This is a theory forum. Although we accomodate posters with a wide variety of backgrounds, some of the discussions can become quite technical. These discussions often draw upon concepts, facts, and words from the field of mathematics known as game theory.

Unfortunately, many words commonly used in game theory have very specific, exact meanings which may or may not line up with their usage in everyday speech. In order that we are able to communicate effectively, it is important to understand the technical meanings of a few terms. Way too many threads have degenerated into arguments between posters who mean different things when they say "optimal".

With that in mind, here's a quick intro to some common terminology, some of which comes from game theory.

range

A range is basically just a group of hands, possibly with some frequency information attached. Ranges are particularly useful for specifying strategies. For example, you might want to say that you take a particular action with all of some particular hands, half of others, etc.

equity

The equity of a hand or a range is the percentage of the pot expected to be won if all betting were stopped and all players checked down to showdown. This frequency is averaged over all the cards that can come and all the ranges involved.

game

There's a few different ways to define a game depending on the situation you want to study. Basically, it's just going to be a sequence of decisions which lead to some payoff depending on what decisions you make and what decisions the other players make. That is, your payoff depends on your decisions as well as other peoples' decisions.

Sometimes you have a situation, and you get to make some decisions, and the results only depend on your own decisions (and maybe some randomness from Nature). These might be interesting problems, but they're not questions for game theory. It's the fact that your payoff depends on other peoples' decisions too that puts us in the field of game theory.

For example, rock paper scissors (RPS) is a game with only one decision point, and three payoffs are possible: win, lose, or tie, depending on your choice and your opponent's choice.

strategy

A strategy specifies how you choose your move at every decision point in a game, that is, at every situation you could possibly be faced with in the game.

pure strategy

A pure strategy is a strategy that specifies exactly how you will play in every situation you could face. In RPS, there are three pure strategies: throw rock, throw paper, or throw scissors.

mixed strategy

A mixed strategy is one which associates some probability with each of the possible pure strategies. For example, in RPS, you could play a mixed strategy by using "throw rock" 60% of the time and using "throw paper" the other 40%. It is a good idea to play a mixed strategy in RPS. It's not as clear that this is necessary to be successful in poker.

dominated strategy

A strategy is (strictly) dominated if there is another strategy possible which is more profitable regardless of your opponents' strategies. This actually does not come up too often in poker, since almost anything can be good if your particular opponent happens to play particularly poorly against it.

One example from hold'em is as follows: any strategy that involves folding the nuts on the river is dominated (at least in a cash game context, ignoring rake). In particular, it is dominated by a strategy which is the same except that the nuts are played somehow other than by folding. This second strategy will be more profitable regardless of how your opponents play.

+EV

+EV is short for "positive expected value" or "higher expected value". This is vague for a couple of reasons. First, expected value? expected value of what? Second, higher? higher than what?

Expected value just means the average value of some random quantity. In poker, the quantity we're interested in (and which we're interested in maximizing) is the size of our chip stack. However, when doing calculations, we don't always calculate the average or expected size of our chip stack. Sometimes we calculate the expected change in our stack size over the course of the hand, starting from the beginning, and try to make choices that maximze that. Or, sometimes we calculate the expected change in stack size starting from some specific point in the hand.

Basically, it doesn't matter which way you do it, as long as you are consistent in your choice and clear about what you are doing when you write a post.

Second, higher than what? Sometimes people say "higher than 0", but that doesn't necessarily mean anything depending on what expectation we're considering. Other times, people are implying "higher than than the EV of folding if I computed it with the same convention", but that may or may not be important in any particular situation (say, if we are really interested in deciding between calling and raising).

From a strategic point of view, it's important to choose the option with an EV higher than all your other possible choices. So if you are arguing that something is "+EV", please say what it is that your favored move has a higher EV than, and if you write down EV equations, make it clear what you're finding the expectation of.

best response strategy or maximally-exploitative strategy or nemesis strategy

Suppose all your opponents' strategies are fixed and you know them. Then, you can compute the very most profitable way to play against them. This best strategy is known as a best response or a maximally-exploitative strategy. An (imaginary) player who automatically knows your strategy and always plays maximally exploitatively in response is sometimes refered to as nemesis.

nash equilibrium or (game theory) optimal strategies or unexploitable strategies

A Nash equilibrium is a set of strategies (one for each player in the game) with a couple properties. These properties are equivalent, they're just different ways of looking at the same thing:
- No player in the game can unilaterally change his strategy to improve his expectation.
- Each player's strategy is maximally exploiting those of his opponent(s), at the same time.

Notice that nothing about these definitions implies that the players will break-even on average. However, it turns out that in poker, if all players are playing their equilibrium strategies, they will break even in the long-term average sense when we average over all positions in the game.

