The Lane-change Model MOBIL

Lane changes takes place, if

the potential new target lane is more attractive, i.e., the "incentive criterion" is satisfied,
and the change can be performed safely, i.e., the "safety criterion" is satisfied.

In our lane change model MOBIL, We base both criteria on the accelerations on the old and the prospective new lanes, as calculated with the longitudinal model, i.e., the IDM in our case.

Model equations

The safety criterion is satisfied, if the IDM braking deceleration acc'=acc'_IDM imposed on the back vehicle B' of the target lane after a possible change does not exceed a certain limit b_save, this means, the safety criterion

acc' (B') > - b_save

is satisfied. In this formula, the dash of the acceleration acc'=acc'_IDM stands for "after a possible change", while the dash of the back vehicle label B' stands for "on the target lane".
To asses the incentive criterion, we weight the own advantage on the target lane, measured by the increased acceleration (or reduced braking deceleration), against the disadvantage imposed to other drivers, again measured by the decrease acceleration or increased braking deceleration for these drivers. Since we tend to be egoistic, we weight the disadvantage imposed on other drivers with a politeness factor p whose values are typically less than 1, resulting in following incentive criterion:

acc' (M') - acc (M) > p [ acc (B) + acc (B') - acc' (B) - acc' (B') ] + a_thr
- As above, acc mean the actual IDM accelerations while acc' mean the accelerations after a possible change. The car labels M and M' mean "Me" before and after a possible change, respectively, while B and B' mean the back vehicle before and after a possible change, respectively.
- The own advantage is measured by "my" acceleration difference acc' (M') - acc (M) after the change, compared to the actual situation.
- The combined disadvatage to the new and old back vehicles is given by the sum [acc (B) + acc (B')] of the accelerations of both vehicles before the change, minus the acceleration sum [acc' (B) + acc' (B')] of these vehicles after the change.
- Note on implementation: In the actual version, we did not consider the disadvantage acc(B) - acc'(B), that the driver of the "old" back vehicle may suffer due to a lane change, so we used the incentive criterion
  
  acc' (M') - acc (M) > p [ acc (B') - acc' (B') ] + a_thr
  
  In fact, acc(B) - acc'(B) nearly always is negative since changing to another lane generally gives an advantage to the driver of B. This means, we ignored "pushy" people urging us to change lanes for their own benefit. This definitely is a correct strategy for drivers on the right-hand lane while the "full" MOBIL should be considered for drivers on the left-hand lane.
- To avoid lane-change maneoeuvres triggered by marginal advantages which can lead to frantic lane hopping, an additional lane-changing threshold a_thr has been added to the balance of the above equation.
- There are some situations where high-frequency lane hopping may occur triggered, e.g., when a faster vehicle follows a slower one and the leader changes due to politeness while the follower changes to move ahead faster with the result that both drivers find themselves in the same situation, and so on. Therefore, I do not allow lane changes if the actual vehicle, or one in the immediate neighbourhood, changed lanes less than vehicle.dt_lastLC or vehicle.dt_lastPassiveLC ago for active and passive lane changes, respectively.

Model Parameters and Typical Values

Parameter	Typical Value	Remarks
Politeness factor p	0 ... 0.5	For details, see below
Maximum safe deceleration b_save	4 m/s²	Must be lower than maximum deceleration of about 9 m/s²
Threshold a_thr	0.2 m/s²	Must be below the lowest acceleration ability (IDM parameter a) of any vehicle type
Bias to the right lane Delta b	0.2 m/s²	Only for European traffic rules (see below)

The lane-change model MOBIL has the following main features:

While other lane-change models typically assume purely egoistic behaviour, i.e., p=0, we can model different behaviours by varying this factor:
- p > 1 => a very altruistic behaviour.
- p in ]0, 0.5] => a realistic behaviour: Advantages of other drivers have a lower priority, but are not neglected: Notice that this feature means that yielding to "pushy" is included into MOBIL.
- p=0 => a purely selfish behaviour. Notice that also selfish drivers do not ignore the safety criterion!
- p<0 => a malicious personality who takes pleasure in thwarting other drivers even at the cost of own disadvantages. This may have some interesting game-theoretic consequences. Of course, even those mischief makers do obey the safety criterion.
A special case is given by p=1 and a_thr=0. In this case, lane changing takes place whenever the sum of the accelerations of all affected drivers increases after the change, or, equivalently, the overall decelerations are minimized. This effect gave birth to the acronym for this model:

MOBIL = Minimizing Overall Braking decelerations Induced by Lane changes

Variants of MOBIL for asymmetric traffic rules or situations

The model defined by the two equations above describes more or less symmetric lane usage where overtaking on the right is not explicitely forbidden. If traffic rules or situations are asymmetric, one needs modifications. In the following, there are some typical situations:

In many European countries, the lane usage rules are explicitely asymmetric, particularly,
- one should keep to the right,
- overtaking to the right is forbidden unless traffic is congested.
For some countries such as Thailand, Britain or Australia, swap "left" and "right". To incorporate the keep-right directive, we added to the incentive criterion an additional bias a_bias in favour of the right lane.
If there are forced lane changes such as
- entering a freeway via an onramp,
- exiting a freeway,
- lane changes as a reaction of a closing of the actual lane ahead,
the lane-usage bias introduced above can be used to treat this situation: Of course, the bias is in favour of the target lane(s):
- For on-ramp traffic, a_bias is negative (modelling a bias to the left)
- For the lane closing szenario 3, a_bias is positive for vehicles on the left lane that is about to be closed.
Implementing the overtaking rule is more difficult and requires a so-called "longitudinal-transversal coupling", see this reference (in German) for details.

References

The original publication (in German):
M. Treiber and D. Helbing, Realistische Mikrosimulation von Straßenverkehr mit einem einfachen Modell , 16. Symposium "Simulationstechnik ASIM 2002" Rostock, 10.09 -13.09.2002, edited by Djamshid Tavangarian and Rolf Grützner pp. 514--520.