Learn the game.
A sixteen-chapter course, beginner to shark — game theory, market math and famous hands included. Work top to bottom, then break each idea in the drills and puzzles.
What beats what
Ten hand categories, best to worst. Ties inside a category go to the higher ranks: a king-high flush beats a queen-high flush, aces up beats kings up. When the category and the ranks match exactly, the pot splits. Kickers settle the rest — with one pair of aces each, the better side card wins, which is why dominated hands bleed money.
The board plays too
Your hand is the best five of seven cards: two in your hand, five on the board. You are never obliged to use your hole cards. If the board is the best five, everyone left splits the pot no matter what they hold — and if the board pairs, every flush and straight should hear alarm bells, because full houses just became possible.
The only free information at the table
Poker is a game of incomplete information, and position is the one edge that costs nothing. Acting after your opponent means you see their decision before making yours, on every street, all hand long. The button acts last after the flop — it is the best seat in the game. The blinds act first and play every pot out of position — they are the worst.
Order after the flop, worst seat to best. UTG is “under the gun” — first to act, blind to everyone's intentions. The cutoff (CO) and button (BTN) see everything before committing a chip.
What position buys you
- Cheaper bluffs. When it checks to you on the button, one small bet often takes it. Out of position you're guessing.
- Bigger value. In position you can size the river bet after seeing their reaction. Out of position you set the price blind.
- Free cards. With a draw in position, check behind and see the next card for nothing. Out of position that option belongs to your opponent.
- Pot control. Medium hands keep the pot small in position. Out of position the pot grows whether you like it or not.
The practical rule: play roughly 3x more hands on the button than under the gun, and give a raise from early position far more respect than the same raise from the button — an UTG raise is built from half the hands a button raise is.
The same hand, two seats
K♦J♠ under the gun: fold. Five players behind you, and every ace-king, ace-jack and
king-queen has you beaten before the flop. When you get action, you're usually dominated —
hitting a king and losing to AK is how stacks disappear.
K♦J♠ on the button: raise. Two players left, both forced in by the blinds, both out of
position for the rest of the hand. Now KJ is a favorite against what calls, and position covers the mistakes.
Think in ranges, not hands
You can't play preflop hand by hand — you'd be guessing forever. You play a range: a fixed set of hands you open from each seat, tight up front, wide on the button. The standard open is 2.2–3x the big blind. If someone raised before you, tighten sharply: flat-call with hands that flop well (pairs, suited connectors), re-raise your monsters and a few of your best suited aces, fold the rest. Calling with dominated offsuit hands like KJo and QTo against an early raise is the most expensive habit in small-stakes poker.
A baseline opening map
Pick a seat. Gold cells open-raise when the pot is unraised; everything else folds. Suited hands sit above the diagonal, offsuit below, pairs on it. This is a sane baseline, not gospel — but if your range looks nothing like it, you have a leak.
Why suited and connected matter
76s is a weaker pair of cards than KJo, but a far better hand: it flops flushes, straights and monster draws — hands that win big pots — and when it misses, it misses cleanly and folds. KJo mostly flops one mediocre pair that wins small pots and loses big ones. Playable garbage beats trap gold. That's also why the button opens 54s and folds Q7o.
The only equation in poker
Facing a bet B into a pot P, a call is break-even when your chance of winning equals B ÷ (P + 2B) — what you pay over what the pot becomes. Every call you ever make is this number against your equity. Memorize the common ones: half-pot needs 25%, two-thirds needs 28.5%, full pot needs 33%.
Worked example: the good call
Pot 120, villain bets 60. You call 60 to win 240, so you need 25%. Nine hearts complete your flush: by the rule of 4, roughly 36% with two cards to come — and the ace might be an out on top. 36 against 25 is a clear call, and it stays a call even counting only the turn (~20%) once implied odds are in the picture: when the flush arrives, his top pair will pay you more.
Worked example: the good fold
Pot 100, villain pots it for 100. You need 33%. The gutshot is four outs — about 17% by the river, half that if he bets again on the turn. Your overcards are dirty outs against the overpair his sizing is shouting about. 17 against 33: fold and keep the hundred. If the ten rolls off next card, nothing about the fold was wrong. You bought a coin flip's worth of equity, refused, and the price was double.
Counting outs honestly
An out is a card that gives you the winning hand, not just a better one. Discount outs that complete your straight while flushing the board, and overcard outs when a pair might not be good. Flush draw = 9, open-ended = 8, two overs = 6 (dirty), gutshot = 4, pocket pair to a set = 2. Then ×4 on the flop, ×2 on the turn for the percentage. The drills tab will burn these into you.
