Over its short tenure, online poker has been an incredibly profitable venture for many professionals. But what might the future hold in store? We could just predict that the future will, on average, be like the recent past. Maybe things will be better, maybe they will be worse, but we could expect those scenarios to even out, producing an average environment like the present.
This method is both too simplistic and too optimistic for my liking. In order to predict the future with a higher accuracy than blind guesswork, we need to make use of a model, a simplified representation of the poker economy that illustrates some core features of reality. The process of simplification has both pros and cons. On the plus, side it allows us to make actual predictions about the future — something we are unable to do in the messy complexity of reality. On the minus side, it means that we will, out of necessity, be ignoring some potentially relevant aspects of the real world.
For this reason, I will make use of two separate models. One is drawn from economics, the other from biology. By looking at the predictions of both models, and with an awareness about significant factors that they might be overlooking, we can make some useful predictions about the future for online poker. Using a model from biology to predict the future of the online poker economy might seem strange, but this sort of reasoning by analogy is rather common. The behavior of stock prices has been explained with a model borrowed from physics of particles randomly moving around while suspended in fluid.160
Using these models, we can hopefully produce some forward looking predictions about the future for online poker. Attempting to anticipate new situations, unlike what has happened in the recent past, is important. One of the causes for the property bubble which precipitated the credit crunch is that individuals and banks bought and lent money on houses based only on recent house price data. They looked at a narrow band of history where housing prices did nothing but go up, and erroneously concluded that serious declines were negligibly probable. The rest is history.
Barriers to Entry Model
The “barriers to entry model” is borrowed from microeconomics — the field of economics that studies the behavior of individual people and firms. Game theory is another branch of microeconomics where the question of interest is how people will behave in strategic situations.
“Rationality” is a key assumption in microeconomics —people will respond to incentives in a way that maximizes their wellbeing. In game theory, the rationality assumption is interpreted in that people will alter their strategic behavior until their payoff is maximized. The game theorist’s chief prediction is that of a Nash equilibrium, a state where no individual can improve upon his strategy (each player is rationally maximizing his payoff given what the other is doing.)
The barrier to entry model is a theory of the firm. A barrier to entry is some device, be it legal, geographical, technological, or an economy of scale, that prevents and hinders competition between firms. The only way for firms to make abnormal profits in the long run is to have some barrier to entry that prevents competition.161 Companies such as Google or Microsoft are able to make large profits year on year because competitors are unable to copy their technology. If no barrier exists, then new competitors are free to enter highly profitable markets, reducing average profits until the market reaches an equilibrium where each firm can only earn a normal profit — a minimum acceptable rate of return.
So, are there barriers to entry in poker? Obviously, yes! Poker is a complex game, and the process of learning how to play, while slowly building up a bankroll, takes years to do. Additionally, the better players presumably have some indefinable edge either in their understanding of the game or how they process information at the table.
These barriers to entry have protected incumbent poker pros’ edges. However, I expect the barriers to be much less significant when viewed over a space of time rather than at the present moment. In other words, the long run barriers to entry are much less than the short run barriers.
An intelligent, hard working player can build his way up in poker, starting from essentially nothing. Either by depositing a trivial sum, or starting off with freerolls, he can start to win at the lowest levels, slowly making his way up. Many of the top high stakes players did this, depositing $50, or so initially, and never having to invest new money again. Of course, this will take time, but in the long run, there is nothing to prevent this player from reaching the higher limits.
Online poker has become one of the world’s truly global industries, engineered by the proliferation of televised poker. Barring legal restrictions, there is nothing to stop a player from Russia building up a bankroll and competing for the same money as an American pro. In addition, whenever poker opens up to a new market, there will be a wave of new players. Many of them will be bad, contributing new money to the online economy, and the majority of these players will lose their money quickly. But the stronger players located in the new market are likely to stick around.
Wage rates are highest in the developed world; it’s likely that professionals from the developing world will have much lower required returns from poker than a player from let’s say the United States. Just as how the world’s production of manufactured goods has shifted to lower wage countries, in a competitive future poker economy, the location of professional poker players could shift towards the developing world too.162 This is what we would expect to happen in a world where good strategy advice is available to all aspiring pros.
Good strategy advice is, at the moment, still a scarce resource, but it’s becoming less so over time. Poker books are being translated into many languages and new forums as well as other resources are springing up for non-English speakers all over the web.
The best players can always make a pact not to share their secrets, but this is likely to be ineffective. As with all cartels, an individual member has much to gain by cheating on the arrangement. Furthermore, information about how a top pro plays can always be collated by observing his play. From this you can attempt to reverse-engineer his strategy.
The spread of online heads-up tables has been a boon for players specializing in this area of poker. With a bad player all to yourself, you can achieve a high win rate, and as mentioned previously in this book, this implies a higher hourly rate and fewer downswings. For a time, heads-up players had a really amazing niche.
However, the good aspects about heads-up poker also serve to reduce the barriers to entry. High win rates imply smaller downswings, in which case moving up in stakes is much easier for new players. The presence of just two players at a table means that your required skill level to compete at a table is much less: the only player you need to beat is the fish. Compare this to six-max where a bad regular can get beaten up by the better pros at the table.
These low barriers to entry are reflected in the current state of heads-up poker. It’s common to see 30 or more professionals waiting for heads-up action at one limit at a time — at most, one, or two of them might be getting action. Heads-up poker has become a gruelling game of who can sit behind their computer the longest to catch a fish.
Barriers to entry are at their most significant in the highest games around. In no-limit poker today, the highest game running could be $500-$1,000, but the next highest game might be $25- $50. This huge gulf in stakes makes it incredibly hard for a $25- $50 pro to take shots at the bigger game unless he sells the majority of his action. Even if the $25-$50 player sits down, the higher game will rarely run for long, so his results will be driven as much by variance as by his win rate.
Additionally, due to the high standard of play, the learning barrier is greatest at the highest stakes — you need to play well to stand a chance of competing. These factors will favor the incumbent high stakes player: His edge is less assailable from competition compared to players at any other limit implying more sustainable long run returns. (This assumes that the high stakes economy remains healthy.)