The existence of a set of strategies like this has some special consequences (at least in heads-up play!). Whenever players are not playing their equilibrium strategies, and one player is making more money than he would at equilibrium, and thus the other is making less, then the guy who is making less has incentive to switch to his equilibrium strategy. However, whenever both players are playing the equilibrium, neither has any incentive to change. Thus, if both players are rational and smart enough to compute the equilibrium strategies, they those are the strategies they will end up playing. It's only then that neither has any incentive to deviate.

Now it is not immediately obvious that such a set of strategies exist, but John Nash proved it for a class of games that includes poker. Of course, knowing that it exists is different than knowing exactly what it is, and in fact the equilibrium strategies are unknown for all "real" poker games. But they definitely exist, and also, equilibriums for greatly-simplified versions of some games are known. For example, if the SB is restricted to playing shove-or-fold preflop in heads up no limit hold 'em, then the game becomes much simpler, and we can find the well-known shove/fold equilibrium which can be useful for short-stacked play.

Any non-equilibrium strategy may also be refered to as exploitable. Notice that, if you were playing unexploitably, but then you changed your strategy to take advantage of some mistakes of your opponent, then you yourself are now playing exploitably, but that's OK if your opponent isn't taking advantage of it.

Nash equilibrium strategies are also known as optimal or game theory optimal or GTO or unexploitable. The fact that "optimal" does not simply mean "maximally exploitative" really seems to trip people up and is thus unfortunate, but that's the language the mathematicians chose, so we're stuck with it.

The usage of "optimal" or "game theory optimal" to refer to the Nash Equilibrium appears to be somewhat unique to poker. It's genesis may be from the book "The Mathematics of Poker" which uses the term this way. This usage does not appear to be common among game theorists and definitely causes some confusion at times.

if all Players would shove and call any two cards they would be break even too, maybe add that point

ty for ya post =)

Looks pretty good, thanks!

sticky at the top imo.

fwiw, I'm not sure if people would be interested in distinguishing between "hand distribution" and "range".

Hand Distribution: All hands for a player before any action is taken on a particular street
Range: All hands used by player when taking an action

For example, on the Flop:
Hand Distribution: All pocket pairs 22+
Betting Range = JJ+
Checking Range = 22 - TT
Check-Calling Range = 66 - TT
Check-Folding Range = 22 - 66

I'm not sure if it's necessary, but I feel that I get confused sometimes on these boards.

thx for doing this.

Perhaps a more clear example of the definition is how folding first to act is dominated by checking.

good post. btw, i come from a stock/futures (and to a lesser extent - options) trading background and i find poker interesting in a lot of ways. a lot of the game theory carries over, but a lot is quite unique to poker (and trading)

the primary difference i see is that while technically any trade affects the market (even lifting one share from the offer), it has far more minimal effects on the other player's actions vs. poker (obviously) since there might be thousands or millions of players in a future contract in a given day, such that one trade doesn't have the effect that one bet, raise, call or fold does in a poker hand. also, while the futures market is zero sum (like poker) , the stock market is not zero sum.

i have found traders who don't "get" that last point ... which boggles my mind, but it's indisputable

the other difference is that in trading, the "cards" so to speak are the market price at the time (as well as how thin or thick the current orders are on the DOM), but the cards to come are affected by the current actions which is very distinguishable from poker

in brief, in trading, you cannot "force" the game and doing the equivalent of a three barrel bluff will just cause you to lose more money. unless you are soros or something, you cannot appreciably move the market

in poker, you CAN - the market is each individual hand.

can you bluff? sure. people do it on the DOM all the time by flashing fake size to try to get others to react. a market maker can create the illusion of demand where none really exists, in order to move the price. otoh, he (unlike the poker player) has to be there to provide liquidity at all times, whereas the poker player can simply fold

but ultimately, it's most similar in bankroll management (poor money management is amongst the #1 downfall of traders, even if they are very talented at trading itself), in not going on tilt (making emotional/revenge trades will KILL you) and especially in being able to say "i'm wrong and i'm getting out". the best traders are more than willing to take their stop loss. if the market turns around and it would have been a winning trade ... oh well. just like the poker player must fold when he thinks he's beat.

the good trader wants to lose small and win big... "cut your losses short and let your winners ride", so often a winning strategy in trading might only have 30% successful trades, but you still make money.

also, the costs of slippage and commission are similar to the costs of rake.

options guys remind me the most of the internet whizzes because the options guys are VERY comfortable with the math, and with probability curves, and some very complex , galfond'ian if you will , math. the futures guys, ESPECIALLY the pit guys (who are a dying breed) are much more like "feel players" who have great intuition and card sense. they know when they are beat, they are more sensitive to physical tells, and especially for a floor trader, they use their physicality to help execute a trade, just like a live player can use his presence, false tells, etc. to sell his bet.

it's interesting stuff i am seeing here in this forum.

