Card Sharks, Money Cards--optimal wagering strategy?
March 7th, 2011 at 6:56:51 PM
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Well, as you may have surmised from this and my "Price is Right" thread earlier...I am an admitted game show fan. I'm always looking for ways to combine my enjoyment of game shows with the topics we talk about on this forum, so I thought I'd throw this one out there.
For those who don't remember the 70's and 80's game show "Card Sharks," just do a search on YouTube for "card sharks money cards" to view tons of clips of the show. The "Money Cards" is the name of the bonus round that the winner of the main game played.
The basic rules: There are three rows of playing cards, 4 on the bottom, 3 in the middle and 1 on top. The player starts with a $200 bankroll and is shown the first card on the bottom row, and must guess whether the next card will be higher or lower in rank (deuces low, aces high), wagering all or part of their bankroll (minimum $50) on the guess. If they're right, they win even money; if not, they lose their wager.
When the contestant reaches the last card in the first row--or if s/he loses all her money on a card before that--that card is moved up to the middle row and she is given an additional $200 (or $400, depending on which version of the show you're watching). If the player makes it to the end of that row without busting out, the end card is moved to the top row and the player must risk at least half their bankroll to that point on one final guess.
There were a couple of rule variations between the different versions, which complicates things a bit...
1. Ties were considered a LOSS on the original version (the one with Jim Perry), just as they were in the main game. However, in the late-80's revival (the one with Bob Eubanks) ties were considered a push.
2. Originally, the player had the option to change only the first card on the bottom row, and replace it with the top card from the deck. Later, this was changed to being able to change a card up to three times, once per row.
Now to the main point of this thread...is there an optimal strategy that can be devised as to how much a contestant should wager on each card? Here's what I know so far:
-If ties are a push as on later versions of the show, then obviously the player should bet all his money on a 2 or Ace. But if ties lose, is going all-in still the right thing to do?
-I remember reading on another website that the player should never bet less than half their bankroll on any card, since they never have less than a 50% chance of being right. Again, this makes sense if ties push, since an 8--the worst possible card to get--would be a 50/50 shot either way. But if ties lose, wouldn't it make more sense to bet less than half on an 8, since the odds are automatically against them no matter which way they choose?
Then there's the matter of card counting to consider--to what extent does considering what cards you've seen already affect the decision making? (I've never been sure as to whether the bonus round is played with a fresh deck of cards, or the same deck that the contestant played with in the main game--if the latter, an observant contestant would have had a HUGE advantage going into the Money Cards, no?)
It's a surprisingly complicated game, and analysis of it would be academic--the show has been off the air for over 20 years, after all. But still, anyone with extra time on their hands care to help satisfy my curiosity?
For those who don't remember the 70's and 80's game show "Card Sharks," just do a search on YouTube for "card sharks money cards" to view tons of clips of the show. The "Money Cards" is the name of the bonus round that the winner of the main game played.
The basic rules: There are three rows of playing cards, 4 on the bottom, 3 in the middle and 1 on top. The player starts with a $200 bankroll and is shown the first card on the bottom row, and must guess whether the next card will be higher or lower in rank (deuces low, aces high), wagering all or part of their bankroll (minimum $50) on the guess. If they're right, they win even money; if not, they lose their wager.
When the contestant reaches the last card in the first row--or if s/he loses all her money on a card before that--that card is moved up to the middle row and she is given an additional $200 (or $400, depending on which version of the show you're watching). If the player makes it to the end of that row without busting out, the end card is moved to the top row and the player must risk at least half their bankroll to that point on one final guess.
There were a couple of rule variations between the different versions, which complicates things a bit...
1. Ties were considered a LOSS on the original version (the one with Jim Perry), just as they were in the main game. However, in the late-80's revival (the one with Bob Eubanks) ties were considered a push.
2. Originally, the player had the option to change only the first card on the bottom row, and replace it with the top card from the deck. Later, this was changed to being able to change a card up to three times, once per row.
Now to the main point of this thread...is there an optimal strategy that can be devised as to how much a contestant should wager on each card? Here's what I know so far:
-If ties are a push as on later versions of the show, then obviously the player should bet all his money on a 2 or Ace. But if ties lose, is going all-in still the right thing to do?
-I remember reading on another website that the player should never bet less than half their bankroll on any card, since they never have less than a 50% chance of being right. Again, this makes sense if ties push, since an 8--the worst possible card to get--would be a 50/50 shot either way. But if ties lose, wouldn't it make more sense to bet less than half on an 8, since the odds are automatically against them no matter which way they choose?
