What ICM Is: Stacks to Dollars by Hand

Why this lesson matters

Defines the independent chip model formally: each player's probability of finishing first equals their share of chips, and finish probabilities for lower places are computed by removing each possible winner and renormalizing the remainder. Works one complete 3-player example entirely by hand: stacks of 5,000 / 3,000 / 2,000 with a 50/30/20 payout on a $1,000 pool, computing every finish-distribution branch and arriving at exact $EV for all three players, then observing that the chip leader holds 50% of chips but well under 50% of the prize money. Uses the worked numbers to state the model's core output — the dollar value of a chip falls as your stack grows — and flags ICM's known blind spots (no position, no skill, no blind pressure) so students treat it as a baseline rather than gospel. The quiz should test executing a small ICM calculation by hand for a new 3-player stack configuration and identifying what the model ignores. This lesson belongs to ICM Theory, so the goal is not to memorize one attractive hand history. The goal is to build a tournament decision process that survives changing blind levels, changing table draws, and changing payout pressure. Cash-game instincts are useful background, but tournament poker adds a clock, a finite stack, and a prize ladder. Whenever those forces conflict with pure chip accumulation, you need a way to name the conflict before you act. The practical question is always: what is my stack in big blinds, who covers whom, what dead money is available, and what happens to my future options if I lose this pot?

Table-time framework

Start every decision by converting the hand into a stack-band problem. A player with 75bb can open, call, 3-bet, and play turns and rivers with meaningful maneuvering room. A player with 35bb is already in a compressed tree: opens are smaller, 3-bets are more linear, and many postflop bets commit stacks by the river. A player with 18bb has reshove pressure and limited flatting room. A player with 10bb or less is usually deciding whether to move all-in, call all-in, or preserve fold equity for the next orbit. The tournament stage then modifies that stack-band answer. Early levels reward chip accumulation against weak ranges. Pre-bubble levels reward pressure on medium stacks that do not want to bust. Bubble and final-table levels introduce risk premium. Bounty and satellite formats can invert the usual conclusion. This is why the same two cards can be a value 3-bet, an open-fold, a reshove, or a mandatory muck in different tournament states.

Quantitative anchor

The quantitative anchor for this lesson is the ICM diff calculation in content/mathlogs/tournament-poker/icm-theory.md. In the logged 5000/3000/2000 stack example with a 50/30/20 payout, the short stack's chip-EV break-even point of 50 percent is not enough; the tool reports an ICM-neutral equity requirement of 53.9 percent because losing chips hurts more than winning the same number helps. This is an exact Malmuth-Harville calculation for the stated stacks and payout, not a solver recommendation for every final table. Use it to price risk premium, then still adjust for position, future fold equity, and opponent range.

Worked hand 1

In the first worked spot, Hero holds As Ks in a formation matching the lesson scope. Before looking for a clever line, Hero writes the state in plain numbers: effective stack, pot before the action, players left to act, and whether losing the pot merely hurts or ends the tournament. If Hero is the covering stack and the opponents behind are medium stacks, aggression gains value because folds arrive more often and Hero retains a playable stack when called and lost. If Hero is covered near a payout jump, the same aggressive action needs more equity because the downside has extra dollar cost. The clean adjustment is to choose hands that keep their equity when called. High cards, pairs, suited aces, and blockers generally perform better than pretty disconnected holdings that needed deep implied odds. The line recommended by this lesson is therefore chosen for its full package: fold equity now, equity when called, and future stack utility after the hand.

Hero

A♠♠A♠

K♠♠K♠

Villainunknown

1. 1.Preflop action creates the tournament formation described in the lesson
2. 2.Hero converts all stacks to big blinds before choosing a line
3. 3.Hero chooses the lesson line and writes down the range and payout assumptions

Analysis

After the action, the review question is not whether Hero won the pot. The review question is whether the assumptions were available before acting. Did the opener really have enough folds? Did the blinds really have stacks that could not fight back? Did the bounty, bubble, or satellite pressure legally belong to Hero rather than to another player? A disciplined tournament player writes those assumptions down because the result will otherwise overwrite memory. When the pot is won uncontested, any raise feels brilliant. When the pot is lost at showdown, even a good shove feels reckless. Neither feeling is analysis.

