06 — Expected Value · Market-Making · The Ladder

The game they make
you play before
they hire you.

Optiver, IMC, Jane Street, SIG, Akuna, DRW — they all run a version of the same interview: "Make a market on this." You compute an expected value, quote a two-way bid/ask, and trade against someone who might know more than you. Win by capturing the spread without getting picked off. Learn the math, read the ladder, then play it below.

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Foundations

Expected value is the whole job.

Every quote, every trade, every spread you've seen in this lab reduces to one idea: the probability-weighted average outcome. E[X] = Σ pᵢ·xᵢ. If you can price the EV faster and tighter than the other side, you win over enough repetitions.

Roll a die → watch EV emerge

One fair die: E[X] = (1+2+3+4+5+6)/6 = 3.5, Var = 35/12 ≈ 2.92. Roll enough and the empirical mean converges to 3.5 — the Law of Large Numbers, the bedrock under every edge.

empirical mean
0
rolls
|mean − 3.5|

Five worked EVs

One die E = 3.5, Var = 35/12 ≈ 2.92, SD ≈ 1.71
Two dice (sum) E = 7, SD ≈ 2.41 — the canonical "make a market on the sum"
Biased coin Win $10 @ 40%, lose $4 @ 60% → EV = 0.4·10 − 0.6·4 = +$1.60
Card pip draw Mean pip ≈ 340/52 ≈ 6.54 (A=1…K=13 across 4 suits)
Quote-as-EV Quote 6.5 / 7.5 on a true-7 instrument → mid = EV, half-spread 0.5 = your edge per uninformed trade
Read the tape

The ladder (DOM) — where quotes live.

A price ladder, or Depth of Market, is the vertical map every futures trader stares at: prices stacked top to bottom, resting bids on one side, asks on the other, size at each level. Your two-way quote is just two rows on this ladder. Learn the vocabulary — it's the language of the game.

A LIVE LADDER (illustrative)
BID
PRICE
ASK
72.10
42
72.09
71
last
72.08
88
72.07
51
72.06

Inside market = 72.07 bid / 72.09 ask, spread = 0.02, mid = 72.08.

13 terms that matter

Bid / Ask best buy / best sell price
Inside market / Top of book the best bid & ask
Spread ask − bid; the MM's raw edge
Mid (bid+ask)/2; proxy for fair value
Depth / Size quantity resting at a level
Lifting the offer buying the ask (aggressive)
Hitting the bid selling the bid (aggressive)
Last most recent traded price
Queue position your place in line at a price
Iceberg hidden size; only a sliver shows
Skew shifting quotes to manage inventory
Scalping the ladder capturing the spread repeatedly
Adverse selection getting filled by someone who knows more
Play it

Make a market. Don't get picked off.

Each round you make a market on the sum of N hidden dice. Quote a bid/ask on the ladder. Uninformed (noise) flow trades at your quote and pays you the half-spread. Informed flow only trades when your price is wrong — and picks you off. Reveal dice to sharpen your EV, re-quote, then settle at the true value. 5 rounds. Start: 350 chips. Beat 500.

How to win: Center your market on the EV (shown live). Keep the spread tight enough that noise flow fills you for profit, but wide enough that small mis-pricing doesn't invite the informed bot. Reveal dice before quoting when unsure — information narrows your risk.
YOUR MARKET ON THE LADDER · EV
350
chips
0 / 5
round
0
position
0.00
open P&L
TRADE LOG
JANE STREET · FIGGIE

The card game that teaches this for real

Figgie is a 40-card, 4-suit game (8 / 10 / 10 / 12 cards). One suit is the hidden goal suit — always the same colour as the 12-card suit, and itself an 8 or 10. Players ante into a $200 pot, then trade cards open-outcry for ~4 minutes, inferring the goal suit from the flow. Holders of the goal suit earn $10/card; the majority holder takes the rest of the pot. It's market-making, Bayesian inference, and adverse selection compressed into a card game.

Rules: figgie.com · [05]

GLOSTEN–MILGROM 1985

Why the spread exists at all

A market maker's bid-ask spread decomposes into three costs:

spread = order-processing
    + inventory risk
    + adverse selection

The third term is the killer: every counterparty might be informed. The MM widens to survive informed flow and earns back the spread from uninformed flow — exactly the tension you just played. Skew your quotes to offload inventory; widen when uncertain.