Then there's the matter of card counting to consider--to what extent does considering what cards you've seen already affect the decision making? (I've never been sure as to whether the bonus round is played with a fresh deck of cards, or the same deck that the contestant played with in the main game--if the latter, an observant contestant would have had a HUGE advantage going into the Money Cards, no?)
It's a surprisingly complicated game, and analysis of it would be academic--the show has been off the air for over 20 years, after all. But still, anyone with extra time on their hands care to help satisfy my curiosity?
"I believe I've passed the age/of consciousness and righteous rage/I've found that just surviving was a noble fight...
I once believed in causes too/I had my pointless point of view/And life went on no matter who was wrong or right..." --Billy Joel
March 8th, 2011 at 6:01:19 AM
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As with some of the threads I have started and contributed to, people have probably seen I am a game show fan as well. Good to have fellow game show lovers to discuss these pressing issues with!
Keep in mind that making it to the bonus round might be a once-in-a-lifetime thing. There is no guarantee that the champion will return to play it again, so going for broke might be the right play, especially if the amounts of money involved are relatively small. If you begin the game with $200, and only take $700 to the final bet, is it worth holding back $50 to make the "correct" bet, or just go double or nothing? Now, if you had $10,000 to bet on the final card, you might not even go with "correct", and may keep $2,000 back as a decent buffer.
Since the "ties push" version was the latest to air (not counting the HORRIBLE revival in 2001), that is the one I would strategize with. This was also the version used in the "Gameshow Marathon" special that aired in 2006, but with different money amounts.
I wrote a Card Sharks analyzer at one point, to try to figure out the best betting strategy. I'll see if I can dig that up, or rewrite it if necessary.
I know there is a minimum bet of $50, and I think players can bet any whole dollar amount, but it was rare that any player deviated from betting in multiples of $50. Caveat: Players had to wager at least half of their bankroll on the last card, which sometimes caused a hanging $25 or $75 on the bet.
One thing I am sure of: You have to be bold in the early stages if you want to have a large bankroll in the later stages.
It is a new shuffled and cut deck, and every card is exposed for viewing (except for cards that are covered when changed). So unfortunately, it is NOT the same deck as from the main game. Sidenote: The same deck is used throughout the front game's match, so observant players CAN get an advantage by remembering cards throughout that portion of the game.Quote: OneAngryDwarfI've never been sure as to whether the bonus round is played with a fresh deck of cards, or the same deck that the contestant played with in the main game--if the latter, an observant contestant would have had a HUGE advantage going into the Money Cards, no?
If ties lose, and a player is faced with an Ace or 2, they will have a 3/51 or 1/17 chance of losing on their first card. This will drop to 3/50, 3/49, 3/48... (assuming no other matching Aces or 2's are on the board) as the game goes on. The worst case would be: On the final card, if a card is changed on each row, an Ace or 2 would have losing odds of 3/42, or 1/14. So holding back that fraction of money would be the "correct bet".Quote: OneAngryDwarfIf ties are a push as on later versions of the show, then obviously the player should bet all his money on a 2 or Ace. But if ties lose, is going all-in still the right thing to do?
Keep in mind that making it to the bonus round might be a once-in-a-lifetime thing. There is no guarantee that the champion will return to play it again, so going for broke might be the right play, especially if the amounts of money involved are relatively small. If you begin the game with $200, and only take $700 to the final bet, is it worth holding back $50 to make the "correct" bet, or just go double or nothing? Now, if you had $10,000 to bet on the final card, you might not even go with "correct", and may keep $2,000 back as a decent buffer.
If ties lose, and a contestant chooses to play an 8 right out of the chute, they would have a 24/51, or 8/17 (~47%) chance of being correct, 9/17 (~53%) chance of being wrong. If possible, betting slightly less than half would be best.Quote: OneAngryDwarfI remember reading on another website that the player should never bet less than half their bankroll on any card, since they never have less than a 50% chance of being right. Again, this makes sense if ties push, since an 8--the worst possible card to get--would be a 50/50 shot either way. But if ties lose, wouldn't it make more sense to bet less than half on an 8, since the odds are automatically against them no matter which way they choose?
Since the "ties push" version was the latest to air (not counting the HORRIBLE revival in 2001), that is the one I would strategize with. This was also the version used in the "Gameshow Marathon" special that aired in 2006, but with different money amounts.
