Market Making Game: Quoting Bid-Ask Spreads

Worked Solution

How to Think About It: This is the classic market-making exercise you see at every trading firm interview. The game tests whether you can think about fair value, manage risk through your quotes, and articulate the economics of why spreads exist. The key insight is that your spread is not just a profit margin -- it is your compensation for bearing uncertainty and adverse selection risk. A good market maker quotes tight when confident and wide when uncertain, and skews quotes to manage inventory.

Key Insight: Your bid-ask spread should be proportional to your uncertainty about fair value, and your quote midpoint should shift to penalize further accumulation of inventory in any direction.

The Method:

1. Estimate fair value. Before quoting anything, figure out $E[V]$ based on whatever information the game gives you. If the asset pays based on a die roll, compute the expected value. If it depends on a card draw, compute the conditional expectation given what you have observed so far.

2. Set the initial spread. Quote: $\text{Bid} = E[V] - \frac{s}{2}, \quad \text{Ask} = E[V] + \frac{s}{2}$ where $s$ reflects your uncertainty. If you are very sure about fair value, quote tight (small $s$). If you have little information, quote wide. A reasonable starting rule: set $s$ roughly equal to one standard deviation of your value estimate.

3. Adjust for inventory. As you trade, you accumulate a position. If you are long (bought more than you sold), you want to encourage others to buy from you, so you lower both bid and ask. If you are short, raise both. The standard approach is to shift the midpoint: $\text{Mid}_{\text{adjusted}} = E[V] - \delta \times \text{Inventory}$ where $\delta > 0$ is a skew parameter. This means your bid and ask both drop when you are long, making it cheaper for others to buy from you and more expensive for others to sell to you.

1. Update fair value. As rounds progress, use new information (trades that occur, signals revealed) to update your estimate of $E[V]$. Bayesian updating is the right framework: if someone aggressively lifts your ask, that is a signal they may know the value is higher than your estimate.

Practical Considerations:

• Adverse selection is the biggest risk. If an informed trader knows the true value is 60 and you are quoting 48-52, they will lift your ask at 52 every time. Your spread needs to be wide enough to survive these losses on average.
• Inventory risk compounds with adverse selection. Being long 100 units when the true value turns out to be lower than your estimate is painful.
• Don't be afraid to quote wide early. In the first round you have minimal information. Quoting a spread of 4-6 on a value you think is around 50 is reasonable. As you learn more, tighten.
• Profit comes from the spread. Every time you buy at your bid and sell at your ask, you earn $s$. Your job is to earn enough spread to cover the occasional adverse selection loss.

Answer: Set your spread proportional to uncertainty about fair value. Skew your midpoint away from your current inventory to manage risk. The spread compensates you for adverse selection (informed traders), inventory carrying cost, and estimation uncertainty. A good market maker constantly updates fair value, adjusts spread width based on confidence, and manages inventory through quote skewing.