*What Would Lance Do?

by Simon Angus, University of New South Wales, s.angus@student.unsw.edu.au

and Horacio R. Trujillo, RAND Graduate School, trujillo@rand.org

By now, you’ve seen Coke’s new Dasani water commercial of Lance trying to take on everyone else in any sport.

…But what if you and Lance decided to show up at the SFI Computational Economics Workshop? What would be your strategy for taking on Lance and this wily pack of crazy (read computational) economists if being the leader of a pack of riders provides a big benefit to everyone in the pack and providing this benefit to others by riding as pack leader gives you a slight advantage over riding solo? To give you some help deciding what you might want to do, we’ve done what all computational economists would do – built you a model.

Model constructed in MatLab:

v Riders(Agents):

· Number of Agents: Variable (manually assigned)

· Fitness: Variable (randomly assigned or manually assigned from 0 to Race Distance)

· Objective: Reach end of race (distance) as quickly as possible (in as few stages as possible)

v Racecourse (Environment):

· Distance: Variable (manually assigned, to100 lengths, 1000 lengths, other lengths)

· Difficulty: Variable (randomly assigned per turn, values from 0.5-2.5)

v Rules of the road (Agent Behavior Rules):

· Riders move forward a specified maximum Length per turn, trying to cover as much distance as possible each turn while preserving enough fitness to finish the race. Riders lose one unit of Fitness per Length traveled.

· The Optimal Effort for each Rider is calculated each turn as the average units of Fitness/Length available to be used by the Rider for the remainder of the race (Total Remaining Fitness / Distance to Go).

· This Optimal Effort then determines the actual Optimal Speed (Lengths traveled in the turn) by inflating Effort by Course Difficulty, Drafting Advantage and Pack Leader Advantage

Speed = Effort / (Course Difficulty *Drafting Advantage * Pack Leader Advantage),

where Drafting Advantage and Pack Leader Advantage depend on position relative to other riders, and Course Conditions (as described below)

· Drafting Advantage: If a Rider is <=1 Lengths behind another, it receives the Drafting Advantage (a manually assigned, constant value 0-1), otherwise its Drafting Advantage is 1.

· Pack Leader Advantage: If a Rider is <=1 Lengths ahead of another, it receives the Pack Leader Advantage (a manually assigned, constant value 0-1), otherwise its Pack Leader Advantage is 1.

· Course Conditions: Value determined randomly at each step in range of (0.5-1.5) or manually set at a constant level, and uniformly applied to all Riders in each step

· If any Riders in the pack attempt to move ahead of others in front of it, the Rider’s success is determined probabilistically, with the Challenger (the Rider attempting to overtake the others) has the advantage. This gives Riders an additional incentive o move forward, rather than to risk being overtaken. (Initially, to establish a baseline example, we gave an absolute advantage to Challengers.) If a Rider succeeds in passing others, it moves ahead as normal. If a Rider attempts and fails to pass others, it will move to the space one Length behind the Rider just in front of it.

· If any Rider attempts to overtake the lead, the Leader is determined probabilistically, with the current Leader having the advantage. This gives Riders an additional incentive to lead, particularly towards the end of the race, rather than to risk not being able to overtake the Leader. (Initially, to establish a baseline example, we gave the Leader an absolute advantage.) If a Rider succeeds in overtaking the Leader, it moves ahead as normal. If a Rider attempts and fails to pass the Leader, it will move to the space one Length behind the Leader.

