Greg Paul
United States Phoenix Arizona

[This is a simplified version of a example I posted to another very long thread.]
Suppose I just invented a simple new variation of poker. The game uses seven cards in a deck, they are numbered 17. Each time we play I will draw a single card, if I draw 1, 2 or 3 I will win. If I draw 4, 5, 6 or 7 you will win. So I will draw a winning card 3/7 times or 43% of the time, while I will draw a losing card 4/7 times or 57% of the time.
The game works like this:
 We each ante $1 before I draw a card, so there's $2 total in the pot.  I will draw a card and look at it, you cannot see which card I have until we are done betting.  I will then be first to act and can check or bet $1.  On your turn you will then have the typical poker options of checking, calling, folding, betting, or raising in increments of $1. If you fold then I win the $2 antes.
 If you bet or raise I will then have the choice of either folding or calling you for the additional $1. I cannot raise you.
It looks like you're in good shape since you will win 57% of the time. Would you want to gamble some money and play this game with me?
Spoiler (click to reveal) The answer: No you don't.
Even though the odds are in your favor that I will draw the card that causes you to win 4/7ths of the time, I can bluff in such a way that you are guarenteed to be a loser in this game. I could tell you before we start the game that I will bluff and explain exactly how I will bluff and you still can't win in the long run.
How does this work?
There are 7 card in the deck. I will take the following actions:
Cards 1,2,3 I will bet $1 with the winning hand. Card 4 I will bet $1 as a complete bluff. Cards 5,6,7 I will check to you and fold if you bet.
You now know exactly what my strategy is, but what can you do to respond to me? First of all recognize if I check to you that there's absolutely no profit for you to bet. Since I know whether I have the winning hand I will never call with a losing hand. Thus if I check to you then you should also check.
A) Let's say you decide to call whenever I bet. What happens:  (13) You will call my bet and lose a net of $2 per hand ($6 total).  (4) You will call my bluff bet and win a net profit of $2 per hand (+$2 total)  (57) You will check and win a net profit of $1 per hand (+$3 total)
Summing up the 7 possibilities we see over an average 7 hands you will show a net result of $1 every 7 hands or an average loss of $0.14/hand.
B) Let's say you decide to fold whenever I bet. What happens:  (13) You will fold to my bet and lose a net of $1 per hand ($3 total)  (4) You will fold to my bluff and lose a net of $1 per hand ($1 total)  (57) You will check when I check and win a net of $1 per hand (+$3 total).
Summing up the 7 possibilities you again show a net result of $1 every 7 hands, again an average loss of $0.14/hand.
Let's say you go with option (C) and raise whenever I bet:
C) You check when I check and raise when I bet.  (13) I will call your raise and you will lose $3 per hand ($9 total)  (4) I will fold to you raise and you will win $2 per hand (+$2 total)  (57) You will win $1 per hand (+$3 total).
The sum is now $4 over 7 hands or $0.57/hand which is even worse than folding or calling.
It looks like magic doesn't it? But this is solid game theory using probabilities. I was able to turn an underdog situation into a positive situation simply by bluffing at an optimal rate. Even better I am bluffing at a frequency that means it makes absolutely no difference in the long run whether you fold or call my betsyou will lose the same amount either way.
Why post this example? The principle is illustrative for many games that allow some opportunity to bluff. If you never bluff or always bluff you're giving too much information away. In fact in this game if I always bluffed or never bluffed you would be a winner. By bluffing at the right frequency I am now the winner and you cannot defeat my strategy even though I've told you exactly what I'm doing beforehand.

Mike K
United States Fairless Hills Pennsylvania

pwn3d wrote: If you fold do I win?
If that is the case I would play this game.
1) If you check, I bet. 2) If you bet, I call.
My EV is positive so why not play this game? I need to study the model more closely, but I can tell you that this reasoning is flawed.
With (1), he folds, and you win $1 (3 times). With (2), you lose $2 if you're wrong. With his strategy, he wins 3 out of 4 times.
EV = (3*$1) + (3*{$2}) + ($2) = 36+2 = 1 (div. by 7) = $.14
There may yet be a combo strategy of callvsraisevsfold that works. Bears researching.

