Risk budgets in a toy model

Apply an explicit loss constraint instead of treating positive expectation as sufficient.

11 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Check a fictional chip-risk constraint and explain what an EV ranking leaves out.

Prerequisites: Repeated bets and dependence

A constraint is part of the decision problem

Suppose a simulation imposes a worst-case loss cap of 30 chips across two positions. If both can lose together, their possible joint loss must fit the cap. A positive mean for each position does not waive the rule.

A risk budget has to say what is limited: stake, worst-case loss, probability of a loss, or another measure. Those are not interchangeable. The exercises here use explicit fictional chip-loss caps, not a recommendation for real bankrolls or financial investments.

Dependence matters to combined loss

If two positions each can lose 20 and simultaneous loss is possible, the combined worst-case loss is 40. Calling them different positions does not establish diversification. If the model instead guarantees one’s gain exactly offsets the other’s loss, the combined outcome changes.

A worst-case constraint can be conservative, and a probability-based constraint requires a joint distribution. Neither can be reconstructed from mean payoffs alone. Before optimizing, identify the objective, feasible actions, loss definition and what assumptions support the joint outcomes.

Worked example

A simulation allows two equal positions, each losing at most b, with both losses possible together. The combined loss cap is 30. What is the largest b allowed?

The worst simultaneous loss is b+b=2b.
The constraint requires 2b≤30.
Divide by two to obtain b≤15.
A higher expected payoff does not change this stated feasibility condition.

The maximum per-position worst-case loss is 15 chips. This is a toy constraint calculation, not an optimal real-world staking rule.

Common mistakes

Optimizing EV before checking the permitted action set.
Treating different labels as proof the positions cannot lose together.
Using a worst-case cap and a loss-probability cap as synonyms.

Keep these ideas

Name the loss metric and the cap.
Check possible joint outcomes, not just separate means.
Positive expected value is one input, not a universal permission.