ICM in poker: chips are not money

At an example final table you hold 5,000 of the 10,000 chips in play. Is your stack worth half of the prize money?

The short answer

No. A tournament, where the players usually start with the same stack and play until one player holds every chip, pays by the place each player finishes in, not by the chips each player holds. At the final table, the last table where the prizes are decided, half the chips gives half a chance of first place, and the lower places pay less, so half the chips is worth less than half the prize money. The same gap means that calling an all-in there, a bet of a player’s whole stack, needs your hand to come out higher more than half the time.

Share of the chips

ALeader, 5,000 chips: 50%.
BSecond stack, 3,000 chips: 30%.
CShort stack, 2,000 chips: 20%.

Share of the prize money

ALeader: 38.4% of the money.
BSecond stack: 32.8% of the money.
CShort stack: 28.9% of the money.

The stacks are the same in both circles, and only the yardstick changes: measured in prize money, the leader’s slice shrinks and the two shorter stacks’ slices grow. The prizes are an example, not those of any room. Amounts are in Canadian dollars.

Leo, in the dealer’s place, touches the end of the longest of three rows of chips laid side by side on a poker table, five, three and two, while Maya looks on with folded arms and one eyebrow raised.

Leo smiling, explaining with an open hand.

Leoformer dealer · rules and maths

At final tables I watched people count their chips as if every chip were a dollar, and then make their calls on that count. The number worth working out is what a stack is worth in prize money, and a chip count does not tell you that number.

Tournament chips are not money

The players in a tournament usually start with the same stack of tournament chips, the chips in front of one player. A player whose chips are gone goes out, and the prizes go to the places in the order the players go out, the last player left taking first place.

The tournament rules of Casino du Lac-Leamy, dated 22 Aug 2019, say this in so many words: the chips each player starts with cannot be cashed, bets are made with chips that have no money value, and tournament chips cannot be passed from one player to another.

A tournament chipCounts how far a player is from going out. Under the rules dated 22 Aug 2019, it cannot be cashed or handed to another player.

A prizePaid in money, by finishing place, out of a prize pool made of the buy-ins after the room’s fee.

The prizes add up to the prize pool, and the prize pool is what is left of the buy-ins after the room takes its fee, which is lesson 7.11. Taken together, the players in a tournament finish behind by that fee: nothing at a final table turns a tournament into income.

From chips to prize money

To turn a stack into prize money we count with a model, not a rule: no live room whose documents we could read names a method. ICM, the independent chip model, is the name players use for working out what tournament chips are worth in prize money, and the version here is the Harville model, after a 1973 paper on the outcomes of competitions with many entries. The steps below are our reading of how the model works.

First placeEach player’s chance of finishing first is that player’s share of the chips.
Each next placeTake away the player who finished above, and run the same rule among the players still left.
Prize moneyEach place’s prize, times the chance of finishing there, added up across the places.

How often each stack finishes in each placeOur count with the Harville model, at the example final table
Stack	First place	Second place	Third place
Leader5,000 chips	50.0%	33.9%	16.1%
Second stack3,000 chips	30.0%	37.5%	32.5%
Short stack2,000 chips	20.0%	28.6%	51.4%

The leader finishes first exactly as often as its share of the chips, and the shorter stacks pick up most of the second and third places.

Leader, 5,000 chips$383.93in prize money.
Second stack, 3,000 chips$327.50in prize money.
Short stack, 2,000 chips$288.57in prize money.
All three stacks$1,000the whole prize pool.

Each figure is the prizes weighted by the chances in the table above, counted exactly over every order in which the three players can finish.

So the leader holds 50% of the chips and 38.4% of the money, and the short stack holds 20% of the chips and 28.9% of the money.

An all-in call at the final table

In the same example, take the second stack’s seat: the leader moves all in against your 3,000 chips. You either call, matching the bet, or fold, giving up the hand. Calling puts all 3,000 at risk: if your hand comes out higher you double to 6,000 chips, and if it comes out lower you are out in third place. Suppose your hand comes out higher exactly half the time; in chips the call is even, because you stand to gain as many chips as you stand to lose.

