The Illinois Lottery offers a Grand Prize each week of several million dollars to the holder(s) of (a) winnning ticket(s). In order to win, a ticket must display exactly six correct winning numbers drawn from a pool of fifty-four possible numbers in the range 1 ... 54. As you can see from 54 x 53 x 52 x 51 x 50 x 49 = 18.6 million, the odds of picking a winning ticket are quite small. The amount of the Grand Prize (for purposes of this problem only) is advertised as $1 for each ticket sold. So if 20 million tickets are sold, then the Grand Prize for that week is $20 million - payable in equal annual installments of $1 million over twenty years. Wagering winnings, however, are subject to both Federal (currently 39% for amounts over $200,000 per year) and Illinois (currently 3%) income taxes; but, are considered unearned income for other tax purposes. Currently, a twenty-year U.S. Treasury Bond can be purchased at a price to yield a 7% return and municipal bonds can be purchased at a price to yield a 5% return.
There are about 12 Million people in Illinois - of whom about 2 Million regularly purchase at least one lottery ticket per week. Another 4 Million resident rarely -if ever- purchase a lottery ticket. The remaining 6 Million residents tend to purchase occasionally (perhaps two tickets a month) and increase their rate of purchasing when rollovers increase the size of the Grand Prize. Non-resident purchases are insignificant.
After a Grand Prize has been won, the next starting Grand Prize is automatically set to $4 Million. When more than 4 Million tickets have been purchased in a week, then the prize is increased automatically as desicribed above. If there is a rollover, then the Grand Prize is automatically increased by another $4 Million on top of the current Grand Prize.
The largest Grand Prize won to date was $80 Million. The average Grand Prize is $20 Million. Ninety-five percent of all Grand Prizes fall within the range of $10 - $30 Million. The average number of weeks until there is a winner is three. Ninety-five percent of all prizes are won between 2-4 weeks inclusive. The $80 Million prize was won on the sixth week. As you would expect, the rate of ticket sales increases as an increasing function of the size of the Grand Prize.
Sixty-six percent of the time, there is only one winning ticket. NInety-five percent of the time there are no more than two winning tickets. Ninety-nine percent of the time there are no more than three winning tickets. On one fluke occurrence caused by improper keying of a ticket vending machine, there were six winning tickets. Other than that, the maximum number of winning tickets for a Grand Prize was four.
After years of work on signal analysis, I developed an algorithm which would allow me to predict with absolute certainty all six winning numbers for a coming week's Illinois Lottery drawing. I waited for a week when the Grand Prize was made unusually large ($80 Million) due to rollovers from previous weeks, purchased my winning ticket and now enjoy an excellent lifestyle. Nevertheless, I am eager to continue to benefit from my work; but, I recognize that if I were to win a second time, I could end up killing the goose that laid golden eggs.
Accordingly, I have devised a plan to enhance my wealth by auctioning off winning numbers (in a manner similar to that used by the FCC) to a select group of wealthy individuals whom I have met as students taking my C96 course at Northwestern University. Each week, each player submits successive bids for between 0 ... 6 of the numbers that will appear on that week's winning ticket in successive bidding rounds until there is no further increase in bids.
A player can elect to bid for exclusive rights or non-exclusive rights. If a bid for exclusive rights wins, then I will not disclose that (those) number(s) to anyone else. Otherwise, I can sell the same number to more than one player.
A player can elect ot bid or not bid on any particular round or on any particular week; however, once a player's bid is accepted, then that player is excluded from further auctions (irrespective of whether the player purchases a winning ticket) and replaced by a new player. A player who fails to win a bid in six weeks is also replaced. The reasons for these rules are a) to prevent a player from winning more than one lottery (thereby placing the venture at risk) and b) to discourage low-ball bidding.
Naturally, I will select that (those) bid(s) which maximize(s) my income and warrant to refund the money you will pay to me at the close of the auction if my prediction is wrong.
For all of the following questions, you may assume that a Grand Prize of $20 Million and other necessary facts are as stated above. If additional facts are necessary for your solution, you should state them clearly and then proceed.