363Z
Yeah I copy pasted and deleted the information sets on the Tablet. Later I'll update. I was in a hurry. BTW, for the information set, you simply draw a dashed line below P, HB and HD.
Matteo Bulgarelli
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2022-10-19T16:21:13.827Z
No, there are two different information sets for the TC, since he observes the report before deciding whether or not to audit.
VARulle
CC BY-SA 4.0
2022-10-19T18:04:18.523Z
@MatteoBulgarelli I solved this part -b according to the answers and comments. What do you think?
studentp
CC BY-SA 4.0
2022-10-19T18:04:36.323Z
I solved this part -b according to the answers and comments. What do you think?  @VARulle
studentp
CC BY-SA 4.0
2022-10-20T08:16:52.697Z
I would brutally answer as I did, with no fantasy at all. But I might be wrong!
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-20T12:20:43.270Z
This information set makes no sense. It includes the decision nodes of nature (!?) and of the TP. Instead, TC should have two information sets, one including the nodes after observing a 0 report and one including the nodes after observing a 1 report.
VARulle
CC BY-SA 4.0
2022-10-20T19:36:54.777Z
I guess this is very complicated question. At first, really thank you for your great efforts! :)
studentp
CC BY-SA 4.0
2022-10-20T19:38:09.727Z
Dear @VARulle if it is possible, can you please show your solution? Because i am preparing an important exam and I really need to understand this question. I will be really glad
studentp
CC BY-SA 4.0
2022-10-20T19:38:57.847Z
Dear @MatteoBulgarelli I am really glad for your great effort and help. Many thanks:)
studentp
CC BY-SA 4.0
2022-10-20T22:10:19.220Z
@studentp, I haven't found the time to solve the game, I can just see that all the suggested game trees are wrong. But this should be the easy part. The only relevant TP-decision is whether to report high or low in the case that he is opportunistic and rich. And the only relevant TC-decision is whether or not to audit, given that low is reported. All other decisions are either fixed (poor reports low, independently of whether honest or opportunistic, and honest rich reports high by definition) or trivial (TC never audits a high report).
VARulle
CC BY-SA 4.0
2022-10-20T22:54:02.717Z
@VARulle I re-draw the tree according to your comment. And I added this new tree in the question part as a last edit. Please take a look. What do you think? At least, please help me to do this question by your comments. Thank you.
studentp
CC BY-SA 4.0
2022-10-20T23:35:46.300Z
@studentp Still wrong. All reports are either "Report High" or "Report Low", don't switch to "true" in the right half. Next, delete the A action on the TC's leftmost node: A high report will never be audited, so this branch can be reduced to lead directly to the NA terminal node. Then: the TC's information set connects only the nodes between which he cannot distinguish, i.e. the 3 nodes after receiving a low report. (Strictly speaking there is also an information set connecting the 2 nodes after receiving a high report, but this can be ignored since there is no decision to be made there.)
VARulle
CC BY-SA 4.0
2022-10-21T01:42:39.607Z
@VARulle thank you for your correction. I corrected my mistakes and I re-draw it and post it as the last edit in the question part. Please take a look. I am really happy for your help:) This is so important for me.
studentp
CC BY-SA 4.0
2022-10-21T07:30:54.033Z
@studentp I'll continue under the question, since this here should be comments to Matteo's answer...
VARulle
CC BY-SA 4.0
2022-10-21T09:38:49.250Z
@VARulle 1) Be polite. I don't know who you are, but please, say "it doesn't make sense to me", or rephrase. I am not your son or your student.

2) It does make sense. It simply conveys the idea that nature and the reporters share the same information, whereas the TC does not have access to such information. However, I should have just included the $P, HB, HD$ nodes in the same set, without the Nature, since also the TC knows the probabilities.

