An online sportsbook is advertising a “Risk Free” promotion for new users: If your first wager loses, you will be awarded a “free bet” in the amount of your wager (up to $1,000).
In reading the terms and conditions of the promotion, you note that unlike a normal wager, a “free bet” does not return the stake along with any winnings. e.g. if you wager $1,000 cash on a selection with odds of 1.95 and win, you’d get back $1,950 (your $1,000 stake is returned, plus $950 in winnings); if you’d placed a $1,000 “free bet” on that same selection, you’d only be paid the $950 in winnings.
For this puzzle make the following simplifying assumptions:
- The sportsbook reduces the fair odds of all of their selections by 2.5%. For example, the true odds on selection with a 50% probability of occurring should be 2.00. However the sportsbook pays only 2 * (1 - 0.025) = 1.95.
- The sportsbook knows the true probabilities of all events occurring, and uniformly prices all selections with the same 2.5% margin.
- The sportsbook offers a large enough variety of markets that any odds you seek are available for you to bet on.
a) Determine a strategy to maximize the expected value of this promotion.
b) Assuming you wanted to make this a truly risk-free proposition, maximize the amount of guaranteed profit you can get out of this promotion.
clc; clearvars
max=0;
for p=.01:.14:.99
fprintf('%5.2f ',p)
for p2=.01:.14:.99
q=1-p; q2=1-p2;
bet=1000;
nomPayoff=bet/p-bet;
nomPayoff2=bet/p2-bet;
payFrac=.975;
expVal=-bet+p*(nomPayoff*payFrac+bet)+q*p2*nomPayoff2*payFrac;
fprintf(' %6.2f', expVal)
if expVal>max
max=expVal;
mp=p; mp2=p2;
end
end
disp(" ")
end
disp([mp mp2 max])
finds the highest expected value when the probability of winning either bet is the lowest, that is has the highest payoff. That's at p=.01 of winning for each game and an expected value overall of 930.8475. The program produces a brief table of expected value:
p2
p1 .01 .15 .29 .43 .57 .71 .85 .99
0.01 930.85 795.71 660.58 525.44 390.31 255.17 120.04 -15.10
0.15 799.21 683.19 567.16 451.14 335.11 219.09 103.06 -12.96
0.29 667.58 570.66 473.75 376.83 279.92 183.00 86.09 -10.83
0.43 535.94 458.14 380.33 302.53 224.72 146.92 69.11 -8.69
0.57 404.31 345.61 286.92 228.22 169.53 110.83 52.14 -6.56
0.71 272.67 233.09 193.50 153.92 114.33 74.75 35.16 -4.42
0.85 141.04 120.56 100.09 79.61 59.14 38.66 18.19 -2.29
0.99 9.40 8.04 6.67 5.31 3.94 2.58 1.21 -0.15
So play the higest odds both times and have the best chance of winning almost a hundred thousand dollars, although it's still more likely you'll lose the $1000.
b) I can't see any way of getting a guaranteed profit. Both bets can lose.
Edited on March 10, 2021, 12:41 pm
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Posted by Charlie
on 2021-03-10 12:39:51 |
computer calculations
