[WIP] Roundabout clone example

[dfridovi]

A roundabout example with cloning like we discussed yesterday. The basic setup is like this:

             <=P2
           /      \
         /          \
        |              - - <=P1
         \          /
           \      /
            P3a/b=>

P1 is uncertain both about the nominal speed and the planned exit of P3, hence (a, b). Players to the rear of another player bear collision-avoidance responsibility, and P1 is "polite" to both P3a and P3b.

[lassepe]

Nice! So you essentially duplicate player 3 to consider both options in the plan for player 1? This would essentially be a continuous time QMDP approach. @zsunberg discussed a similar idea.

[dfridovi]

Yeah it's like a game theoretic version of that - hopefully pretty neat behaviors come out

[lassepe]

There might be some weird coupling going on though. In this formulation P2 and P3 may change their behavior since they know about P1's uncertainty about P3's objectives. Thus, P3a indirectly responds to decisions of P3b though they live in different realities. But the behavior emerging form this might still be very useful and interesting (and maybe P3a and P3b being coupled is even the right thing; it's just not what you would see in QMDP). I am wondering what happens if you evaluate this strategy against P2 and P3 using strategies computed from a "fully observed" version of the game.

[dfridovi]

Yeah so I was planning to do that comparison actually, like in a receding time horizon. There's interesting coupling all over the place here, so I'm very curious to see what happens there. The trouble is that to simulate things realistically I'll need some infrastructure that's really heavy. Like I'll need to simulate how beliefs would reasonably evolve over time, in addition to people (sometimes not) following their own Nash strategy from a game with ambiguity. Anyway, my own feeling is that for now, such a comparison would be pretty meaningless (without building out this extra infrastructure), so I'm more interested in seeing how well the idea is able to generate different strategies depending on the belief about whether it's P3a or P3b.

[dfridovi]

Tentative results look like this (P1 starts top right, P2 top, P3(a/b) bottom)

[dfridovi]

Added plotting overlays for problems with different initial beliefs. If we believe with 10% chance P3 is going to the top, that is shown in blue and if we believe it with 90% chance then the result is in green.