Greg Jones
United States Washington

Without doing the math, I would say no, since you've got all the information and that probably outweighs a slight disadvantage in odds.

Mike K
United States Fairless Hills Pennsylvania

carpejugulum wrote: [This is a simplified version of a example I posted to another very long thread.] Suppose I just invented a simple new variation of poker. The game uses seven cards in a deck, they are numbered 17. Each time we play I will draw a single card, if I draw 1, 2 or 3 I will win. If I draw 4, 5, 6 or 7 you will win. So I will draw a winning card 3/7 times or 43% of the time, while I will draw a losing card 4/7 times or 57% of the time. The game works like this:  We each ante $1 before I draw a card, so there's $2 total in the pot.  I will draw a card and look at it, you cannot see which card I have until we are done betting.  I will then be first to act and can check or bet $1.  On your turn you will then have the typical poker options of checking, calling, folding, betting, or raising in increments of $1. If you fold then I win the $2 antes.  If you bet or raise I will then have the choice of either folding or calling you for the additional $1. I cannot raise you. It looks like you're in good shape since you will win 57% of the time. Would you want to gamble some money and play this game with me? Spoiler (click to reveal) The answer: No you don't.
Even though the odds are in your favor that I will draw the card that causes you to win 4/7ths of the time, I can bluff in such a way that you are guarenteed to be a loser in this game. I could tell you before we start the game that I will bluff and explain exactly how I will bluff and you still can't win in the long run.
How does this work?
There are 7 card in the deck. I will take the following actions:
Cards 1,2,3 I will bet $1 with the winning hand. Card 4 I will bet $1 as a complete bluff. Cards 5,6,7 I will check to you and fold if you bet.
You now know exactly what my strategy is, but what can you do to respond to me? First of all recognize if I check to you that there's absolutely no profit for you to bet. Since I know whether I have the winning hand I will never call with a losing hand. Thus if I check to you then you should also check.
A) Let's say you decide to call whenever I bet. What happens:  (13) You will call my bet and lose a net of $2 per hand ($6 total).  (4) You will call my bluff bet and win a net profit of $2 per hand (+$2 total)  (57) You will check and win a net profit of $1 per hand (+$3 total)
Summing up the 7 possibilities we see over an average 7 hands you will show a net result of $1 every 7 hands or an average loss of $0.14/hand.
B) Let's say you decide to fold whenever I bet. What happens:  (13) You will fold to my bet and lose a net of $1 per hand ($3 total)  (4) You will fold to my bluff and lose a net of $1 per hand ($1 total)  (57) You will check when I check and win a net of $1 per hand (+$3 total).
Summing up the 7 possibilities you again show a net result of $1 every 7 hands, again an average loss of $0.14/hand.
Let's say you go with option (C) and raise whenever I bet:
C) You check when I check and raise when I bet.  (13) I will call your raise and you will lose $3 per hand ($9 total)  (4) I will fold to you raise and you will win $2 per hand (+$2 total)  (57) You will win $1 per hand (+$3 total).
The sum is now $4 over 7 hands or $0.57/hand which is even worse than folding or calling.
It looks like magic doesn't it? But this is solid game theory using probabilities. I was able to turn an underdog situation into a positive situation simply by bluffing at an optimal rate. Even better I am bluffing at a frequency that means it makes absolutely no difference in the long run whether you fold or call my betsyou will lose the same amount either way.
Why post this example? The principle is illustrative for many games that allow some opportunity to bluff. If you never bluff or always bluff you're giving too much information away. In fact in this game if I always bluffed or never bluffed you would be a winner. By bluffing at the right frequency I am now the winner and you cannot defeat my strategy even though I've told you exactly what I'm doing beforehand.
Based on my early trials ... I agree with you 100%.
First off, I can ignore the 'raise' option altogether. I can't do any better raising that I can calling; if you don't have it, you fold (so I get the same amount), and if you do have it, I only lose more. Thus (assuming you bet) I must either call or fold.
I tried a 7525 strategy: fold 75% of the time, and call 25% of the time (matching the odds that you do have it). I ended up coming up with the exact same expected value of $.14 that you got with either 'extreme' strategy (either always call or always fold).
It's actually a straightforward algebraic exercise to show that, with any call/fold probability strategy, the EV is $.14.
(Of course, poker is so much more complex that this example; still, the idea shows what all good poker players know; you need to be able to bluff successfully.)
Something to show my colleagues and students this year. Thank you!



