Goldilocks in Skid Row (Act II)

Entering shelters...or leaving them

quandt_004

In the kitchen of homelessness policy, ‘Housing First’ and ‘Treatment First’ sound like magic bullets—until you ask, “When won’t this work?” Housing First advocates say give people a key and watch the rest simmer down. Treatment First advocates counter that the key requires drug or alcohol tests, therapy, job-readiness training, and budget management. Not everyone is ready to manage a lease.

Advocates prefer magic bullets. They are salesmen who’d lose every sale if they hawked fridges with caveats: “This fridge works so long as it stays at 72°F, the door opens fewer than 25 times a day, and your breaker never dips by more than two watts.” But every policy—like every appliance—has breaking points and sweet spots. So let’s ditch the slogans, leave the PR battle for others, and nail down the thresholds.

In the last Quandt, we tracked Joe’s daily choices—street, shelter, or a housing lease—and why crowding, risks, and zero penalties can leave even a free bed looking uglier than sleeping under the stars. That was about the flow into, or out of, unsheltered homelessness.

Act II drops Joe inside the system. When does he show up for counseling, take his meds, or buckle down on sobriety—what makes him leave homelessness for good? What puts him back on the streets?

In this post, I look at the policy levers that keep Joe in the program as well as the crime risk dynamics that push him out.

Post-Entry Model

Let me stress, ‘Joe’ isn’t the flesh-and-blood guy panhandling when you get off the 35. He’s a statistical average of thousands of panhandlers, tent dwellers, shelter stayers, couch surfers, backyard encroachers, and whoever is in that van parked you know where.

We begin with Joe sitting in the manager’s office of a shelter or for housing placement, and the manager steps out. Joe thinks, ‘Now what?’ He takes stock: How much money is on him, how crowded is this place, and what’s the wait time till he gets a cot or apartment? We bundle these concerns into a tidy state vector,

\(s_t=\bigl(a_t,\;n_t,\;w_t\bigr), \)

where a is his liquid assets, n is the crowding, and w is the wait time, and all of these are indexed by time (the little t). We’ll assume that tomorrow’s cash is independent of today’s decision (so it’s “exogenous”) and there’s also a taste shock that inserts unpredictability to even our oh so average Joe’s decisions.

Now Joe’s gotta act. Think of his day as a three-button console:

1. Stay or leave?

   • 0 = street

   • 1 = stay in shelter or housing placement

2. How to spend the day (if he stays)?

   • 0 = idle (hang around, no job hunt)

   • 1 = job search or work

   • 2 = crime (panhandling that risks arrest, small theft, etc.)

3. Treatment?

   • 0 = skip

   • 1 = show up for counseling, meds, or job training.

That’s it. Three buttons, eight possible profiles. In math,

\(\underbrace{x_t}_{\substack{\text{exit}\\[2pt]\text{decision}}} \;=\; \begin{cases} 0 & \text{street (exit)}\\ 1 & \text{stay in program} \end{cases} \qquad\qquad \underbrace{a_t}_{\substack{\text{activity}\\[2pt]\text{choice}}} \;=\; \begin{cases} 0 & \text{idle}\\ 1 & \text{job search}\\ 2 & \text{crime} \end{cases} \qquad\qquad \underbrace{m_t}_{\substack{\text{treatment}\\[2pt]\text{participation}}} \;=\; \begin{cases} 0 & \text{skip}\\ 1 & \text{attend} \end{cases} \)

To keep the Bellman equation tidy, we fold these into one decision vector

\(d_t = (x_t, a_t, m_t)\)

So if Joe stays at the shelter (x = 1), spends the day looking for day labor (a = 1), and shows up for counseling (m = 1), his decision vector is

\(d_t = (1, 1, 1).\)

Swap any of those slots and you have a different day in Joe-land.

Take the decision vector and Joe’s state, and we can score how well off Joe thinks he’ll be from his panoply of options. Say Joe decides, “I’m out,” and hits the streets. Here’s what he should expect:

\(u_t = a_t-\phi_{\text{exit}} + \varepsilon^x_t\)

His benefits/harms are made up of his liquid assets, the headache of finding somewhere copless, sleeping on a bench, or avoiding rain, and the random benefit or harm—maybe returning to the streets gives him euphoria (+) or depresses him more (-). Add and subtract as needed, and you’ve got a score for returning to the streets.

