A new (fictitious) group of casinos, called credit-friendly casinos—CFCs for short—allow gamblers to enter a $100 bet on credit. The penalty to a gambler who does not pay in one of these casinos is forbidding that gambler from playing in there ever again. However, the gambler can still play at another credit-friendly casino and can place bets in several at the same time.
For each even-money $100 bet (the only ones we will consider), a gambler g receives $100 when g wins and owes the casino $110 when g loses. All losses must be paid by the end of each week for g to be able to continue to bet at that casino.
The first CFC model is that since the expected value for each gambler is negative, the CFCs allow gamblers to start betting without a deposit.
Warm-up: Suppose there are two CFCs: c1 and c2. How much can gambler g guarantee to win in just one $100 bet at each of the two casinos?
Solution to Warm-Up: $100. Gambler g makes some arbitrary bet at casino c1 that says some team t1 will beat t2 by at least p points. Then g, at casino c2, makes the "opposite" bet: it is not true t1 will beat t2 by at least p points. One of those bets will win. Gambler g will collect from one casino and abandon the other one.
Warm-Up 2: Suppose the CFCs loosen the rules so new gamblers can make two $100 bets at each casino on credit. Gambler g will make two bets at each of c1 and c2 where the bets at c2 are opposite to those at c1. Suppose that g will abandon any casino at which g loses. What are g's expected winnings, assuming all bets are even-money bets?
Solution to Warm-Up 2: Gambler g will win nothing if g wins one bet and loses one bet at c1 (because then the same thing will happen at c2). The likelihood of that is 1/2. However, if g wins both bets at one of the casinos, then g will win $200 by collecting from that one and abandoning the other one. So, g's expected winnings are half of $200 or $100.
This second warm-up suggests more bets per casino do not help. Let's see whether that is true.
Question: Compute the expected winnings for g using the strategy of warm-up 2, but with three bets at c1 and c2. What are the minimum winnings of g, the maximum and the expected winnings?
Solution: There are eight win/lose patterns at c1 (where 1 represents a win and 0 a loss). The 0s and 1s are flipped at c2.
000—g wins $300 at c2 and abandons c1
001—g wins $200 - $110 = $90 at c2 and abandons c1
010—g wins $90
011—g wins $90
100—g wins $90
101—g wins $90
110—g wins $90
111—g wins $300
So, the minimum is $90, the maximum is $300, and the expected value is (0.75 * $90) + (0.25 * $300) = $142.5. End of solution.
Question: Do the gains of g keep increasing as the number of bets increase? At what point are the expected gains more than $280?
The CFCs allow gamblers to start betting without a deposit.
Solution: Yes. By 20 bets, the gains are expected to be more than $280.
The CFCs look at their mounting losses and decide they must require any gambler g who makes k bets in a week to deposit (or "post" in gambling parlance) some fraction of g's maximum loss (which is k * $110). In such a case, if g loses at a particular casino, the casino can recover anything up to the post amount. Gambler g could still abandon the casino if the amount g owes is greater than the post amount.
Question: If the post fraction is 0.5, what is the expected gain of gambler g by making one bet at casino c1 and the opposite at c2?
Solution: Gambler g will win $100 at one of the casinos but will lose the deposit of $55 (0.5 * 1 * 110) at the other casino. So the net gain is $45.
Question: If the post fraction is 0.5, for which values of k (number of bets per week) does gambler g still have an expected advantage?
Solution: Only up to three bets (based on my simulator).
Question: What if the post fraction is 0.25?
Solution: Up to nine bets.
Upstart: Find the minimum post fraction for each value of k such that g never has an expected benefit by ceasing to gamble.
All are invited to submit their solutions to email@example.com; solutions to upstarts and discussion will be posted at http://cs.nyu.edu/cs/faculty/shasha/papers/cacmpuzzles.html
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