Brilliant.
The key is that because you have the information, you can turn 4 from a losing to a winning card.
If he was the one who gets to see the information, he can increase his chances from 4/7 to something even higher by bluffing on all 3s right? Hell he can bet on every single round and still win, although probably not as much if he only bluffed on 1 or 2 of the 3 cards.



Quote: I tried a 7525 strategy: fold 75% of the time, and call 25% of the time (matching the odds that you do have it). I ended up coming up with the exact same expected value of $.14 that you got with either 'extreme' strategy (either always call or always fold).
I was thinking the exact same thing but it turns out, it's just playing "fold" for 3 games and playing "call" for 1 game. Your expected gain is negative in each of those 4 games.

Greg Paul
United States Phoenix Arizona

For me what's so interesting about this example is if I employed a strategy of only betting when I draw the winning 13 card or the opposite extreme of betting every single hand regardless of which card I draw then I would be an overall loser if you knew how I played. By bluffing the right amount I swing from an overall loser to a winner even though you know exactly how I play.
Returning to the game, suppose I always bet no matter which card I draw.
A1) If you fold whenever I bet you'll lose $7 over 7 hands or $1/hand since I will bet every time.
A2) If you call whenever I bet then  (13) You will call my bet and loose a net of $2 per hand ($6 total)  (47) You will call my bluff and win a total of $2 per hand (+$8 total)
Summing up the 7 possibilities you show a net profit of +$2 or +$0.28/hand. So if I bet every single time you should call every single time and you will be an overall winner.
Now, suppose I only bet when I draw the winning card (13) but will never bluff.
B1) If you always call when I bet, then  (13) You will call my bet and lose $2 per hand ($6 total)  (47) You will check and win a net of $1 per hand (+$4 total)
Summing up, if I never bluff and you call me you will lose $2 or $0.28/hand. Since I never bluff you should not call when I bet.
B2) But, if I never bluff and you always fold when I bet, then  (13) You will fold to my bet and lose $1 per hand ($3 total)  (47) You will check and win a net profit of $1 per hand (+$4 total)
Summing up the 7 possibilities if I never bluff then you can safely fold whenever I do bet and you will show a net profit of +$1 or +$0.14/hand.
In short, if I bet every single time regardless of which card I draw then you would be an overall winner by calling every time. If I never bluffed and only bet when I draw the winning card then you would be an overall winner by folding whenever I bet.
By finding a strategy between these two extremes I've switched from being an underdog to a profitable EV situation.

