John asks his friend, Paul, to borrow some money to make it to payday. Paul would like to help John, but John is known for not paying his debts. At the same time Paul is making his decision, John is deciding whether to buy lottery tickets. He really likes the lottery. If he buys them, he will not be able to pay Paul back.
If Paul decides not to loan John the money, Paul gets 0. If John then buys lottery tickets, he gets 0. If he decides not to buy lottery tickets, he gets -1. If Paul decides to loan John the money and John plays the lottery, Paul gets -2 and John gets 8. If Paul decides to loan John the money and John does not play the lottery, Paul gets 3 and John gets 6. If this game is played just once, determine the equilibrium. What is John’s expected payoff?
Paul talks to John about making this a repetitive offer. It turns out that John has some rich relatives who might help him out regardless of what happens in the previous games. John has a 70% chance of becoming independently wealthy after each round and never returning to ask for a loan. Considering all of this new information, what is John’s decision now? What is his expected payoff?
Finally, a benevolent aunt feels sorry for her family. Any time John is denied a loan by a friend, she gives him some worth 2, bringing his payoff up to 1 if he does not play the lottery and 2 if he does. Now what is John’s decision? What is his payoff? (Note: we are still working off of the discount factor for a repetitive offer)
In which of these three scenarios is John the best off?