The Geometry and the Game Theory of

ChasesCharlene S. Ahn

Edward BoasBenjamin Rahn

Harvard UniversityPresented by: Alonzo Genelin

Velociraptor vs. Thescelosaur• Givens:• Vv = 60 km/hr = 16.7 m/s (Velociraptor speed)

• Velociraptor can maintain this speed for a time T=15 seconds• Vt = 50 km/hr = 13.9 m/s (Thescelosaur speed)

• Thescelosaur can maintain this speed for a comparatively infinite amount of time• Rv = 1.5 m (Velociraptor turn radius = 3x hip height of .5 m)

• Assume maintaining top speed with this turn radius is most beneficial for both • Rt = .5 m (Thescelosaur turn radius)

• Assume it can make 180 in roughly .1 s (average human reaction time)• Assume both dinosaurs reaction time is somewhere between .005 s and .05 s• Thescelosaur detects Velociraptor at distance 15 m < D < 50 m

• Assume that D is not dependent on angle of approach from Velociraptor

• Additional Assumptions:• Velociraptor must rest at least T(vv – vt )/vt = 3 s after 15 s chase; implying the

thescelosaur will gain more distance from the velociraptor than the velociraptor will be able to close in an additional 15 s chase • δv =.6 m (anything within a circle of radius .6 m will be caught by the

velociraptor) • δt = .2 m (thescelosaur is represented as circle of radius .2 m, as it is .4 m wide

at the torso)• If these two circles come into contact, the thescelosaur is caught

One Velociraptor• Thescelosaur needs to elude velociraptor for (T-t) s, or however much

endurance velociraptor has left • 1st case (Thescelosaur notices Velociraptor from > 42m):

• Distance from Thescelosaur to Velociraptor = d > (vv – vt )(T-t)• Thescelosaur can run directly away for remainder of the chase

• 2nd case (Thescelosaur notices Velociraptor from < 42m):• Distance from Thescelosaur to Velociraptor = d < (vv – vt )(T-t)

• Thescelosaur will get caught if it runs directly away for length of chase

Encounter Strategy A• When Velociraptor is at a

distance k, the Thescelosaur can use smaller turn radius to get out of way

Encounter Strategy B• At a distance l>k, turn all the way

around and approach Velociraptor head on, and at a carefully chosen distance of m, dodge out of the way of the Velociraptor

Endgame Strategy A• If d > (vv – vt )(T-t), the Thescelosaur will have enough distance to

evade the Velociraptor for the remainder of the chase and can simply run away

Endgame Strategy B• If d < (vv – vt )(T-t), instead of running, use superior agility to get

directly behind Velociraptor • Due to Velociraptor’s superior speed, the thescelosaur will not be

able to do this for the entirety of the chase and at some point will have to resort to Endgame Strategy A

Velociraptor Metric

• Due to turning radius constraints, Velociraptor cannot necessarily travel in straight lines

Velociraptor Metric• Figure 4 shows a density plot when we assume the velociraptor is a

point• Figure 5 shows a density plot when we take into account the

velociraptor’s grabbing radius (minimizing distance needed to travel)

• In both, the darker areas are the shortest distances for the velociraptor to travel

• In both we assume the velociraptor is facing the positive y direction

Velociraptor Metric• The origin in the density plots represents the starting position of the

velociraptor’s center of gravity, (.6 m behind the center of grabbing radius)• Using the figure below and treating the velociraptor as a circle, we will

get a turning radius of 1.6 m and a grabbing radius of .8 m (δv + δt )

Computer Simulated Hunts• In their first simulation, the

thescelosaur survives using Encounter Strategy A

Encounter Strategy A Simulation Conclusions

• Thescelosaur: Metric 2 best• Metric 2 the thescelosaur knows to avoid the catching radius of the

velociraptor whereas with Metric 1, it just avoids the center of it

• Velociraptor: Metric 1 best• Using metric 2 the velociraptor consistently came up just short of catching the

thescelosaur

• Using given speeds and turning radii:• Thescelosaur survives if grabbing radius, δ<.4• Thescelosaur is caught if grabbing radius is, δ>.5• Therefore, Encounter Strategy A is useless as δ=.8

Encounter Strategy B Simulation

Encounter Strategy B Simulation Conclusions

• If the thescelosaur dodges out of the way at a distance of exactly 2.15 meters, it will not be caught by a .6 m grabbing radius• After the first escape, there will be a distance of 7 m, and after it

makes a full 180, the velociraptor will gain 1.9 m, and the thescelosaur will have ample distance to perform the strategy again• Therefore with this encounter strategy, the thescelosaur will always

escape

Velociraptor Anticipation• If the thescelosaur can consistently implement Encounter Strategy B

successfully, the velociraptor must then guess which way the thescelosaur will dodge• Conversely, the thescelosaur can anticipate the velociraptor’s

anticipation, and can dodge either left or right, or even stay going straight if the velociraptor will move either left or right• Each dinosaur must then choose which of the three directions to

choose in a life-death sort of version of rock-paper-scissors

Thescelosaur Evasion Payoff• If velociraptor chooses correctly, it catches the thescelosaur, resulting in

a payoff of 1 for the velociraptor and a payoff of 0 for the thescelosaur• If one dinosaur swerves either direction and the other stays straight,

the thescelosaur has a payoff of p, and the velociraptor, 1-p• In this case, it is a small miss and there will be a shorter distance between the

two than in the next case, resulting in potentially more encounters

• If the thescelosaur swerves one way and the velociraptor anticipates the opposite direction, the thescelosaur has a payoff q, and the velociraptor, 1-q• In this case it is a larger miss for the velociraptor and there will be a greater

distance between the two, resulting in fewer encounters

Game Theory• Let a= probability velociraptor goes L• Let b= probability thescelosaur goes L• The probability of going L=probability of going R, therefore:• Probability of C are 1-2a and 1-2b respectively

Game Theory• Each dinosaur wants to maximize its payoffs, and minimize its

opponent’s• This occurs when expected payoff of its opponent is equal for all

options• Let Pt (V|L) be expected payoff if thescelosaur chooses left

Results• Then Pt (V|L) = Pt (V|R) = (1-2b)q+bp and • Pt (V|C) = 2qb• And Pv (V|L) = Pv (V|R) =a+(1-q)(1-2a)+a(1-p) and • Pv (V|C) = 2a(1-q)+(1-2a)• Setting Thescelosaur payoffs equall and Velociraptor payoffs equal, we

find that a=b=q/(4q-p)

Conclusion• To find a and b we must find p and q• If there is only enough time for one encounter, p=q=1 and a=b=1/3• Thescelosaur survives 2/3 of the time

• Now assume if there is a small miss there is time for another encounter, but if it is a large miss there is not• Then p=1 and q=2/3 and a=b=2/5• Thescelosaur survives 3/5 of the time