The barrier to entry model paints a bleak picture of the future for many. The forward looking barriers to entry should be much less than the advantages incumbent players have enjoyed to date. A more competitive poker economy will imply lower win rates all around which will in turn result in larger downswings and breakeven stretches than players have been used to. Due to the higher complexity of their games, deep stack big bet players have a stronger barrier protecting their edges than limit players, but this assumes that shortstacking, which involves a very simple optimal strategy, is not allowed in the big bet games. For some long time professionals, the poker industry might hold significant barriers to exit — playing the game for a long time, will, no doubt, reduce your employability in many other sectors in the economy. Lack of outside options might force these professionals to keep playing long after poker has become a much less attractive vocation.
We can also use the barrier to entry model to predict the areas of poker which might be most profitable in the future. We have already mentioned high stakes, although this will be inaccessible to most. Legal barriers might create opportunities for players inside the barrier. This has been the case for many European players who have been able to play on a broader range of sites than their American counterparts. If more governments create regulated poker networks, the professionals on those sites might have an attractive niche created for themselves assuming the rake charged is not too high. Live game players have another barrier to entry. Only other pros in the immediate geographical vicinity can compete away their edge.
The other area is mixed games; learning to play multiple games well is more difficult than specializing in one type of poker. A jack of all trades will have an advantage here, and it will take other players a relatively long time to catch up. And having a theory driven approach to learning poker should make playing multiple games easier to do.
Predator-Prey Dynamics
This model is used in biology to assess the long run population dynamics between two populations: a species of prey and the species of predator that feeds on them.163 This is a good description of the relationship in online poker: fish provide the food source and sharks need to feed on them or else they will ultimately starve. Fish are assumed to reproduce in the absence of sharks (either through new deposits or new players taking up the game). Sharks rely on fish for their food source; in the absence of fish, there is a natural loss rate among the shark population.
The interaction between the two population sizes is of interest. With a plentiful quantity of fish, a small group of sharks can quickly grow — either padding their bankrolls for a higher level or by allowing new sharks to enter the system. The number of sharks will grow, until at one point, the number of sharks will begin to deplete the quantity of fish. The fish population will then decrease, until at one point, a number of the swollen contingent of sharks will no longer be able to survive. The shark population will then rapidly decrease; fewer predators will allow the fish population to grow again until the system comes full circle, reaching a situation like the starting point.
Figure I: Predator-Prey Cycles in Action
Prey–Predator Cycles
These population dynamics are illustrated in Figure I above. Notice how each population is in a constant state of flux, the system never settles on an equilibrium value, and even though it’s always drawn toward a fixed point, it can never remain on it.
The poker boom created a situation with an abundance of prey. Many new players took up the game and there was a relatively low number of established professionals ready to take their money.
It must also be remembered that online poker in its current form is still a relatively young industry. In terms of Figure I, that history would represent less than one entire cycle. The level of fish boomed for a while, more sharks were created, and now the fish population is on a downward arc.
Looking forward, the model predicts that the less abundant food source will cause the shark population to decrease — it will be the survival of the fittest. I have witnessed this in my own games. The large quantity of fish allowed many pros to survive despite errors in their play — tilting, bad strategy, or poor game selection. With fewer fish around today, these error-prone professionals have seen their win rates become especially squeezed, and a number have stopped playing entirely.
The near term prediction from this model is that we can expect the population of professionals to contract, which fits with my general observations. Of course, nobody wants to be the one forced out of the industry, even if it’s inevitable that some must leave. My thesis in this book has been that playing well is a necessary part of playing for a living, but that is not enough —intelligent poker requires a broad range of attributes and skills. For one thing, it can be psychologically difficult coming to terms with the reality of a lower hourly rate and a resulting lower standard of living.
Looking on the brighter side, the model does predict sunnier times in the future for those who remain. At some point, the drag on poor players’ bankrolls from the rake and good players will be less than the replenishment rate, causing the quantity of fish online to cycle up. Hopefully, happier times, more like the start of the online boom, will be somewhere around the corner. My personal opinion is that this won’t occur until the global economy becomes more robust than in recent years.
We can use the variables in the model to compare the health of various poker games. First, there is the death rate among predators due to a combination of the rake and the tendency for marginal professionals to get pushed out of the game as the number of fish decreases. There is no reason to assume that this variable would be more significant in some poker games than others.
Then there is the predation rate at which the sharks consume the fish. This will be highest in heads-up and big bet poker.
In heads-up, there is a higher frequency of significant decisions meaning that sharks can attain a higher win rate there. In big bet poker, there are more opportunities for fish to make errors in choosing their strategy. All else equal, a high predation rate will speed up the boom bust cycle in the two populations.
Then there is the growth rate amongst the fish, and this should vary between different games. The growth rate has been highest in no-limit hold ’em, particularly in tournaments, due to the high public profile of this form of poker. This growth rate has exceeded the higher predation rate, making it a more profitable form of poker so far. However, if the growth rate dips then the higher predation rate could make big bet poker more susceptible to a downturn in the poker economy. In this case, limit poker could become more profitable as the fish will be able to play longer and it’s my understanding that this is exactly what happened in the late 1970s and early 1980s.
Conclusion
This chapter may seem overly gloomy to many of you. However, this is just the impression I get from analyzing the predictions of these two models. There are always exogenous factors (elements outside the models) which could easily take the poker economy off on some unpredictable course.
There are also many optimists predicting that the second poker boom is just around the corner. The reason for this used to be the emergence of the Far East as a factory for creating new fish. The current rationale is that a repeal of gambling laws in the United States will do it.
For one thing, I think the current argument is flawed. Up until recently, gambling laws have been lax all across Europe, yet the same pattern of declining games is occurring there. Second, this sort of narrow thinking can easily get you into trouble; a plausible sounding argument lies behind mistaken ideas.164 Therefore, it’s much better to take a broader perspective on things by looking at a range of models and considering their limitations compared to the real world.
I have a couple more concerns about online poker. First, it seems like the majority of fish are located in some narrow geographical areas. In Europe, players from the Mediterranean countries (such as Italy) are in my opinion the worst; whereas the Scandinavian countries and Germany are filled with professionals. Looking at the broader world, South American players appear to be much worse than North Americans. This makes the system much more vulnerable to shocks. A ban, or a severe recession in a high-fish country, could have a severe knock-on effect for the rest of the poker economy.165
The “trickle up effect” is a well known piece of poker jargon. Poor players can win some money at the lower limits, giving them the confidence to move up where they donate their bankroll to the poker economy. Unfortunately, I think we are currently seeing a “trickle down effect” amongst professionals. The highest games are amongst the hardest hit by the recent troubles in poker. Therefore, the best players are having to move down a few rungs in order to play. This is making the mid-high games much tougher than they were since they are now filled with more expert players. This causes the next layer of professionals to play more, and at lower stakes than they are used to, in order to try to maintain their overall income. This process continues until all levels of poker are affected.
