Abstract: A counter tapered diblock copolymer composition comprises a monovinyl aromatic monomer A polymerized with a conjugated diene monomer B to form the structure A-[A/B], where the [A/B] block is a copolymer of the A and B monomers that is counter tapered, which means the ratio of B to A is lower proximal to the A block relative to the ratio of B to A distal to the A block. The [A/B] block preferably has a vinyl content that is higher proximal to the A block relative to the vinyl content distal to the A block. The diblock copolymer is preferably partially coupled or formed with a multifunctional initiator to provide a mixture of the diblock copolymer composition and a linear (A-[A/B])-X-([B/A]-A) structure and/or a coupled radial and/or multiarm structure (A-[A/B])n-X, where X is the residue of a coupling agent or an initiator. Asphalt and adhesive compositions are made with the diblock copolymer composition and with the mixture.