Diego has $6,400.He plans to bet on a soccer game.Team A is a favorite to win.Assume no ties can occur.For $.80 one can buy a ticket that will pay $1 if team A wins and nothing if B wins.For $.20 one can buy a ticket that pays $1 if team B wins and nothing if A wins.Diego thinks the two teams are equally likely to win.He buys tickets so as to maximize the expected value of lnW (the natural log of his wealth).After he buys his tickets, team A loses a star player and the ticket price moves to $.50 for either team.Diego buys some new tickets and sells some of his old ones.The game is then played and team A wins.How much wealth does he end up with?

Question

Quizplus · Accepted Answer

To maximize the expected value of lnW, Diego should allocate his wealth in proportion to the odds. Initially, he splits his $6,400 into $1,600 for team B (at $.20 per ticket, buying 8,000 tickets) and $4,800 for team A (at $.80 per ticket, buying 6,000 tickets). After the odds change to $.50 for either team, to maximize expected utility, he should have equal bets on both teams. He sells some of his old tickets and buys new ones to balance his bets. However, the question doesn't provide enough information to calculate the exact transactions. Since team A wins, and assuming he adjusted his bets to have equal amounts on both teams after the odds changed, he would end up with double the amount bet on team A at the new odds. If he perfectly balanced his bets, he would have $3,200 on each team. Since team A wins, he doubles the $3,200 bet on team A, ending up with $6,400 from the bet plus his remaining wealth, leading to a total greater than his initial wealth but not specified exactly how much. The only option greater than $6,400 and feasible under these conditions is $10,000.

Diego Has $6,400