I think the lower levels frequented by professionals, say $3- $6 and $5-$10 limit hold ’em, are more delicately poised than most people think. There has always been an abundance of fish here, causing many people to believe that games this low will always be good. However, the rake is a massive factor at low stakes — 
or more per player.166 And due to the high rake, there needs to be a steady supply of fish just to maintain a positive win rate.
However, as things stand, many professionals at these limits rely on rakeback for the vast majority of their earnings. If the trickle down effect disrupts this balance, then the high rake could make it difficult to maintain a living in these games even with a good rakeback deal.
Ultimately, you will need to have a competitive advantage to do well for a long time in this career. That could be finding a pond full of below average players, playing a niche form of poker better than anyone else, or finding a low-rake deal (or even better, getting paid to play). Most importantly of all, you should stay open to new opportunities as they come about.
Zero-sum games (like poker) comprise just one branch of game theory. These are games of strict competition where any party’s gain is mirrored by an identical loss elsewhere. Conventional wars, such as the Second World War, can be seen as a zero-sum game: All that mattered was winning, and either the Allies would or the Nazis would. But war changed with the invention of the atom bomb, and nuclear conflict is one example of a non-zero-sum game: When both sides get bombed to annihilation, everyone loses. In these games, cooperation between the competing parties can increase the size of the pie for everyone.
The Sitting Out Dilemma
The prisoners’ dilemma is a famous problem in game theory.167 Two suspects are arrested for a crime and are held in separate cells. The police are interrogating both separately in order to try and extract a confession from each suspect. If both refuse to confess against the other, (labelled “Cooperate” in the table), each will get six months in jail. If only one confesses (labelled “Not Cooperate” in the table), then he will go free; the one who remained silent will go to prison for two years. If both confess, then each will serve a term of one year in jail. Since spending time in jail is undesirable, we can represent these situations with the numerical payoffs (on an ordinal scale) below:
Table I: Normal Form
of The Prisoners’ Dilemma*
Notice that the best payoff for the two suspects as a group happens when both players cooperate with each other, giving each prisoner a payoff of three units. However, the best payoff for each prisoner individually is to not cooperate against a cooperative partner: This solution yields a payoff of five units to the non-cooperative prisoner.
The only Nash equilibrium in this game is for both suspects to not cooperate (confess). This can be seen using the principle of strict dominance. That is if Player 2 chooses to cooperate, then Player 1 will receive 5 from not cooperating, and 3 from cooperating. So Player 1 should not cooperate if Player 2 cooperates.
If Player 2 chooses to not cooperate, then Player 1 will receive -1 from Cooperate and 1 from Not Cooperate. So Player 1 should not cooperate if Player 2 chooses to not cooperate. Therefore, Not Cooperate strictly dominates Cooperate, so Player 1 should not cooperate. Also, since the game is symmetric, Player’s 1 optimal play is also Player’s 2 optimal play — the Nash equilibrium is for both to choose Not Cooperate.
The dilemma is that the Nash equilibrium of the game pulls the players toward a payoff schedule that is socially inefficient. As things stand, they are both confessing (their dominant strategy), subsequently spending a year in jail. If both remained silent, then they would both be better off — spending just six months in jail.
Despite acting in full accordance with the wisdom of game theory, these two players are each spending an additional six months in jail. In terms of the payoff table in Figure I, they are receiving one unit each when the top-left cell would provide them with three units each.
Although poker is a zero-sum game, the behavior of the professionals can alter the willingness to gamble amongst the fish, essentially creating a non-zero-sum game amongst the pros, where a friendly gambling atmosphere benefits them all more than a hostile one. This can be shown with what I call the sitting out dilemma — a situation that is similar to the prisoners’ dilemma.
In today’s online environment, the most usual line-up at a high stakes six-max game is for there to be one outright fish who the game is built around; in the other seats are five professionals of variable skill. Suppose the fish sits out while remaining at the table; it’s unclear if he’s sat out for a cigarette or toilet break, or whether he’s finished gambling for the immediate future. What is each professional’s optimal strategy with regards to sitting out?
None of the players will want to post a blind if the game will immediately end after the hand, and even the best player at the table will have a negative expectation in the blinds. The worst professional’s strategy is obvious: He should sit out against the four better players. The second worst player knows this, so instead of playing a game that will soon only feature three better players, he will also choose to sit out. The same argument applies to the next two players; they should also choose to sit out given they know the players worse than them will be sitting out. This leaves the best player on his own, but he should also choose to sit out! Since he has an edge over everyone, he would like to post his blind and continue the game. However, given that nobody else will be posting after him, his best choice is to also sit out (saving the minus-EV blind). The Nash equilibrium is for all of the players at the table to immediately sit out.
This presents a similar situation to the prisoners’ dilemma where the Nash equilibrium pulls the group of players towards a socially inefficient situation. This is because having an entire table of players sit out every time the live one takes a coffee break creates a really bad atmosphere for gambling, and soon it becomes obvious that all of the players are out there for one thing only. Notice that this will hardly encourage the fish to keep on playing with them, especially when he has a ready array of rival gambling opportunities at his fingertips.
Acting as individuals, each professional maximizes his payoff by sitting out. If the game will stop as soon as the fish leaves, then by sitting out you avoid posting an unnecessary blind — this corresponds to the (Confess, Confess) cell in the prisoners’ dilemma. Therefore, the best situation (since you get the benefits from cooperation without bearing any of the costs) is to sit out and have some of the other pros continue the game for you, making it more likely the fish will return — this produces your highest payoff, corresponding to (Confess, Silence). So whatever the other pros do, your best strategy is to defect on them by sitting out. But when everyone does this, it leads to the socially inefficient equilibrium — just like in the prisoners’ dilemma.
