Variable Time Step Linear Movement Approach

While we have shown that the DDF is an effective method for incorporating position into a MT-DWTA model, it is not without problems. One significant problem arises when the units of a team are spread out over a large distance. In this case the distance from the centre of all units to a particular enemy unit can vary vastly from the distance of a given unit. For example, consider the scenario shown in Figure 5.11. Here we see that Blue unit 3 is very close to Red unit 3, yet Red 3 would be regarded as the most distant by the DDF. Also, we see that the Blue units are very similar in distance to the Red Team’s centre meaning Red would likely assign a DDF value near 1 for each of the Blue units. The end result being that the Red Team’s units would be assigned as if there was no DDF.

B1 R1 R2

Centre of Red Min Dist

Figure 5.11 – Illustration of the DDF for Two Dispersed Teams

Another problem that arises when considering the DDF is the effects of planning a strategy over multiple battle steps. Clearly, the DDF is defined at step k, but it is difficult to determine what the DDF should be at step and higher. This effect becomes problematic when planning battles over multiple battle steps. While the DDF has a significant advantage over a MT-DWTA implementation that does not consider the effect of position, it is not suited to be used as the sole representation of distance when incorporating position into the MT-DWTA model. A valid implementation must not only be capable of properly penalizing targets that are far away, it must also predict where units will be after a given battle step.

1 k+

To solve these problems we introduce the Variable Time Step Linear Movement approach.

This method incorporates position, speed and weapon range using a rudimentary path planning algorithm. Consider a case where Team Blue, composed of units is attacking Team Red, composed of units. We assume that each battle step is defined over a duration of time

NB

NR ^{t k}

( )

^.

Centre of Blue

R3

B3 B2

Min Dist R4

Now say that the i^th unit on Team Blue located at lB_i

( )

k and having maximum velocity of is assigned to target the

( )

B i

ϖ j^thunit of Team Red at the k^thbattle step, with the j^thRed unit

located at. This targeting will be carried out with the i^th Blue unit’sω^th weapon having a range of ρ ω . Depending on these values, the Blue unit will behave in one of three possible B_i

( )

ways:

ith

1. If , or the case where the unit on Team Blue can reach its target in allotted time , then

( ) ( )

unit on Team Blue will move to the location of the

ith j^th unit at battle step . Upon

reaching this position, the unit on Team Blue will launch its

k

ith ω^th weapon upon the j^th

unit of Team Red.

2. If ϖB

( ) ( )

i t k <d l

(

Bi

( ) ( )

k ^,lRj k

)

≤ϖB

^{( ) ( )}

i t k +ρ_Bi

^{( )}

ω , or the case where the unit on Team Blue can not reach its target in allotted time, but can move to within weapon range of its target, then

ith

unit will move in a direct line to its target at maximum speed for the entire duration of the battle step. At the end of the battle step, the Blue unit will launch its ω^th weapon at the

like the previous case, the Blue unit will move in a direct line to the j^thunit on Team Red’s location at battle step k. The difference being that at the end of the battle step, the Blue unit will not launch any weapons as it is not within the necessary range.

Several facts should now be noted about the above variable time step linear movement approach. A battle step is divided into two distinct stages, a beginning and an ending. Movement occurs at the beginning of a battle step while sensing, weapon launches and battle damage calculation occurs at the end of a battle step. This has the curious effect of assuming that units launch weapons from their position at the end of a battle step and take damage from their position at the beginning of the same battle step. This can create problems when dealing with pursuit and evasion type scenarios as the pursuer will target the evader’s previous location while the evader won’t see any benefit from running away. However, when considering this is a TDT level reasoner, the variable time step linear movement approach does have an advantage with regards to solution stability. If a unit were to be able to change its current position, the action - reaction search would not function. Units would always move toward units that were not targeting them while moving away from units that did target them. This would have the effect of moving units out of the range of targeting units. The other side would behave similarly. The net effect would be a continuous loop of unintelligent strategies that were solely a result of modeling errors. We should also note that it is not practically to compare the DDF approach to the variable time step linear movement approach. When calculating the performance of the ULTRA algorithm, 10,000 to 20,000 runs were needed before the results stabilized enough to validate our results. However, where ULTRA is fast, capable of calculating solutions in fractions of a second, the OEP is slow, taking as much as an hour for a single run.

In document Game Theoretic Target Assignment Strategies in Competitive Multi-Team Systems (Page 126-130)