Hassiba Belbachir committed suicide in McHenry County Jail, a local Illinois jail in which she was confined at the request of federal authorities pending a removal hearing. An Algerian citizen, aged 27, she had entered the United States as a visitor in November 2004. She overstayed and in February of the following year flew from Chicago to England (we don’t know why), where the immigration authorities detained her for about a week and then returned her to Chicago. U.S. immigration authorities took her into custody upon arrival and placed her in the McHenry County Jail, which has a contract with the federal government to house persons detained by order of those authorities. She was to remain there until her removal hearing, anticipated to take place within a couple of weeks. She planned to ask at the hearing for asylum in the United States on the ground that she had a well-founded fear of being persecuted should she be returned to Algeria.
*978Her first day in the jail was March 9; she killed herself on March 17. Her estate brought suit against a variety of defendants under both 42 U.S.C. § 1983 — arguing that they had deprived Belbachir of her life without due process of law — and Illinois tort law. The district judge relinquished the supplemental state law claims when he granted summary judgment in favor of all the defendants with respect to the section 1983 claims. The plaintiff has appealed from the dismissal only of six of the defendants: McHenry County itself; the county sheriff and the director of the McHenry County Jail; and three employees of the Centegra Health System, a private firm that the County had hired to provide medical services at the jail.
The defendants argue to begin with that the doctrine of the law of the case requires their dismissal because the ground on which the district court dismissed the other defendants is equally applicable to them. Even if it is, the argument fails. The doctrine of law of the case “never blocks a higher court from examining a decision of an inferior tribunal.” Payne v. Churchich, 161 F.3d 1030, 1038 n. 9 (7th Cir.1998) (internal quotation marks and citations omitted); see also Marseilles Hydro Power LLC v. Marseilles Land & Water Co., 481 F.3d 1002, 1004 (7th Cir.2007). A plaintiffs decision to abandon a defendant on appeal, when other defendants remain in the case, is not an acknowledgment that the dismissal was sound, and is therefore not a basis on which the remaining defendants can plead waiver or forfeiture.
The plaintiff rightly bases her federal claims on 42 U.S.C. § 1983, which imposes tort liability on state and local employees, and sometimes their employer, and sometimes other state and local agents, for violating federal rights. Had the contract between the federal government and McHenry County to house aliens suspected of being forbidden to enter or remain in the United States made the county jail a federal instrumentality and its personnel (maybe including Centegra’s employees, though they were not employees of the jail) federal officers, the jail staff would be suable for federal constitutional violations under the doctrine of Bivens v. Six Unknown Named Agents of Federal Bureau of Narcotics, 403 U.S. 388, 91 S.Ct. 1999, 29 L.Ed.2d 619 (1971), rather than under section 1983. But the contract did not federalize McHenry County Jail, which continued to house nonfederal as well as federal prisoners. Cases similar to this, allowing section 1983 claims by federal prisoners against county or city employees, are legion. See, e.g., Hunter v. Amin, 583 F.3d 486 (7th Cir.2009); Lewis v. Downey, 581 F.3d 467, 471 n. 3 (7th Cir.2009); Ortiz v. Downey, 561 F.3d 664 (7th Cir.2009); Grieveson v. Anderson, 538 F.3d 763 (7th Cir.2008); Porro v. Barnes, 624 F.3d 1322 (10th Cir.2010); Wilson v. Blankenship, 163 F.3d 1284 (11th Cir.1998). And cf. Logue v. United States, 412 U.S. 521, 529-30, 93 S.Ct. 2215, 37 L.Ed.2d 121 (1973), a jail-suicide suit under the Federal Tort Claims Act. See 28 U.S.C. § 1346.
Although Centegra’s employees are not public employees, they rightly do not deny that in performing functions that would otherwise be performed by public employees, they were acting under color of state law and therefore could be sued under section 1983. See, e.g., West v. Atkins, 487 U.S. 42, 49-54, 108 S.Ct. 2250, 101 L.Ed.2d 40 (1988); Rice ex rel. Rice v. Correctional Medical Services, 675 F.3d 650, 670-73 (7th Cir.2012); Rodriguez v. Plymouth Ambulance Service, 577 F.3d 816, 824-26 (7th Cir.2009); Conner v. Donnelly, 42 F.3d 220, 224-25 (4th Cir.1994). Otherwise state and local government could immunize itself from liability under *979section 1983 by replacing its employees with independent contractors.
