# Please check for payoffs which I calculated in the incomplete and imperfect 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)?

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i posted the extensive form game tree as follows. Please look at whether this tree is correct? If correct, then I will try to  solve the part (b) and (c) by myself. But I cannot be sure about the payoffs. Please check for payoffs especially. Thank you.

Actions of tax collectors ={Audit(A), Not audit(N)}

types of tax payers ={honest (h), oppurtunist ($$\phi$$} }

Lor inceme (L)

High inceme (H)

Tax payer with honest type is donated by $$TP^h$$

Tax payer with opportunist type is donated by $$TP^{\phi}$$

Sorry for adding hand-written picture but I could not the tree in latex format.

I posted the whole question, but I only want you to check my tree and payoffs.

If there is unreadable part, then please let me know, I will explain.

In the payoff calculation, the first one is the payoff for tax payer, the second one is the payoff for tax collector.

Edit: (the corrected version is as follows)

Edit -2 : I also changed the tree in the extensive form as follows

• Please make comment even if my answer is true. Thank you. Aug 24, 2022 at 21:50
• This game tree doesn't have any actions of the TP, it looks just like an optimization exercise for the TC. Apart from that, the payoffs in the bottom right outcome seem wrong. The TP has low income there and shouldn't owe any tax. Aug 25, 2022 at 11:00
• Thank you @VARulle Can you please write down correct answer? Because ı am confused too much writing and drawing this tree. Based on the correct tree, I can solve other parts by myself. I will appreciated your correct answer. Aug 25, 2022 at 12:16
• Dear @VARulle I edited. I also tried to correct the payoff which you said. But again I am not sure. And I change the actions and types of taxpayer (TP) in the second edited photo. Please share your ideas with me. I am studying a final exam, and I need to learn this type of questions. Thank you a lot. Aug 25, 2022 at 12:59
• I still don't understand the setting. If honest and opportunistic are actions, then the choices might depend on the type (high or low income). But then you cannot write $q$ for the probability of choosing honest, since this probability might differ between the two types. Aug 25, 2022 at 13:43