Observations:

Graphs that follow are representative only. Titles indicate `y versus x’. `Distance v Time’ plots show how the riders move through the race distance (up the screen), splitting and re-joining. Time increases left-to-right. Thus, races start at the bottom-left and finish at the top-right.

v Baseline(20 Riders; Distance = 200; Fitness = Uniform Random * Distance * (1.5,1.6);Max Speed = 5; Challenger always passes other Rider(s); Leader holds lead with80% probability; Course Conditions = 1; Pack Leader Advantage = 0.95; Drafting Advantage = 0.75). Click here to view graphs: [Baseline] [Baseline Quad]

Final Rider position highly determined by fitness initial fitness assignments

Minimal divergence of Speed and Distance until final sprints

With challenger always passing other Rider(s), these results hold even with different values for Drafting Advantage or Pack Leader Advantage

v Lower Odds of Successfully Passing in Pack (Challenger now passes each other Rider with 50% probability). Click here to view graphs: [Overtake] [Overtake Quad]

Total breakdown of fitness-determined final positions

Large divergence of Riders into many packs with considerable distances between packs.

v No Pack Leader Advantage (with Challenger still passing each other Rider with 50%probability). Click here to view graphs: [Leader] [Leader Quad]

Fitness does not determine final positions

Less divergence of Riders into fewer packs with lesser distances between packs

v Beneficial Course Conditions (Course Conditions = 0.5 (constant); with Pack Leader Advantage reset to 0.95% and Challenger still passing each other Rider with 50%probability). Click here to view graphs: [Tailwind] [Tailwind Quad]

Fitness does not determine final positions

Minimal divergence of Speed and Distance until final sprints; all Riders stay in a single close pack

Riders complete race more slowly (~150periods)

Final sprints start earlier than in Baseline case

v Harsh Course Conditions (same as previous, except Course Conditions = 1.5(constant)). Click here to view graphs: [Headwind] [Headwind Quad]

Fitness does not determine final positions

Divergence of Speed and Distance among Riders begins halfway through race and broadens through final sprints

Riders complete race more quickly (~90periods)

Significant position changes in final sprint

v Variable Course Conditions (same as previous, except Course Conditions are now determined as a random walk within range of (0.5, 1.5)). Click here to viewgraphs: [Variable Cond] [Variable Cond Quad] [Course Conditions]

Fitness does not determine final positions

Divergence of Speed and Distance among Riders is highly dependent on variability of Course Conditions – greater variability causes greater dispersion of Riders

At beginning of race, initially beneficial or harsh Course Conditions can cause race to evolve similarly to case of constant Course Conditions until variability begins to cause divergence among Riders

Further questions yet to be explored with the model:

v Does a winner lead or race in the pack? What is the range of optimal strategies for leading or staying in the pack?

v What are the critical values of the variables that cause significant changes in the race results? Why are these values critical?

v How consistent are the observations? How many steps need to be included in the model or how many runs of the model are needed to generate exceptional observations?

v How does endogenizing the strategies of players affect outcomes? – For example: model Rider strategy as a function of individual overtake probabilities of artificial adaptive agents (AAA).

v How do rider strategies change if cooperation is explicitly incorporated into the model, such as by the designation of “teammates” who support each other’s efforts to win?

v How would the inclusion of more strategic behavior rules (riders decelerating to store energy in order to be able to accelerate to and travel at maximal speeds at critical points in the course rather than always riding at their maximum speed as determined by their current fitness level)?

v What factors determine pack sizes and spacing between packs?

v And, many, many, more…

Possible applications of this type of model to social science questions:

v Technology/Innovation Development (greater investment is needed for a single company to develop new technology than for multiple companies collaborating – however, at some point, companies want to take the lead in development in order to be first to patent or to market the development)

v New Market Development (similarly, greater investment is needed for a single company to develop new geographic or product markets than for multiple companies developing new markets – however, at some point, companies want to take over and build market leadership/dominance)

v National Security Investments of Allies (nation-states can free-ride off of national security investments of allies, but will want to stay within a short “distance” of the allies making larger investments in order not to be completely out-run or unable to vie for leadership if an end-game were to arise)

v Essentially, any quasi-competitive situation in which a resource (such as knowledge, technology (broadly defined), weapons capabilities) cannot be perfectly protected and yet an advantage can be gained from maintaining a certain closeness to other players

Many thanks to all of the other at the SFI Comp Econ workshop who identified the many holes in our first model and provided us with invaluable suggestions and examples from their models to improve our own model.