I wrote a Card Sharks analyzer at one point, to try to figure out the best betting strategy. I'll see if I can dig that up, or rewrite it if necessary.
I know there is a minimum bet of $50, and I think players can bet any whole dollar amount, but it was rare that any player deviated from betting in multiples of $50. Caveat: Players had to wager at least half of their bankroll on the last card, which sometimes caused a hanging $25 or $75 on the bet.
One thing I am sure of: You have to be bold in the early stages if you want to have a large bankroll in the later stages.
-Dween!
March 8th, 2011 at 6:53:29 AM
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New consideration: Betting strategies aside, what about card-changing strategies?
If playing by the old rules, where only the very first card can be changed, or only the first card in each row can be changed, the decision is easy: If a player has a 5 through Jack, they should change. Note the chances for improving or breaking even/improving for each card:
The above assumes 51 cards remain in the deck. As more cards are revealed, these percentages will change.
The 5/J change results in a breakeven (another 5 or J shows up), or improvement (2,3,4 or Q,K,A) 63% of the time. Changing with a 4 or Q may end up being feasible, as it is nearly a 50/50 split.EDIT: Updated second column with correct %, changing a 4/Q would be rare, and might only occur on the top level if only one card of the set 2, 3, K, A showed up in the game.
However, with the most up-to-date rules, a player can change 1 card per line, with the first two lines being 3 cards in length. Should a player change a card early to increase bankroll? Or wait til the end of the line? This may take more analysis than I can handle.
If playing by the old rules, where only the very first card can be changed, or only the first card in each row can be changed, the decision is easy: If a player has a 5 through Jack, they should change. Note the chances for improving or breaking even/improving for each card:
| Current Card | % Improve | % BE/Impr |
|---|---|---|
| 8 | 94% | 100% |
| 7 or 9 | 78% | 92% |
| 6 or 10 | 63% | 76% |
| 5 or J | 47% | 61% |
| 4 or Q | 31% | 45% |
| 3 or K | 16% | 29% |
| 2 or A | 0% | 14% |
The 5/J change results in a breakeven (another 5 or J shows up), or improvement (2,3,4 or Q,K,A) 63% of the time. Changing with a 4 or Q may end up being feasible, as it is nearly a 50/50 split.EDIT: Updated second column with correct %, changing a 4/Q would be rare, and might only occur on the top level if only one card of the set 2, 3, K, A showed up in the game.
However, with the most up-to-date rules, a player can change 1 card per line, with the first two lines being 3 cards in length. Should a player change a card early to increase bankroll? Or wait til the end of the line? This may take more analysis than I can handle.
-Dween!
March 8th, 2011 at 2:01:59 PM
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Card Sharks (Jim Perry version) was my absolute favorite game show as a young kid, maybe that planted the gambling seed in me? Perry was a far more exciting host (see also his work on $ale of the Century, particularly the speed round).
If I recall correctly only 1 person ever made it thru the Money Cards winning the maximum. GSN showed that run in their "Greatest Game Shows of all time" series.
Trivia for you game show buffs: The theme music for the Jim Perry version was originally used on a different game show..what was it?
If I recall correctly only 1 person ever made it thru the Money Cards winning the maximum. GSN showed that run in their "Greatest Game Shows of all time" series.
Trivia for you game show buffs: The theme music for the Jim Perry version was originally used on a different game show..what was it?
March 8th, 2011 at 3:23:47 PM
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Quote: slyther
Trivia for you game show buffs: The theme music for the Jim Perry version was originally used on a different game show..what was it?
"Double Dare"...not the Nickelodeon kids' game show, but a short-lived series that aired on CBS in 1976.
Now, to tie that show into something a bit more popular...what does that show have in common with the "Pirates of the Caribbean" movie series?
BTW, I agree with you about Jim Perry...Bob Eubanks worked well for the Newlywed Game, but he was out of his element on CS.
"I believe I've passed the age/of consciousness and righteous rage/I've found that just surviving was a noble fight...
I once believed in causes too/I had my pointless point of view/And life went on no matter who was wrong or right..." --Billy Joel
March 9th, 2011 at 5:01:24 AM
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Agree with the sentiment that Perry was a better fit for Card Sharks than Eubanks. What about Bill Rafferty? He hosted a syndicated version in 1986, with the only noticeable difference being "Prize Cards" added in the main game.
