Finding PBE in incomplete game

Question

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)?

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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!

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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) <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.

Answer 1

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...)

Answer 2

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.