Dry boards, wet boards
A board is dry when it offers no draws — unconnected, rainbow, paired. Whoever was ahead on the flop tends to stay ahead, so small bets do the work and big hands can slow-play. A board is wet when straights and flushes are one card away. Equities run close together, leads change street to street, and made hands must charge draws now — bet big, and don't give free cards.
Whose board is it?
Ask which range the flop favors. A♠K♦7♣ hits the preflop raiser — aces and kings live in raising ranges, so a continuation bet prints whether it connected or not. 7♥6♥5♣ hits the caller — suited connectors call raises, big cards don't flop straights on it. On the caller's boards, the raiser should check more and bluff less. Betting “because you raised preflop” on a board that smashed their range is lighting money on fire.
Every bet needs a reason
There are only two: value — a worse hand calls — and bluff — a better hand folds. Before the chips leave your hand, name which one this is and name the hands that call or fold. If you can't, check. “Finding out where I stand” is not a reason; the information costs exactly what you bet.
What sizes say
| size | bluff must work | caller needs | use it for |
|---|---|---|---|
| ⅓ pot | 25% | 20% | dry boards, range bets |
| ½ pot | 33% | 25% | standard value & bluffs |
| ⅔ pot | 40% | 28.5% | wet boards, charging draws |
| pot | 50% | 33% | polarized: nuts or air |
| 2x pot | 67% | 40% | maximum pressure rivers |
Big bets say “nuts or nothing”; small bets speak for your whole range. Against most opponents, bet bigger with value than theory says — people call too much — and save the huge sizes for when the story holds up.
Bluff like an accountant
A bluff of B into P profits when villain folds more than B ÷ (B + P). Half-pot needs a fold a third of the time; pot-size needs half. Prefer semi-bluffs — draws that win the pot now or complete later, two ways to the money. Bluff hands that block their calls (holding the ace of the flush suit when the flush misses) and give up when the target is a calling station: bluffing someone who never folds is charity with extra steps.
Defend or be farmed
Facing a bet, the minimum defense frequency = P ÷ (P + B). Fold more often than that and any two cards profit by betting at you. Vs half-pot, continue with 67% of your range; vs pot-size, 50%. You don't need to call with trash — you need to not fold two-thirds of everything the moment chips move.
The continuation bet
You raised preflop, one player called, the flop missed everyone — which is most flops. The c-bet is the raiser cashing in the story they started: “I had a hand before the flop and I still have it.” It works because the caller whiffs two times in three, and because your range holds all the big pairs their calling range doesn't. On dry, ace-high or king-high boards, c-bet small (a quarter to a third pot) and often — the cheap bet does the same job as an expensive one when there's nothing to draw to. On wet, connected boards the caller's range hits too; bet bigger but pick fewer hands.
One barrel is a question. Two is an argument.
A c-bet gets called by pairs, draws and stubbornness. The second barrel on the turn is where those hands start paying real money or letting go. Barrel when the turn card helps your story — an overcard to their likely pair (a K or A over a middling flop), a card that completes draws YOU could hold, or a brick while you still hold equity (overcards, a gutshot, a flush draw). Check when the turn helps their range or your hand has showdown value that only a worse hand calls. Value hands and busted stories barrel; medium hands shut up and see a showdown.
The triple barrel & the overbet
Three barrels is a whole-hand story: it says you've had them beat since the flop, and the river bet asks for the stack. Fire the third barrel only when the busted draws are on their side of the range and the scare cards are on yours. And when the river nails your range and misses theirs — the flush completes and you were the one raising flush draws — the overbet (125–175% pot) extracts maximum from their bluff-catchers and prices your rare bluffs perfectly. Overbets belong to polarized ranges: the nuts, air, and nothing in between.
When the c-bet is a leak
- Multiway: with two callers the auto-c-bet burns money — someone connected.
- Vs a calling station: c-bet value only. The bluff half of your range goes on strike.
- Boards that smash the caller (middling connected: 8-7-6, T-9-8 when they defended a blind).
- When you never check strong hands: observant players attack your checks. Protect them.
Every extra player taxes your hand
Heads-up, top pair is a value hand. Four ways, it's a bluff-catcher. Each added player cuts every hand's share of the pot and multiplies the chance that somebody flopped two pair or better. The math is brutal: AA wins ~85% heads-up but only ~56% against four callers. The first law of multiway pots: the threshold for every action goes up — value bets need stronger hands, bluffs need to fold out several players, and “probably good” is probably second best.