You don't need to make two sets.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-21T11:04:50.053Z
I'm sorry for my blunt language. To rephrase: Your red "information set" is wrong. It is not an information set at all. By definition, an information set is a subset of the decision nodes of a single player, and each decision node included necessarily has to branch out into the same moves. Your red set violates both these conditions, and if it "just included the P, HB, HD nodes", as you now seem to suggest, then it would still violate the second condition. (And it's not about who "knows the probabilities", since these are common knowledge anyway.)
VARulle
CC BY-SA 4.0
2022-10-19T06:43:31.840Z
2022-10-20T08:13:48.427Z
2022-10-20T08:13:48.427Z
Matteo Bulgarelli
<p>Taxpayers may either have a high income or low income, and they may be either opportunistic or honest.</p>
<p>The tax collector cannot observe any of these characteristics, but after receiving a report from the taxpayer, it may choose to conduct an audit (at cost c) to determine the taxpayer's income.</p>
<p>To simplify, assume low income taxpayers have no income, and high income taxpayers have income equal to 1. High income taxpayers owe a tax t, where 0 &lt; t &lt; 1, while low income taxpayers
owe 0. Honest taxpayers report their true income while opportunistic taxpayers are optimizers who report an income level (0 or 1) that maximizes net (after tax and penalties) income.</p>
<p>The ex-ante probability a tax payer has high income is p, and independently, the probability that a taxpayer is honest is q. If the tax collector audits and finds the taxpayer underreported their income, the taxpayer is fined a predetermined penalty f in [0, 1] in addition to any tax due.</p>
<p>The tax collector's payoff is equal to expected revenue including any penalties) minus audit costs.
(a) Draw the extensive form for this game (don't forget Nature's move).</p>
<blockquote>
<p>(b) Compute a perfect Bayesian equilibrium for this game. How does the equilibrium
vary with f, t, c, p, q.</p>
</blockquote>
<blockquote>
<p>(c) Suppose the tax collector is considering changing the penalty f to maximize expected tax revenue (keeping all other parameters fixed). What f should they choose (f cannot exceed 1 - t)?</p>
</blockquote>
<p>————————</p>
<p>I understand the part (a). I could not solve for the part (b) and part (c). Please help me to do these two parts as well. All helps will be appreciated. Many thanks!</p>
<hr />
<p><strong>According to answers and comments, I solved this question in the following way;</strong></p>
<p>I don't know whether my solution for part b is true or not. I did this according to the answers. Please make a comment on my solution. And I could not do the part c.</p>
<p><strong>Last Edit</strong>
<a href="https://i.sstatic.net/KkfMg.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/KkfMg.jpg" alt="enter image description here" /></a></p>
<p>The transformed tree is as follows:</p>
<p><a href="https://i.sstatic.net/BzmcB.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/BzmcB.jpg" alt="enter image description here" /></a></p>
<p><strong>I tried to solve the part b with the help of user @VARulle</strong></p>
<p>First, I define the belief <span class="math-container">$\mu$</span> for the information set of TC for low-report.</p>
<p>TC's optimal strategy is</p>
<p><span class="math-container">$$EU_{TC}(A|\mu)=\mu(t+f-c)+(1-\mu)(-c)= \mu(t+f) -c $$</span></p>
<p><span class="math-container">$$EU_{TC}(NA|\mu)=\mu(0)+(1-\mu)(0)= 0$$</span></p>
<p>So, we have three cases</p>
<p>(i) <span class="math-container">$\sigma_{TC}(A)=1$</span> if <span class="math-container">$\mu(t+f) -c &gt;0$</span> or, <span class="math-container">$\mu(t+f) &gt;c $</span></p>
<p>(ii) <span class="math-container">$\sigma_{TC}(NA)=1$</span> if <span class="math-container">$\mu(t+f) -c &lt;0$</span> or, <span class="math-container">$\mu(t+f) &lt;c $</span></p>
<p>(iii) <span class="math-container">$\sigma_{TC}(A)\in (0, 1)$</span> and <span class="math-container">$\sigma_{TC}(NA)\in (0, 1)$</span> if <span class="math-container">$\mu(t+f) -c =0$</span> or, <span class="math-container">$\mu(t+f) =c $</span></p>
<p>Let's look at the Tax-payer's optimal strategy (TP)</p>
<p><strong>Case i:</strong>  <span class="math-container">$\sigma_{TC}(A)=1$</span> if <span class="math-container">$\mu(t+f) -c &gt;0$</span> or, <span class="math-container">$\mu(t+f) &gt;c $</span></p>
<p>For the type of dishonest, High income (DH);</p>
<p><span class="math-container">$$U_{TP}^{DH}(RL, A)=1-t-f$$</span></p>
<p><span class="math-container">$$U_{TP}^{DH}(RH, NA)=1-t$$</span></p>
<p>since <span class="math-container">$(1-t) &gt; (1-t-f)$</span>, <span class="math-container">$\sigma_{TP}^{DH}(RH)=1$</span></p>
<p>so, this type of TP deviates! So, there is no PBE for this case.</p>
<p><strong>Case ii:</strong>  <span class="math-container">$\sigma_{TC}(NA)=1$</span> if <span class="math-container">$\mu(t+f) -c &lt;0$</span> or, <span class="math-container">$\mu(t+f) &lt;c $</span></p>
<p>For the type of dishonest, High income (DH);</p>
<p><span class="math-container">$$U_{TP}^{DH}(RL, NA)=1$$</span></p>
<p><span class="math-container">$$U_{TP}^{DH}(RH, NA)=1-t$$</span></p>
<p>Then, <span class="math-container">$\sigma_{TP}^{DH}(RL)=1$</span></p>
<p><span class="math-container">$$\mu = \frac{(1-q)*p*1}{(1-q)*p*1+ (1-q)*(1-p)*0}=1$$</span></p>
<p>So, <span class="math-container">$\{(RL, NA), \mu =1, c&gt;(t+f)\}$</span> is pure Strategy PBE.</p>
<p><strong>Case iii:</strong>  <span class="math-container">$\sigma_{TC}(NA)\in (0, 1)$</span> and <span class="math-container">$\sigma_{TC}(A)\in (0, 1)$</span>  if  <span class="math-container">$\mu(t+f) = c $</span></p>
<p><span class="math-container">$$U_{TP}^{DH}(RL)=\sigma_{TC}(NA)*1 +\sigma_{TC}(A)*(1+t-f)=1+(t-f)\sigma_{TC}(A)$$</span></p>
<p><span class="math-container">$$U_{TP}^{DH}(RH)=1-t$$</span></p>
<p>we have 2 sub-cases</p>
<p>Subcase-1: <span class="math-container">$1+(t-f)\sigma_{TC}(A) \ge 1-t$</span> Then, <span class="math-container">$\sigma_{TP}^{DH}(RL)=1$</span></p>
<p>Subcase-2: <span class="math-container">$1+(t-f)\sigma_{TC}(A) \le 1-t$</span> Then, <span class="math-container">$\sigma_{TP}^{DH}(RH)=1$</span></p>
<p>Let's continue with <strong>Subcase-1</strong></p>
<p><span class="math-container">$\sigma_{TC}(A) \ge -t/(t-f)$</span></p>
<p><span class="math-container">$\sigma_{TP}^{DH}(RL)=1$</span></p>
<p>Then, <span class="math-container">$\mu =1$</span></p>
<p>Then, Mixed strategy PBE = <span class="math-container">$\{ (RL, \sigma_{TC}(A) \in [-t/(t-f), 1]), \mu = 1, c=t+f \}$</span></p>
<p>Let's start with <strong>Subcase-2</strong></p>
<p><span class="math-container">$\sigma_{TC}(A) \le -t/(t-f)$</span></p>
<p><span class="math-container">$\sigma_{TP}^{DH}(RH)=1$</span></p>
<p>Then, <span class="math-container">$\mu =0$</span> which implies <span class="math-container">$\mu (t+f)=c \to c=0$</span>. But, <span class="math-container">$c&gt;0$</span></p>
<p>So, this case is not possible.</p>
<p>As a result, there are one Mixed Strategy PBE and one pure strategy PBE.</p>

