The influence of ligand size on electrostatic binding to DNA in a background of competing counterions is analyzed with the planar Poisson-Boltzmann (P-B) equation, which is analytically integrated in two screening layers near the polyion surface : one sterically accessible only to the smaller counterions and the other accessible to both species. We obtain an explicit expression for the dependence of the electrostatic binding constant on the difference in counterion radii Delta delta. This dependence is approximately exponential with decay length lambda/z(2), where lambda is the thickness of the screening layer due to the smaller species and z(2) is the ligand valence. lambda is determined by the surface charge density of the polyion and the Bjerrum length, and is a few angstroms for double-helical DNA. This approach agrees well with detailed P-B and Monte Carlo calculations for mixtures of hard sphere counterions, and accounts for NMR results on the competitive binding of monovalent and divalent cations. The distribution of charges on the ligand is shown to affect the power S through which the salt concentration enters the ligand binding constant. If the counterion species have similar radii (Delta delta < lambda), their competition is well described by a point charge model, which gives the conventional S = z(2)/z(1). If the Ligand is much larger than the salt counterions (Delta delta >> lambda), it does not participate in screening at all, so S approximate to 0. An intermediate difference in counterion radii (Delta delta approximate to lambda) results in an effective ligand charge z(eff) different from z(2). If the ligand is larger than the salt counterion, then z(eff) < z(2). The opposite situation is also possible depending on ligand structure. We propose a simple method for the approximate prediction of the effective charge in competitive electrostatic binding of a ligand with distributed charges.