# Finding PBE in incomplete game

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

The transformed tree is as follows:

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

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)

(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)

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 = $$\{ (RL, \sigma_{TC}(A) \in [-t/(t-f), 1]), \mu = 1, c=t+f \}$$

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

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

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.

• 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? Commented Oct 20, 2022 at 14:00
• 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 Commented Oct 20, 2022 at 14:37
• 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 Commented Oct 20, 2022 at 14:38
• 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. Commented Oct 21, 2022 at 7:33
• Cases i and ii look good, but in case iii, the factor $(1+t-f)$ should be $(1-t-f)$... Commented Oct 23, 2022 at 16:33

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:

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...)

• The TC's payoff in the rightmost outcome is $t$, not $0$. Also, the information sets are missing. Commented Oct 19, 2022 at 11:23
• 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. Commented Oct 19, 2022 at 12:30
• No, there are two different information sets for the TC, since he observes the report before deciding whether or not to audit. Commented Oct 19, 2022 at 16:21
• @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. Commented Oct 21, 2022 at 9:38
• 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.) Commented Oct 21, 2022 at 11:04

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.

• 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:) Commented Oct 18, 2022 at 11:56
• It needs to be done from the top to the bottom, otherwise it has no sense. Where are the stages of the game, here ? Commented Oct 18, 2022 at 16:19
• 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 Commented Oct 18, 2022 at 16:37
• Additionally, when the collector discovers the dishonest, she earns $t + f - c$, not just $f -c$. Commented Oct 18, 2022 at 19:12
• 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
Commented Oct 18, 2022 at 19:23