675 F.3d 743 (2012)
Mark GEINOSKY, Plaintiff-Appellant,
CITY OF CHICAGO, et al., Defendants-Appellees.
United States Court of Appeals, Seventh Circuit.
Argued January 13, 2012.
Decided March 28, 2012.
Lawrence V. Jackowiak (argued), Attorney, Law Offices of Lawrence V. Jackowiak, Chicago, IL, for Plaintiff-Appellant.
Christopher S. Norborg (argued), Attorney, City of Chicago Law Department, Chicago, IL, for Defendants-Appellees.
Before POSNER, WOOD, and HAMILTON, Circuit Judges.
HAMILTON, Circuit Judge.
Plaintiff Mark Geinosky received twenty-four bogus parking tickets and made a federal case out of it. At first blush, this case might seem like a good candidate for the summary Rule 12(b)(6) dismissal granted by the district court. A closer look at the alleged facts, however, reveals a disturbing pattern. Absent a reasonable explanation, and none has even been suggested yet, the pattern adds up to deliberate and unjustified official harassment that is actionable under the Equal Protection Clause of the Fourteenth Amendment. We therefore reverse the dismissal of Geinosky’s “class-of-one” equal protection claim as well as his related civil conspiracy claim. We affirm the dismissal of Geinosky’s substantive due process claim, however. He has not pled facts suggesting a deprivation that meets the high threshold for such claims established by the Supreme Court and by our precedents.
All of Geinosky’s parking tickets concerned the same Toyota and were received over a fourteen-month period beginning in October 2007. All arrived in the mail, typically in batches of three or four. All were written by officers of Unit 253 of the Chicago Police Department. Some of the tickets were inconsistent with others received at the same time, implying, for example, that the Toyota was in two places almost at once or was simultaneously double-parked and parked on the sidewalk. All thirteen of the tickets attributed to Officer Wilkerson and issued on May 2, July 8, August 27, and October 7, 2008 had sequential citation numbers and concerned alleged violations that occurred on each date at exactly 10:00 p.m. Ten of the tickets were issued while Geinosky’s estranged wife was in possession of the Toyota. Another ten were issued after he got the car back, and the last four were issued even after Geinosky sold the Toyota. Because none of the twenty-four tickets were legitimate, Geinosky had all of them dismissed, but he had to go to court seven times to do so.
Before filing this lawsuit, Geinosky tried other paths to stop the harassment. He complained to police supervisors in Unit 253, to the police department’s Internal Affairs Division, and to the Independent Police Review Authority. After Internal Affairs closed the case without investigating, Geinosky contacted the Chicago Tribune, which ran several stories about his plight as part of its Problem Solver column. The Internal Affairs Division then reopened the case and began an investigation. Geinosky also filed this lawsuit. While this appeal was pending, the Tribune reported that the internal investigation had resulted in a recommendation to fire several defendant officers from Unit 253. See Jon Yates, Problem Solver: Police cite 3 officers over fake tickets; man vindicated, Chicago Tribune, Nov. 7, 2011, available at http://www.chicagotribune.com/business/problemsolver/ct-biz-1107-problem-geinosky-20111107,0,7221534. column (last visited Mar. 23, 2012).
Geinosky sued the City of Chicago and eight of the officers then assigned to Unit 253 under 42 U.S.C. § 1983. He alleges “class-of-one” discrimination in violation of the Equal Protection Clause, denial of substantive due process rights, and an unlawful civil conspiracy. The district court dismissed all claims on defendants’ Rule 12(b)(6) motion for failure to state a claim. We affirm only with regard to the substantive due process claim. The district court, relying on McDonald v. Village of Winnetka, 371 F.3d 992, 1009 (7th Cir.2004), held that Geinosky’s equal protection claim failed because he did not specifically identify a similarly situated individual who was treated differently. In selective investigation or prosecution cases such as McDonald, a “meaningful application of the `similarly situated’ requirement” serves to distinguish between constitutional claims for discrimination and ordinary tort claims, and the plaintiff must eventually offer evidence of a similarly situated person. Id. But as we explain below, in a straightforward official harassment case like the allegations here, forcing the plaintiff to name a person not so severely harassed serves no such purpose (and in any event certainly is not necessary in the complaint itself). Are there people in Chicago who have not received more than a dozen bogus parking tickets from the same police unit in a short time? Geinosky could find hundreds of those people on any page of the Chicago phone book.