If the professionals could find a way of deviating from the Nash equilibrium, then they could keep the gambling spirit alive and encourage the fish to lose more of his money to them as a group. The trouble is that no player can do this alone. If one player attempts to resolve the sitting out dilemma, then he will simply cost himself money posting a pointless blind as the game will end immediately after.
In a one shot game, the optimal play is to defect against the group interest in each dilemma, but what about when the situation is played repeatedly? In the prisoners’ dilemma, there are strategies that can be used to enhance your payoff when the game is played a number of times. If cooperation can somehow be enforced on all rounds, then the two players will treble (given the numbers in the chart) their collective payoff compared to the Nash equilibrium.
The prisoners’ dilemma was played in a computer tournament where the players strategies were created by some of the world’s best game theorists.168 The winning strategy was also the simplest — called Tit for Tat. This strategy involves cooperating on the first round; on subsequent rounds it copies its opponents move in the last game. Tit for Tat will cooperate with a sharing opponent, but it will immediately punish a player that tries to take advantage of this by defecting on the next round and again until its opponent starts cooperating. If an opponent is intent on always defecting, then Tit for Tat will only lose on the first round compared to a strategy of always defecting. If Tit for Tat finds a willing collaborator, then the two can enhance their payoff on every round of the game. This simple strategy frequently beats all other candidates in a round robin tournament.
Tit for Tat points at a way of resolving the sitting out dilemma. We want to find collaborators who can keep a game going when the fish sits out, preventing the penny from dropping, to the benefit of each pro, but we don’t want to get abused by serial defectors. If the fish on the table runs out of money, then attempt to keep the game alive if the other players are unknown, or if at least one of them will play on for a while. This will increase the probability of the fish returning with a reload. If, however, you are at a table full of known defectors, then you should also defect in order to avoid the sucker payoff, but you should give unknown players one chance to cooperate before you defect. Repeated play is the element that makes Tit for Tat so successful. If the game involved is truly one-shot, perhaps you are leaving town and are done playing for the night as soon as the fish leaves, then the optimal strategy is to defect.
As the proportion of outright fish in the high stakes economy continues to fall, norms of behavior such as Tit for Tat will become more important in enhancing the gambling experience among the losing players — enhancing the size of the pie for all professionals.169
The No Trade Theorem
The presence of high stakes heads-up matches amongst professional players is somewhat of a puzzle; such games are generally inconsistent with fully rational behavior on the part of both players. The rake makes most poker games negative-sum —causing the average player in each game to lose money.
Taxes compound this effect since the after-tax winnings from the game will be considerably less than the total losses, and this will be most damaging at the highest limits where games are infrequent and the law of large numbers has less of a role. Suppose you and I are evenly matched but I happen to run better this year and win $1 million — $600,000 after tax. Suppose next year we play some more sessions, and you win your $1 million back which converts to another $600,000 after tax; the high stakes poker community has dropped $800,000 to the taxman.
In “Risk Preferences: From EV to EU,” starting on page 278, we saw the importance of risk aversion in building and maintaining a poker bankroll — this should be even more important at the highest level where life changing sums of money are being passed across the table.170 But some of the players at the highest level don’t seem to behave in this way.
Another important element is that of rationality. That is, we should strive to make decisions that are not systematically wrong in any way, and utilize all available information. Suppose it’s common knowledge among the players that each is rational: I know you are rational, you know I am rational, I know that you know that I am rational, and so on.
If we are both rational, and this is common knowledge, then there is no way we can ever play together. But suppose I spend some time working on my game and make some improvements which should give me and edge, so I sit down with you. But you should know there must be a good reason for me to sit with you, and thus should factor this into your calculus on whether to play. If you deem playing worthwhile, it must be because of some additional private information you know. And I’ll notice your willingness to play; perhaps you know something I don’t? This process will go back and forth, until eventually, one of us agrees with the other’s assessment and deems the game a bad spot. This is known as the no trade theorem. 171 You should always reject a zero or negative-sum bet a fully rational individual is willing to take against you as he must be privy to some information you don’t know.172Hence, the source of the puzzle: Why do so many professionals engage in high stakes heads-up matches? The theorem results in much intrigue in finance too: Why is there so much purely speculative trade in markets? For instance, the daily volume of trade in the foreign exchange market is far more than the daily trade in international goods and services. If each trader was rationally impounding the beliefs of other traders into their assessment of fair value, there would never be a reason to trade. The puzzle is resolved with the presence of noise traders. In the words of Fischer Black:
“People who trade on noise are willing to trade even though from an objective point of view they would be better off not trading. Perhaps they think the noise they are trading on is information. Or perhaps they just like to trade.” 173
Even though noise traders are behaving irrationally by trading on irrelevant information, their presence creates a source of liquidity in the market, allowing trades to be quickly and cheaply conducted. Without noise traders there would be almost no trade in financial markets, and there would also be no poker games.
The fact that we can explain the functioning of many real world institutions only by relying on the presence of many irrational agents came as a shock to many in economics — a field that boasts of its explanatory power on the basis of rational thought. Poker players have known this fact for a lot longer.
We have touched on some of the factors that might cause you to act like a “noise trader” already. Maybe you have an overconfident appraisal of your own abilities (“Psychological Biases”), or, maybe you just like to gamble, either due to a natural inclination towards risk or a desire to reverse some recent losses (“Risk Preferences”). Another factor is that you might not be giving sufficient weight to new information — in either the beliefs of others or you haven’t updated your beliefs about your personal skill level in the face of recent results.174 If two players are involved in a negative-sum game of poker, then at least one of them would be better off not in the game.
There are two factors that could make these high stakes heads-up games rational for all involved — sponsorship and learning from the best. Sponsorship can create a positive-sum game for players since a sponsored player can play a better player using the sponsorship money as compensation for his expected loss. This argument has theoretical appeal, but doesn’t seem to have much support in practice. Phil Ivey is probably the most highly compensated sponsored player online, but he is also the biggest winner in all of online poker history with winnings of nearly $20 million to date.
The expected profits from a potential sponsorship deal could make taking on all comers at the highest limit a good strategy. We’ve seen this recently with Tom Dwan signing a lucrative contract with Full Tilt Poker on the back of his high profile proposition bets and general play against other elite players.
