We provide a simple model that simulates the ranking of a cycle race. Our cyclists have bounded rationality and simple cycling rules that result in interesting and realistic looking ranking dynamics. Cyclists end up clustering together into groups, with the fastest cyclists forging ahead of their competitors. Cyclists begin with randomly given stamina and energy levels, and reciprocation is assumed. A Java program of our model is available as two files ( 1) and ( 2).

I. Model and Key Assumptions:

Each agent has a comfort speed at which they can remain in a steady state, without tiring or reducing their energy stock.  All cyclists want to ride at a speed that doesn’t draw down their energy during the race but stay as far towards the front of the race as possible.  Stores of energy are desired at the end when they deplete them to sprint to the finish line.  Energy is a function of comfort speed CS, stock of energy E, and position a and try to keep dE/dt = 0.

E = f( CS, E, a)

The specifications of our model are:

• 20 Cyclists that take turns in deciding their speed asynchronously.
• Race space is flexible - range is from last cyclist to first cyclist, the finish is always far away.
• Each cyclist can decide to race at actual speeds S[t] at time t of
  1= Slow speed
  2 = Medium speed
  3 = Fast speed
  4 = Very Fast speed
  5 = Breakneck speed

• Speed is represented by each cyclist moving S spaces ahead, in their turn.
• Each agent has a randomly endowed comfort speed CS ~ {1,2,3}.  This is the speed at which a lone rider has constant energy levels.
• Each agent has an initial endowment of energy E[0] ~U[1,20]
• Energy level E[t] of each cyclist is dependent the previous energy level, how fast they are going S[t], relative to their nominal comfort speed CS, and their position in the pack at a[t] which determines their real comfort speed level.

• Along with energy values, agents use local information from the cyclists directly in front of them and behind them to make their relative speed decision: speed up, stay at the same speed or slow down.  The value of alpha represents the extra momentum given to a cyclist depending on their position relative to other cyclists. The leader uses additional information, looking behind him calculating how many cyclists n are benefiting from his draft.

Sequence of Movements

Cyclists take their turn in deciding their speed.  The order of sequence is from the front rider to the last, in each round.  Their decision choice is based on their position, energy levels and surrounding agent speed levels.

Each cyclist has the same goals:

• Loner will decide his speed first based on the speed of the agent in front of him – 2 or more spaces away. If his CS+1 is greater than or equal to the agent in front’s speed, then sprint ahead to catch up with the group, else look at the cyclist behind them.  If the loner’s CS+1 is greater than the speed of the agent behind, then they remain a loner. If they are running out of energry (E<1) they slow down by 1 each period until they can reach a group to renew their energy.
• Followers, cyclists who are behind another cyclist, have an increasing energy of 0.2 per turn, because their energy levels are given a boost from the draft.  They choose to travel at CS+1
• Leaders, that agent which is out in front, depletes their energy losing 0.5 per turn and also choose to travel at CS+1.  If the leader remains at the front of the group, he does so basing his decision on reciprocity in relation to the other free riders in his group.   He will decide to stay in front drawing down his energy by E[t]/n[t].  At this point he will decide to slow down and move only CS spaces ahead, with the cyclist behind him becoming the new leader.

Thus the simple heuristics of our model are:

We expect that agents with the same comfort speeds will group together and race at a group speed of CS+1, with cooperation taking place where each agent takes their turn at being at the front.

II. Key Results:

We run a simulation of 100 time periods. Blank lines are gaps that occur between cyclists with varying speeds.  Comfort speeds of our ranked agents are presented in table 1.  Cyclists are sorted by their race rank from left to right, with the fastest person on the far left.  Table 2 presents the changing rank of our cyclists, which are assigned numbers 1 to 20. 

Simulation 1.