We review de novo the district court’s dismissal for failure to state a claim, presuming the truth of the facts alleged in Geinosky’s complaint and drawing all reasonable inferences in his favor. E.g., London v. RBS Citizens, N.A., 600 F.3d 742, 745 (7th Cir.2010). We discuss each of his claims below, starting with equal protection.
I. Equal Protection “Class-of-One”
The Equal Protection Clause of the Fourteenth Amendment, ratified to help protect the equality that had been won in the Civil War, is most familiar as a guard against state and local government discrimination on the basis of race, national origin, sex, and other class-based distinctions. E.g., Loving v. Virginia, 388 U.S. 1, 8-12, 87 S.Ct. 1817, 18 L.Ed.2d 1010 (1967); see also United States v. Virginia, 518 U.S. 515, 533-34, 116 S.Ct. 2264, 135 L.Ed.2d 735 (1996) (heightened scrutiny for legal classification based on sex). The Equal Protection Clause has also come to be understood to protect individuals against purely arbitrary government classifications, even when a classification consists of singling out just one person for different treatment for arbitrary and irrational purposes. To state a so-called “class-of-one” equal protection claim, Geinosky must allege that he was “intentionally treated differently from others similarly situated and that there is no rational basis for the difference in treatment.” Engquist v. Oregon Dept. of Agriculture, 553 U.S. 591, 601, 128 S.Ct. 2146, 170 L.Ed.2d 975 (2008), citing Village of Willowbrook v. Olech, 528 U.S. 562, 564, 120 S.Ct. 1073, 145 L.Ed.2d 1060 (2000). We have held that class-of-one claims can be brought based on allegations of the irrational or malicious application of law enforcement powers. E.g., Hanes v. Zurick, 578 F.3d 491, 495 (7th Cir.2009) (holding that Engquist exemption for public employment decisions does not extend to law enforcement decisions). Although the police are necessarily afforded wide discretion in performing their duties, that discretion does not extend to discriminating against or harassing people. The district court correctly found that nothing bars a properly pled class-of-one equal protection claim in this context, but it erred when it found that Geinosky’s pleadings were insufficient to state such a claim.
Courts have understood that if class-of-one claims are not defined appropriately, they might turn many ordinary and inevitable mistakes by government officials into constitutional violations and federal lawsuits. One element of a proper class-of-one claim is a wrongful act that necessarily involves treatment departing from some norm or common practice. McDonald, 371 F.3d at 1009. But “the purpose of entertaining a `class of one’ equal protection claim is not to constitutionalize all tort law nor to transform every claim for improper provision of municipal services or for improper conduct of an investigation in connection with them into a federal case.” Id. The appropriate limiting principle must be tailored to the type of government action at issue.
For example, because the government traditionally is given even more discretion in its role as employer than in its role as enforcer of the law, public employees simply do not have recourse to class-of-one claims if they are singled out for firing. Engquist, 553 U.S. at 607, 128 S.Ct. 2146 (“we are guided, as in the past, by the `common-sense realization that government offices could not function if every employment decision became a constitutional matter'”). To bring an equal protection claim, public employees aggrieved by their firing must be able to allege and later prove discrimination against a protected class. Under Engquist, the prohibition on class-of-one claims in the public employment context is categorical. In contrast, in the context of complicated government investigations or prosecutions, we have relied on careful application of the similarly-situated requirement to distinguish between unfortunate mistakes and actionable, deliberate discrimination. E.g., McDonald, 371 F.3d at 1009. When the parties raise a serious question whether differences in treatment stem from a discriminatory purpose or from a relevant factual difference, the key evidence is often what was done in the investigation or prosecution of others in similar circumstances. The Supreme Court also relied on the similarly-situated prong in Olech, in the context of providing municipal services. 528 U.S. at 565, 120 S.Ct. 1073.
