Each of two individuals receive a ticket on which there is an integer from one to five indicating an amount of money he may receive. The individuals' tickets are assigned randomly and independently. The probability that an individual receives any of the five numbers is positive.
After receiving his ticket, each individual is asked independently and simultaneously with the other whether he wants to exchange his prize for the other individual's prize. If both individuals agree, then the prizes are exchanged, otherwise each individual keeps his own prize and receives its attached sum of money.
- How to model this situation as a Bayesian game?
- I have to show that in any pure strategy Bayesian equilibrium, the highest prize that either individual is willing to exchange is the smallest possible prize
- Are there some mixed strategies Bayesian equilibria?
As a player, if I receive a $5$, I definitely won't offer to exchange it, because I can't possibly get a higher number and might wind up with a smaller one. But this means that, unless I know my opponent is some kind of fool, I shouldn't offer to exchange a $4$ either: My (rational) opponent won't offer a $5$, so I can't improve on a $4$ but could do worse. The same reasoning now says I shouldn't offer to exchange a $3$, or a $2$, or, finally, a $1$, except that in the final case you can't do worse than a $1$ so you might as well offer to exchange it, on the off-chance you are playing a fool. This line of reasoning holds no matter what the probabilities are.