Health Inspection Games

The Bear: S1, E2 “Hands”

Game Theory: Mixed Strategies

The Beef receives a surprise visit.

“Nancy Chore. Chicago Board of Health. I’m here to inspect the property.”

Nancy is here to inspect. And she is not pleased with her findings: a former gas line not properly sealed and caulked, no hot water in the hand-washing station, a pack of cigarettes left near a burner on the stovetop. As a result, she brings down the hammer on the sandwich shop.

“Unfortunately, these violations leave me no choice. I award you a C.”

The health inspection grade, which explains to customers that the restaurant does not comply with minimum sanitary standards, goes in the front window. Not only will the restaurant lose some potential customers, but they will also have to take on additional expenses to meet the standards before Nancy is back in 30 days.

Did The Beef make a mistake by being unprepared for the inspection? Maybe, but only because an inspection occurred. If Nancy had visited other restaurants that day, it may have been in their best interest to push off the maintenance costs. But if The Beef can’t foresee every health inspection, what should they do?

This situation can be modeled with game theory, because the best course of action for The Beef depends on what the Board of Health chooses to do. And vice versa. To model this, assume the following:

The Beef has two choices: be spotless or be dirty. If inspected, spotless restaurants receive an A and dirty restaurants a C.
The Beef makes $10,000 in profit if they can skip maintenance costs (being dirty) and aren’t inspected. However, if they fail an inspection, that deducts $5,000 from their profit due to lost customers and fines. To avoid that risk, they can choose to be spotless, lowering profit by $1,000 due to increased costs.
The Board of Health has two choices: inspect or not inspect.
It costs the Board of Health $100 to inspect a restaurant because they pay the inspector for their time and have other miscellaneous inspection costs. But if they inspect a dirty restaurant, they collect a $400 fine.

This game is modeled in the matrix below. Each of the four cells represents a potential outcome of the game. The payoff on the left (in blue) is the monthly profit for The Beef. The payoff on the right (in orange) is the monthly profit for the Board of Health (at least as it relates to the inspection of this one restaurant).

The actions that occur in the episode, where The Beef is dirty and the Board of Health inspects, are not a Nash equilibrium: a pair of actions where each is a best response to the other player’s action. The Beef’s best response to an inspection is to be spotless.

In fact, none of the four combinations of actions constitute a Nash equilibrium. In each of the four, one of the players could improve their payoff by switching their action.

The Beef is spotless, and the Board of Health inspects? The Board of Health will have wished they didn’t inspect, and they could save money on paying the inspector or could have sent her elsewhere.
The Beef is spotless and no inspection comes? The Beef could have skipped the maintenance costs and been better off.
The Beef is dirty, and no inspection comes? The Board of Health could do better by inspecting and collecting the fine.

This game has no Nash equilibrium in pure strategies: where each player chooses one action and does this 100% of the time. And because neither player has a dominant strategy, it makes sense for both players to use mixed strategies: each player applies probabilities to their possible actions.

Suppose the Board of Health inspects 10% of the time. In other words, The Beef has a 10% chance of getting inspected in a given month. How would The Beef respond to this probability? Calculate expected payoffs.

Spotless: The Beef collects $9,000 in profit regardless of whether an inspection occurs.
Dirty: The Beef has a 10% chance of getting inspected, receiving a fine and a harsh score, and collecting $5,000 in profit for the month. That’s scary, but it’s very unlikely to happen. It has a 90% chance of skating by and saving $1,000 in cleaning costs for a $10,000 profit. That’s enticing, and it’s the most likely outcome. Combining both outcomes and their probabilities, The Beef would expect to receive a profit of $9,500 by being dirty and taking the risk.¹¹ Expected value is a way to calculate the average outcome when there is uncertainty. In this scenario, The Beef has a 10% chance of making $5,000 and a 90% chance of making $10,000. To find the expected value, multiply each outcome by its probability and add them: $9,500 = (0.1 × 5,000) + (0.9 × 10,000). This gives the average profit The Beef would expect to earn over time if the inspection rate stays at 10%.

Because being dirty has a higher expected payoff, The Beef would respond to 10% inspection rates by always being dirty, exploiting the Board of Health’s insufficient inspection rate. And this cannot be a Nash equilibrium. If The Beef is always dirty, the Board of Health would want to inspect every month and collect fines. A 10% inspection rate is not its best response to The Beef being dirty.

Similar logic would lead one to conclude that a 90% inspection rate cannot be part of a Nash equilibrium. The Beef would get a higher expected payoff from being spotless. And if The Beef is always spotless, the Board of Health would wish to never inspect.

The Nash equilibrium inspection rate is one that makes The Beef equally profitable whether they are spotless or dirty. The Beef’s expected payoffs across their two possible actions should be equal.

Let the Board of Health’s inspection probability be a variable p. Then, for The Beef, expected payoffs are as follows:

Spotless:²² This is simplified from 9,000(p) + 9,000(1-p). $9,000
Dirty: $5,000p + $10,000(1-p)

Set the two expected payoffs equal and solve for p to find p* = 0.20, a 20% inspection rate. With a 20% chance of a random drop-in, The Beef has no advantageous response to exploit the Board of Health’s inspection probability.

And how often should The Beef be spotless? With 20 percent inspection probabilities, it doesn’t matter. However, if they are dirty every month in response, the Board of Health may catch on and start inspecting more often. The Beef similarly needs to be spotless with a probability that the Board of Health cannot exploit.

Let The Beef’s spotless probability be a variable q. Then, for the Board of Health, expected payoffs are as follows:

Inspect: -$100(q) + $300(1-q)
Don’t Inspect: $0

Set the two expected payoffs equal and solve for q to find q* = 0.75. With a 75 percent spotless rate, the Board of Health has no response that can exploit the restaurant’s cleanliness level.

The mixed strategy Nash equilibrium is then that The Beef is spotless 75 percent of the time and the Board of Health inspects The Beef 20 percent of the time. It’s the only set of strategies in which both are best responding to each other. We can see this by mapping out the best responses for each player in the game to the actions of the other.

If the Board of Health shows up more than 20% of the time, The Beef’s best response is to be always spotless. This is a common result of games where one player monitors another player who is being monitored. The party that monitors, in this case the Board of Health, could always get perfect compliance with sufficiently intense monitoring. But monitoring is costly!

As a result, it may be optimal for the Board of Health that some restaurants are not spotless. It may be similarly acceptable to a police department that some drivers speed. And it may be acceptable from a manager’s point of view that some workers show up late.

For the party being monitored, it may be optimal to cut corners occasionally. Perfect compliance would carry very high costs. Sometimes drivers may optimally speed. Sometimes workers will optimally show up late for work. And sometimes restaurants will leave packs of cigarettes near the burners. “Sometimes” is the key word.

“No funny business, Ms. Chore.”

¹ Expected value is a way to calculate the average outcome when there is uncertainty. In this scenario, The Beef has a 10% chance of making $5,000 and a 90% chance of making $10,000. To find the expected value, multiply each outcome by its probability and add them: $9,500 = (0.1 × 5,000) + (0.9 × 10,000). This gives the average profit The Beef would expect to earn over time if the inspection rate stays at 10%.

² This is simplified from 9,000(p) + 9,000(1-p).