Arriving at the merits, we meet at the threshold the question of the proper standard for determining liability. The plaintiff argues that for want of any judicial or even quasi-judicial determination that Belbachir was or might be a criminal — her detention in the jail was a civil commitment — the proper standard of liability is that of reasonableness, the standard under the Fourth Amendment (made applicable to state action by interpretation of the due process clause of the Fourteenth Amendment) for the seizure of a person. See, e.g., Ortiz v. City of Chicago, 656 F.3d 523, 530 (7th Cir.2011); Sallenger v. City of Springfield, 630 F.3d 499, 503 (7th Cir.2010); Aldini v. Johnson, 609 F.3d 858, 866 (6th Cir.2010). The defendants, seconded by the district judge, contend that the proper standard is “deliberate indifference” (to the risk of suicide), a standard they interpret as requiring knowledge (not just suspicion or reason to know) that the risk is “substantial.” And there are cases that say that too. E.g., Estate of Miller v. Tobiasz, 680 F.3d 984, 989 (7th Cir.2012); Porro v. Barnes, supra, 624 F.3d at 1325-26; Edwards v. Johnson, 209 F.3d 772, 778 (5th Cir.2000).
The Fourth Amendment forbids unreasonable searches and seizures. The immigration authorities, reasonably believing that Belbachir was inadmissible to the United States, were entitled to detain her pending the hearing on her application for asylum; in fact the applicable regulation required them to do that. 8 C.F.R. § 235.3(b)(4)(ii). As the validity of the regulation, and therefore of her detention, is not challenged, she was in the same position as a lawfully arrested pretrial detainee, Porro v. Barnes, supra, 624 F.3d at 1325-26; Edwards v. Johnson, supra, 209 F.3d at 778, who cannot complain of an unreasonable seizure in violation of the Fourth Amendment but is entitled, by the due process clause of the Fifth or Fourteenth Amendments, to at least as much protection as convicted criminals are entitled to under the Eighth Amendment, City of Revere v. Massachusetts General Hospital, 463 U.S. 239, 244, 103 S.Ct. 2979, 77 L.Ed.2d 605 (1983); Rice ex rel. Rice v. Correctional Medical Services, supra, 675 F.3d at 664; Gibson v. County of Washoe, 290 F.3d 1175, 1187 (9th Cir.2002); Boswell v. County of Sherburne, 849 F.2d 1117, 1120-21 (8th Cir.1988) — namely protection from harm caused by a defendant’s deliberate indifference to the detainee’s safety or health. See, e.g., City of Revere v. Massachusetts General Hospital, supra, 463 U.S. at 244, 103 S.Ct. 2979; King v. Kramer, 680 F.3d 1013, 1017 (7th Cir.2012); Padilla v. Yoo, 678 F.3d 748, 759 (9th Cir.2012); Ramos v. Patnaude, 640 F.3d 485, 489 (1st Cir.2011). In Forrest v. Prine, 620 F.3d 739, 744 (7th Cir.2010), we added to “at least” — “and probably more.”
Belbachir was detained only because an alien determined to be inadmissible upon arrival in the United States who seeks asylum is required to be detained until his or her asylum application is ruled on. She was not a convicted criminal; and “persons who have been involuntarily committed are entitled to more considerate treatment ... than criminals whose conditions of confinement are designed to punish.” Youngberg v. Romeo, 457 U.S. 307, 321-22, 102 S.Ct. 2452, 73 L.Ed.2d 28 (1982). True, Romeo had been committed because of profound mental retardation rather than because of any violation of law. He may therefore have deserved greater protection than a person who may well be an illegal alien, which was Belbachir’s status, though she had a shot at asylum. She might be analogized instead to a person who having been lawfully arrested and failed to make bail is being held in jail *980pending trial. But that person will have received a probable cause hearing before a judicial officer within 48 hours of arrest. County of Riverside v. McLaughlin, 500 U.S. 44, 56, 111 S.Ct. 1661, 114 L.Ed.2d 49 (1991). Belbachir was detained indefinitely without any hearing.
By depriving her for an indefinite time of any source of protection (including protection against herself) other than persons employed by or working at or managing the jail, the jail personnel became obligated by ordinary tort principles to provide her with reasonable care. Dezort v. Village of Hinsdale, 35 Ill.App.3d 703, 342 N.E.2d 468, 472-73 (1976); Gordon v. City of New York, 70 N.Y.2d 839, 523 N.Y.S.2d 445, 517 N.E.2d 1331, 1332 (1987) (per curiam); Thornton v. City of Flint, 39 Mich.App. 260, 197 N.W.2d 485, 493 (1972); Restatement (Second) of Torts § 314A(4) (1965). Neither the guards nor the members of the medical staff were involved in the decision to detain her; but they were her custodians, responsible therefore for her safety.