Second, the element of learning by testing yourself against the best can make these games positive-sum. Playing the best players might be minus-EV in the short-run, but if you can learn from the sessions and improve your strategy because of it, playing the best might increase your overall win rate sufficiently to compensate.175If this is the case, it would seem the challenger would benefit from playing the game at as low a stake as possible, but this does not fully fit the observed behavior at the high stakes. Perhaps this is why heads-up games amongst regulars are more common in big bet than in limit poker. Big bet poker is more complicated shorthanded or heads-up, so there is more of an opportunity for continual learning, even for the best in the world, at this type of poker. The no trade result holds much better at middle stakes heads-up limit hold ’em where matches between two professionals are rare as any benefits from playing are subsumed by the rake and the loser’s expected loss.
These two rational factors can explain some high stakes heads-up matches, but by no means all of them. A lot of the time, otherwise skilled professionals will be putting themselves in high stakes matches without enough compelling reason — having a slight perceived edge is not enough as presumably nobody would play high stakes without one.
Consequently, the enduring health of the very high stakes does create a puzzle. If these games are solely contested amongst professionals and the game is negative-sum after rake and taxes, then the average nosebleeds pro is a losing player! In fact, many players at these stakes are willing to risk enough money to live comfortably for the rest of their life without any average profit to show for it. Perhaps this is the price one pays for a shot at poker immortality.
“Perhaps the hardest thing for any triumphant manager is to continue evolving, to dismantle what has brought trophies and make it new in response to changes in the evolution of the game.”
— Jonathan Wilson 176
The above quote is about English football, but it works equally well for poker. The two are actually quite similar: poker and sports are both zero-sum, there can only be one winner of the World Series of Poker, and only one team can win the World Cup. Another similarity is the understated role of chance. Sure, the winner of the World Series of Poker has to be lucky enough to win through a field of thousands, but there are a lot of hidden elements to luck that we fail to spot by making mistaken attributions to skill.
We tend to think that over time the variance in poker will run even. That isn’t the case: Your poker career is path dependent. On an elementary level, novice players who run exceptionally well in their first few sessions are much more likely than others to continue playing the game. How well you run when you take shots is also incredibly important. Hitting the ground running at a new level can give a huge boost, allowing you to build a psychological and financial buffer against downswings. On the other hand, a 200 big bet downswing at twice your normal stakes would equate to a 400 big bet downswing at your normal limits — even in the best case scenario this will hamper your progress. In the worst case, such a downswing, or bigger, could lead to a downward spiral of tilt and discouragement with the game. High stakes players will move up many times in their poker career, in which case their progress will be highly dependent on how well they ran in a few 5,000 hand samples at each new limit.
As a demonstration of this, the $500-$1,000 no-limit hold’em tables online first appeared in 2008. A number of professionals, who regularly played $100-$200 and $200-$400, started playing in the games. Some of them ran well, but others racked up downswings in the millions of dollars right away. Before the fact, there was no way to successfully predict which ones would initially run well and which wouldn’t. Nowadays, the ones who ran well are considered some of the best in the world, whereas the ones who ran badly had no way to make their losses back without a significant risk of ruin, and have largely dropped out of the high stakes economy. Observers will attribute the survivors’ success to their great skills, but this causal attribution fails to account for the incredibly high volatility and the people of equal skill who dropped out solely due to luck.177
Estimating the win rates of high stakes players by just looking at the current crop is an example of survivorship bias. The survivors are likely to have achieved above average returns, so taking a random sample of only the survivors will greatly underestimate the variability of returns.
Our natural underestimation of the role of luck leads to two adjustments. First, you should temper your enthusiasm after a period of high winnings. Yes, your skill level contributed to you winning money, but there will always be a hidden or visible element of luck to your results too. Second, you should not become too disheartened after a period of losing since it’s likely you have been the benefactor of various bits of good luck along your poker career, even if that doesn’t currently seem like the case.
On a personal note, my $150,000 downswing in my first 5,000 hands at $250-$500 limit hold ’em was really disappointing. This was a very low probability downswing given I game selected absolutely diligently, but I suffered a number of large losing sessions in a row against some poor opponents.
So my point is that such swings are part and parcel of being a poker player. You just have to accept it and move on, and as you move up these swings will have more and more influence on your lifetime results. That $150,000 downswing has been seared into my memory for a long time, but what is less memorable to me is that I made about half of it back shortly after running way above expectation in some $100-$200 limit games. My distribution of luck was a little less than optimal, my preference would have been to run better at the higher game, which would have given me the confidence to push smaller edges in those games, but my overall pattern of results wasn’t that bad, both results-wise and in dollars.
Furthermore, I certainly benefitted from some good luck at earlier stages in my poker career, particularly in my first full week of $25-$50, where I ran extremely well, winning over $20,000. My transitions to $50-$100 and $100-$200 were slightly bumpier, but in both cases my results were boosted by the presence of lots of games just one step below which allowed me to quickly replenish any losses.
The jump to $250-$500 was relatively large, and by this time good $100-$200 games had become rather infrequent, making the transition unsuccessful when the big downswing came. But most importantly of all, the biggest piece of luck I benefitted from was getting into poker right around the boom in the early noughties. Matching my peak year earnings in today’s tougher environment would be much, much harder.
Another area of struggle was not giving sufficient weight to the other players’ information. One thing I really love about poker is spending time thinking about my strategy. However, I became far too attached to my own way of playing, failing to really take on board the ways in which the top players played differently from me. If all the best players in the world play a spot one way, while your strategy was different, then you are most likely playing it wrong, and the quicker you can accept this and try and find out why, the faster your game will develop. The cost of keeping an open mind about other ideas and trying different things is very low. In the worst case, if you play a spot sub-optimally by temporarily experimenting with different lines, the most it can cost is a few bets. But if you can find a more profitable line to take, then your win rate can be increased forever. If I could learn poker again, this is the biggest thing I would change.
Learning to play well is the first and most crucial element of becoming an intelligent poker player. Game theory appears as complex and mysterious to the uninitiated. Nevertheless, I think it’s the most simple and holistic framework to approach poker. The old school art of exploitive poker is often seen as being far removed from the game theoretic approach, but nothing could be further from the truth. Preventing exploitation is the central pillar in game theory; due to the symmetry of poker, all you need to do is to flip your frame of reference to get from one style of playing to the other. And playing a game theoretic style doesn’t require a great knowledge of mathematics. The only equation you should really know is the optimal bluffing ratio, and even that has an intuitive explanation when you express it in the form of the well known concept of pot odds. (See “Appendix B: Game Theory” starting on page 383.) Poker is such a complex game that complete solutions to real poker situations are not even on the foreseeable horizon, so there is little need for attempting to work out optimal solutions to common spots to the third decimal. Instead, think about the really big game theoretic ideas: balance, information, and hand ranges — and how you can work them into your game. Approaching poker in this way can lead to many easier decisions.
