Play: make a market on five dice

Reading about market making only gets you so far. This lesson is a game in the style firms use: a market on a sum of dice, revealed one at a time, against a counterparty who knows a little more than you.

The rules

• Five dice are rolled and hidden. You are making a market on their sum.
• Before each die is revealed, show a two-sided market, bid @ ask, between 1 and 6 wide.
• The trader has peeked at the next die. They buy one lot at your ask if it is at least 1 below their estimate, sell one lot at your bid if it is at least 1 above it, and otherwise pass. Now and then they trade at random.
• After the fifth die, everything settles at the true sum.

Your theoretical value

Each hidden die has mean 3.5 and variance $35/12$. With $s$ the sum of the revealed dice and $h$ dice still hidden:

At the start that is $17.5 \pm 3.8$. The readout shows it live; the quote boxes are pre-filled with fair value ± 2, which you are free to change.

How to play it well

• Width follows uncertainty. Early, with five dice hidden, you know little and the trader knows a lot relative to you. Late, the uncertainty is small, so you can tighten.
• Trades are information. If they buy, the die they’ve seen is likely 5 or 6. If you are now short, raising your market both protects you and reflects the news.
• Watch your position. Ending long 3 when the last die comes up 1 is how games are lost. Skew to get flat before the end.
• You will lose sometimes. An informed trader takes money from a market maker on average. The skill is losing small when you’re picked off and earning the spread from the random trades.

In code

import numpy as np
rng = np.random.default_rng(15)

def play(width=4.0, games=20_000):
    pnl = 0.0
    for _ in range(games):
        dice = rng.integers(1, 7, 5)
        pos = cash = 0.0
        for shown in range(5):
            fair = dice[:shown].sum() + 3.5 * (5 - shown)
            bid, ask = fair - width / 2, fair + width / 2
            trader = dice[:shown + 1].sum() + 3.5 * (5 - shown - 1)   # sees one die ahead
            if trader - ask >= 1:  pos -= 1; cash += ask
            elif bid - trader >= 1: pos += 1; cash -= bid
        pnl += cash + pos * dice.sum()
    return pnl / games

for w in (2, 4, 6):
    print(w, round(play(w), 3))      # wider markets lose less to the informed trader