Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-97-07

May 1997

Minimax Triple Rendezvous on the Line: Corrections and Extensions

Steve Alpern and Wei Shi Lim


We reconsider the problem which we introduced in an earlier paper, in which three players, placed randomly at adjacent integer positions on the line, seek to all meet together. We assume that they move with unit speed, are placed facing in random directions, and do not know the placement of the others. Prior to the game they choose their strategies together to minimize the maximum time it may take (which depends on the random initial placements) to all meet together. In our earlier paper, we proved that no strategy guarantees a triple meeting earlier than 3.5, and subsequently V. Baston has found a strategy which has 3.5 as the maximum meeting time.

Our earlier paper claimed (incorrectly) that no such strategy existed. In correction this error in the earlier paper, we are now able to continue that analysis in such a way that gives a complete solution to the problem. In doing so we derive Baston's strategy, thus showing that it is uniquely optimal. We also show that a post-meeting strategy called the up-down strategy optimizes any pre-meeting strategy.

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