Poker for traders: pot odds, ranges and bluffing frequency

GAME · Chapter 313 min readAsked at SIG, Jane Street, Optiver, Citadel

Assumes Game theory: dominance, mixing and the indifference condition.

After this lesson you should be able to

  • Convert a bet size into the equity you need to call.
  • Think in ranges rather than in single hands.
  • Derive the bluffing frequency that makes a caller indifferent.

SIG built its training around poker for a reason: it is decision-making under uncertainty with immediate scoring, incomplete information and an opponent who adapts. You do not need to be a strong player, but you are expected to compute pot odds without hesitating and to understand why a balanced strategy has to bluff.

Equation 3.1

Pot odds

The break-even win probability for calling. Above it, calling is profitable; below, folding is.

equity needed=callpot after the bet+call\text{equity needed} = \frac{\text{call}}{\text{pot after the bet} + \text{call}}
call\text{call}
What you must put in.
pot after the bet\text{pot after the bet}
Including the bet you are facing, but not your call.
Bet as a fraction of the potEquity needed to callBluffs as a share of the betting range
14\tfrac14 pot16.7%16.7\%20%20\%
12\tfrac12 pot25%25\%25%25\%
34\tfrac34 pot30%30\%30%30\%
Pot33.3%33.3\%33.3%33.3\%
2×2\times pot40%40\%40%40\%
Table 3.2 · Bet size and the equity it demands. The two right-hand columns coincide, and that is not an accident: the frequency at which you bluff is exactly the equity your bet demands. A bigger bet demands more equity and must therefore be backed by more bluffs, or it becomes trivially foldable.

Example 3.3

The pot is $100\$100 and your opponent bets $50\$50. You believe you win 30%30\% of the time. Call or fold?

Show the worked solutionHide the worked solution

Worked solution

  1. Formula
    equity needed=50150+50\text{equity needed} = \frac{50}{150 + 50}
  2. Substitute
    =50200= \frac{50}{200}
  3. Solve
    =25%= 25\%
  4. 30%>25%⇒call30\% > 25\% \Rightarrow \text{call}
  5. Answer
    Call: +EV by 0.30×150−0.70×50=$10\text{Call: } +\text{EV by } 0.30 \times 150 - 0.70 \times 50 = \$10

Sanity check. Check the expected value directly: you win $150\$150 three times in ten and lose $50\$50 seven times, for 45−35=$1045 - 35 = \$10. The pot-odds shortcut and the direct calculation must always agree.

Proposition 3.4

Think in ranges

A strong player never asks "what does he have?" but "what could he have, and in what proportions?". Every action narrows the range: a raise from early position removes most weak hands, a call on a draw-heavy board keeps them in. Your decision is against the whole distribution, weighted — which is exactly how a market maker reasons about who is on the other side of a trade.

Holds when

  • A hand that beats half the range and loses to half is a call at short odds and a fold at long ones.
  • Ranges are the reason position is worth so much: acting last means you have seen one more narrowing.

Derivation 3.5

How often to bluff

Bet bb into a pot of PP. Choose the bluff frequency that leaves the caller indifferent.

  1. Callersˊ equity needed=bP+2b\text{Caller\'s equity needed} = \frac{b}{P + 2b}

    The pot-odds formula, with the bet counted on both sides.

  2. Set P(bluff∣bet)=bP+2b\text{Set } P(\text{bluff} \mid \text{bet}) = \frac{b}{P + 2b}

    Then calling and folding have identical value, so the caller cannot exploit you.

  3. b=P⇒P3P=13b = P \Rightarrow \frac{P}{3P} = \tfrac13

    A pot-sized bet is one third bluffs, two thirds value.

bluff frequency=bP+2b\text{bluff frequency} = \frac{b}{P + 2b}

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