Fold$327.50Your 3,000 chips, as they are.
The call holds$410Your stack doubles to 6,000 chips.
The call loses$200You go out in third place.
The call, on average$305Half the hands one way, half the other.

The chips at stake are the same either way, and the prize money is not: going out costs more than doubling up adds, so the average of the call falls below the fold.

How often your hand comes out higher50%
Needed, counting chips50%
Needed, counting prize money60.7%

The third bar is the share of hands your call would need to be worth as much as the fold in prize money; the chips ask only for half.

With two players left the gap closes: the second prize is certain, and the rest of the prize money moves in a straight line with the chips, so chips and prize money give the same answer. A call worth more in prize money can still lose the hand: these figures are averages over many such hands, and one hand is decided by the cards. How often one hand comes out ahead of another before the flop is lesson 7.6.

Who sets the prizes, and who may change them

How the prize pool is split by place is the room’s payout table, and payout tables differ from room to room and with the number of entries. The example prizes here, $500, $300 and $200 out of a pool of $1,000, are what the payout table on page 3 of the Elements Casino Brantford tournament sheet gives the first three places when 30 entries or fewer play. How a pool is built and paid is lesson 7.11.

Players at a final table sometimes agree to split the prize money among themselves. Under the terms of Elements Casino Brantford tournaments that is the room’s decision and not the players’ right: if all the remaining players agree on a formula, the room may choose to accept the agreement and change the published payouts.

The playersMay all agree on a way to split the prize money among themselves.

The roomDecides whether to accept that agreement and change the published payouts.

ICMThe model on this page, for what a stack is worth in prize money; no live room we could read names it.

Rules in some rooms forbid players to split a pot by agreement; those rules are about a pot at a cash table, which is lesson 7.5, not about a deal at a final table.

Maya with a hand on her chest, speaking gently and seriously.

Mayaformer floor supervisor · manners, money and breaks

A final table is exciting, and excitement is when limits slip. The buy-in and any re-entries were decided before the first hand, and what your stack is worth now does not change that amount: the money is already spent, and the prize is not yet paid.

If the evening has started to run on the hope of a final table, stop and look at the plan you made at home. Losing the tournament after a long run is the moment lesson 7.13 is about.

If it stops being fun

If gambling stops being entertainment, help is available. 19+ in most of Canada; 18+ in Alberta, Manitoba and Quebec. More on responsible gaming.

1-866-531-2600Ontario · ConnexOntario · 24 hours

1-800-461-0140Quebec · Jeu: aide et référence · 24 hours

1-888-795-6111British Columbia · Gambling Support BC · 24 hours

1-866-332-2322Alberta · Addiction Helpline (Recovery Alberta) · 24 hours

1-800-463-1554Manitoba · Problem Gambling Helpline · 24 hours

1-800-306-6789Saskatchewan · Problem Gambling Help Line · 24 hours

1-888-429-8167Nova Scotia · Mental Health and Addictions Crisis Line · 24 hours

1-800-461-1234New Brunswick · Gambling Information Line

1-855-255-4255Prince Edward Island · PEI Gambling Support Line · 24 hours

811Newfoundland and Labrador · Gambling Help Line · 24 hours

1-844-533-3030Yukon · Reach Out Support Line (CMHA Yukon)

If you are thinking about suicide, call or text 9-8-8

Practice: what are your chips worth?

Level 1 sets an example final table, with stacks and example prizes, and asks what your stack is worth in prize money. Level 2 has a player move all in against you, says how often your hand comes out higher, and asks whether to call or fold: in most of its moments the chips say one thing and the prize money another.

Nothing is bet here, and no card is dealt: each moment stops at the decision. The stacks and prizes are examples, not those of any room.

The practice table needs JavaScript. Here is one moment from it, with the answer.

Level 1 · What a stack is worth

AN EXAMPLE FINAL TABLE

YOU5,000 chips
PLAYER B3,000 chips
PLAYER C2,000 chips

EXAMPLE PRIZES

1ST$500
2ND$300
3RD$200

You hold 5,000 of the 10,000 chips at this example final table. In prize money, what is your stack worth right now?