Table 1. Rank of Cyclists for each t, by comfort speed

Table 2. Rank of Cyclists for each t, by actual speed

Table 3. Rank of Cyclists for each t, by ID tag

Table 4. Rank of Cyclists for each t, by energy levels

Actual Speed:

-The group speed heuristic is active for almost every rider in almost every turn since riders begin in a group; do much of their filtering still in a big pack; and are already members of their final group by the time they break away from the initial pack. This causes most riders to choose comfort_speed + 1 in almost every turn.
-The next most important heuristic in choosing speeds is that group leaders reduce their speeds when their energy level drops. This leads to people cycling from the front to the back of their groups.
-The low energy heuristics and group joining heuristics are sometimes important.
-Sometimes the group speed heuristics and the group joining heuristics lead to irrational "yo-yo" patterns where a fast leader uses the group member heuristic to move at comfort speed +1 and breaks away from a slower follower; only to drop back. An excerpt from a different simulation is presented below, taken from periods 6-10, this "yo-yo" result happened to the race leader before there was good sorting. The image for this example shows each speed decision after it has had its effect.

    4 3 3 2 4 4 3 3 4 4 3 2 _3 3 3 __2 2 2 2 2 (here 4 moved #11 to the front)
    4 _3 3 4 4 3 3 4 4 3 3 3 _3 3 3 ___2 2 2 2 2 (now 4 causes #11 to break away
    2 3 4 4 3 3 4 4 3 3 3 3 _3 3 3 ____2 2 2 2 2 (he slows down to 2 and meets 
    the group again)
    4 _4 4 3 3 4 4 3 3 3 3 3 __3 3 2 _____2 2 2 2 1 (here, 4 leads him to break 
    away; meanwhile other fast guys have also passed the slow riders)
    4 4 2 3 4 4 3 3 3 3 3 3 __3 3 3 _______2 2 2 2 1 (here, choosing 2 lands him 
    in third since the fast riders pass him when he slows down)

    11 17 16 18 7 8 14 13 3 2 19 20 _6 9 4 __15 12 10 5 1
    11 _17 16 7 8 18 14 3 2 13 19 20 _6 9 4 ___15 12 10 5 1
    11 17 7 8 16 18 3 2 14 13 19 20 _6 9 4 ____15 12 10 5 1
    11 _7 8 17 16 3 2 18 14 13 19 20 __9 4 6 _____12 10 5 1 15
    7 8 11 17 3 2 16 18 14 13 19 20 __9 4 6 _______10 5 1 15 12
  

Comfort Speed:

These heuristics lead to a general and rapid sorting by comfort speed in table 1 where the first line are the initial ordering:

After the first decisions, comfort speeds are fairly randomly distributed, where initial ordering and time period 1 are shown below:

2 2 2 2 1 2 2 2 2 2 3 3 3 2 2 2 1 3 1 3
2 2 2 2 _2 2 2 2 3 3 3 2 2 2 2 3 1 3 1 1

By the 14th time period, this run led to perfect sorting and significant clustering by comfort speed.

3 3 3 __3 _3 2 2 2 2 ____2 2 2 2 2 2 2 2 __________1 1 1

However, there can be cases when the sorting is slightly imperfect. In the example below, from a different simulation, there was a similar random order in the first turn; by the 11th time period, there was almost perfect sorting, except that an actor with a comfort speed of 2 is leading the group of people with comfort speeds of 1. An examination of the energy data shows that he has low energy. His tired of leading heuristic seems to have been activatived, which has slowed him to his comfort speed which is the same as the followers' group speed (comfort_speed +1).

2 2 1 3 3 3 2 1 1 3 2 3 1 3 2 3 3 2 3 3
3 3 3 3 ____3 3 ___3 3 3 3 2 2 ___2 2 2 ___2 1 1 1 1

Jockying for Position:

Agents become ranked by comfort speed quite early, and teamwork rotation occurs with each pack. Looking at the rank of cyclists in ID table 3 we can see a very realistic teamwok formation within each pack. If one looks at the end of our time series all three packs have the leader slowing down and falling to the back of the pack, and riders rotating through the pack to maximize group speed and energy levels. This can be seen when comparing IDs with energy from table 4 listed for each cyclist in each period. The leader of a pack slows down to his comfort speed, moves to the back of the pack and then regains speed and energy to keep with the pack benefiting from the draft created by the bike rider in front of them. As the next leader does the same, each follower moves up one position through the pack each time period.