Update on betting strategies for the Money Cards:
The following table shows 5 different betting strategies, the average amount won, the % of bust outs, and the biggest win. These were run through 100,000 trials each. This was also using the strategy of never changing cards. In all cases, the minimum bet was $50, all bets were in multiples of $50, and never exceeded the current bankroll.
Rules assumed: $200 to start, +$400 for second row, change 1 card per line anywhere on line.
What do those betting strategies mean?
Always bet x%: Rounded up to the nearest $50, always bet this % of the bankroll on every card.
Bet the odds: Bet a fraction of the bankroll, where numerator is # cards that will lose,. denominator is # cards that will win. Example, if the first card is a 10, then there are four 2's, 3's, 4's... up to 9's, or 32 winning cards. There are four J's, Q's, K's and A's, or 16 losing cards. 16/32 = 1/2. This fraction is then subtracted from 1, and the resulting fractional amount is bet from the bankroll. After spelling this method out in English, I just realized it is not quite the way I wanted the bet to work. Fix to come later.
Bet odds+50: Use the above, but add an extra $50 to the result, not to exceed the bankroll.
Update on betting strategies for the Money Cards:
The following table shows 5 different betting strategies, the average amount won, the % of bust outs, and the biggest win. These were run through 100,000 trials each. This was also using the strategy of never changing cards. In all cases, the minimum bet was $50, all bets were in multiples of $50, and never exceeded the current bankroll.
Rules assumed: $200 to start, +$400 for second row, change 1 card per line anywhere on line.
| --- | Always bet 50% | Always bet 75% | Always bet 100% | Bet the odds | Bet odds+50 |
|---|---|---|---|---|---|
| Busts | 0.003% | 1.182% | 69.361% | 0.455% | 3.075% |
| Avg. Win | $1,912.93 | $3,040.35 | $5,503.08 | $3,352.36 | $3,761.12 |
| Max. Win | $5,650 | $13,650 | $32,000 | $28,900 | $30,800 |
What do those betting strategies mean?
Always bet x%: Rounded up to the nearest $50, always bet this % of the bankroll on every card.
Bet the odds: Bet a fraction of the bankroll, where numerator is # cards that will lose,. denominator is # cards that will win. Example, if the first card is a 10, then there are four 2's, 3's, 4's... up to 9's, or 32 winning cards. There are four J's, Q's, K's and A's, or 16 losing cards. 16/32 = 1/2. This fraction is then subtracted from 1, and the resulting fractional amount is bet from the bankroll. After spelling this method out in English, I just realized it is not quite the way I wanted the bet to work. Fix to come later.
Bet odds+50: Use the above, but add an extra $50 to the result, not to exceed the bankroll.
-Dween!
March 9th, 2011 at 10:42:01 AM
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Excellent analysis Dween, thanks! You are right though, the long run doesn't really mean much if you're actually a contestant on a game show. Since you only get one chance to play the vast majority of the time, sometimes it works to just go with that "gut feeling." Doubling up every single time may indeed be best for 100,000 trials...but if you try it on your first shot, bust out, then lose the next main game, it didn't get you anywhere...
"I believe I've passed the age/of consciousness and righteous rage/I've found that just surviving was a noble fight...
I once believed in causes too/I had my pointless point of view/And life went on no matter who was wrong or right..." --Billy Joel
March 9th, 2011 at 10:58:20 AM
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I'm finding bugs in my simulation, which I am squashing as they are found. Some information above may be incorrect.
I'm upping the simulations to 1,000,000 trials, and implementing "change card if..." rules of:
* Change on 5-J
* Change on 6-10
* Change on 7-9
* Change on the following conditions:
----- First card in row, change if 7-9
----- Second card in row, change if 6-10
----- Last card in row, change if 5-J
Any other "change card if..." or wagering scenarios you can come up with, I'd love to hear.
I'm upping the simulations to 1,000,000 trials, and implementing "change card if..." rules of:
* Change on 5-J
* Change on 6-10
* Change on 7-9
* Change on the following conditions:
----- First card in row, change if 7-9
----- Second card in row, change if 6-10
----- Last card in row, change if 5-J
Any other "change card if..." or wagering scenarios you can come up with, I'd love to hear.
-Dween!
March 9th, 2011 at 4:20:19 PM
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Quote: OneAngryDwarf"Double Dare"...not the Nickelodeon kids' game show, but a short-lived series that aired on CBS in 1976.
Now, to tie that show into something a bit more popular...what does that show have in common with the "Pirates of the Caribbean" movie series?