Bluffing dies in a crowd
A bluff that folds one player 60% of the time folds two players 36% of the time and three players 22%. Multiway, the c-bet bluff all but disappears from correct play: check your air, bet your made hands, and pick bigger sizes with real hands because someone in the crowd always pays. The corollary when facing bets: a bet into three people is honest far more often than a bet into one. Respect multiway aggression — fold the hands that live on fold equity.
What goes up in value
- Nut draws — the ace-high flush draw, the top-end straight draw. Multiway pots pay them a crowd's money when they hit, and hitting second-best costs stacks.
- Sets — hidden, huge, and happy to see five players see a flop cheap.
- Suited aces — implied odds scale with the number of wallets at the table.
What goes down: offsuit broadways, weak top-pair hands, dominated aces — every hand that wins small pots heads-up loses big pots multiway.
Position matters double
Multiway, the player who closes the action sees everyone's intentions and pays no tax for it. In early position you're guessing about three ranges instead of one. Tighten your opens up front precisely because pots go multiway from there — and squeeze (a big 3-bet over an open plus callers) when the dead money and the capped calling ranges make the pot worth stealing before a flop ever falls.
Look for deviations, not gestures
Movie tells — the eye twitch, the trembling hand — are mostly noise. The professional method is baseline first: watch how a player stacks chips, talks, and times their actions when nothing is at stake. A tell is a deviation from that norm under pressure. The chatty player who goes quiet, the frozen player who starts chatting, the instant actor who suddenly tanks — the change is the information, not the behavior itself.
The classics, decoded
- Strong means weak, weak means strong. The dramatic sigh followed by a raise is nearly always a monster. Aggressive chip-slamming is usually a bluff propping itself up.
- Timing — instant calls are draws and medium hands (no decision to make); long tanks then raises lean value. Snap-raises rep decisions made before your bet: big hands.
- The speech — an opponent who starts narrating (“I think you missed your flush…”) while betting is talking himself into a bluff or goading a call from a monster. Either way: a big hand decision, so slow down and use the other evidence.
- Breathing & stillness — bluffers freeze. Humans under threat go quiet, stop blinking, hold the exhale. Value bettors relax; some even lean back.
- Glance at chips — a player who flops big instinctively checks their stack. The look lasts half a second and it's worth the whole session.
Concealing your own
Give them nothing: same motion, same timing, same posture for every action. Decide what you'll do before the action reaches you, count to three, then do it identically whether you hold the nuts or napkins. Online, the same rule wears different clothes — randomize your act timing, because insta-checks scream weakness and full-clock tanks followed by jams get called.
Reads are a bonus, not a foundation
Tells adjust a close decision; they don't replace the math. If the price says fold and his nostrils say bluff, you're still gambling on nostril data. Build the decision from range and pot odds first, then let the live read move you one notch. The players who invert that order donate with style.
Ranges narrow, they never widen
Hand reading is subtraction. A player starts with every hand they'd play that way preflop; every action after that deletes hands and can never add them back. The UTG raiser who checks the flop deleted his monsters (mostly); the caller on a flush board who suddenly raises the river deleted everything but flushes and boats. By the river, a story told street by street leaves a range you can almost read like cards face-up. Practice one habit: name the range out loud at every action, even approximately. “Big pairs, big aces, some suited broadway” beats “he could have anything.”
The level war
Level one thinks about its own cards. Level two thinks about your cards. Level three thinks about what you think their cards are — and that's where the real game lives. Play one level above your opponent, exactly one: bluff the player who's reading your range; value bet the player who only sees his own hand, because he isn't folding anyway. Outleveling a level-one player is called outplaying yourself — the fancy river bluff sails past a man who never asked what you had.
Consistency is the lie detector
Every bet claims something. Check each claim against the whole story: the turn check-raise that reps a slow-played set — did he play the flop like a set? The river jam repping the backdoor flush — would that hand really have called the flop? When a line is internally inconsistent, it's a bluff or a misplayed medium hand, and your bluff-catchers go up in value. When every street rhymes, believe the story and fold the pretty hand. Blockers refine the last step: holding the ace of the feared suit yourself makes his claimed flush measurably less likely.
Bet sizes talk
Most players size by hand strength without noticing: small when unsure, big when comfortable, weird when bluffing. The tiny river bet is a hand that wants a cheap showdown; the sudden 3x overbet from a passive player is the nuts finding its voice. Against unbalanced sizers, read the size like a subtitle. Just remember the good players know this too — their sizes are chosen for your range, not their hand.
Two games wearing one deck
Cash chips are money; tournament chips are a claim on a prize pool, and the exchange rate moves all night. In cash you can rebuy, so every decision is judged purely by chip EV. In a tournament, busting ends the equity you've built, so chips you might lose are worth more than the identical chips you might win — that asymmetry (formalized as ICM) bends every decision near the money and at final tables. Same cards, different game.