CC BY-SA 4.0
2022-10-20T14:00:11.003Z
I don't understand the game tree. Honest or dishonest and high or low income are independent decisions of nature, so there should be 4 different types of TP, right?
VARulle
CC BY-SA 4.0
2022-10-20T14:37:12.080Z
Yes you are right! But, I eliminate the dishonest with low income agents. Since the question says, honest agents always report true, I only consider the honest taxpayer with low income report low income (LR) and the honest tax payer with high income report ( HR). That is, I reduced the types @VARulle
studentp
CC BY-SA 4.0
2022-10-20T14:38:22.670Z
How do you solve such a question? Can you please show your solution? This question is very important for me to understand because I am preparing an important exam. Many thanks @VARulle
studentp
CC BY-SA 4.0
2022-10-21T07:33:17.457Z
The last edit is correct (except that N should be NA). You could further simplify the tree by noting that the 2 rightmost branches have identical terminal nodes, so they could be collapsed to a single branch labelled "low income", followed by "Report low" and the TC's final decision.
VARulle
CC BY-SA 4.0
2022-10-21T07:39:41.073Z
After that, first check the case where $c>t+f$. You will see that then auditing is prohibitively costly and the only NE is (RL, NA). The more interesting case is $c<t+f$. There is no pure-strategy NE in this case, but a completely mixed one. You should be able to calculate it by writing down the usual indifference conditions, which is a bit messy because of the many parameters involved. (I haven't done it myself.)
VARulle
CC BY-SA 4.0
2022-10-21T13:17:03.443Z
Thank you so much for your checking my attempts. I added the expression of pure strategy PBE. And also, I tried to do the mixed strategy PBE. (I hope I am in a correct way). What do you think? @VARulle
studentp
CC BY-SA 4.0
2022-10-22T11:30:53.730Z
Dear @VARulle I also did the mixed strategy in the case-iii. I hope it is correct? And what is your suggestion for the solution of part c? I will be happy if you inform me. Many thanks:)
studentp
CC BY-SA 4.0
2022-10-23T16:33:07.637Z
Cases i and ii look good, but in case iii, the factor $(1+t-f)$ should be $(1-t-f)$...
VARulle
<game-theory>
Finding PBE in incomplete game
studentp
# Finding PBE in incomplete game