But ordinary tort principles do not govern suits charging federal constitutional torts. If we assimilate Belbachir’s case not to that of Romeo or McLaughlin but to that of a prison inmate, then the plaintiff must prove not only that the defendants by detaining Belbachir deprived her of an alternative source of protection from danger but also that in failing to protect her (as only they could do) they acted with deliberate indifference to the danger, though if the danger was obvious the defendants’ deliberateness (that is, their knowledge of the risk) could be inferred. Farmer v. Brennan, 511 U.S. 825, 842-43, 114 S.Ct. 1970, 128 L.Ed.2d 811 (1994); Doe v. St. Francis School District, 694 F.3d 869, 871 (7th Cir.2012); Estate of Miller v. Tobiasz, supra, 680 F.3d at 990; Prude v. Clarke, 675 F.3d 732, 735 (7th Cir.2012); Vaughn v. Gray, 557 F.3d 904, 909 (8th Cir.2009).
We do not have to choose among standards of care in this case, however. Even under the toughest possibly applicable standard, that of deliberate indifference, one defendant, Vicki Frederick, should not have been dismissed from the suit; and even under the most lenient standard, that of Youngberg v. Romeo, which resembles though is not identical to a negligence standard, see 457 U.S. at 319-25, 102 S.Ct. 2452, the other defendants were properly dismissed. (As we’ll see, they may have been negligent, but no causal relation between their negligence and the suicide was shown.)
We begin with deliberate indifference and defendant Frederick — a licensed clinical social worker employed by Centegra — noting first that the application of the standard of deliberate indifference varies with circumstances, as this case illustrates. Had Belbachir been a known violent criminal, security needs might have made it difficult for the guards and medical personnel to provide her with the same level of protection, including protection from herself, as would be possible in a different environment. If the risk of suicide is enhanced by isolation, nevertheless regard for the safety of other prisoners may preclude allowing a prisoner to have a cellmate. But this is not such a case.
The suicide rate in jails has fallen a great deal in recent years; it was estimated to be 9 times the rate in the general population in 1988 and “only” 3 times that rate in 2005. Lindsay M. Hayes, National Study of Jail Suicide: 20 Years Later 46 (National Center on Institutions and Alternatives, April 2010). Yet even the lower ratio is frighteningly high. And the risk of a jail suicide is concentrated in the first week of detention — 48 percent of jail suicides occur then. Christopher J. Mumola, *981Suicide and Homicide in State Prisons and Local Jails 8 (U.S. Bureau of Justice Statistics, Aug. 2005). Belbachir killed herself on the eighth day.
She was not a criminal and was no danger to any person in the jail, whether staff member, detainee, or visitor. She was an obvious suicide risk who should have been hospitalized or at least placed on suicide watch, during which a guard would have glanced into her cell every 10 minutes. There are more elaborate forms of suicide watch (see “Suicide Watch,” Wikipedia, http://en.wikipedia.org'wiki/Suicide_wateh (visited Aug. 8, 2018)) — for it needn’t take 10 minutes to kill oneself. But there is no contention that the jail’s method of suicide watch was inadequate, let alone constitutionally deficient.
We don’t have to recite the depressing facts that culminated in her strangling herself with her socks. It is enough simply to reproduce at the end of this opinion the report of the intake screening by a guard and the mental health progress notes by defendant Frederick — the only potentially culpable Centegra employee, as we’ll see.
The guard who filled out the intake report changed Belbachir’s answer “yes” to the question “Are you currently extremely depressed or feeling suicidal?” to “no.” He stated in his deposition that he had simply made a mistake initially, in circling “yes” when he meant “no.” Maybe so, but a jury would not have to believe him. It might find that he’d been afraid of getting into trouble by having answered “yes” but then having failed to place Belbachir on suicide watch, and so he changed “yes” to “no.” There is conflicting evidence on when he changed it. He is not, however, a defendant.