Joe Grundy
Australia Sydney NSW

That is an outstanding post!
To add complexity (so skip this if you don't want complexity)...
I will continue the labels that You are the dealer, and I am player two.
Of course You don't necessarily need to bluff in such precise circumstances. You could, for instance, say you will bluff "half the time" when it's a 4. In the particular game you propose it turns out that bluffing all the time on one particular losing card (such as 4) gives you the highest return, but that's sort of coincidental.
Change The Game #1 If you change the game so that You win on 1,2,3,4 and I lose on 5,6,7 and then force me to play ($1 ante, $1 raise) you can still lift your expected return by throwing in some bluffing! ... So how much do you win? If you never bluff: I fold when you bet and you win $1 per 7 hands If you bluff on a 5: I fold when you bet and you win $3 per 7 hands If you bluff on 5 or 6: I call when you bet and you win $3 per 7 hands If you always bet: I call when you bet and you win $1 per 7 hands
To be precise, if You bluff 4/9 times when You have a "losing" card, you hit that sweet spot and whatever I do you will win $3.67 per 7 hands.
Change The Game #2 Can You make the return positive if You only win with 1 or 2? No, but You minimise your losses (at $1.67 per 7 hands) if you bluff 2/3 of the time when you have a 3.
Change The Game #3 What if the ante is twice the betting increment? So we each ante $2, and then you can raise by $1 if you like. You win on 1,2,3 and I win on 4,5,6,7 ... In this case, the best You can do is force us to break even. If Your strategy is to bluff on a 4, then I will call every bet and we'll break even.
Dynamic Strategy And finally, if You don't tell me your exact strategy and I have to work it out from your behaviour?
Well, You can start with the "You win" strategy of bluffing on every 4. (It turns out this is the only way that you make a profit.) Whether I always call or always fold when You bet, you still expect to make $1 per seven hands.
Eventually, I figure out what you are doing. I have three options:  Keep on always calling  Stop calling  Mix them (since it makes no difference to the expected loss) If I keep always calling, you can stop bluffing on 4s. I will never know, and You now expect to make $2 per seven hands If I stop ever calling, you can bluff much more frequently... if you bluff on 4s and 5s (and I always fold) I may well not notice, and your take goes up to $3 per seven hands If I mix and match it's more interesting. For example, You can start bluffing on 4s and 5s until I happen to catch You bluffing on a 5. (Except of course You won't really strictly bluff on 4s or 5s, you'll bluff a certain fraction of the time and it will take me much longer to gain statistical confidence in Your current strategy.) Then presumably I'll start calling most or all of your bets again. If I do, you can stop bluffing (and I'll never know) until I fold again some time.
Or You could start out being completely honest, as an investment, and once I give up calling You can introduce bluffing.
This is a really cool line of thought. Alas I don't have time at the moment to work out the best concealed dynamic strategies for player 1 and player 2.

Greg Paul
United States Phoenix Arizona

pwn3d wrote: How much of a draw advantage could the better give away that and still have the upper hand?
I specificly choose the # of cards in the game to demonstrate optimal bluffing. Optimal bluffing is bluffing at a frequency that means it doesn't matter whether the other player calls or folds, he will have the same longterm expectation either way.
It can be shown that optimal bluffing frequency is exactly equal to the pot odds you are offering the other player. In this game when I bet the other player is getting 31 odds to call (there will be $3 in the pot and he can call another $1 to try and win). Since I bluff with the #4 card and value bet with the #1#3 there is exactly a 31 chance I am bluffing.
Optimal bluffing in any particular game doesn't mean the bluffer is automatically a winner. It is possible that an optimal bluffing strategy could simply result in the bluffer losing the same amount whether the other player calls or folds.
For example, let's say I modified the game to now include nine cards #1#9 however everything else remains the same: I win on #1#3 and will bluff #4.
A) If you call everytime I bet then you end up with:  (13) You will call and lose $2/hand ($6 total)  (4) You will call and win $2/hand (+$2 total)  (59) You will check and win $1/hand (+$5 total)
You will win a net of +$1 over 7 hands or +$0.14/hand.
B) If you fold everyime I bet then you end up with:  (13) You will fold and lose $1/hand ($3 total)  (4) You will fold and lose $1/hand ($1 total)  (59) You will check and win $1/hand (+$5 total)
You will again win a net of +$1 over 7 hands or +$0.14/hand.
Again I have employed an optimal bluffing strategy, but it only means I lose the same amount whether you fold or call me.
For optimal bluffing to swing a losing situation to a winning the combined odds of winning naturally plus the bluff need to exceed 50%. I could probably come up with variations of this game that give up larger draw advantages but change the size of the antes and bets after the draw to still show a +50% money win rate.