(Asked by: username_0 on 2022-08-23)

Taxpayers may either have a high income or low income, and they may be either opportunistic or honest.

The tax collector cannot observe any of these characteristics, but after receiving a report from the taxpayer, it may choose to conduct an audit (at cost c) to determine the taxpayer's income.

To simplify, assume low income taxpayers have no income, and high income taxpayers have income equal to 1. High income taxpayers owe a tax t, where 0 < t < 1, while low income taxpayers
owe 0. Honest taxpayers report their true income while opportunistic taxpayers are optimizers who report an income level (0 or 1) that maximizes net (after tax and penalties) income.

The ex-ante probability a tax payer has high income is p, and independently, the probability that a taxpayer is honest is q. If the tax collector audits and finds the taxpayer underreported their income, the taxpayer is fined a predetermined penalty f in [0, 1] in addition to any tax due.

The tax collector's payoff is equal to expected revenue including any penalties) minus audit costs.
(a) Draw the extensive form for this game (don't forget Nature's move).

> (b) Compute a perfect Bayesian equilibrium for this game. How does the equilibrium
> vary with f, t, c, p, q.

> (c) Suppose the tax collector is considering changing the penalty f to maximize expected tax revenue (keeping all other parameters fixed). What f should they choose (f cannot exceed 1 - t)?

————————

I understand the part (a). I could not solve for the part (b) and part (c). Please help me to do these two parts as well. All helps will be appreciated. Many thanks!

---

**According to answers and comments, I solved this question in the following way;**

I don't know whether my solution for part b is true or not. I did this according to the answers. Please make a comment on my solution. And I could not do the part c.

**Last Edit**
[![enter image description here](https://i.sstatic.net/KkfMg.jpg)](https://i.sstatic.net/KkfMg.jpg)

The transformed tree is as follows:

[![enter image description here](https://i.sstatic.net/BzmcB.jpg)](https://i.sstatic.net/BzmcB.jpg)

**I tried to solve the part b with the help of user @username_1**

First, I define the belief $\mu$ for the information set of TC for low-report.

