Suppose a gambler makes a series of independent bets. In each bet, there's a 50% chance that he loses $\$100$.

If he wins, he has an option to take $\$100$ or run a "winner's bet", instead of taking that reward. In the winner's bet, there is a 60% chance to win 50 dollars, a 30% chance to win 150 dollars, and a 10% to win 200 dollars.

What should he do in order to maximize the expected reward?

Note: This is not homework. I've been out of school for a decade. :)

  • $\begingroup$ Which browser and rendered funny how? It rendered ok on both the ones I tried. I'm guessing maybe the If he wins, he has an option... looks cramped (like $If he wins, he has an option$) in which case it would be because $ characters are considered special here. They are used to indicate the start and end of math equations. $\endgroup$ – cardinal Oct 1 '11 at 23:56
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    $\begingroup$ The expected gain from the winner's bet is $0.60 \cdot 50 + 0.3 \cdot 150 + 0.1 \cdot 200 = 30 + 45 + 20 = 95$, so the winner's bet is a loser... $\endgroup$ – cardinal Oct 2 '11 at 0:01
  • $\begingroup$ Does that reformatting help? $\endgroup$ – cardinal Oct 2 '11 at 0:03
  • $\begingroup$ Thanks for the reformatting! How do you enter a dollar sign here? $\endgroup$ – Tom Tucker Oct 2 '11 at 0:09
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    $\begingroup$ The edits will show you how. But, briefly, $\$$ will get you a dollar sign. :) $\endgroup$ – cardinal Oct 2 '11 at 1:24

@Cardinal hasn't elaborated on his/her correct answer so I'll do so...

To learn the expected value of the whole we calculate the expected value of the parts. Let's start with the optional winner's bet. Expected winnings = Value of each prize * the probability of each prize.
$\$$50 * 0.6 = $\$$30
$\$$150 * 0.3 = $\$$45
$\$$200 * 0.1 = $\$$20

30+45+20 = $\$$95

The winner's bet has expected winnings of $\$$95 but the gambler always has a choice between the winner's bet and a guaranteed $\$$100. $\$$100 > $\$$95 so a gambler maximizing their expected winnings will never take the winner's bet. Knowing that the winner's bet is a bad idea, we can ignore it when calculating the expected value of the overall game. The game reduces to a 50% chance of winning $\$$100 and a 50% chance of losing $\$$100, resulting in expected winnings of $\$$0.

Therefore, the gambler can either play the game (and always decline the winner's bet) or not play the game at all, and have the same amount of money in expectation.

  • $\begingroup$ While it's not immediately relevant to this particular game, it's worth mentioning that even in games in which the bettor has an edge, maximizing expectation is a very poor strategy to use in repeated play. $\endgroup$ – cardinal Oct 9 '11 at 21:31
  • $\begingroup$ I think that what @cardinal is getting at is that there are some games with positive expected winnings in which the player nonetheless has a non-trivial chance of losing a lot of money. One needs to consider the entire distribution of possible outcomes, not just the mean winnings, to make rational decisions. Especially since most people find losses hurt more than equal size gains feel good. $\endgroup$ – Michael Bishop Oct 9 '11 at 22:14
  • $\begingroup$ Well, you're on the right track. In repeated play of even the simplest of games with an edge, maximizing expected value leads to a probability of ruin (i.e., going broke) of 1 in the long run. Strategies to optimize other sensible objective criteria do not have this (rather serious) flaw. $\endgroup$ – cardinal Oct 9 '11 at 22:17

Agree with @Michael Bishop. Given the current probabilities on the winner's bet there is no incentive to opt for it. Given the your data, if the gambler opts for the winners bet his payoff on an average is 95, while his payoff is deterministically 100 if he does not opt for the winners bet.

These problems are best solved using a decision tree. Unfortunately I am not being allowed to post a picture showing the payoff choices for this problem.

However, even a minor change in probabilities or the payoffs could change the choice the gambler needs to make to maximize his payoffs. You may play around with the numbers (probabilities / payoffs) and do some sensitivity analysis. For instance with either of the two changes mentioned below it would make more sense for the gambler to opt for the Winners bet.

  1. If the probability of wining 50, 150 and 200 were changed to 0.5, 0.25, 0.25 (keeping the payoffs same)
  2. If the maximum payoff is increased from 200 to 255, while the probabilities remained the same.
  • $\begingroup$ Welcome to the site. If you'd like to edit your post and add a link to the picture you want displayed, then I or someone else can follow-up with an additional edit to actually show it. Cheers. :) $\endgroup$ – cardinal Oct 8 '11 at 15:43

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