Cash game law: always value, never limp
Deep-stacked cash rewards patience and punishes flair. Raise or fold preflop — limping caps your range and invites the crowd multiway. Go for thin value relentlessly: the extra half-bet extracted with second pair, hand after hand, is the actual win rate. Keep your stack deep (top up to the max buy-in) so your edges get paid in full, target the reckless, and quit the session the moment your decisions start following your emotions instead of the price.
Tournament law: the blinds are chasing you
Escalating blinds mean the game changes every level. Early and deep, play cash-game poker. By 25 big blinds, implied-odds hands lose value and stealing blinds becomes the profit engine. Under ~12 big blinds the whole game collapses to shove or fold — solved charts, no flops, fold equity doing the work. Near the bubble, ICM makes big stacks bullies and medium stacks hostages: attack the players who can't afford to call, and don't become the hostage who folds into oblivion.
Which numbers change
| cash | tournament | |
|---|---|---|
| Stack depth | 100bb+, stable | shrinks all night |
| Chip value | linear — a chip is a chip | nonlinear — ICM near payouts |
| Marginal calls | take every +chipEV edge | pass thin edges when busting costs equity |
| Blinds | fixed — patience is free | rising — patience is taxed |
| Bankroll rule | 20–30 buy-ins | 75–100+ entries (variance is savage) |
Judge the decision, not the river
A 3-to-1 favorite loses one time in four. That is not bad luck, that is the game working as designed. You will make the perfect fold and watch your card roll off; you will get it in ahead and lose three times running. The only question that matters: given what you knew, was the price right? Get that answer right relentlessly and the results follow — the money in poker flows from the players judging by outcomes to the players judging by decisions.
Bankroll is a seatbelt
Variance at one table swings 10 buy-ins without anything being wrong. Keep 20–30 buy-ins for your cash-game stake and never carry more to a session than you'd calmly lose. Moving up because you ran hot for a week is how winners become broke players with good stories.
Tilt has a smell
You know yours: calling “to see it” without doing the math, raising because you're annoyed, playing hands you folded an hour ago. The moment you catch a decision made by mood instead of price, the session is over — the game runs tomorrow, and the players who made you tilt will still be there, still making the same mistakes.
How to actually improve
- Review the big pots you lost: cooler or leak? Coolers you salute; leaks you fix.
- Run the drills here until pot odds are reflex, not arithmetic.
- Play the puzzles cold, then replay the ones you missed a day later.
- One concept per session at the table. Today: position. Tomorrow: sizing.
01 · Expected value is the only scoreboard
Every GTO number in poker comes out of one equation. An action is worth the probability-weighted sum of what it wins and loses:
Checking and folding are the zero points: a fold is worth 0 from here on (the chips already in the pot are gone either way; that's why sunk cost is a fallacy). A call of B into a pot of P (the pot already includes his bet) with equity e:
Set it to zero and you get the price of a call: e* = B / (P + 2B). A pure bluff of B into P that works with probability f:
Those two break-evens are the raw material. The next section turns them into strategy.
02 · The two indifference equations
Take the classic river spot. The bettor has a polar range: nutted hands and pure bluffs. The defender holds a bluff-catcher: it beats every bluff and loses to every value hand.
Equation 1, the bettor makes the caller indifferent. If a share α of the betting range is bluffs, calling is worth α(P + B) − (1 − α)B. Setting that to zero:
Equation 2, the caller makes the bluff indifferent. If the defender continues with frequency d, a bluff earns (1 − d)P − dB. Setting that to zero:
Notice what's strange: the bettor's bluff frequency is set by the caller's payoffs, and the caller's defense frequency is set by the bettor's payoffs. In a mixed equilibrium you choose your frequencies to neutralise your opponent, not to maximise your own hand. Move the size and watch both curves cross zero exactly at the equilibrium numbers.
Gold: the caller's EV (per pot) of calling with a bluff-catcher, as the bettor's bluff share rises. Crimson: the bettor's EV of a bluff as the caller's defense frequency rises. Each crosses zero at its indifference point; the dashed markers are the solved values.