TC's optimal strategy is

$$EU\_{TC}(A|\mu)=\mu(t+f-c)+(1-\mu)(-c)= \mu(t+f) -c $$

$$EU\_{TC}(NA|\mu)=\mu(0)+(1-\mu)(0)= 0$$

So, we have three cases

(i) $\sigma\_{TC}(A)=1$ if $\mu(t+f) -c >0$ or, $\mu(t+f) >c $

(ii) $\sigma\_{TC}(NA)=1$ if $\mu(t+f) -c <0$ or, $\mu(t+f) <c $

(iii) $\sigma\_{TC}(A)\in (0, 1)$ and $\sigma\_{TC}(NA)\in (0, 1)$ if $\mu(t+f) -c =0$ or, $\mu(t+f) =c $

Let's look at the Tax-payer's optimal strategy (TP)

**Case i:** $\sigma\_{TC}(A)=1$ if $\mu(t+f) -c >0$ or, $\mu(t+f) >c $

For the type of dishonest, High income (DH);

$$U\_{TP}^{DH}(RL, A)=1-t-f$$

$$U\_{TP}^{DH}(RH, NA)=1-t$$

since $(1-t) > (1-t-f)$, $\sigma\_{TP}^{DH}(RH)=1$

so, this type of TP deviates! So, there is no PBE for this case.

**Case ii:** $\sigma\_{TC}(NA)=1$ if $\mu(t+f) -c <0$ or, $\mu(t+f) <c $

For the type of dishonest, High income (DH);

$$U\_{TP}^{DH}(RL, NA)=1$$

$$U\_{TP}^{DH}(RH, NA)=1-t$$

Then, $\sigma\_{TP}^{DH}(RL)=1$

$$\mu = \frac{(1-q)\*p\*1}{(1-q)\*p\*1+ (1-q)\*(1-p)\*0}=1$$

So, $\{(RL, NA), \mu =1, c>(t+f)\}$ is pure Strategy PBE.

**Case iii:** $\sigma\_{TC}(NA)\in (0, 1)$ and $\sigma\_{TC}(A)\in (0, 1)$ if $\mu(t+f) = c $

$$U\_{TP}^{DH}(RL)=\sigma\_{TC}(NA)\*1 +\sigma\_{TC}(A)\*(1+t-f)=1+(t-f)\sigma\_{TC}(A)$$

$$U\_{TP}^{DH}(RH)=1-t$$

we have 2 sub-cases

Subcase-1: $1+(t-f)\sigma\_{TC}(A) \ge 1-t$ Then, $\sigma\_{TP}^{DH}(RL)=1$

Subcase-2: $1+(t-f)\sigma\_{TC}(A) \le 1-t$ Then, $\sigma\_{TP}^{DH}(RH)=1$

Let's continue with **Subcase-1**

$\sigma\_{TC}(A) \ge -t/(t-f)$

$\sigma\_{TP}^{DH}(RL)=1$

Then, $\mu =1$

Then, Mixed strategy PBE = $\{ (RL, \sigma\_{TC}(A) \in [-t/(t-f), 1]), \mu = 1, c=t+f \}$

Let's start with **Subcase-2**

$\sigma\_{TC}(A) \le -t/(t-f)$

$\sigma\_{TP}^{DH}(RH)=1$

Then, $\mu =0$ which implies $\mu (t+f)=c \to c=0$. But, $c>0$

So, this case is not possible.

As a result, there are one Mixed Strategy PBE and one pure strategy PBE.

**Tags:** game-theory

> **Comments:**
> **username_1**: I don't understand the game tree. Honest or dishonest and high or low income are independent decisions of nature, so there should be 4 different types of TP, right?  
> **username_0**: Yes you are right! But, I eliminate the dishonest with low income agents. Since the question says, honest agents always report true, I only consider the honest taxpayer with low income report low income (LR) and the honest tax payer with high income report ( HR). That is, I reduced the types @username_1  
> **username_0**: How do you solve such a question? Can you please show your solution? This question is very important for me to understand because I am preparing an important exam. Many thanks @username_1  
> **username_1**: The last edit is correct (except that N should be NA). You could further simplify the tree by noting that the 2 rightmost branches have identical terminal nodes, so they could be collapsed to a single branch labelled "low income", followed by "Report low" and the TC's final decision.  
> **username_1**: After that, first check the case where $c>t+f$. You will see that then auditing is prohibitively costly and the only NE is (RL, NA). The more interesting case is $c<t+f$. There is no pure-strategy NE in this case, but a completely mixed one. You should be able to calculate it by writing down the usual indifference conditions, which is a bit messy because of the many parameters involved. (I haven't done it myself.)  
> **username_0**: Thank you so much for your checking my attempts. I added the expression of pure strategy PBE. And also, I tried to do the mixed strategy PBE. (I hope I am in a correct way). What do you think? @username_1  
> **username_0**: Dear @username_1 I also did the mixed strategy in the case-iii. I hope it is correct? And what is your suggestion for the solution of part c? I will be happy if you inform me. Many thanks:)  
> **username_1**: Cases i and ii look good, but in case iii, the factor $(1+t-f)$ should be $(1-t-f)$...  