Worked hand 2

The second worked spot uses Qh Qd and deliberately changes one variable. Maybe Hero no longer covers the opponent. Maybe the blind level is about to rise. Maybe a payout jump is close. Maybe the table has redrawn and the aggressive big stack is now on Hero's left. This is where many tournament mistakes happen: players remember the previous rule but not the condition that made the rule true. A hand that was a profitable open at 40bb can become a fold at 22bb because flatting invites squeezes and opening cannot comfortably call a jam. A hand that was a profitable reshove away from the money can become a fold at a final table because busting transfers too much prize equity to everyone else. A hand that was a freezeout fold can become a PKO call when Hero covers and the bounty is large enough.

Hero

Q♥♥Q♥

Q♦♦Q♦

Villainunknown

1. 1.Hero faces the tempting alternative line
2. 2.Hero checks whether stack depth, fold equity, and payout pressure support it
3. 3.Hero rejects the line when the tournament condition changes the cash-game instinct

Analysis

The discipline check is to state the competing line honestly. Calling looks attractive because it keeps weaker hands in, but calling with a shallow stack often removes the only weapon short stacks possess: fold equity. Open-raising looks attractive because it risks less than shoving, but at some depths raise-folding too much burns chips and raise-calling too wide becomes accidental gambling. Jamming looks attractive because it denies realization, but jamming into players who are priced in or bounty-incentivized to call can be worse than making a smaller raise with the top of range. Folding looks tight, but folding is the highest EV action when the pot cannot be won often enough or when the dollar downside overwhelms a thin chip edge.

Common mistakes

The first mistake is importing a 100bb cash-game range without asking what the current stack can realize. Small pairs and suited connectors are profitable deep because they can win stacks when they hit disguised hands. At 25bb, the call-to-stack ratio shrinks and those hands lose much of their reason to enter. The second mistake is treating every tournament chip as though it has a fixed cash value. Early in an event, chip EV is a close working model. Near bubbles, final tables, and satellites, dollar EV can diverge sharply. The third mistake is overusing words like Nash, solver, and standard to hide a guess. Unless the exact tree was computed for this game, stack, ante, payout, and range set, the honest label is baseline, approximation, or chart-guided heuristic. The fourth mistake is failing to update after a redraw. A great steal seat with two nits on the left becomes a bad steal seat when an 80bb regular appears directly behind.

Practical rules

Use this lesson as a repeatable checklist. First, mark your stack band at the start of every level and after every pot: 60bb plus, about 40bb, about 25bb, about 15bb, and 10bb or less. Second, mark cover relationships, because a player who covers you can threaten your tournament life while a player you cover cannot bust you. Third, ask what happens if everyone folds, what happens when one opponent calls, and what range is likely to call. Fourth, separate chip EV from dollar EV. If the payout structure is flat, the bubble is close, the final table has large jumps, or seats are equal, the price for risking elimination rises. Fifth, when a bounty is present, ask whether Hero can actually collect it and whether the bounty is large relative to the pot and blind level. Sixth, keep the language modest: a chart can guide the decision, but a chart is not permission to stop thinking.

Review drill

Before moving on, replay two hands from your own database or memory that match this lesson. For each one, write the stack sizes in big blinds, the pot size, the open or shove size, the players left to act, and the relevant prize pressure. Then write the range you believe Villain opens, continues, or calls. If you cannot write that range, the hand was not a precision spot; it was a judgment spot, and your review should focus on whether the judgment was reasonable. Finally, compare your action to the lesson rule. If you deviated, name the exploit trigger. A trigger can be a player overfolding, calling too wide for bounties, refusing to risk a medium stack near the bubble, or defending the big blind too tightly against small ante-era opens. A trigger cannot be simply wanting to win the pot.

Takeaway

What ICM Is: Stacks to Dollars by Hand is a tournament lesson because the correct answer depends on more than card strength. The best players are not memorizing one heroic shove or one famous fold. They are translating each hand into stack depth, dead money, position, range interaction, and payout pressure, then choosing the action that performs best across the whole tree. That is the standard you should use in game and in review.

Module connection

Within ICM Theory, this lesson should be studied alongside the surrounding lessons rather than as a standalone trick. The neighboring topics define the other side of the same pressure. If this lesson emphasizes aggression, another lesson in the module tells you when aggression becomes spew. If this lesson emphasizes survival, another lesson tells you when survival becomes surrendering too much equity. Good tournament strategy is rarely one maxim. It is a hierarchy: stack depth first, position second, opponent range third, payout and format adjustment fourth, and only then the exact hand. That hierarchy keeps decisions stable when the lobby, table draw, or blind clock changes.