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River achieved a career-high singles ranking of World No. 57 in 1988.
During his career Rive finished runner-up in 4 doubles events. His highest doubles ranking was World No. 48, reached in 1989.
Doubles runner-ups (4)
|Outcome||No.||Date||Tournament||Surface||Partner||Opponents in the final||Score in the final|
|Runner-up||1.||1986||Båstad, Sweden||Clay||Craig Campbell|| Sergio Casal
|Runner-up||2.||1988||Sydney Outdoor, Australia||Grass||Bud Schultz|| Darren Cahill
|Runner-up||3.||1989||Johannesburg, South Africa||Hard (i)||Kelly Jones|| Luke Jensen
|Runner-up||4.||1990||Hong Kong||Hard||Kevin Curren|| Pat Cash