Why Chips ≠ Cash in Tournaments
In a cash game, chips have a direct 1:1 relationship with money. Double your chips, double your money. In a tournament, this linear relationship breaks down entirely. Doubling your stack does not double your prize equity.
The reason is the prize structure. First place pays far more than second, and second pays more than third, but you can only win one prize. Chips are merely a tool to survive and reach higher prize tiers — and their marginal value decreases as you accumulate more.
The Independent Chip Model (ICM) quantifies this. ICM converts chip counts into dollar equity (\$EV) by calculating the probability each player finishes in each prize position, weighted by the prize amounts.
The ICM Formula
The Malmuth-Harville model is the standard ICM approximation. The probability that player \(i\) finishes first is simply their chip share:
$$P_i(\text{1st}) = \frac{s_i}{S}$$where \(s_i\) is player \(i\)'s chips and \(S\) is total chips. For finishing second, we calculate:
$$P_i(\text{2nd}) = \sum_{j \neq i} P_j(\text{1st}) \times \frac{s_i}{S - s_j}$$This continues recursively for all positions. The ICM dollar equity is:
$$\text{ICM}_i = \sum_{k=1}^{n} P_i(\text{finish } k) \times \text{Prize}_k$$Note the key limitation: ICM assumes skill is irrelevant and winning probability is proportional to chips only. It is an approximation, but a powerful one for tournament decision-making.
Step-by-Step ICM Calculation
Setup: 3 players remain. Prize pool: 1st = \$500, 2nd = \$300, 3rd = \$200 (total \$1000).
| Player | Chips | Chip % |
|---|---|---|
| Alice | 6,000 | 60% |
| Bob | 3,000 | 30% |
| Carol | 1,000 | 10% |
| Total | 10,000 | 100% |
Step 1: P(1st) for each player
$$P_A(1st) = 0.60,\quad P_B(1st) = 0.30,\quad P_C(1st) = 0.10$$Step 2: P(2nd) for Alice
$$P_A(2nd) = P_B(1st) \times \frac{6000}{10000-3000} + P_C(1st) \times \frac{6000}{10000-1000}$$ $$= 0.30 \times \frac{6000}{7000} + 0.10 \times \frac{6000}{9000}$$ $$= 0.30 \times 0.857 + 0.10 \times 0.667 = 0.257 + 0.067 = 0.324$$Step 3: ICM equity for Alice
$$\text{ICM}_A = 0.60 \times 500 + 0.324 \times 300 + (1-0.60-0.324) \times 200$$ $$= 300 + 97.2 + 0.076 \times 200 = 300 + 97.2 + 15.2 = \$412.4$$Performing the same calculation for all players:
| Player | Chips | Chip % | ICM \$EV | vs. Chip % |
|---|---|---|---|---|
| Alice | 6,000 | 60% | $412 | –$188 |
| Bob | 3,000 | 30% | $340 | +$40 |
| Carol | 1,000 | 10% | $248 | +$148 |
Notice Alice has 60% of the chips but only 41.2% of the prize equity. Carol has 10% of chips but 24.8% of equity. Short stacks have disproportionately high ICM equity — they are already "locked in" to the prize pool.
ICM Pressure: Chip EV vs. $EV
Here is a classic example showing a spot where chip EV is positive but \$EV is negative.
Situation: Using the same 3-player setup. Alice (6,000) faces an all-in from Carol (1,000). Alice has a coin flip (50% equity).
Chip EV for Alice:
$$\text{chipEV} = 0.50 \times 7000 + 0.50 \times 5000 - 6000 = 3500 + 2500 - 6000 = 0$$Chip EV is exactly 0 — a neutral call in chip terms. But what about \$EV?
If Alice wins (has 7,000 chips): Recalculate ICM → Alice ≈ \$466
If Alice loses (has 5,000 chips): Recalculate ICM → Alice ≈ \$388
$$\text{\$EV} = 0.50 \times 466 + 0.50 \times 388 - 412 = 233 + 194 - 412 = +\$15$$In this case the call is actually slightly positive in \$EV too. But notice how much smaller the gain is versus chip terms — and the analysis flips negative with a worse edge.
Bubble Factor
The Bubble Factor (BF) quantifies how much more ICM pain you feel losing chips versus the ICM gain from winning the same chips:
$$BF = \frac{\Delta\text{\$EV}_{\text{loss}}}{\Delta\text{\$EV}_{\text{gain}}}$$A BF of 2.0 means losing a given amount of chips hurts twice as much in \$EV as winning the same chips helps. This directly affects the equity required to call profitably.
The required equity to call with a given Bubble Factor:
$$\text{Required Equity} = \frac{BF}{1 + BF}$$With BF = 2.0, you need \(\frac{2}{3} = 66.7\%\) equity to call profitably — much more than the 50% needed in a cash game. On the money bubble with large stacks, BF can reach 3–5, requiring 75–83% equity to call.
| Bubble Factor | Required Equity to Call | Typical Situation |
|---|---|---|
| 1.0 | 50% | Cash game / early tournament |
| 1.5 | 60% | Mid-tournament, near bubble |
| 2.0 | 67% | Bubble / final table |
| 3.0 | 75% | Large stack on bubble |
| 4.0 | 80% | Extreme bubble pressure |
Practical Usage
You cannot run full ICM calculations at the table, but you can develop ICM intuition with these simplified rules:
- Short stacks call wider: They have little ICM equity left to protect. A shove with 10BB is almost always correct when ICM equity is already near min-cash.
- Big stacks fold tighter: Your stack is your weapon. Losing chips costs more in \$EV than winning the equivalent chips gains.
- ICM matters most at pay jumps: The bubble, final table entry, and heads-up are the three spots where ICM has the largest impact.
- ICM matters least early: Deep in the field with many eliminations before the money, play closer to chip EV (maximize chip accumulation).