Frederick’s mental health progress notes, composed on March 14, three days before Belbachir’s suicide, on the basis of an interview that Frederick had conducted with Belbachir earlier that day, stated that Belbachir was suicidal and suffering from a “major depressive disorder” — a warning sign for suicide — and described in detail a host of deeply alarming symptoms. Frederick denigrates the gravity of her notes by arguing-what in another context would be comical — that Belbachir had denied having a specific suicide plan. The implication is that if Belbachir had said, “I intend to commit suicide as soon as I formulate a specific plan for how to do so, but not before,” Frederick would have replied, “okay, but tell us when you have devised your plan so that we can prevent your carrying it out,” but that otherwise Frederick was entitled to do nothing.
The intake report is dated March 9; the mental health progress notes are dated five days later. The deterioration in Belbachir’s mental state is apparent from a comparison of these documents, which we append to this opinion. Frederick did not report the contents of her mental health progress notes to any of the guards. And despite the alarming contents of those notes, neither she nor any other member of the medical staff had any further contact with Belbachir after March 14, three days before her death. Frederick denies having seen the intake report, although it was in Belbachir’s medical file, to which Frederick had access.
The risk of suicide was obvious to Frederick. The defendants argue that the risk was small. That is imprecise. The risk, in the sense of the probability that it would materialize (5 percent, 20 percent, etc.), was unknown. But when an adverse consequence is very great, the failure to take a simple, inexpensive, obvious, and indeed prescribed measure to avert it is inexcusable. In economic terms, an expected loss can be expressed as P x L, where P is the probability that the loss will occur and L is the magnitude of the loss if it occurs. Even if the probability is low, if the loss if *982the probability materializes is very great, as it was here, and the burden (i.e., cost), B, of preventing the event causing the loss (the suicide) is very low, as it seems to have been here, as well, the failure to take preventive action may be negligent (if B is smaller than P x L — the famous Hand formula for negligence, United States v. Carroll Towing Co., 159 F.2d 169, 178 (2d Cir.1947)), or worse than negligent if B is much lower than P x L, as appears to have been the case here too. And even if the variables cannot be quantified, often they can at least be approximated.
The defendants could have placed Belbachir in a mental hospital or at least on suicide watch. These were simple and obvious precautions against a risk of suicide. A severely depressed person who has hallucinations, acute anxiety, and feelings of hopelessness and helplessness and who cries continually, talks incessantly of death, and is diagnosed as suicidal, is in obvious danger, and if the danger (known to a defendant) can be averted at slight cost, the failure to try to avert it is willful. Cavalieri v. Shepard, 321 F.3d 616, 620-21 (7th Cir.2003); Sanville v. McCaughtry, 266 F.3d 724, 737-38 (7th Cir.2001); Clouthier v. County of Contra Costa, 591 F.3d 1232, 1244-45 (9th Cir.2010);Coleman v. Parkman, 349 F.3d 534, 539 (8th Cir.2003); Comstock v. McCrary, 273 F.3d 693, 704, 709-10 (6th Cir.2001).
The district judge reviewed the facts in detail but regarding the medical staff concluded simply that none of them “subjectively knew [that Belbachir] was at substantial risk of committing suicide.” But as the judge didn’t explain what he meant by “substantial” in this context we can give no weight to his conclusion. Nor are the facts so one-sided as to justify taking from a jury the issue of defendant Frederick’s refusal to act in the face of a significant risk of suicide known to her.
The other Centegra defendants, however, were properly dismissed, and would have been even under a negligence standard. The nurse who found Belbachir dead in her cell, and whom the plaintiff accuses of having exhibited willful disregard for Belbachir’s welfare by failing to attempt cardiopulmonary resuscitation, cannot be held liable. The attempt would have done no good. Belbachir was already dead, as confirmed by a prompt attempt to resuscitate her by means of a defibrillator, which registered no pulse. Thus an elementary requirement of liability — causation — was missing.
As for Centegra Nurse Manager Kaplan (Frederick’s boss), there is no evidence that she knew that Belbachir was suicidal. She treated Belbachir for panic attacks and anxiety, and recommended both that she be given a roommate (which she was, initially) and that she be transferred to the “med pod” (the part of the jail in which detainees requiring medical treatment were confined); and that was also done. But Frederick didn’t tell Kaplan that Belbachir was suicidal. Kaplan didn’t review Frederick’s notes until after Belbachir’s death, but there is no evidence that her failure to do so was a dereliction of duty.
