OPTIMAL STRATEGIES FOR A TWO-RUNNER MODEL OF MIDDLE-DISTANCE RUNNING

It is common practice in middle-distance running to try to position oneself behind but within striking distance of the leader for most of the race and then overtake the leader near the finish line. One of the reasons for this is so that the runner behind can take advantage of the slipstream of the r...

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Bibliographic Details
Published in:SIAM journal on applied mathematics 2009-01, Vol.70 (4), p.1032-1046
Main Author: PITCHER, ASHLEY B.
Format: Article
Language:English
Subjects:
Air resistance
Applied mathematics
Bicycling
Boundary conditions
Control theory
Cycles
Finishes
Game theory
Mathematical models
Optimal control
Optimal strategies
Optimization
Race
Running
Slipstreams
Strategy
Studies
Trajectories
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Summary:It is common practice in middle-distance running to try to position oneself behind but within striking distance of the leader for most of the race and then overtake the leader near the finish line. One of the reasons for this is so that the runner behind can take advantage of the slipstream of the runner in front. Slipstreaming plays a huge role in cycling because of the fast speeds involved. However, despite slower speeds, middle-distance running speeds are still believed to be fast enough to make slipstreaming beneficial. In this paper, we investigate a mathematical model of a two-runner race and examine optimal strategies using optimal control theory. Assuming Runner 1 runs the race as if he or she was running alone, we determine the corresponding optimal strategy for Runner 2. We find that if Runner 2 is the stronger runner, then, instead of running at a higher constant speed (as would be the optimal strategy in the absence of another runner), Runner 2 should run at a speed close to that of Runner 1 and then use any extra energy to run faster at the end. This is to avoid giving Runner 1 the benefit of the slipstream for the entire race. If Runner 2 is the weaker runner, extra energy needs to be exerted to remain in the slipstream of the runner in front for as long as possible. In the case where Runner 2 is the stronger runner, we look at Runner 1's optimal strategy in response to the new strategy of Runner 2 and find that there is a strategy that allows Runner 1 to win the race even though Runner 2 is the better runner. We conclude that changing strategy from the solution of the one-runner optimal control problem is very risky.
ISSN:0036-1399
1095-712X
DOI:10.1137/090749384