Strategic dates

Two contestants play at exchanging dates. (They do not take the year into account.) The first must name a day in January, for example, January 14. Then each player in turn picks a later day, retaining either the month or the date just named by his adversary. For example, after January 14, it is possible to name
January 16 or January 20, April 14 or October 14. One contest might begin as follows: January 14, April 14, April 27, April 30, June 30 . . . The winner is the first to reply December 31. Which of the players is able to clinch the victory by an unbeatable strategy and how ?

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