In chapter 11 of Play Optimal Poker 2, Andrew Brokos explains how tournament-specific factors—especially antes, short effective stacks, and the non-linear value of chips—change optimal decision-making compared to cash games. Although foundational concepts (equity, range construction, polarization, board coverage, equilibrium reasoning) remain the same, tournaments introduce constraints and incentives that alter what “maximizing EV” means. The chapter proceeds in two phases: first, applying equilibrium logic to a standard postflop scenario with antes; second, introducing ICM and solving a final-table variant of the Clairvoyance Game.
Why Tournaments Require Adjustments
Tournament chips do not scale linearly in value because survival carries utility. Losing your last chips is far worse than gaining an equivalent amount is good. This affects equilibrium only in certain phases (notably the bubble and final table), but antes and shorter stacks affect every hand by widening ranges and reducing stack-to-pot ratios.
Brokos emphasizes:
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Early and middle stages mostly play like cash games with different stacks.
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The biggest adjustments occur when survival has direct monetary value.
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Wider ranges in tournaments frequently make marginal hands stronger and bluffing more profitable.
Scenario 1 — Playing With Antes (UTG vs BB on 9♥ 7♦ 6♥)
Antes greatly increase the pot size before cards are dealt, which widens both players’ preflop ranges—but not equally.
Why the BB Adds More Hands Than UTG
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UTG still faces eight players behind who can punish loose opens.
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BB closes the action and receives significantly improved pot odds.
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Therefore BB adds many marginal holdings; UTG adds only borderline opens that now become slightly profitable.
Equity, Nuts Advantage, and EV Capture
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Equity advantage: UTG (Ivan) benefits from a stronger preflop range and captures slightly more equity than in non-ante play.
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Nuts advantage: BB (Opal) gains straights (e.g., T8o) but also gains large amounts of weak holdings—so her nut density is diluted.
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EV capture: Ivan earns an even larger fraction of the pot than in the no-ante version because Opal’s weaker range is harder to defend.
Should UTG Have a Checking Range?
Yes. Even with an equity edge, UTG benefits from keeping pots smaller with marginal hands and taking free cards with weak ones. Wide ranges and low SPRs create more incentive for pot-control from out of position.
Preferred C-Bet Size: Large Over Small
This is the first scenario in the book where the preflop raiser prefers a larger continuation bet size.
Reason:
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BB’s range contains many weak but non-folding hands (7x, 6x).
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A larger bet stresses these marginal holdings and extracts more value with overpairs.
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UTG’s wider range gives him abundant natural bluffs to balance large bets.
Increased Flop and Turn Aggression
With antes:
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UTG c-bets wider.
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BB check-raises wider.
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This opens the door for UTG to three-bet the flop, which did not occur in the non-ante simulation.
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Sets (like 99) and robust drawing hands become natural 3-bet candidates.
Which of UTG’s Newly Added Hands Bet More Often?
Hands that:
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Value folds,
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Retain good equity when called.
Frequent bets:
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A9s, K9s, A8s.
Frequent checks:
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Weak pairs (44, 55),
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Medium-strength A7s/A6s.
These hands either preserve their equity better through checking or do poorly when facing a check-raise.
The Non-Linear Value of Chips and ICM
Brokos introduces the Independent Chip Model as the standard way to translate stack sizes into $EV.
Key ideas:
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Winning chips rarely increases $EV proportionally.
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Losing chips—especially the last chips—can drastically reduce $EV.
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Medium stacks are the most risk-averse; short stacks often must gamble; big stacks can apply pressure.
ICM does not replace equilibrium logic; it modifies the EV functions players use when choosing frequencies.
Scenario 2 — Clairvoyance at the Final Table
Using the original Clairvoyance Game (A vs Q against a known K), Brokos re-solves the game under ICM considerations at a 3-handed final table.
Key Differences from the Cash-Game Clairvoyance Game
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Opal calls far less often because losing her last chips is extremely costly.
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Ivan bluffs more often to compensate, because Opal folds too much by default.
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Equilibrium still requires mutual indifference, but the payoffs are now ICM-weighted.
Important Results
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Opal must call only about 13–14% of the time (down from 67% in the cash-game version).
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Ivan must bluff so that ~36% of his bets are bluffs (up from 25% originally).
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Even though bluffing is riskier under ICM, the caller is even more risk-averse, which increases bluff frequency.
Who Gains When They Clash?
When Opal calls Ivan’s equilibrium betting range:
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Opal’s $EV = 0 (by design).
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Ivan loses $EV.
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The “lost” EV is gained by the uninvolved player, Nia, who benefits whenever others put their stacks at risk.
This illustrates the powerful incentive to avoid marginal confrontations at final tables.
Practical Tournament Lessons
1. Tournaments are won one decision at a time
Avoid big-picture thinking like:
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“I need chips now”
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“It’s too early to take risks”
Choose the most +EV play measured in the correct unit (cEV or $EV).
2. Wider ranges change hand strength
With antes or shallow stacks:
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Top pair becomes stronger.
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Medium-strength one-pair hands support more betting.
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Bluffing becomes more frequent because opponents have more air.
3. ICM is an approximation, not a perfect model
It ignores:
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Skill differences,
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Future positional advantages,
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Table dynamics.
Yet it’s the best practical tool for evaluating final-table EV.
4. Being out of position is even more damaging in tournaments
When calling ranges widen, the defender has more weak hands. Aggressors should bet more thinly for value.
5. Risk aversion affects callers more than bettors
Shoving is lower variance than calling:
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The bettor wins the pot without showdown often.
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The caller always realizes variance.
6. Medium stacks must be the most conservative
Big stacks can pressure;
Short stacks must gamble;
Medium stacks risk losing huge $EV by busting prematurely.
7. Opponent mistakes can reduce your own $EV
At final tables, bad calls by other players benefit the uninvolved stacks.
Exploit cautiously so their mistakes don’t drag you down.
In Summary
Chapter 11 blends tournament-aware thinking with game-theoretic reasoning. Most poker principles remain the same, but antes and ICM reshape incentives radically:
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Antes widen ranges and justify larger flop bet sizes.
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Shallow stacks boost aggression and reduce the utility of checking marginal hands.
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As survival becomes valuable, equilibrium shifts toward conservative calling, but surprisingly, this often increases bluffing frequencies.
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ICM allows equilibrium reasoning in multi-player payout structures, revealing how EV is redistributed among all surviving stacks.
If you’d like, I can also create a condensed cheat sheet for Chapters 10–11 covering optimal tournament adjustments, bet sizing, and ICM-driven strategy shifts.