Sure, but at that point, each player can improve his expectation by changing his strategy, something which is not true at the equilibrium.

I just wanted to make the point about the value of the equilibrium being 0 only on average over all positions because it seems like that's something people get confused about. I've even seen people solve for the "equilibrium" in a subgame by setting the EV equal to 0, as if EV=0 is the definition of Nash equilibrium.

Thanks for the positive feedback guys.

Ah, to be honest, I think I use the terms "range" and "hand distribution" almost interchangably. In particular, to specify the range at the beginning of a street, I usually go with 'turn starting distribution' or 'turn starting range'.

If anything, 'range' is sort of a grouping of hands without any context, whereas 'hand distribution' usually implies a distribution over the various possible hand strengths, which means that there's opponent(s)' range(s) and a particular board involved.

And of course the 'equity distribution' is something specific and different.

But anyway these are kind of poker-specific (not game theory) words, so I wouldn't expect this thread to be authoritative about their meanings. But I am happy to talk about it.

Yup, it's true that it's another example, and it's probably more intuitive. However, it's a bit harder to construct a strategy that dominates it. I guess you could take the strategy which is the same except that instead of open-folding, you check and then proceed by only putting any more money in the pot if you get to the river and have the nuts. It should be clear that this strategy does at least as well as the one that involved open-folding, and so it weakly dominates it. (Weakly because it doesn't do strictly better.. it might do the same.)

Interesting stuff, thanks for the response.

Very nice post, yaqh, but I just wanted to comment on the above. If you talk to a game theorist (my brother is an economics professor specializing in game theory, so I did ), his or her understanding of the word 'optimal' is completely different from the meaning used in poker theory. An optimal strategy in game theory is not typically an equilibrium or unexploitable strategy; it's a max EV strategy when the game parameters are completely specified (including opponent responses). When people talk about GTO strategy in poker, they really mean minimax strategy, and the misuse of the word 'optimal' has somehow crept into poker-speak and has stuck. It's unfortunate, not only because it's confusing, but ironically because it contradicts the usage from game theory itself.

If you simply google the phrase 'game theory optimal' in quotes, you won't find a single usage of the phrase in any mathematical literature. The only hits you'll get are from poker.

Gah, I didn't want it to be so, but you might be right. Chen and Ankenman definitely use 'optimal' to mean 'equilibrium' in The Mathematics of Poker, though, which may be where the confusion began. I wonder why they did that. I'm going to keep looking for uses of the word optimal in the math literature. I don't trust economists .

Well, of course they don't call it game theory optimal in game theory.

K, here's two game theory books that use 'optimal' to mean 'equilibrium'

http://books.google.com/books?id=orO...sec=frontcover
http://books.google.com/books?id=fep...sec=frontcover

edit: 2 more:

http://www.amazon.com/Introduction-T.../dp/0486428117
http://www.amazon.com/Games-Theory-A.../dp/0486432378

Just search inside the book for 'optimal' to verify. If I had to guess, it seems to be a difference between pure mathematicians and economists.

Gilpin and Sandholm also used 'optimal' when they solved Rhode Island Holdem, which was before MoP was published.

http://www.cs.cmu.edu/~sandholm/RIHo...roceedings.pdf

Good job, should be sticked or at least linked in the FAQ page for me.

As a side note, another possible "dominated strategy" is folding preflop hand XYsuited while raising hand XYoffsuited. If you start folding some of the offsuited and raising the suited hand you gain on every possible spot.
Or folding AA preflop (in cash game, in certain spot in sat tournament can be right), opposed to go all-in.

We should just outlaw the use of the word optimal in this forum

When I spoke to him a couple of days ago, I asked my brother that exact question - is it possible that mathematician game theorists use the word 'optimal' differently from economist game theorists, and he said "I don't think so". I'll point him to the links you posted and see what he has to say!

Is this actually the correct definition of Equity? I admit I've always been a little confused about it. Most knowledgeable poker experts and mathematicians will tell you that your hand equity is actually your expected "share" of the pot. But in EV calculations (and on TV!) it's treated just like it's the percentage chance that it will will by showdown.

Per the Pokerstove FAQ, for instance, it says:
"The values generated are all-in equity values. This is not the chance that a hand will win the pot. Rather it is the fraction of the pot that a hand will win on average over many repeated trials, including split pots. The equity for a hand is calculated by dividing the number of "pots" that the hand won by the number outcomes considered. Because two players can split a pot, a player can win fractional pots. Thus, it is possible for a hand to have non-zero equity despite the fact that it cannot win."

So which is it?

To further complicate matters, when we talk about fold equity, most players are actually trying to express the probability that their opponent will fold to a raise.