Best I can come up with is Double Dare was hosted by Alex Trebek, who also hosted "High Rollers" which is a dice game..and they played Liar's Dice in the 2nd pirates movie?
I loved the kids Double Dare (i was a teen when those type of shows were airing), and I even had a filmed audition at an affiliate for "Fun House" but alas was not selected.
March 9th, 2011 at 4:29:51 PM
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Good guess...the real connection is that Jay Wolpert, creator of the series, co-wrote the screenplays.
"I believe I've passed the age/of consciousness and righteous rage/I've found that just surviving was a noble fight...
I once believed in causes too/I had my pointless point of view/And life went on no matter who was wrong or right..." --Billy Joel
March 10th, 2011 at 12:20:00 PM
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Here is my final analysis of the Money Cards on Card Sharks. Each mini-system ran 1,000,000 times. Here is what each term represents:
RightCards is how many cards left in the deck would result in a correct call.
WrongCards is how many cards left in the deck would result in incorrect call.
RightCardsInclTies is how many cards left in deck would result in a correct OR push call.
TotalCards is how many cards are left in the deck.
* The strategies tend to improve moving down and to the right (save for the 100% method, which is high-risk).
* The "Bet 100%" method busts over half the time, so even though it has a higher average, the RISK value is lower.
* "Bet 50%" almost never busts, but has a low average, and you'll never win more than $5,650.
* If you keep your bet consistent with the probability of making a correct call, busting out stays at or below 1%.
* The "best" method, which is to the "Btie/Tot,+4", and changing with the above described "7,6,5" rule, works out with the highest average win, $6,272, but has an 8% change of busting.
Longer explanation of "Btie/Tot, +4"...
The idea behind this strategy is to pretend your card is one rank better, and bet based on the odds of your call being right.
Say you have a 10 as the first card. That means a call of 2 through 10 would be correct or push... that's 35 cards. (Btie)
The total number of cards unseen is 51. (Tot)
The amount bet would be: (35+4/51), or 39/51, or just over 75% of the bankroll. Since we're rounding up, this would be all $200 on the first card. Rounding down, we might have less busts and a lower average.
If there is a different strategy to changing cards that you think might do better, let me know. I can only go by my own ideas at this point, and the 7,6,5 strategy seems like a decent enough idea. It seems to do better than changing the first 6-10 seen.
Same goes for betting strategy. Betting with the odds works well, but adding $50 or pretending you have a better card gives a push to the average win, but carries a higher risk.
OneAngryDwarf, I hope this answers your original question. This was a question I've always wanted to answer for myself, and I am glad I had a push to do it.
| 50% | Always bet half the bankroll, rounded up to nearest $50 |
|---|---|
| 75% | Always bet 75% of the bankroll, rounded up to nearest $50 |
| 100% | Always bet the whole bankroll |
| HiLo Frac | Bet fraction of bankroll: WrongCards÷RightCards |
| HLFrac+50 | Bet fraction of bankroll: WrongCards÷RightCards, plus $50 |
| Btie/Tot | Bet fraction of bankroll: RightCardsInclTies÷TotalCards |
| Btie/Tot+50 | Bet fraction of bankroll: RightCardsInclTies÷TotalCards, plus $50 |
| Btie/Tot, +4 | Bet fraction of bankroll: (RightCardsInclTies+4)÷TotalCards |
| Busts | Number of times player busts out |
| Avg Win | Average amount won |
| Avg(nobust) | Average amount won, disregarding busts |
| Max win | Best score for algorithm |
| Risk | Average win * Bust % |
| Change: None | Never change cards |
| Change: Any 5-J | Anytime a card comes up that's between 5 and J, change it |
| Change: Any 6-10 | Anytime a card comes up that's between 6 and 10, change it |
| Change: 7,6,5 | Change 1st card on line if 7-9, change 2nd if 6-10, change last if 5-J |
RightCards is how many cards left in the deck would result in a correct call.
WrongCards is how many cards left in the deck would result in incorrect call.
RightCardsInclTies is how many cards left in deck would result in a correct OR push call.