$500 That is the prize pool shared out in proportion to the chips. Prizes are paid by finishing place and not by chips, so a stack this large is worth less than its share of the chips.
$383.93 Each place pays its prize, weighted by this stack’s chance of finishing there: the chance of first place is the share of the chips, and each next place runs the same rule among the players still left.
$333.33 That is the prize pool split equally among the players left, as if every stack were the same size. The stacks are not the same, and a stack’s chance at each place follows its share of the chips.
$200 That is the smallest prize, the least this stack can finish with. The stack still has a chance at every place above it, and each of those chances is worth prize money as well.

Your 5,000 chips are 50% of the chips in play. In prize money they are worth $383.93. That is 38.4% of the prize pool. First prize is only part of the pool, and the other stacks keep a claim on every place, so a large stack is worth less in money than in chips.

Level 2 · Call or fold

AN EXAMPLE FINAL TABLE

PLAYER A ALL IN5,000 chips
YOU3,000 chips
PLAYER C2,000 chips

EXAMPLE PRIZES

1ST$500
2ND$300
3RD$200

Player A has moved all in.

Player A, with 5,000 chips, moves all in. Calling puts all 3,000 of your chips at risk. Against Player A’s hand, yours comes out higher 50% of the time, with no ties. Call or fold?

Call In chips the call comes out even, but in prize money it is worth $305 on average, against $327.50 for folding. Losing the 3,000 chips at risk would cost more prize money than taking 3,000 chips would add.
Fold Folding keeps your stack worth $327.50; calling is worth $305 on average.
Either: calling and folding are worth the same here In chips the call does come out even at this share, but prize money is not shared out like chips: calling is worth $305 and folding $327.50.

By chips the call needs your hand to come out higher half the time. By the prizes it needs 60.7%. Here it comes out higher 50% of the time. If the call loses, you go out in third place for $200; if it holds, your stack is worth $410. Going out costs more prize money than doubling up adds, so the prizes ask for more than an even share.

Every moment in the practice run is worked out the same way: the stacks and the example prizes go through the Harville model, counted exactly over every order in which the players can finish.

How the practice run works

Where the answers come from. Every stack and every prize goes through the Harville model when the page is built, counted exactly over every order in which the players can finish. On level 1 the wrong answers are named mistakes: the prize pool shared out by chips, the first prize, the prize for the place a stack holds in chips right now, the pool split equally among the players, or the smallest prize. On level 2 the right answer is the choice worth more in prize money on average; where chips and prize money disagree, the share of hands is set between half and the share the prizes need.

One strip, and no rounds. Every square on the strip is one moment of this lesson, and all of the squares are there from the start. Your first answer fills the square it belongs to; when the squares of one level are full, the next level begins by itself. Practising a moment you got wrong turns its cross into “fixed in practice” and does not change the count, because only first answers count.

Your progress stays in this browser, and nothing is sent anywhere.

Check yourself

Maya with a finger raised and a knowing look.

Mayaformer floor supervisor · manners, money and breaks

Three quick ones. Pick an answer, then check it. No timer, no points: just whether it stuck.

1Stacks of 5,000 chips, 3,000 chips and 2,000 chips play for prizes of $500, $300 and $200. The leader holds 50% of the chips. What share of the prize money is the leader’s stack worth?

Less than half: about 38.4%More than half, because the leader is aheadExactly half, the same as the chips

Show the answer

Answer: Less than half: about 38.4%. Counted with the Harville model, the leader’s stack is worth $383.93 of the $1,000 pool.

Less than half: about 38.4% Half the chips gives half a chance of first place; the rest of the value comes from second and third place, which pay less.
More than half, because the leader is ahead A larger stack is worth less than its share of the chips, not more: the shorter stacks keep a claim on every place below first.
Exactly half, the same as the chips That would hold only if chips were paid out like money. A tournament pays by finishing place, not by chips.