03 · The AKQ game, solved on a 2×2 matrix
Any spot where each player picks between two options is a 2×2 zero-sum matrix game. Write the row player's payoff for each pair of choices, then solve. If a cell is the smallest in its row and the largest in its column it is a saddle point: both players play one pure strategy. Otherwise both mix, and with payoffs a b / c d the closed form is:
Load a preset or type your own payoffs. The AKQ preset is the toy game from the Game theory chapter written as a matrix: the numbers are the bettor's average share of a 100 pot with 100 bets.
| call | fold | |
|---|---|---|
| bluff Q | ||
| check Q |
04 · Fictitious play: watching an equilibrium emerge
How would two players who start clueless ever find these numbers? In 1951 George Brown proposed fictitious play: each round, every player best-responds to the average of everything the opponent has done so far. Julia Robinson proved the averages converge to the equilibrium in every two-player zero-sum game. The naive alternative, best-responding only to the opponent's last move, never settles: it's the exploit, counter-exploit spiral you see at real tables.
Here it runs on the bluff-catcher game for the bet size below: the bettor chooses whether to bluff his air, the caller whether to call. Solved values: bluff share of the air …, call frequency ….
05 · Von Neumann poker: the whole range at once
John von Neumann's own model (1944). Each player gets a hand that is a uniform random number between 0 and 1, higher wins. The pot is P; player 1 may bet B or check to showdown; player 2 calls or folds. Solving the three indifference conditions (the weakest bluff, the weakest value bet and the weakest call are each indifferent) gives thresholds a < c < b:
Player 1 bluffs the bottom of his range [0, a], checks the middle and value bets the top [b, 1]. Player 2 calls with [c, 1]. The two lessons real solvers repeat: your worst hands bluff, not your medium hands (medium hands win showdowns by checking), and there's an optimal bet size.
Player 1's equilibrium share of the pot for every bet size (checking everything earns exactly half). The gold marker is your slider; the peak is the best size in this model.
06 · Bayes: what a bet tells you
A bet is evidence. Before it, some share of his range is value; after it, the share changes by how much more often value hands take that line than bluffs do. That is Bayes' theorem:
Your bluff-catcher's equity is then just the posterior bluff share, and you call when it beats B / (P + 2B). Most players skip the update and call on the prior. Drag the dials.
07 · Equity realization, implied & reverse implied odds
Raw equity assumes you see every card for free. You don't. Realization R scales it: in position with a playable hand R can exceed 1 (you win more than your share), out of position with a weak draw it sits well below 1. The call becomes:
Implied odds add the money you expect to win later when you hit (X). Reverse implied odds subtract what you pay off later when you make a second-best hand (Y). The future money you need to justify a call with equity e:
08 · Solve heads-up push/fold yourself
A real equilibrium, computed in your browser. Small blind (0.5bb posted) either shoves or folds; big blind (1bb posted) calls or folds. With effective stack S big blinds:
EVBB(call) = evs shove · 2S − S ≥ −1 ? call ⇒ needs e ≥ (S − 1) / 2S
Here q is the share of hands the big blind calls with. The solver runs fictitious play: every round it Monte-Carlos all 169 starting hands against the opponent's average range, takes the best response, and folds it into the average. Cell brightness is how often the hand is played. Expect small Monte-Carlo wobble near the edges of each range.
09 · ICM: why tournament chips aren't money
In a tournament your chips convert to prize money non-linearly: doubling your stack never doubles your payout. The Independent Chip Model (Malmuth-Harville) prices a stack by assuming your chance to finish first equals your share of the chips, then recursing for second, third and so on:
The consequence is risk premium: losing chips costs more prize money than winning the same chips earns, so calling an all-in needs more equity than pot odds say. The ratio is the bubble factor. Your tournament mode at the table plays winner-take-all, where ICM is linear; set payouts below to see how a real prize ladder changes the math.
10 · Math check
Ten questions that need a pencil, not a feeling. Each comes with the worked solution.
Train the math.
Answer, get graded, see the worked solution. Speed comes with reps.
Game theory.
Six chapters on why the right play is right — indifference, sizing, ranges, blockers, exploitation. Every dial is live.
The engine of GTO: make them indifferent
Rock-paper-scissors has one unbeatable strategy: throw each option a third of the time. You win nothing extra — but nothing your opponent does can beat you. Poker's equilibrium works the same way. A balanced river bettor mixes value and bluffs in exactly the ratio that makes your bluff-catcher indifferent: call or fold, your EV is identical, zero. A balanced caller defends exactly often enough that his bluffs earn nothing. Every GTO frequency you'll ever memorize is just one of these two indifference conditions solved for a bet size.
The AKQ game — all of poker in three cards
The cleanest toy game in poker theory. Deck of three: A, K, Q. You each ante; the pot is 100. Your opponent gets one card and bets 100 (pot) or checks; you hold the K — the eternal bluff-catcher: it only beats a bluff, only loses to value. He always bets the A. The whole game is two dials: how often he bluffs his Q, and how often you call. Drag them.