The district court here invoked the similarly-situated requirement and faulted Geinosky for failing to identify and describe any such individuals. But in this case, requiring Geinosky to name a similarly situated person who did not receive twenty-four bogus parking tickets in 2007 and 2008 would not help distinguish between ordinary wrongful acts and deliberately discriminatory denials of equal protection. Such a requirement would be so simple to satisfy here that there is no purpose in punishing its omission with dismissal. Here, the pattern and nature of defendants’ alleged conduct do the work of demonstrating the officers’ improper discriminatory purpose. Geinosky’s general allegation that defendants “intentionally treated plaintiff differently than others similarly situated” is sufficient here, where the alleged facts so clearly suggest harassment by public officials that has no conceivable legitimate purpose. To require more would elevate form over substance. Geinosky’s complaint states a class-of-one claim in light of the pattern of unjustified harassment he has alleged.
Geinosky does not know for certain why he was targeted. He suspects a connection between his estranged wife and officers of Unit 253, but his case would be just as strong if the officers picked him to harass for no reason at all. The complaint clearly tells a story in which Geinosky was targeted. Reason and common sense provide no answer to why he was targeted that could be considered a legitimate exercise of police discretion. Somewhere between the first several and the twentyfourth bogus tickets from officers of the same police unit, Geinosky’s grievance rose to the level of an actionable class-of-one discrimination claim.
We do not credit the city’s assertion that allowing this suit will open the floodgates to a wave of ordinary malicious prosecution (or other tort) cases brought as constitutional class-of-one claims. The extraordinary pattern of baseless tickets that Geinosky received will remain rare, we trust, particularly now that the Police Board and the courts are involved. The litigation floodgates should not open for the additional reason that truly random law enforcement, as when an officer picks one of many speeding cars to stop and ticket, provides a rational basis for the selection even if the ticketed driver feels she was unfairly singled out. Officers have discretion for powerful reasons, not the least of which is the impossibility of ticketing all traffic or parking violations and the ineffectiveness of ticketing none. Because officers must choose among violators, random selection is certainly rational. We are not inviting every driver with a couple of parking tickets (even invalid ones) to sue in federal court. See Engquist, 553 U.S. at 604, 128 S.Ct. 2146 (“But allowing an equal protection claim on the ground that a ticket was given to one person and not others, even if for no discernible or articulable reason, would be incompatible with the discretion inherent in the challenged action.”).
But the pattern of conduct alleged here, as the district court correctly wrote, “is not a legitimate exercise of discretion.” No one has suggested, let alone demonstrated as a matter of law, a rational and proper purpose for the ticketing. On these unusual facts — many baseless tickets that were highly unlikely to have been a product of random mistakes — Geinosky’s general assertion that other persons were not similarly abused does not require names or descriptions in support. We reverse the dismissal of plaintiff’s equal protection claim.
II. Civil Conspiracy
The district court also dismissed Geinosky’s section 1983 civil conspiracy allegations because they depended on his dismissed substantive constitutional claims. Because we reverse the dismissal of the equal protection claim, we also reverse the dismissal of the dependent conspiracy claim.
Citing Redd v. Nolan, 663 F.3d 287, 292 (7th Cir.2011), defendants argue that the complaint fails to allege a conspiracy under the standards of Bell Atlantic Corp. v. Twombly, 550 U.S. 544, 127 S.Ct. 1955, 167 L.Ed.2d 929 (2007), and Ashcroft v. Iqbal, 556 U.S. 662, 129 S.Ct. 1937, 173 L.Ed.2d 868 (2009). We disagree. While the complaint makes only rather conclusory direct allegations of conspiracy, the complaint also alleges a pattern of harassment by several officers over a period of months. It is a challenge to imagine a scenario in which that harassment would not have been the product of a conspiracy. Under Twombly, all plaintiff needed to allege was a plausible account of a conspiracy. See 550 U.S. at 556, 127 S.Ct. 1955. This complaint goes well beyond that. In this regard, the complaint here is fundamentally different from the complaint in Redd, which contained “not a whiff of a conspiratorial agreement.” 663 F.3d at 292. Iqbal calls on us to apply our “judicial experience and common sense.” 556 U.S. at 679, 129 S.Ct. at 1950. If several members of the same police unit allegedly acted in the same inexplicable way against a plaintiff on many different occasions, we will not dismiss a complaint for failure to recite language explicitly linking these factual details to their obvious suggestion of collusion. Geinosky’s allegations of a conspiracy among the officers of Unit 253 to harass him by issuing bogus parking tickets go well beyond the required threshold.