TotalCards is how many cards are left in the deck.
| Wager | 50% | 75% | 100% | Hilo Frac | HLFrac+50 | Btie/Tot | Btie/Tot+50 | Btie/Tot,+4 |
|---|---|---|---|---|---|---|---|---|
| Change | None | None | None | None | None | None | None | None |
| Busts | 45 | 11010 | 643234 | 4172 | 29452 | 10479 | 66990 | 70607 |
| Avg Win | 1911 | 3026 | 4768 | 3350 | 3650 | 3713 | 4024 | 4152 |
| Avg(nobust) | 1911 | 3059 | 13366 | 3364 | 3761 | 3752 | 4313 | 4468 |
| Max win | 5650 | 13650 | 32000 | 30400 | 31000 | 30500 | 32000 | 32000 |
| Risk | 1911 | 2992 | 1701 | 3336 | 3543 | 3674 | 3755 | 3859 |
| Change | Any 5-J | Any 5-J | Any 5-J | Any 5-J | Any 5-J | Any 5-J | Any 5-J | Any 5-J |
| Busts | 23 | 5522 | 546110 | 3931 | 26312 | 8822 | 53407 | 80165 |
| Avg Win | 2320 | 3997 | 6824 | 4961 | 5377 | 5404 | 5846 | 6128 |
| Avg(nobust) | 2320 | 4019 | 15036 | 4981 | 5523 | 5452 | 6176 | 6662 |
| Max win | 5650 | 13650 | 32000 | 31600 | 32000 | 32000 | 32000 | 32000 |
| Risk | 2320 | 3974 | 3097 | 4942 | 5236 | 5357 | 5534 | 5636 |
| Change | Any 6-10 | Any 6-10 | Any 6-10 | Any 6-10 | Any 6-10 | Any 6-10 | Any 6-10 | Any 6-10 |
| Busts | 7 | 4692 | 538156 | 3979 | 28303 | 9397 | 54793 | 78774 |
| Avg Win | 2360 | 4087 | 7042 | 4999 | 5449 | 5469 | 5952 | 6216 |
| Avg(nobust) | 2360 | 4106 | 15248 | 5019 | 5608 | 5521 | 6297 | 6747 |
| Max win | 5650 | 13650 | 32000 | 31600 | 32000 | 31650 | 32000 | 32000 |
| Risk | 2360 | 4068 | 3252 | 4979 | 5295 | 5418 | 5625 | 5726 |
| Change | 7, 6, 5 | 7, 6, 5 | 7, 6, 5 | 7, 6, 5 | 7, 6, 5 | 7, 6, 5 | 7, 6, 5 | 7, 6, 5 |
| Busts | 7 | 4644 | 534124 | 4077 | 27968 | 9387 | 55220 | 80642 |
| Avg Win | 2371 | 4118 | 7129 | 5057 | 5520 | 5526 | 6046 | 6272 |
| Avg(nobust) | 2371 | 4137 | 15304 | 5077 | 5679 | 5578 | 6399 | 6822 |
| Max win | 5650 | 13650 | 32000 | 32000 | 32000 | 31650 | 32000 | 32000 |
| Risk | 2371 | 4099 | 3321 | 5036 | 5366 | 5474 | 5712 | 5766 |
* The strategies tend to improve moving down and to the right (save for the 100% method, which is high-risk).
* The "Bet 100%" method busts over half the time, so even though it has a higher average, the RISK value is lower.
* "Bet 50%" almost never busts, but has a low average, and you'll never win more than $5,650.
* If you keep your bet consistent with the probability of making a correct call, busting out stays at or below 1%.
* The "best" method, which is to the "Btie/Tot,+4", and changing with the above described "7,6,5" rule, works out with the highest average win, $6,272, but has an 8% change of busting.
Longer explanation of "Btie/Tot, +4"...
The idea behind this strategy is to pretend your card is one rank better, and bet based on the odds of your call being right.
Say you have a 10 as the first card. That means a call of 2 through 10 would be correct or push... that's 35 cards. (Btie)
The total number of cards unseen is 51. (Tot)
The amount bet would be: (35+4/51), or 39/51, or just over 75% of the bankroll. Since we're rounding up, this would be all $200 on the first card. Rounding down, we might have less busts and a lower average.
If there is a different strategy to changing cards that you think might do better, let me know. I can only go by my own ideas at this point, and the 7,6,5 strategy seems like a decent enough idea. It seems to do better than changing the first 6-10 seen.
Same goes for betting strategy. Betting with the odds works well, but adding $50 or pretending you have a better card gives a push to the average win, but carries a higher risk.
OneAngryDwarf, I hope this answers your original question. This was a question I've always wanted to answer for myself, and I am glad I had a push to do it.
-Dween!
March 10th, 2011 at 2:29:36 PM
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Very cool Dween! Now if only someone could figure out how to play "Pay The Rent" on TPIR! :)