The strategy sections earlier in this book were presented primarily as my approach to limit hold ’em. I hope these sections will be useful for all players. However, short-handed (and heads-up) limit hold ’em is a relatively simple game, and shares links with the other popular games — making it a good model to illustrate core principles. Players of more complicated games can use these ideas as a starting point by grafting on higher-order concepts. Some things will be the same, poker is still poker, but often a more complicated game can lead to radically different play.
In addition, playing multiple games to a high level might be an ever more important skill in the future. In that case, I hope my approach will help simplify the problem at hand by emphasizing theoretical similarities between different games.
My aim with this book has been to highlight all of the elements that go into making a successful poker player. Playing well is important, but there are a number of other things that can change a successful career into an unsatisfying use of time. One of these elements is your attitude towards risk.
Early in my poker career, I was far over towards the risk averse side of the spectrum. This kept me out of trouble, but it also lead to some huge opportunity costs as I moved up much slower than was possible. In my eyes at the time, my results at $3- $6 were terrific, and it was a lot of fun being one of the best players at that limit. In fact, the only thing that spurred me on to moving up in stakes was seeing players with less skill than me succeeding higher up. This lighted up my competitive streak and allowed me to take a plunge at some higher limits that I was more than ready for.
Other players might have a great love of risk, and this can be their greatest strength when games are good or when they are running well, and their greatest weakness otherwise. If this describes you, then it’s probably time to start imposing some structure on your career to try and avoid the downsides of risk taking. The best way to do this is to impose external limitations and checks on your risk taking.
Thomas Schelling presents a good example of this with a drug clinic.178 The clinic makes patients write a signed letter of confession about their drug habit and address it to their place of work. If the clinic finds any traces of drug use during their weekly blood tests, the clinic is bound to send the letter to the addressee. This imposes a huge external cost on taking the drug which then helps the patient to finally kick the habit for good. Poker players can use similar devices such as banning themselves from poker sites with blackjack rooms, giving a friend authority to terminate their sessions, and investing spare cash in illiquid investments.179
I’ve suffered a number of the psychological biases, described previously, first-hand. Some time after controlling my excessive fear of risk, I started having some good results and moved up to the higher games. At this point, overconfidence set in and I began to attribute my results to how great my game had become. In truth, my strategy was somewhat ahead of my opponents, but that doesn’t say much, I still had a lot more to learn. With hindsight, it’s clear that my results were as much to do with being in the right place at the right time as they had to do with skill. Some awful opponents and some good luck explained a lot of it, and the trouble with overconfidence is that it blinkers an unbiased assessment of your game. This means you will cost yourself money by improving at a slower rate than if you had remained honest about your true level of skill.
There is a saying that all political careers end in failure. I suppose this is also true about poker. Even if they don’t end in absolute failure, poker players usually have plateauing results for a long time before they finally call it a day. I’ve experienced this firsthand. My winnings have been decreasing for the last two years despite improving my style of play. The games have become tougher and more infrequent, and it’s been difficult to keep pace. The dollar amounts are still good, but it’s really difficult psychologically to deal with a decreasing income stream. Of course, this is quite a luxurious problem to have, but I think it’s a universal constant that any poker player who has made $x is less happy with their lot than they expected they would be after winning that amount. In addition, moving down in stakes, if necessary, can be as difficult as moving up in stakes as you need to take the game seriously even though the amounts are smaller, and take care to minimize rake paid, game select well, and optimize the number of tables you play — all common pitfalls for the low stakes.
For me, the goal of professional poker is to increase my standard of living by more than other available alternative jobs. Playing 40 hours a week is not done for the fun of it, (although it helps that I enjoy the game), but instead, for the money it gives me. This makes spending and investing wisely just as important a part of professional poker as successfully playing the game.
The thing with money is that it’s incredibly easy to make really stupid decisions when you are young. On a personal note, I lost my entire poker bankroll at the age of 18 with some really bad decisions outside of poker, and this is a common occurrence. For instance, one of the largest winners in online poker history has lost most of his winnings with naive property speculation, a high stakes player recently lost $300,000 because he allowed a third party to see his holecards during a match, and I’ve heard of one economics graduate student getting $200,000 in debt by borrowing money on credit cards and investing in a leveraged portfolio shortly before the credit crunch.180 These were all young people who needlessly lost huge sums of money relative to the size of their bankrolls. Don’t be one of them.
The message of “Part Two: Other Topics” might seem quite negative to you. A whole host of factors such as problem gambling, loss aversion, cognitive dissonance, or core personality traits can shape how well you do in poker. More optimistically, I prefer to subscribe to the view laid out in Geoff Colvin's Talent is Overrated that the main factor separating top performers from the rest of us is not innate skill, but persistent hard work — an option that is available to us all.
It’s my view that poker will remain an incredibly rewarding career for the people willing to work harder and more intensely on their game than anyone else. Good luck!
This appendix provides recommendations for further reading on some of the topics covered in this book. And most of the academic papers referenced in the footnotes of the main text can be found using Google Scholar: http://www.google.com/scholar.
Poker
There are many resources to learn poker from, such as training videos and private coaching sessions. I still think a lot can be learned from books; here are my favorites:
In that respect, this book is truly timeless as its goal is to teach you how to think about poker — something that is relevant in any game.
Chapters 1 to 3 are essentially a refresher course on statistics. The material is useful to know, but if you are not mathematically inclined, you can skip these chapters without losing anything from the rest of the book.
Chapters 4 to 9 cover exploitive play. These chapters are an important part of the book, so it’s vital that you understand what is going on before moving on. Fortunately, this section of the book contains the least amount of math.
Chapters 10 to 21 cover optimal play, and is where the real value added lies. This section is just under 200 pages long, and is worth knowing well. You do not need to understand how the authors come to their results, only what they mean.
The most important concept in this section is “alpha,” which is introduced in Chapter 11, and recurs throughout the rest of the section. Limit players can skip Chapters 12 and 14 which cover concepts in big bet poker.