In another simulation that we ran, we have the interesting phenomenon of the trailing pack having a cyclist with comfort speed 2 leading the pack. This allows followers behind him to stay in their same sequence and free ride off the leader. Unfortunately, for this leading cyclist, he did not catch up with the faster speed group in time.

The position of a cyclists within his pack, is more constant the smaller the pack, easily seen in comparing the groups coming second and third. A smaller group In the pack coming last cyclists change leadership every time period. In the middle pack with the much larger group of

In our simulation at line 12, we can see that a gap opens up as the leading pack with the faster speed breaks away, forming a gap between the leading pack and cyclist number 20, who has a medium comfort speed of 2. Cyclists 1 and 3 further back, have fast comfort speeds and come up through the pack to overtake the leader - cyclist 20. They must sprint to catch up with the leading pack, increasing their speeds to breakneck speed and depleting their energy.

Cyclist Energy Levels:

Cyclist lose energy as they become leaders and then gain energy as they move to the back of the feild, slowing down for one period to CS which takes them to the back of the pack and then returning to CS+1 in the pack.

Cyclist number 1 came from the very back of the pack, last in the starting line, with a randomly selected energy level of 13 units and comfort speed of fast. By the time he makes his way up to the leading pack, having to sprint between packs, his energy is now down to 3.5. Eventually he takes his turn at the lead and by the end of our simulation he has an energy of 16.1 units. If the size of our bicycle race was larger, then one being able to catch up to the front pack, which is increasingly getting away, may not be possible, e.g if cyclist number 1's energy had run out to 1 before reaching the leading pack. This happened in other simulations that we performed, and it results in the faster rider leading a slower pack, with no teamwork involved, all riders behind him can afford to be complete free riders, and attempts to break away from the pack usually lead in the decline in energy and then return to the slower pack.

III. Possible Future Directions:

• We could extend this model by adding greedy agents who choose not to lead, along with the reciprocating agents currently included.
• Cyclists with large accumulating energy levels might decide to sprint ahead and join a group riding at CS+2 for a time period until his energy stock falls below a certain level and then he will move back to join his original slower group, or ride as a loner.
• Energy levels could be made into real numbers instead of natural numbers.
• The range of comfort levels could be increased
• Distance from the leader might also be taken into account

IV. Applications to other social science processes

The two primary features of this model are variable transaction costs depending on ones relation to other agents, free riding, the provision of a public good or positive externality through intentional or unintentional cooperative behavior, and fragile alliances.  Some examples of how such a process may be applicable to social sciences include:

Innovation

• There may be excessive startup costs to innovating a new product, but once it has begun, it may generate spin offs through information and increasing skill base of workers, leading to declining costs for those that follow.

Entrepreneur e.g. Bill Gates, might have initial large stock pile of intelligence or cash, and can hop from group to group restoring energy each period, giving spin offs to others, eventually becoming the leader of the pack, and he may even decide to pull away from the pack (form a monopoly) if his comfort speed is higher than everyone elses, therefore denying his intellectual property to others, removing his public good.

Team work

• Union protests. One person has to risk the initiation of a protest and may get punished for his individual action by the employer, but if enough other workers strike or protest along with him for the same benefits, then as a group they will gain and individual won’t be punished. 

• It may be that some peoples comfort level to never cause trouble, others might be inclined to initiate trouble, and others might be indifferent, but can be persuaded to join if the costs of doing over the long run are low compared to the returns.

• Army formation, attacking an enemy on the ground, front line may be rotated back.

• Political formation where ‘comfort speed’ is endogenous.  Eg. Ralph Nader’s, political bid, leads to his own considerable transaction costs to increase his support base. But each year green or independent parties are increasing.  Ralph might ultimately use up all his energy and fall back but a new reenergized leader can take his place.