Then again, Joe can stay in the shelter or housing placement. Then there are more dials:

\(u_t^{\text{stay}} = \underbrace{(a_t - k\,w_t)}_{\text{cash on hand} \; - \; \text{wait‐time fatigue}} \\[6pt] \quad -\;\underbrace{\phi}_{\text{baseline communal hassle}} -\;\underbrace{\nu_t}_{\text{victimization risk}} -\;\underbrace{\psi\,t_t}_{\text{counseling / treatment sting}} \\[6pt] \quad -\;\underbrace{\chi\,\mathbf 1\{\ell_t = 2\}}_{\text{moral and legal cost of crime}} +\;\underbrace{g(n_t)}_{\text{illicit gain if crime is chosen}} +\;\underbrace{\varepsilon^{\ell_t}_{t}}_{\text{idiosyncratic taste shock}}\)

Walking left to right, the cash is likely safer indoors, especially if Joe gets an apartment, but if a shelter bed hasn’t been cleared or an apartment found, there’s the lethargy of waiting.

Still, in a shelter you’re subjected to communal living. A bunkmate snores, the bathrooms are occupied, dishes in the sink. And did someone check under Joe’s mattress when he was gone?

Cue the risk of being victimized. The more money and possessions Joe has, the higher the risk. There’s more to gain by robbing him. So, too, with crowding. More people increases the odds someone is scoping Joe out and that he (the bad actor) is less likely to be found after robbing Joe.

Face it, counseling and substance abuse treatment isn’t fun. It will pay off in the long run, sure, but the cold chill of sobriety pales in comparison to gleeful oblivion. So let’s assume here that treatment casts imminent shade over Joe’s day.

Maybe Joe isn’t Bundy, but he’s in dire straits. Pardon the truism, but people need money to get things. If strapped, the most honest person may be tempted to steal. After all, what if someone stole Joe’s things? If he commits a crime, he faces the pain of conscience or anxiety of being caught, and he gets the reward of whatever he snags.

And, just as before, we stick a final epsilon on the backend, which is the random benefit or harm that Fate decides.

The two-line dynamic can be put in one line and solved:

\(V_t(s) \;=\; \max\Bigl\{ \underbrace{u_t(\ldots,x_t=1)}_{\text{bolt to street}} \;+\;\beta\,\mathbb E\bigl[V_{t+1}(s^{\text{street}})\bigr], \; \max_{\ell,\,t} \Bigl\{ \underbrace{u_t(\ldots,x_t=0)}_{\text{stay in program}} \;+\;\beta\,\mathbb E\bigl[V_{t+1}(s')\bigr] \Bigr\} \Bigr\}. \)

Unlike above, this equation calculates the best future score starting from today’s (t) state (s). There is the one day pay-off from hitting the street, and the day’s pay-off if staying in the program. The Greek beta is a discount factor for tomorrow’s pay-off: a dollar tomorrow is worth less than a dollar today. And the fancy E is an expectation of the payoff for tomorrow’s starting state. If Joe hits the street, he wakes up on a park bench tomorrow.

Solving this recursion backwards gives us a ‘stay in housing’ threshold. Here’s what falls out:

1. Stay threshold. Joe keeps the bed only if the cash in his pocket is worth the pain of waiting for a bed and navigating crowds.

2. Conflicting effects. A shock to bed capacity shortens wait time and eases crowding, on one hand, but also may increase the risk of victimization and increase the payoff of crime, on the other hand.

The bottom line is that our Bellman equation crushes details into a simple lever, the cash cut-off, a. If Joe has a modest amount of cash, less crowding and shorter wait times may still push him back on the street. Yet bringing those variables down improves the chances he stays. A shock to funding and bed capacity, however, will not necessarily solve our problems. There’s crime risk, too, which is an empirical question—something we need data to crack into.

Policy Implications

Equations aside, there are broad take aways for public policy.

1. Beds aren’t enough. Shorten the queue, great, but if the dorm still feels sketchy or the lull between intake and an apartment drags, people will often bail to the street.

2. Make it safe. Crowding is tolerable when residents trust the rules: curfews, visible staff, quick response to theft. We need to get the risk gap between ‘inside’ and ‘outside’ rightly ordered. Why would someone stick in shelter when they keep getting robbed?

3. Nudge treatment. Counseling and meds suck (at least at first). Tiny perks, like bus passes and grocery cards may raise uptake some, but stronger nudges (and, dare I say, coercion) often matter more: conditional cash for milestone tests, step-down privileges, and time-limited rent subsidies that vanish or diminish if treatment stops.

4. Housing First in tension. If you want a “low barrier” ethos that shies away from sanctions, fine. But then there needs to be creative, non-punitive ways to make sticking with the program an easy choice.

In short, capacity is the floor, not the fix. Pair beds with safety guarantees and meaningful incentives, or the revolving door revolves.

We’ve had our speculative flights. In the next Quandt, Goldilocks pokes in her head to announce what we can learn from the data.