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## 2 answers

### Answer by username_3 on 2022-10-19. Score: 4 | Accepted answer

a) Without loss of generality, assume that every poor is honest. Indeed, form the perspective of the collector, the expected utility of Auditing given that someone is poor and honest or poor and dishonest is the same.

The tree takes the following form:

[![enter image description here](https://i.sstatic.net/ZsQ0E.png)](https://i.sstatic.net/ZsQ0E.png)

b) First, It's clear that everytime a rich reports 1, the tax collector won't audit. If the rich reports 0, the Bayesian belief of the tax collector that he has high income given that he reported 0 is:

$\sigma(p,q) = Pr(H|0) = \frac{Pr(0|H)Pr(H)}{Pr(0|H)Pr(H) + Pr(0|\bar H)Pr(\bar H)} = \frac{p(1-q)}{p(1-q) + 1-p}$

Hence the collector audits if and only if

$E(U(A|0)) > E(U(T|0)) \iff (t+f-c)\sigma(p,q) -c(1-\sigma(p,q)) \geq 0$

Thus iff $ \sigma(p,q) \geq c/(t+f)$, she audits when she receives a 0 report.

The PBEa are thus $[0,-c],[1-t,t-c],[1-t,t-c]$ if $\sigma(p,q) \geq c/(t+f)$ and $[0,0],[1-t,t],[1,0]$ if $\sigma(p,q) < c/(t+f)$

c) $t+f = 1$ maximizes the tax revenue (has no economic sense, but I can't find out any mixed strategy that solves this paradox...)

> **Comments:**
> **username_1**: The TC's payoff in the rightmost outcome is $t$, not $0$. Also, the information sets are missing.  
> **username_3**: Yeah I copy pasted and deleted the information sets on the Tablet. Later I'll update. I was in a hurry. BTW, for the information set, you simply draw a dashed line below P, HB and HD.  
> **username_1**: No, there are two different information sets for the TC, since he observes the report before deciding whether or not to audit.  
> **username_0**: @username_3 I solved this part -b according to the answers and comments. What do you think?  
> **username_0**: I solved this part -b according to the answers and comments. What do you think?  @username_1  
> **username_3**: I would brutally answer as I did, with no fantasy at all. But I might be wrong!  
> **username_1**: This information set makes no sense. It includes the decision nodes of nature (!?) and of the TP. Instead, TC should have two information sets, one including the nodes after observing a 0 report and one including the nodes after observing a 1 report.  
> **username_0**: I guess this is very complicated question. At first, really thank you for your great efforts! :)  
> **username_0**: Dear @username_1 if it is possible, can you please show your solution? Because i am preparing an important exam and I really need to understand this question. I will be really glad  
> **username_0**: Dear @username_3 I am really glad for your great effort and help. Many thanks:)  
> **username_1**: @username_0, I haven't found the time to solve the game, I can just see that all the suggested game trees are wrong. But this should be the easy part. The only relevant TP-decision is whether to report high or low in the case that he is opportunistic and rich. And the only relevant TC-decision is whether or not to audit, given that low is reported. All other decisions are either fixed (poor reports low, independently of whether honest or opportunistic, and honest rich reports high by definition) or trivial (TC never audits a high report).  
> **username_0**: @username_1 I re-draw the tree according to your comment. And I added this new tree in the question part as a last edit. Please take a look. What do you think? At least, please help me to do this question by your comments. Thank you.  
> **username_1**: @username_0 Still wrong. All reports are either "Report High" or "Report Low", don't switch to "true" in the right half. Next, delete the A action on the TC's leftmost node: A high report will never be audited, so this branch can be reduced to lead directly to the NA terminal node. Then: the TC's information set connects only the nodes between which he cannot distinguish, i.e. the 3 nodes after receiving a low report. (Strictly speaking there is also an information set connecting the 2 nodes after receiving a high report, but this can be ignored since there is no decision to be made there.)  
> **username_0**: @username_1 thank you for your correction. I corrected my mistakes and I re-draw it and post it as the last edit in the question part. Please take a look. I am really happy for your help:) This is so important for me.  
> **username_1**: @username_0 I'll continue under the question, since this here should be comments to username_3's answer...  
> **username_3**: @username_1 1) Be polite. I don't know who you are, but please, say "it doesn't make sense to me", or rephrase. I am not your son or your student.  2) It does make sense. It simply conveys the idea that nature and the reporters share the same information, whereas the TC does not have access to such information. However, I should have just included the $P, HB, HD$ nodes in the same set, without the Nature, since also the TC knows the probabilities.  You don't need to make two sets.  
> **username_1**: I'm sorry for my blunt language. To rephrase: Your red "information set" is wrong. It is not an information set at all. By definition, an information set is a subset of the decision nodes of a single player, and each decision node included necessarily has to branch out into the same moves. Your red set violates both these conditions, and if it "just included the P, HB, HD nodes", as you now seem to suggest, then it would still violate the second condition. (And it's not about who "knows the probabilities", since these are common knowledge anyway.)  