What the toy game teaches
- Value bets need bluffs: if he never bluffs, you fold every K and his aces win only antes.
- Bluffs need value: if he bluffs every Q, you call every time and collect.
- At equilibrium (bluff half the Qs, call half the time) neither dial matters anymore — every deviation by one player is pure profit for the other. That mutual best-response is a Nash equilibrium.
- Real poker is this game with 1,326 cards in each range — the structure never changes.
Bet size → defense & bluffs
Every bet size implies three numbers. Move the slider: for a bet of that size, MDF is how often the defender must continue so the bettor can’t profit by bluffing any two cards; the required equity is the price the caller is getting; and the value-to-bluff ratio is how the bettor balances a polar range so a bluff-catcher is exactly indifferent. Bigger bets need fewer bluffs and let the defender fold more — that is the whole tension of poker in one dial.
Bluffs expire street by street
The ratios above are river numbers. Earlier streets allow more bluffs because a bluff raised on the flop still has two cards of life in it — solvers bet the flop with roughly 2 bluffs per value hand, the turn near 1:1, and the river at the ratio the slider shows. That's why barrels feel relentless on early streets and honest on late ones, and why calling down gets safer the later the story gets told.
| street | bluff : value (≈, pot-size bets) | why |
|---|---|---|
| Flop | 2 : 1 | bluffs still have equity + two streets of fold pressure left |
| Turn | 1 : 1 | one card of life left |
| River | 1 : 2 | pure polarization — the math above, nothing else |
Preflop opening ranges
A solver opens tight up front and wide on the button, because position lets you realize equity. Pick a seat to light up an example unexploitable-ish opening range on the 13×13 grid — pairs on the diagonal, suited above it, offsuit below.
Combinatorics: the currency of range reading
Ranges are counted in combos: 6 per pocket pair, 4 per suited hand, 12 per offsuit hand, 1,326 total. Every card you can see deletes combos from your opponent's range — that's a blocker. Hold one ace and AA drops from 6 combos to 3; an ace hits the board and it happens again. The best river bluffs hold the exact card that makes his calling hands impossible: bluffing with the bare A♠ on a three-spade board works because he can't have the nut flush — you're holding it hostage.
| you're facing a river jam on K♠ 9♠ 4♦ 2♠ 8♣ — his range | combos |
|---|---|
| Flushes (suited spade combos that got there) | ~10 |
| Sets that filled up? none — board unpaired... sets | ~9 |
| Busted straight draws (QJ, JT no spade) | ~12 |
| Same jam while you hold the A♠: flush combos | ~4 |
One card in your hand halved his value region. Count combos, not vibes.
Polarization — why bets are big or checks, rarely in between
When your range is stronger than your opponent’s (a range advantage) you bet; when it’s capped, you check. A polarized range is nuts-or-air and bets big — it wants folds from bluffs and max value from the nuts, and medium hands have nothing to gain by bloating the pot. A merged range (lots of medium-strength value) bets small, targeting worse pairs that will call a cheap price. Rule of thumb: size up when polar, size down when merged, check your bluff-catchers.
- Value : bluff shifts with the street — you can bluff more on the flop because there are two cards to improve; you must tighten by the river.
- Equity realization is why suited and connected hands out-perform their raw equity: they flop draws and play well in position.
- Capped ranges get attacked: the player who "can't have it" (he'd have raised earlier with his monsters) faces overbets his bluff-catchers hate.
Range advantage vs nut advantage
Two different weapons. Range advantage — your whole range has more equity (A-K-x flop after you raised preflop) — licenses small bets at high frequency: everything in your range pokes at everything in his. Nut advantage — only you can hold the monsters (the 4-straight board where you raised turn) — licenses huge bets at low frequency: polar, violent, rare. Misreading which advantage you hold is how players end up betting medium into strength or min-betting the nuts.
Equilibrium is the shield. Exploitation is the sword.
GTO guarantees you can't be beaten — it does not maximize your winnings against players who make mistakes, and everyone you'll play makes mistakes. The money move is the maximal exploit: deviate from balance exactly as far as their error invites, and accept that you're now exploitable in a direction they'll never find. Against a player who folds too much, bluff every hand that can't win a showdown. Against one who never folds, delete your bluffs entirely and value bet third pair. Balance is for opponents who are watching. Most aren't.
