Game theory - Finding Nash Equilibria for a cartel game

Challenge: A Good Deal

You are currently learning some important aspects of collusion and cartels. This challenge puts you in the position of a bad guy, namely a price-fixing sales manager. Suppose that you find yourself in a so-called “smoke-filled room” to fix prices for the upcoming year with the sales manager of a competing firm, Snitch Inc.. Suddenly the door opens and one of the employees of Snitch Inc. enters the room. The employee knows that price-fixing is illegal and immediately grabs his cellphone to inform the competition authority. You are fully aware that you are now facing a serious risk of getting a fine or even a jail sentence. Whereas your fellow sales manager is simply in shock, not knowing what to do, you as a University School of Business and Economics alumni are quick to react and you try to save the situation. Your idea is to bribe the employee. You expect that the employee requires a bribe of at least €100 to remain silent. In other words, the reservation price of the employee is €100. For simplicity, assume in the following that this expectation is correct. At the same time, suppose that you are not willing to offer more than €200 for otherwise you would rather save your money and spend it on a good lawyer instead. In other words, your reservation price is €200. You consider your chances and are thinking about making an offer. For simplicity, assume in the following that all parties aim to maximize the gains from trade and that offers can be any positive real number (all values weakly above zero, that is).

Try to provide a clear and concise answer to the following four questions. For the first two questions suppose that the employee is still slightly in shock and therefore can only respond by either accepting or rejecting your offer.

1. How many Nash equilibria does this game have, if any? Explain your answer.
2. How many subgame perfect Nash equilibria does this game have, if any? Explain your answer.

Now suppose that you are dealing with an, somewhat cocky, employee who is brave enough to start negotiating with you. That is, in questions 3 and 4 below, the employee, instead of simply accepting or rejecting the offer, may now make a counteroffer instead. For simplicity, assume that both parties get a zero payoff in case of “no deal”.

1. Suppose that the employee indeed does make a counteroffer and that you will either accept or reject (in other words, the game ends after your response to the counteroffer). What is the subgame perfect Nash equilibrium in this case?

You fear that these negotiations may take quite some time and as a business man you know that time is money. Suppose that your impatience as well as that of the employee is given by a common discount factor 0 < δ < 1 (which is known to you and the employee). The interpretation is that the utility of a money amount K “tomorrow” is equal to the utility of an amount of δK “today”. Your goal is to make an acceptable offer the first round while at the same time saving as much money as possible.

1. What is the subgame perfect Nash equilibrium outcome in this case? Do you (i.e., the sales manager) or the employee benefit from your impatience? Explain your answer.
• This looks like it is copied from a homework set. What have you tried so far? – Trurl Sep 26 '17 at 18:15

I have the same exercise as homework. Spent a few hours on this with a few people, but we are still very doubtful of our answers. @trurl could you please correct/confirm:

Assumption: Only Sales Manager(SM) knows the reservation price of the Employee not vice-versa.

1. The game has 100 Nash equilibria: All points between 100 and 200.
2. Subgame perfect Nash equilibrium: 100 (SM makes first offer at any number between 0 and 99. Employee rejects and responds with 100, which is acceptable to both.)
3. Subgame perfect Nash equilibrium: 100 (Since SM doesn't want him to reject the deal, he offers 100, which maximises his gain but is also acceptable to the employee.
4. Subgame perfect Nash equilibrium: 200 (Since negotiation is allowed, the employee now has a chance to figure out the reservation price of the SM. In order to minimize the number of offers that go back n forth, the SM directly offers 200. To which counter offers will be unacceptable and thus the employee can deduce the SMs reservation price.) We could not figure out how to use the discount rate...
1. How many Nash equilibria does this game have, if any? Explain your answer.

• There is a Nash equilibrium when:

o The participants correctly guess the others’ choice

o Each participant maximizes profit given this guess

• The employee clearly wants to maximize gains and will always play the dominant strategy by accepting accept any bribe > €100.

• The manager knows that a bribe of minimum €100 will keep the employee silent.

o The manager maximizes profit by offering a €100 bribe, which is €100 less than his reservation price.

o The costs of the €100 euro bribe are lower than the higher value bribed up to €200 and definitely less severe/costly than the risks when the authorities are informed.

• Both participants now do not have any incentive to change his/her strategy.