2Why do players near the end of a tournament call an all-in more carefully than they would with chips alone?

Because a rule of the room asks for itOut of politeness to the players with short stacksLosing the chips costs more prize money than taking the same chips adds

Show the answer

Answer: Losing the chips costs more prize money than taking the same chips adds. A tournament pays by place, so a stack’s prize money grows more slowly than its chips: the chips at risk in a call are worth more than the chips on offer.

Because a rule of the room asks for it No rule we could read asks for it. The caution comes from the arithmetic of the prizes, not from any rule.
Out of politeness to the players with short stacks Politeness has nothing to do with it: the prize money does, because prizes are paid by finishing place.
Losing the chips costs more prize money than taking the same chips adds In the example, going out drops a stack worth $327.50 to $200, and doubling up raises it only to $410.

3Two players are left with equal stacks, and the prizes still to be paid are $600 and $400. What is each stack worth? new situation

$600$500$400

Show the answer

Answer: $500. With two players left, the second prize is certain, and each stack adds its share of the chips times the gap between the two prizes.

$600 That is the first prize, and first place is not settled yet: each player can still finish second.
$500 Equal stacks have an equal chance of first place, so each is worth the average of the two prizes.
$400 That is the second prize, which each player already holds for certain; the chance of first place adds more prize money to it.

Back to the question we opened with. Half the chips at the example final table is worth $383.93 of the $1,000 prize pool: less than half, because the prizes are paid by place.

Help, tools and further reading

Tournaments: buy-in, structure and M Where a buy-in goes, how the levels run, and how many orbits a stack can sit out. Tilt, and when to leave the room What to do after a painful loss, and a rule for leaving made in advance. Game plan The money and the time for an evening, written down at home before the first hand. Helplines by province The line for each province and the crisis line.

19+ (18+ in AB, MB, QC) · Play responsibly · the helplines are further up this page

How it’s worked out

Model, numbers, sources and hosts

Where this happens. Every final table on this page is a tournament in a room you can walk into, with players playing one another and not the house. The stacks and prizes are examples; the prizes of a real tournament are set by its room.

Numbers. What a stack is worth is our count with the Harville model: the chance of first place is the share of the chips, and each next place runs the same rule among the players still left. The paper of 1973 is named by its title, which is all of it we read; the description of the method is ours, from the way our engine counts it. The chances are exact fractions over every order of finish, printed rounded. An all-in call compares what folding keeps with the average of the call over the hands where it holds and the hands where it loses; the share of hands is a condition of each moment, with no ties.

What rests on the model alone. No live room whose documents we could read uses ICM or any other method to value tournament chips or to settle a deal, so every figure in prize money here is a model, not a rule of any room.

What is deliberately absent. There is no formula for a deal and no advice on whether to agree to one: the room decides, and the players choose. Where a buy-in goes, the room’s fee, the structure of levels, M and payout tables in full are lesson 7.11. The rake and the swings of cash games are lesson 7.10, and a pot at a cash table, with its side pots, is lesson 7.5. How often one hand comes out ahead of another before the flop is lesson 7.6, and the share a call needs against the pot is lesson 7.8. Tilt and leaving the room are lesson 7.13.

Sources. Every claim on this page carries a button with the sentence it rests on, the publisher who wrote that sentence and the date we read it.

Hosts. Leo and Maya are illustrated characters. Their backstories, a former dealer and a former floor supervisor, are written for the course: neither is a real person, and neither speaks for any casino. The rules and numbers on this page come from the sources it cites, not from their experience.

No promise. Knowing what a stack is worth in prize money makes an all-in call easier to weigh before you make it; nothing on this page makes a tournament pay, and the room’s fee leaves the players, taken together, behind by that fee.

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Chips are not money: the final table

The short answer

Tournament chips are not money

From chips to prize money

An all-in call at the final table

Who sets the prizes, and who may change them

If it stops being fun

Practice: what are your chips worth?

Lesson done

Level 1 · What a stack is worth

Level 2 · Call or fold

Check yourself

Help, tools and further reading

Further reading

How it’s worked out