The exploit table
| their leak | your deviation | what it costs you |
|---|---|---|
| Folds too often to barrels | bluff far above the ratio | exploitable by re-bluffs you'll never see |
| Calling station | zero bluffs, relentless thin value | your checks become honest — nobody's checking |
| Never bluffs rivers | fold every bluff-catcher, ignore MDF | a bluffer would print — this player isn't one |
| Opens too wide preflop | 3-bet linear (more value hands, lighter) | flatting range thins — fine, his range is junk |
| Bets small when weak, big when strong | read the size like a subtitle | nothing — unbalanced sizing is free money |
| Tilts after losing a pot | attack the very next hands | variance — his range is wide but angry |
Who's exploiting whom
The uncomfortable half of the lesson: every deviation you make opens a door behind you. Bluff the fold-happy player long enough and the observant third man starts check-raising your barrels. The skill isn't picking exploit or balance once — it's knowing which opponents merit which, table by table, and noticing the moment someone starts farming your adjustment. When in doubt, or out-classed, retreat to balance: the shield never rusts.
Concept check
Twelve questions on the theory — no cards, no luck, just the ideas. Score persists.
Run the hand.
Twenty scripted hands played on the real table controls — slider, steppers, the fold button. Every street is graded like a chess puzzle: bot reactions and runouts are predetermined, and the wrong move (or the wrong size) gets punished. Full breakdown when the hand ends.
Puzzle breakdown
The same game.
Poker is a market: incomplete information, priced risk, an edge you compound or squander. Four calculators that speak both languages.
A poker hand and a trade are the same object. You commit capital under uncertainty, you have some edge or you don’t, and the long run is decided not by any single outcome but by expected value repeated through variance. The pros who last in both games are the ones who size correctly and survive the downswings. Below: the machinery that connects them.
Rosetta stone
| At the table | In the market | The shared idea |
|---|---|---|
| Expected value (EV) | Expected return | Probability-weighted payoff |
| Variance / swings | Volatility (σ) | Dispersion around the mean |
| Pot odds | Break-even / implied probability | The price the market quotes a bet |
| Fold equity | Optionality | Getting paid when others quit |
| Your range | A distribution of outcomes | Think in sets, never one hand |
| Bankroll management | Position sizing / Kelly | Bet size vs edge and risk of ruin |
| GTO / equilibrium | Efficient-market / minimax | Unexploitable baseline strategy |
| Exploitative play | Alpha | Profiting from others’ mistakes |
| Tilt | Behavioral bias | Emotion taxing the edge |
| Bluff-catching | Bayesian updating | Reading a range from an action |
01 · Edge, variance & the long run
A winning player and a profitable strategy both have a positive edge per bet and a standard deviation many times larger. That gap is why results lie in the short run. Set an edge and a swing size, then run a few thousand hands and watch the equity curve — green ends up, but look at the drawdowns on the way.
02 · Kelly — how much to bet
You have an edge. How much of your stack do you commit? Bet too little and you leave growth on the table; bet too much and variance ruins you even while you’re right. The Kelly criterion maximizes the long-run growth rate: f* = (bp − q) / b, where you win b-to-1 with probability p. Most pros bet a fraction of it — half-Kelly gives up ~¼ of the growth for far less variance.
Long-run growth rate vs the fraction of bankroll wagered. The peak is full Kelly; past it, growth falls and risk climbs. Bet past the right-hand zero and you lose money with a winning hand.
03 · Risk of ruin
Even a lifelong winner goes broke if the bankroll is too thin for the swings. Risk of ruin ≈ e^(−2 · edge · bankroll / variance). It falls exponentially as you add buy-ins — which is why bankroll rules (and stop-losses) are not caution, they’re survival math. Same formula a desk uses to size a book against its Sharpe.
Risk of ruin against bankroll depth. Your current bankroll is the gold marker. Notice how quickly the curve flattens — the first twenty buy-ins buy most of the safety.
04 · A bet is a priced probability
When a market quotes 3-to-1 on an event, it is stating an implied probability of 25% — exactly how a pot lays you a price. Your edge is the gap between that implied number and your true estimate. Bet only when your read beats the price; the size of the gap is your expected value per unit staked.
The takeaway. Nobody wins every hand or every trade — that was never the game. You find spots where your estimate beats the price, you size them so variance can’t bury you, and you repeat the process a few thousand times. Discipline over drama. The edge does the rest.
Sit behind the pros.
Step through the hand street by street. Win probabilities are computed live, ties split.
Speak the language.
Every term you'll hear at the table or in the classroom. Search it, or browse by family.
The numbers that matter.
Everything the drills train, on one page.
Hand rankings, top down
Straight flush › quads › full house › flush › straight › trips › two pair › one pair › high card. The ace plays high and low (A-2-3-4-5 is the wheel). Board pairs = full houses possible. Three suits on board = flushes possible. Best five of seven, always.
