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Logic and rhetoric
A checkerboard rule is a logically defective type of rule, law, or policy. It is easiest to define by example, because there are competing theories as to exactly what makes checkerboard rules logically defective.
Ronald Dworkin, who initially coined the term, also provided a classic example: say that a legislature is trying to figure out what rule to make in respect to abortion, in a jurisdiction where abortion is highly controversial (that is, where roughly 50% of the citizens are strongly in favor of criminalizing abortion, and roughly 50% are strongly opposed to criminalizing it).
To make a law that accurately reflects and respects the extreme split of public opinion, the legislature makes the following rule ("Rule 1"): women who were born in odd-numbered years can legally obtain abortions, but women who were born in even-numbered years can't. This is a quintessential checkerboard rule.
By contrast, consider these alternative rule on abortion: ("Rule 2") abortion is illegal for all women unless the pregnancy is the result of rape; and ("Rule 3") : abortion is legal until the third trimester of the pregnancy. These rules both result in abortion being legal sometimes and illegal other times, but they are not checkerboard rules.
As Dworkin pointed out, people intuitively reject checkerboard rules. In the example of the abortion checkerboard rule above, both the pro-life and pro-choice people in the jurisdiction would say that making abortion illegal or legal depending on what year the woman was born in is a lousy law.
Most interested legal theorists agree that checkerboard laws are unacceptable, although there is some disagreement about why there are bad. Generally, there are two different theoretical explanations for why checkerboard rules are unacceptable:
1. For Dworkin, the reason is that we want our legal systems to set forth a coherent "scheme of justice." That is, legal systems operate by declaring certain things impermissible, and other things permissible. If a law takes the position that something is both acceptable and unacceptable, that law comes across as essentially disinterested in the rightness or wrongness of the behavior it regulates, and people intuitively find that "disinterested" quality offensive and unpalatable.
The upshot of Dworkin's checkerboard abortion rule is that is takes the position that abortion is both permissible and impermissible. In so doing, the law comes across as disinterested — the law doesn't really take a side about whether abortions are just or unjust. The rule makes it seem like the legislature just wanted to make a compromise on a highly controversial issue, without actually putting any thought into what the controversy was about.
By contrast, lets look at our non-checkerboard abortion rules: (2) abortion is illegal except for in cases of rape, and (3) abortion is legal until the third trimester. Both of these rules offer compromises on the issue, but the compromises aren't as disinterested-seeming as the "even-or-odd-numbered birth year" rule. In other words, rules (2) and (3) seem to take a more thoughtful position on the issue.
Thus, for Dworkin, Rule (1) lacks "integrity," but Rules (2) and (3) would not.
2. Ofer Raban has argued  that the real reason checkerboard laws are unacceptable is simply that they violate the principle of "equality before the law", by treating similarly-situated people differently. Checkerboard laws classify people into categories, and impose different requirements on the different categories, but there is no logical connection between the classification and the reason for the classification.
In Dworkin's illustrative checkerboard abortion rule, the law uses "even-numbered birth year versus odd-numbered birth year" as the trait for categorization. And the law imposes criminality, which attaches to the act of having an abortion. But being born in an odd-or-even-numbered year has absolutely nothing to do with having an abortion.
By contrast, lets look again at one of our comparator rules: abortion is illegal for all women unless the pregnancy is the result of rape. That would not be a checkerboard rule, because the distinction of "voluntary pregnancy versus pregnancy by rape" has some logical connection to abortion. Thus, this second contrasting rule would result in some women being allowed to get abortions, but others not, but it would not be a checkerboard rule.
- Law's Empire by Ronald Dworkin (1986) Belknap Press of Harvard University Press. ISBN 0674518365.
- The Rationalization of Policy: On the Relation between Democracy and the Rule of Law by Ofer Raban (2015) New York University Journal of Legislation and Public Policy 18:45-66.