Joe Grundy
Australia Sydney NSW

carpejugulum wrote: It can be shown that optimal bluffing frequency is exactly equal to the pot odds you are offering the other player. In this game when I bet the other player is getting 31 odds to call (there will be $3 in the pot and he can call another $1 to try and win). For some reason that took me a while to figure out (maybe 'cause it's 3am) but yes $1 ante each plus 1$ opening bet = $3 in the pot. Silly me.

Mark C
United States Ypsilanti Michigan

Bluffing is not what enables you to win this game. You win because you have perfect information. By having perfect information, you use that information to limit your loss on the hands that you are going to lose, raising your expected value on the payouts.
I think most people would say bluffing entails risk, whereas this has certain outcomes in all cases, known to one side before the bet.



Gamer_Dog wrote:
Bluffing is not what enables you to win this game. You win because you have perfect information. By having perfect information, you use that information to limit your loss on the hands that you are going to lose, raising your expected value on the payouts.
I think most people would say bluffing entails risk, whereas this has certain outcomes in all cases, known to one side before the bet.
?

Mark C
United States Ypsilanti Michigan

Quote: ?
You know the result. Your opponent doesn't. This gives you a large advantage when placing a bet. There aren't many games where this kind of setup wouldn't net you a positive return.



Gamer_Dog wrote: Quote: ? You know the result. Your opponent doesn't. This gives you a large advantage when placing a bet. There aren't many games where this kind of setup wouldn't net you a positive return.
I am pretty sure you can apply the same concept in games where you don't know the result any more than your opponent does to optimize your winning percentage. Bluffing, along with your information, lets you win this game.
In games where both players have equal information, one can adopt a randomnized formula for bluffing to make his chances 50%.
The situations aren't totally analogous but the idea is that in a bluffing game, you can actually tell your opponent your gameplan and not lose. Kind of ironic since the whole idea behind a bluffing game is to be able to better read the other player's mind. It is actually better to give up reading your opponent's mind altogether and just "roll a dice" to see whether you should bluff or not.

Seriously, turn off Facebook. You'll be happier.
United States Riva Maryland

chrisohaver wrote: We need more posts like this.
BGG Admins  Can we create a Forum Category : "Probability and Game Theory"
This has nothing to do with probability, but it would indeed be an interesting category.
This game is like that old Monty Hall riddle  it looks like it's a numbers issue, but it's more about how the other person plays you. Learn to read the guy, and you clean up. Don't pull that off, and you get crushed. That's poker.
Sag.

Marion Jensen
United States Taylorsville Utah

Excellent post. Correct me if I'm wrong, but the reason you can turn a 'losing situation' into a winning one is because the game is lopsided. At no point does your opponent get to go first. If you took turns getting to bid or check first, then your opponent could employ a similar strategy, and bring the odds back into their favor.
Game theory is fascinating, and I've seen threads hundreds of posts long arguing over Monty Hall. You should ALWAYS take door number three.
Good times.

Mark C
United States Ypsilanti Michigan

Quote: I am pretty sure you can apply the same concept in games where you don't know the result any more than your opponent does to optimize your winning percentage
I don't believe so.
Take the instance of drawing a card from a deck face down. Say the outcomes are unequal if it's a spade, I win, if it's anything else I lose. We reveal. There is no opportunity (in the long run) for me to improve my take by bluffing. If the roles are reversed, the same is true. Obviously if the outcomes are equal, the same applies. Bluffing becomes valuable when you have an information advantage. If you know nothing your opponent doesn't (or as has been pointed out, you use randomness as a surrogate to act as though you know nothing) then there is no bluffing strategy to improve your outcomes.

Oil Slider
United States Illinois

An interesting variation of this puzzle:
http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challeng...
Obviously, whoever goes first has an advantage, but neither side knows for sure who the winner is. The correct strategy for the first player was pretty unintuitive, for me, at least.

Marc Hartstein
United States North Plainfield New Jersey

chrisohaver wrote: I am pokertarded. What happens when both players check. Does the pot stay, and next hand dealt?
The hands are revealed (the one card in this example), and the best hand wins the antes.


