1) A single judge from those biased against you (we'll call these by the fictitious name "Democrats");
2) A single judge from those biased for you (we'll call these "Republicans");
3) A panel composed of two judges, a Democrat biased against you and a Republican for you;
4) A panel composed of three judges, one Democrat biased against and one Republican for and one judge chosen by the two judges according to their discretion.
Obviously, for the man concerned with the justice of his case, both 1 and 2 are equally problematic. One would not want either to lose his case without need, if he believed in the justice of it, nor to benefit from bias.
A fair man, wanting only justice but acknowledging the plain facts of bias, perhaps even thinking bias inevitable, might pick either 3 or 4. If an active resolution of the dispute were needed, then he would pick 4 to rule out a tie. The fourth option is, in fact, a common form of private-arbitration agreements, where the parties want equity but need a resolution of their dispute. But it is ruled out by the U.S. Constitution as a model because there is no provision for Supreme Court judges picking Supreme Court judges.
(N.B. This is not necessarily ruled out as a means of picking lower court judges under art. 2, sec. 2:
"but the Congress may by law vest the appointment of such inferior officers [including, arguably, federal judges inferior to the Supreme Court], as they think proper, in the President alone, in the courts of law, or in the heads of departments."By acts in 1891 and 1948, Congress chose not to pursue this option but to give the appointment of inferior court judges to the President with senate confirmation, although it is worth considering the possibility of placing appointment in the courts of law themselves, if we could resolve our troubles with the apparent bias of the Supreme Court as Republican or Democrat picks predominate.)
What about the third option, an even balance of Justices? It's certainly permitted by the Constitution and has occurred during three periods.
It's worth remembering that the Constitution does not set the number of Supreme Court Justices. Hamilton wanted 12 Justices; this was also the traditional number of judges on the British Court of King's Bench. The first number set, in the Judiciary Act of 1789, was six; then amended in 1801 to 5; then in 1802, 6; then 1807, 7; then in 1837, 9; in 1863, 10; in 1866, 7; then, finally in 1869 to its current number of 9. In 1937, a famous measure was proposed under Roosevelt's auspices to increase the number to 15, but it failed as the justices capitulated to his imperial demands. So, for three periods, between 1789-1801, 1802-1807 and 1863-1866, the Court had an even number of Justices.
Given the ideological split on the Court, or at least the overwhelming appearance of it to the public, perhaps those concerned with justice should urge a negotiated version of number 3, above. The "Republicans" and the "Democrats" could agree that the Court will remain composed of an even numbers of justices. They further agree to allow one another to compose their side of the bench with complete freedom, subject to constitutional restraints. Both sides give up trying to dominate the Court, and they allow their chosen champions to work on each other with argument through deliberation. (They would have to wait until Kennedy is gone to begin the deal, or provide as part of the deal that when one of the champions has turned traitor, the size of the Court can be increased to diffuse the traitor's influence. Increasing the size of the Court would also decrease the performing to the public of the Justices and encourage deliberative engagement. And, since the Court has turned into a kind of super-legislature, it would allow a more representative Court, e.g., as Scalia noted even one evangelical Christian, someone born outside of New York City or California, someone who is not a tall-building attorney, someone whose sense of justice didn't begin at Harvard or Yale Law School but in the fields of the great plains or the deserts of the west.)
The need for resolution is not as intense in the Supreme Court because lower courts stand ready to decide cases, and ties on the Supreme Court will simply allow the lower decision to stand. There would be more circuit splits, but another way of putting this is that a certain amount of judicial federalism would prevail. Obergefell for the coastal elites, but not for the Christian circuits. The circuits could even be redrawn to follow natural cultural divides. Even without a "deal," any given Senate or President can accomplish this simply by refusing appointment or confirmation beyond an even number. This may be another way of understanding what is likely to emerge naturally if the Senate and the Presidency remain ideologically split and Scalia's empty chair is not filled. Perhaps, virtue can be made of necessity.