When they solve a game for an arbitrary pot size P, use the equations to derive the optimal play for a certain pot size — e.g. 4 bets. Then see how varying the pot size will affect the optimal solution. Make notes or highlight parts of the text as you go along. The most important lessons will be repeated in the “key concepts” section at the end of each chapter.
Game Theory
I have a couple of problems with much of the game theory literature. For one, there are many different types of games, and a lot of them have nothing to do with poker. Technically, poker is best modelled as a two-person zero-sum stochastic game of incomplete information. If you go across all of these dimensions, then there are going to be a lot of branches of game theory that are entirely irrelevant to poker.
Second, a number of game theory books are very mathematically involved, making them inaccessible to a wide audience of poker players. This is a shame since a lot of the main reasoning and results behind game theory can be very intuitive when presented without the mathematical dressing.
Investing and Finance
Finance is one of my biggest interests outside of poker. In a career as various and uncertain as an online poker pro, putting a part of your bankroll aside to invest can never be a bad idea. In poker, success is measured by one yardstick: money. Therefore, learning at least the fundamental basics of finance should be a requisite for the professional player.
Interest levels in finance vary greatly over the population; for most people, dealing with investments can be a real chore. Fortunately, investing well can be incredibly easy. The key pillars are low costs and picking an appropriate plan (which you will stick to no matter the market conditions). Both of these are attainable through target-date retirement funds which will automatically rebalance your assets as your investment horizon shortens.
Despite it being recently written, much has changed in the short space of time since. The market prices of risky assets have risen, decreasing the expected returns on offer. Paradoxically, this leads to harder decisions than when the markets were trading at historic lows.
A game can be defined as any situation where the payoff to each player (either in money or in utility) depends on the actions chosen by each player. The simplest possible game is with two players, each choosing between two options (a 2x2 game). For example:
Game No. 1: Saddle Point
Each player simultaneously chooses between their two options, with Player 1 choosing between T and B, and Player 2 choosing between L and R. Player 1’s payoff is written first in each cell. Player 1’s payoff is maximized at 7 when choosing B and when Player 2 chooses L. Player 2’s payoff is maximized at 1 when choosing R and when Player 1 chooses B.
Like poker, this game is zero-sum: when Player 1 and Player 2’s payoffs are added together in each cell, they sum to zero. A zero-sum game is logically equivalent to a constant-sum game: where Player 1 and Player 2’s payoffs sum to any constant number, for example, 10. Poker is a zero-sum game because one person’s winnings must be another’s losses (before the rake is factored in). An example of a constant-sum game might be the division of a government grant between two charities, where they are in direct competition for the money at stake (this is modelling the division as being just between the two charities, including the government in the game would make the three player game zero-sum).
So how should Player 1 play the above game? Choosing B is a high risk strategy for him. If Player 2 chooses L, then Player 1 achieves his maximum payoff of 7, but if Player 2 chooses R, then Player 1 achieves his worst possible payoff of -1. On the other hand, if Player 1 chooses T, then he can guarantee a minimum payoff of at least 5.
Looking at Player 2’s choices, if he picks R, then the maximum he can lose is 5 units compared to L where he can possibly lose up to 7. Taken together, the play of (T, R) has a special significance. It maximizes Player 1’s minimum gain, and it minimizes Player 2’s maximum loss. This is known as the saddle point of the game, and it represents the optimal choice for each player.
Neither player can improve his position by deviating from the choice of (T, R). If Player 1 selects B, he does much worse at (B, R) — now losing 1 unit. If Player 2 deviates by selecting L, then he’s worse off by one additional unit at (T, L). There is no need for either player to mix his strategy between their two options, they are both better off sticking to the saddle point (otherwise known as a pure strategy equilibrium).
In this game, we can derive the optimal solution via another route. For Player 2, R strictly dominates L. If Player 1 picks T then R gives a higher payoff to Player 2, but R also pays off more when Player 1 picks B. No matter what Player 1 does, Player 2’s situation is optimized by selecting R, and we can therefore delete L from consideration:
All that is left is for Player 1 to choose — the decision is an easy one as T is much better than B.
Domination also occurs in poker. The most well known example would be holding the absolute nuts on the river where raising can never be worse than calling. Another example is in bluff-raising. When you are going to be folding some hands, it makes sense to bluff with the best hands you would otherwise fold instead of the worst hands. This applies on early streets where you would rather bluff-raise a one out hand than one with zero outs, but domination also occurs on the river where bluff-raising with slightly stronger hands will take the most advantage of any errors in your opponent’s strategy.
Domination in game theory parlance is very different to the well-known concept of “domination” in poker. When poker players say that ace-king dominates ace-queen, they are referring to the fact that the king kicker trumps the queen kicker on an ace-high flop. But in game theory, domination is a more stringent concept, and comes in two flavors: weak domination and strict domination.
A strategic option is said to strictly dominate another option if it always leads to a higher payoff. In this game, you can see that R does indeed strictly dominate L for Player 2. A weakly dominant strategic option is always at least as good as another option and sometimes better. If Player 2’s payoff in the cell (B, L) was 1, then R would weakly dominate L (since if Player 1 picked B then Player 2 would get the same payoff with either option). But notice that in poker, ace-king does not either strictly or weakly dominate ace-queen since ace-queen is a much stronger hand on a queen-high flop. So when poker players talk about domination, unless they are referring specifically to game theory, their definition of domination is not the same.
Zero-sum games without saddle points involve mixed strategy equilibria. Consider the following game:
Game No. 2: Mixed Strategies
The payoffs in each cell sum to 10, so this constant-sum game is equivalent to a zero-sum game. (We could make the payoffs zero-sum by subtracting 5 from each payoff.) But there is no pure strategy equilibrium in this game: If Player 1 picks B, then Player 2 should pick R, but then Player 1 should pick T and Player 2 should pick L, which takes us back to Player 1 playing B.