Geinosky concedes that he is not seeking damages for the first ten tickets, which were issued outside of the applicable two-year statute of limitations period (before March 3, 2008) by officers Hegewald, Whelehan, Reidy, and Roque. It would be premature to dismiss those officers from this case at this early stage. If Geinosky can prove that they were part of a conspiracy, they can still be held liable for the later actions of co-conspirators that harmed the plaintiff during the limitations period. See, e.g., In re Brand Name Prescription Drugs Antitrust Litigation, 123 F.3d 599, 616 (7th Cir.1997) (defendant could be held jointly liable for later acts of co-conspirators where it did not withdraw from conspiracy); Hoffman-La Roche, Inc. v. Greenberg, 447 F.2d 872, 874 n. 2 (7th Cir.1971) (parties to a civil conspiracy are jointly and severally liable for injuries to plaintiff); see also, e.g., Nader v. Democratic Nat’l Comm., 567 F.3d 692, 697 (D.C.Cir.2009) (under District of Columbia law, civil conspiracy is method of establishing vicarious liability for underlying tort). Also, even if Geinosky cannot recover damages for those discrete acts outside the limitations period, those earlier acts may still be used to prove that the conspiracy existed both before and during the limitation period and harmed him during that period. See, e.g., Scherer v. Balkema, 840 F.2d 437, 442 (7th Cir.1988); see also National R.R. Passenger Corp. v. Morgan, 536 U.S. 101, 105, 122 S.Ct. 2061, 153 L.Ed.2d 106 (2002) (stating same rule with regard to analogous “pattern or practice” and hostile work environment claims). Geinosky has pled a conspiracy and discrete actions that furthered that conspiracy both before and during the limitations period. The district court’s dismissal of Geinosky’s conspiracy claim is reversed with regard to all defendants.
III. Substantive Due Process
The district court correctly dismissed Geinosky’s due process claim. Substantive due process claims can address harmful, arbitrary acts by public officials. E.g., County of Sacramento v. Lewis, 523 U.S. 833, 845-46, 118 S.Ct. 1708, 140 L.Ed.2d 1043 (1998). But such claims must meet a high standard, even when the alleged conduct was abhorrent, to avoid constitutionalizing every tort committed by a public employee. Id. at 846, 854, 118 S.Ct. 1708 (tactics in high-speed chase not redressible under section 1983 even though they unintentionally resulted in fleeing suspect’s death); Tun v. Whitticker, 398 F.3d 899, 902-03 (7th Cir.2005) (expulsion from school not actionable). Although his allegations of harassment are troubling, Geinosky has not pled facts suggesting a deprivation that “shocks the conscience” in the sense required in due process cases. See id., quoting Rochin v. California, 342 U.S. 165, 172, 72 S.Ct. 205, 96 L.Ed. 183 (1952) (forcible stomach pumping to retrieve swallowed evidence was a due process violation). We affirm the district court’s dismissal of Geinosky’s substantive due process claim.
* * *
Law enforcement is a difficult and dangerous job and, in the case of parking and traffic tickets, often an especially thankless one. Society grants considerable discretion to the public servants charged with that job, with the understanding that limited resources and factual nuances will require that some people are sometimes treated differently than others. But a grant of discretion is a grant of trust. If officers betray that trust by deliberately targeting someone for harassment, that discrimination can violate our constitution, and section 1983 provides a remedy. Plaintiff Geinosky alleged a pattern of deliberate and unconstitutional harassment, even though accomplished through the relatively mild mechanism of bogus parking tickets. His First Amended Complaint states class-of-one equal protection and civil conspiracy claims upon which relief can be granted. The judgment of the district court is AFFIRMED in part and REVERSED in part, and the case is REMANDED for further proceedings consistent with this opinion.