Pot odds
Facing a bet: required equity = call ÷ (pot after you call).
Villain bets B into pot P: you need B ÷ (P + 2B).
⅓-pot → 20% · half-pot → 25% · two-thirds → 28.5% · pot-size → 33% · 2x pot → 40%.
Rule of 4 and 2
Equity from outs: on the flop ×4 (two cards to come), on the turn ×2. 9-out flush draw on the flop ≈ 36%, on the turn ≈ 18%. Slightly overshoots above 8 outs; close enough at the table.
| draw | outs | flop → river | turn → river |
|---|---|---|---|
| Pocket pair → set | 2 | ~8% | ~4% |
| Gutshot straight | 4 | ~17% | ~9% |
| Two overcards | 6 | ~24% | ~13% |
| Open-ended straight | 8 | ~32% | ~17% |
| Flush draw | 9 | ~35% | ~20% |
| Flush + gutshot | 12 | ~45% | ~26% |
| Flush + open-ended | 15 | ~54% | ~33% |
Bluff break-even & MDF
Your bluff of B into P must work B ÷ (B + P) of the time:
⅓-pot → 25%, half-pot → 33%, two-thirds → 40%, pot → 50%, 2x pot → 67%.
Facing a bet, the minimum defense frequency = P ÷ (P + B) keeps you unexploitable:
vs ⅓-pot defend 75%, vs half-pot 67%, vs two-thirds 60%, vs pot-size 50%.
Preflop equities worth memorizing
- Overpair vs underpair (AA vs KK) ≈ 82 / 18
- Pair vs two overcards (QQ vs AK) ≈ 57 / 43 — the classic flip
- Pair vs one overcard (TT vs A9) ≈ 71 / 29
- Dominated hand (AK vs AQ) ≈ 73 / 27
- Two overs vs two unders (AK vs 76s) ≈ 60 / 40
- AA vs a random hand ≈ 85 / 15 · AA vs 4 random hands ≈ 56% — aces want few opponents
Combos & blockers
Any unpaired hand = 16 combos (4 suited + 12 offsuit). A pocket pair = 6. Each card you see removes combos: holding one ace cuts AA from 6 combos to 3; an ace on the board does the same. That's blocker logic — you bluff best holding the cards that make their calling hands impossible.
| holding | combos |
|---|---|
| AK (any) | 16 |
| AKs | 4 |
| AKo | 12 |
| 6 | |
| QQ when a Q is visible | 3 |
| QQ when two Qs are visible | 1 |
Bet sizes, translated
| size | your bluff needs folds | their call needs | MDF vs it |
|---|---|---|---|
| ⅓ pot | 25% | 20% | 75% |
| ½ pot | 33% | 25% | 67% |
| ⅔ pot | 40% | 28.5% | 60% |
| pot | 50% | 33% | 50% |
| 2x pot | 67% | 40% | 33% |
Game theory formulas
| quantity | formula (bet B into pot P) | pot-size bet |
|---|---|---|
| Caller's required equity | B / (P + 2B) | 33.3% |
| Balanced bluff share α | B / (P + 2B) | 33.3% |
| Value : bluff | (P + B) : B | 2 : 1 |
| Bluff break-even folds | B / (P + B) | 50% |
| Minimum defense frequency | P / (P + B) | 50% |
| EV of a call | e(P + 2B) − B | · |
| Implied odds needed | (1 − e)B / e − (P + B) | · |
| Bayes posterior | pLv / (pLv + (1 − p)Lb) | · |
| 2×2 mixed equilibrium | p = (d − c)/(a − b − c + d) | · |
| Push/fold BB call needs | (S − 1) / 2S | 45% at 10bb |
Position, one line
Acting last is information for free. Play more hands on the button, fewer up front, and give late-position raises less credit than early ones. Rough opening width (matching the lesson-3 grid): UTG ~11% · MP ~14% · CO ~21% · BTN ~35%+.
Stack-to-pot ratio
SPR = stack ÷ pot on the flop. Under ~3: one pair is happy to stack off. Around 6–10: one pair wants pot control; sets, straights and big draws want the money in. Deep (13+): implied-odds hands (small pairs, suited connectors) go way up in value — and one pair becomes a bluff-catcher, not a stack-off hand.
The five leaks that pay everyone else
- Calling preflop raises with dominated offsuit hands (KJo, QTo, A8o).
- Chasing draws without checking the price — count outs, then the pot odds, every time.
- Betting “to see where you're at” — every bet is value or bluff, name which.
- Paying off huge river bets with one pair “because you came this far”.
- Judging decisions by the river card instead of the price you were offered.