I'm so confooooosed....

The key phrase in that Pokerstove quote is including split pots. This should probably be mentioned here as well. I also think that it should be called "showdown equity", "all-in equity" or "hot'n'cold equity" instead of just "equity". It's the expected share of the pot if the hands went to showdown with no more betting.

It's the "expected share" of the pot if there is no more betting or folding. This is close to the percent chance to win except for the slight difference (as you point out below) caused by the chance of splitting the pot.

Oh, you might be pointing out that some people use 'equity' to refer to a dollar value as opposed to a percent. For example, if you have 60% equity in a $100 pot, they'll say your equity is $60. So they're still refering to your expected share in the pot, they're just using the dollar value instead of the percent of the pot size.

Yea you're right, I was a bit sloppy (i.e. wrong ) in OP since I neglected split pots. We should say that equity is the chance of winning plus half the chance of chopping.

I'm not sure that this is actually true, due to card elimination effects if nothing else. Here's an example that's even kind of realistic:

Consider the heads up no limit game where the SB minraises or folds his button and then, if he minraised, the BB shoves or folds.

In this case, at small/medium (say, 20BB) stack sizes, BB might be shoving a range with mostly big cards and smaller suited connectors (but not small, unsuited cards). In this case, minraising, say, 65o is better for the SB than raising 65s. The reason is that they're both going to be folding to a shove, but BB's slightly less likely to shove if SB holds 65o since 65o blocks twice as many of the BB's 54s,56s,67s combos.

So, the SB's steal is slightly more likely to work with 65o than 65s which makes raise/folding 65o better than raise/folding 65s. And so, if it's close, there definitely could be a situation where SB should be raising 65o but open-folding 65s.

65o is better than 65s only because it blocks one more combo of 65s (2 instead of 1).
vs 54s or 76s every 65 (off or suited) has the same blocker effect.
Eg. if I hold 6h5s or 6s5s there exist 3 combo of 54s in any case (5h4h, 5c4c, 5s4s).

The minor effect of blocking one more combo can change the equilibrium in toy game restricted without post-flop game, but I think that in complex game having more equity if you saw a flop is a lot better.

But you are right, this is only "almost dominated", so not so usefull.

Ah, yea you're right.

Anyway, as far as the definition of 'dominated strategy' goes, the thing to keep in mind is that it really means 'better (or at least as good) versus every possible opponent strategy'. So you have to consider even the really wacky ones where, say, Villain is folding all hands preflop except for 32o.

As far as the opening suited vs unsuited hands example, maybe Villain plays well in general except that he's ******edly terrible on 4-flush boards: he open-folds his flushes but goes all-in with all non-flush hands. In this case, your unsuited hands are probably more valuable than your suited ones since you have twice as many flushdraws.

Anyhow, removing dominated strategies from consideration can help you make a lot of progress towards solving some games, but I don't think it's particularly useful in poker.

Someone mentioned Liv Boeree, and I was reminded of my favorite example of a game that can be solved simply by eliminating the dominated pure strategies.



Scary stuff.

Okay, I showed these links to my brother, and he said roughly the following (this might turn into a bit of a game of 'broken telephone' since I'm repeating things second-hand, but I hope it's more or less accurate )

* at a glance, the first book never defines how it's using the word 'optimal'; the second is a translation from Russian and uses all kinds of funny language; and the last two are archaic and use a lot of outdated terminology;

* in economics, the word 'optimal' would be reserved for two situations. The most common usage would be when discussing a well-defined optimization problem (not poker) in which there would be an optimal solution. The other would be a more casual usage of the term in a game where there is a clear best strategy;

* instead economist game theorists would use more precise language to describe a strategy - they would talk either about a 'best response' strategy' (that would be what we call an exploitative strategy) or a 'minmax strategy' (which is what we call a 'game theory optimal strategy')

* in contemporary game theory, there are almost no pure mathematicians working in the field, outside of mathematical biology. Most research is done in computer science and economics. So while he conceded that there may be some specialized vernacular that is only used among pure math game theorists, that isn't the group that is writing most of the papers about the field;

* the one caveat he mentioned is that zero-sum games like poker are of very limited theoretical interest in contemporary game theory. So if there is some specific terminology in use for that one class of games, he might not be familiar with it. But I think he'd be surprised if that were true.

Anyway, long story short, he still views the usage of the word 'optimal' in poker theory with a lot of suspicion, and says that his game theory colleagues would never use the word the way we do. Take that for what you will!

Sounds good, thanks for looking into it. I think I'll just try to avoid using the word in the future myself. It has led to a lot of confusion around here...

I've stickied this. I'll be monitoring it from time to time. If there are any questions about additions or modifications feel free to put them in this thread and we'll try to address concerns.

Might be nice to add links to some of the better game theory threads also?