 Perhaps our reference to this recent newspaper story might raise an eyebrow, but it should not. The defendants moved to dismiss for failure to state a claim under Rule 12(b)(6). A motion under Rule 12(b)(6) can be based only on the complaint itself, documents attached to the complaint, documents that are critical to the complaint and referred to in it, and information that is subject to proper judicial notice. See Fed.R.Civ.P. 10(c) (written instrument that is exhibit to pleading is part of pleading for all purposes); Papasan v. Allain, 478 U.S. 265, 268 n. 1, 106 S.Ct. 2932, 92 L.Ed.2d 209 (1986) (judicial notice of public records); 188 LLC v. Trinity Industries, Inc., 300 F.3d 730, 735 (7th Cir.2002) (documents referred to in complaint and central to claim); Wright v. Associated Ins. Cos., 29 F.3d 1244, 1248 (7th Cir.1994) (same). If a moving party relies on additional materials, the motion must be converted to one for summary judgment under Rule 56. Fed.R.Civ.P. 12(d).
A plaintiff, however, has much more flexibility in opposing a Rule 12(b)(6) motion and in appealing a dismissal. A party appealing a Rule 12(b)(6) dismissal may elaborate on his factual allegations so long as the new elaborations are consistent with the pleadings. See Chavez v. Illinois State Police, 251 F.3d 612, 650 (7th Cir.2001); Highsmith v. Chrysler Credit Corp., 18 F.3d 434, 439-40 (7th Cir. 1994) (reversing dismissal in relevant part based on such new elaborations); Dawson v. General Motors Corp., 977 F.2d 369, 372 (7th Cir.1992) (reversing dismissal based on new elaborations). In the district court, too, a party opposing a Rule 12(b)(6) motion may submit materials outside the pleadings to illustrate the facts the party expects to be able to prove. Thomas v. Guardsmark, Inc., 381 F.3d 701, 704 (7th Cir.2004) (denying motion to strike new materials submitted on appeal); Early v. Bankers Life and Casualty Co., 959 F.2d 75, 79 (7th Cir.1992) (reversing dismissal; plaintiff is free to assert new facts in brief opposing motion to dismiss); Roe v. Bridgestone Corp., 492 F.Supp.2d 988, 1007 (S.D.Ind.2007) (“Such documents are not evidence, but they provide a way for a plaintiff to show a court that there is likely to be some evidentiary weight behind the pleadings the court must evaluate.”). In the turmoil concerning civil pleading standards stirred up by Ashcroft v. Iqbal, 556 U.S. 662, 129 S.Ct. 1937, 173 L.Ed.2d 868 (2009), and Bell Atlantic Corp. v. Twombly, 550 U.S. 544, 127 S.Ct. 1955, 167 L.Ed.2d 929 (2007), a plaintiff who is opposing a Rule 12(b)(6) or Rule 12(c) motion and who can provide such illustration may find it prudent to do so. (It may also be prudent to explain to the district court that the materials are being submitted for illustrative purposes and should not be used to convert the motion into a Rule 56 motion for summary judgment.)
 Several of our cases have suggested that a showing of subjective ill will can be useful in distinguishing between ordinary errors and discriminatory denials of equal protection. E.g., Hilton v. City of Wheeling, 209 F.3d 1005, 1008 (7th Cir.2000). The differential treatment alleged in Olech was “irrational and wholly arbitrary,” apart from its subjective motivation, and the Supreme Court declined to decide whether a showing of subjective animus is required in every case. See 528 U.S. at 565, 120 S.Ct. 1073. Our court recently heard argument en banc on this issue. Further discussion is unnecessary here, as Geinosky’s allegations tell a story that clearly suggests a vindictive or harassing purpose.
 Even in a case where a plaintiff would need to identify a similarly situated person to prove his case, like the McDonald case cited by the district court on this point, we see no basis for requiring the plaintiff to identify the person in the complaint. McDonald was decided on summary judgment, not on the pleadings. See 371 F.3d at 994, Rule 8(a)(2) requires only “a short and plain statement of the claim showing that the pleader is entitled to relief.” Even the more demanding pleading requirements under Iqbal and Twombly do not require a plaintiff to identify specific comparators in a complaint.
 With regard to Officer Aguilar, the complaint alleges only that he twice looked up the Toyota’s license plate number on police department computers at roughly the time some of the bogus tickets were issued. We might speculate that perhaps he was investigating the conspiracy rather than participating in it, but we cannot resolve the question of Officer Aguilar’s alleged involvement on the pleadings merely because we can imagine an innocent explanation for the alleged actions.
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