Last we consider whether McHenry County may also be liable for failing to prevent Belbachir’s suicide. None of the guards is a defendant, and the county sheriff is sued only in his official capacity, which means that the suit against him is really against the County, the government agency that employs him. Kentucky v. Graham, 473 U.S. 159, 165-66, 105 S.Ct. 3099, 87 L.Ed.2d 114 (1985); Grieveson v. Anderson, 538 F.3d 763, 771 (7th Cir.2008). The jail’s director (the only other County employee named as a defendant) is sued in his personal capacity. But there is no evidence of his complicity in the acts (or *983rather omissions to act) that failed to reduce the risk of suicide to Belbachir, so we affirm his dismissal from the ease.
The judge said that the County could not be liable unless one of its employees was. That’s incorrect. What’s true is that if a County employee were liable, the County itself would not be liable merely by virtue of the doctrine of respondeat superior; the doctrine is inapplicable in section 1983 cases. But an institution can violate section 1983 just as an individual can.
The plaintiff complains that the County failed to provide annual training to its jail staff on how to recognize the risk of suicide (the members of the staff did receive such training when they were hired), failed to have a written suicide-prevention policy, and lacked policies governing communication between the medical staff and the guards. Granted that the sheriff, the County’s policymaker regarding the jail, knew of the suicide risk to prisoners, especially new prisoners-there had been a number of suicide attempts in the previous 10 years (though precisely how many we don’t know) — nevertheless the grounds on which the plaintiff accuses the sheriff of deliberate indifference to the risk of suicide are thin. The facts in cases in which a county jail or other local jail (or its sheriff) have been held liable for deliberate indifference to prisoners’ safety or health are much more favorable to the plaintiff than the facts in this case. See Thomas v. Cook County Sheriff's Dep’t, 604 F.3d 293, 302-04 (7th Cir.2010); Duvall v. Dallas County, 631 F.3d 203, 208-09 (5th Cir.2011) (per curiam); Long v. County of Los Angeles, 442 F.3d 1178, 1188-89 (9th Cir.2006); Blackmore v. Kalamazoo County, 390 F.3d 890, 900 (6th Cir.2004).
It’s difficult to see why formal policies would be necessary to assure adequate communication between the guards and the members of the medical staff, since they work side by side. More troublesome is the failure either to have provided annual training or to have had a written policy; with neither, a lot of weight was being placed on guards’ memory, though no doubt the suicide attempts jogged their memory. But at worst the sheriffs failure to take the additional measures that the plaintiff thinks necessary was negligent, and maybe not even that. For while the immigration authorities, who monitor the jail’s treatment of its federal prisoners, rated the jail’s suicide-prevention policy “deficient” in 2002, they rated it acceptable in both 2003 and (the last evaluation before Belbachir committed suicide) 2004. Shortly after the suicide, it is true, the immigration authorities, in what may be an example of hindsight bias, found fault with the jail’s suicide prevention policies for (1) forgoing annual training of the jail staff in suicide prevention; (2) having only an incomplete or inconsistent policy governing intake screenings; (3) lacking written procedures on when to place detainees on, or remove them from, suicide watch; and (4) sometimes allowing guards to perform the required 10-minute checks of prisoners on suicide watch by intercom rather than visually. But there is no evidence that the sheriff was aware of any of these failures except the first.
And even if the sheriff was culpable for failing to discover and correct the deficiencies argued by the plaintiff, there is no evidence that correcting them before Belbachir arrived at the jail would have prevented her suicide. Once again a causal relation between fault and injury is missing. The critical failure to prevent Belbachir’s suicide occurred when defendant Frederick, having decided that Belbachir was not suicidal, failed to tell the guards to put her on suicide watch. Two of the three deficiencies of which the plaintiff complains — the absence of annual training and of a written suicide-prevention policy — are addressed to the guards rather *984than to the medical staff. The third — the absence of a policy governing communication between the medical staff and the guards — could not have played a causal role either, because Frederick, not thinking that Belbachir was a suicide risk, had no warning to communicate to the guards.
The judgment must be reversed and the case remanded for trial, limited however to defendant Frederick. The dismissal of the supplemental state law claims against Centegra and its employees is vacated as well, since the premise of the dismissal was the dismissal of all the federal claims, and we are restoring one of them.
Affirmed in part, Reversed in Part, and Remanded.
*985APPENDIX: INTAKE QUESTIONNAIRE AND MENTAL HEALTH Progress Notes
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*98616 No. 13-1002
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*987No. 13-1002 17
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*98818 No. 13-1002
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