So what strategy should each player use? To answer this, let’s call p the probability that Player 2 chooses L and (1-p) the probability Player 2 chooses R. Therefore the payoff for Player 1 choosing T is:
2 p + (9)(1 − p)
And the payoff for Player 1 choosing B is:
8 p + (3 )(1 − p)
Now Player 2 has to choose p so as to minimize the value of Player 1’s strategy. Player 2 doesn’t want to give Player 1 an easy strategic choice, if Player 2 sets p so as either T or B is better than the other, then Player 1 can maximize the value of his strategy by playing a pure strategy. Therefore, Player 2 has to set the payoff for T = B, (otherwise known as making Player 1 indifferent between the two options) or:
2 p + (9)(1 − p) = 8 p + ( ) 3 (1 − p) ⇒
− 7 p + 9 = 5 p + 3 ⇒
6 = 12 p ⇒
p = 0.5
So Player 2 should play L and R with equal probability.
Calling q the probability that Player 1 chooses T, the payoff for Player 2 choosing L is:
8 q + (2)(1 − q)
And the payoff for Player 2 choosing R is:
q + (7)(1− q)
Setting the two equal:
8 q + (2)(1 − q) = q + (7)(1 − q) ⇒
6 q + 2 = − 6 q + 7 ⇒
12 q = 5 ⇒
We see that Player 1 should play B slightly more often than T. The game has a mixed strategy equilibrium of
Note that each players’ optimal strategy depends on their opponents’ payoffs. Player 2 picks a strategy by setting Player 1’s payoffs equal to each other — and vice versa. If we changed the game so as Player 2’s payoffs changed but Player 1’s were unaffected, then Player 1 would be the only player to change his mixed strategy.
Once we allow players to use mixed strategies such as these, then any zero- or constant-sum game with a finite number of actions has an equilibrium (either as a saddle point in pure strategies or one using mixed strategies). This is known as the minimax theorem, and for two-person zero- or constant-sum games, it’s equivalent to the more general Nash equilibrium theorem which also applies to other games.
Let’s now consider a simple poker game:
Game No. 3: Poker
Player 2 should always bet with H. Since the hand will always win a showdown, betting dominates checking. With L, Player 2 should use a mixed strategy between checking and betting. If he never bluffs, then Player 1 could play a pure strategy of always folding. Likewise, Player 1 has to use a mixed strategy between calling and folding. If he always folded, then Player 2 should always bluff, and if he always called, then Player 2 should never bluff.
Let q be the probability that Player 1 calls and (1-q) the probability that he folds. The payoff for Player 2 from bluffing is:
− q + ( P)(1 − q)
The payoff from not bluffing is 0 (L always loses). Setting the two equal produces:
Player 2 always bets H for value and some proportion of L as bluffs. Let r be the proportion of bluffs to total bets. If Player 1
folds, his payoff is zero. If he calls, the payoff to (Player 1) is:
If P = 5, then Player 1 should call five-sixths of the time, and Player 2 would bluff one-sixth of the time with L. This means that Player 1 will fold one-sixth of the time — the same proportion as Player 2 bluffs at. This is not a coincidence, working through the algebra you can see that:
Player 1 folds the same proportion of hands that can beat a bluff as the ratio of bluffs to value bets Player 2 bets. As the pot size grows, Player 1 folds with less of his hands that beat a bluff, and Player 2 bluffs less often. This is exactly like the previous mixed strategy equilibrium where players change their strategies when their opponent’s payoff changes. Player 2 bluffs with less hands in a big pot because Player 1 is getting much better odds to call; whereas Player 1 is calling more because Player 2 is getting a better price on his bluff.
For a more practical implementation of these results at the poker table, you should think of the optimal bluffing and folding ratios as pot odds. In this example, the size of the bet was 1, which was then added to an initial pot of P, therefore the caller was receiving pot odds of (1 + P)-to-1. So if you bet $100 into a $100 pot, then the caller will be receiving pot odds of 2-to-1 to call: therefore, the odds of you value betting to bluffing should also be 2-to-1. When the odds of winning with a bluff-catcher are equal to the pot odds on offer, the caller is made indifferent between calling and folding (both yield the same EV). We saw in Game No. 2 that setting your opponent’s strategic options to equal value is the method of playing a mixed strategy equilibrium.
You can use the pot odds analogy to extend the bluffing ratio to bluff raises and bluff reraises. If your opponent has bet 1 on the river, then he will be receiving pot odds of (P + 1 + 2)-to-1 to call your raise, so your bluffing ratio is 1 . If the pot is initially five 
big bets when your opponent bets the river, a GTO strategy would have you bluff-raise once for every eight value-raises.
In this simple game, the players were using mixed strategies because there were only three different cards in the deck. In real poker, there are many more individual hands, and so you do not often have to mix strategies on an individual hand. The best (undominated) way is just to split your hand range into sections, and then take an action with the top or bottom part of your range. So facing 2-to-1 odds on the river, you would call with the best half of bluff-catchers and fold the weaker ones, rather than mixing on individual hands (since your opponent would have bet the pot and was receiving odds of 1-to-1 on his bluff).181 When initiating the bluff, you would bet with the very worst of your hands until you attain the right bluff-to-value-bet ratio.
Most hands will use pure strategies; hands that mix will either be on the border between two options or will mix in order to balance the strength of your ranges on certain community cards.
Table I shows Polaris’s 3-betting strategy from the big blind in heads-up limit hold ’em:182
Hands with the lightly shaded background always 3-bet; hands with the dark background mix between 3-betting and calling. Polaris always 3-bets pairs, many suited hands, and the majority of aces. Notice that the 3-betting range is strongly tilted towards suited hands. For example, queen-seven suited will always 3-bet and queen-seven offsuit will never 3-bet. This allows Polaris to connect with a range of different flops whether it 3-bets or calls despite mixing on relatively few discrete hand combinations. Therefore, quite a large fraction of the starting hands 3-bet at least sometimes, but suited hands and pairs are dealt less often than offsuit hands — 12 combinations for offsuit, 4 combinations for suited, and 6 combinations for pairs.
The bluffing and calling ratios hold exactly on the river; things are a little more complex before that. That’s because hands are not strictly ordered, worse hands have a chance of drawing out on better hands, and so you could call or bluff with more hands than the ratios indicate since these hands could still improve. A clear example would be when you have a range of the nuts and some very strong draws. Here you will be betting the draws as bluffs (since they are your worst hands), but your bluffs have a strong chance of winning and your opponent has little chance of drawing out on your nut hands. In this case, you would bluff with many more draws than the ratio
. Nevertheless, the ratios hold as a good first approximation to proper play.