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### Answer by username_2 on 2022-08-26. Score: 4

These are the errors in the original diagram:

1. Your game tree does not show the opportunistic-type tax payer's income-reporting choices. They choose between reporting an income of 0 or 1 (you need two extra tax-payer decision nodes and four extra branches at the bottom of the tree). You appear to have assumed that high-income opportunistic tax payers always report low income. It may be they want to report truthfully. If the probability of audit is sufficiently high, then they will report truthfully. Otherwise they will under-report.
2. Also, you appear to have assumed low-income opportunistic tax payers report low income (optimally they will, but the decision should be shown in the tree) and are always taxed $t$ (the same amount as for high income) and fined if audited. There should be no fine for under-reporting and there should be no tax for a low-income opportunistic tax payer (who reports truthfully).
3. You should have dashed lines between all tax-collector decision nodes where the same income is reported (rather than between all nodes where the tax payer has the same income).

The top part of the tree (for the honest tax payer) appears to be correct.

Below is the corrected tree. Note I have not drawn the decision of the opportunistic taxpayer with low income as they optimally choose not to report regardless of the tax collector's choices. Note also that for nature you could have just one decision node with four branches.

[![enter image description here](https://i.sstatic.net/g8lVn.jpg)](https://i.sstatic.net/g8lVn.jpg)

> **Comments:**
> **username_0**: Dear @username_2 thank you for your help. I see how to draw the tree. I tried to solve it, but I could not do. I am preparing an exam, can you please help me to solve for the part b and c. I will be really glad. Many thanks:)  
> **username_3**: It needs to be done from the top to the bottom, otherwise it has no sense. Where are the stages of the game, here ?  
> **username_0**: In the tree, two natures are used. What you mean? That’s I need to start with assign beliefs in the nodes $(\mu_1, \mu_2, 1-\mu_1-\mu_2)$ and $(\beta, 1-\beta)$ and then I will calculate the ecpected payoffs of Tax collector(TC) @username_3  
> **username_3**: You can remove the branch where the poor are dishonest.  
> **username_3**: Additionally, when the collector discovers the dishonest, she earns $t + f - c$, not just $f -c$.  
> **username_2**: Yes, it should be $t+f-c$.   In response to the previous comments:   Yes, the game tree could have been labelled better (the first node is the move of nature "N" between the words "Honest " and "Opportunistic"). I could also have consolidated the two moves of nature into one.  The branch where the low income opportunistic tax payer is dishonest has not been drawn (i.e. has already been removed as suggested).  
> **username_0**: I solved this part -b according to the answers and comments. What do you think? @username_2  

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