Staff icebreaker - is stasis ever attained?

Solution 1:

Doesn't this game converge to a point, assuming smallest possible equidistance? Assume that it doesn't. Instead it converges to some equilibrium position L. Then there is a unique convex polygon joining the points on the outside of L. But on the next step , each of these points on the outside must move inside L, hence L cannot be an equilibrium.

I would add that I think the step size rule should be that a point always moves a percentage of the distance to the target midpoint, to make the process more continuous.

Solution 2:

Following the OP's explanation in the comments, I would like to add some considerations, leaving for clarity's sake my initial clarification requests at the end.

Firsty, I believe the algorithm used for the simulations might introduce some complications compared to the description given, as the distance step taken by each point at each time step is bounded by $1$.

I tried to get a simplified, more tractable description, by modelling the problem as a system of differential equations. Denoting the coordinates $(x,y)$ of the $i$-th player as $p_i$, and its chosen pair $p_k$ and $p_j$, an ODE such $$ \dot{p_i} = \gamma \bigg(p_k - p_i + \frac{p_j - p_k}{2} \bigg)$$ could be enforced for each $i = 1,2 \dots n$ (where $n$ is the total number of players) translating the condition that one player instaneously directs itself to the middle point between the two chosen neighbours.

Setting time derivatives to zero one verifies that the trivial solution whereby all the players collapse to a point exists.

As a matter of fact, it seems intuitive to conjecture even that $\max_{i,j} {\vert p_i - p_j \vert}$ is exponentially vanishing: so the "collapsed" configuration will be reached in exponential time, and the players will progressively occupy an ever smaller region of the playing ground.

For sure a mathematical description better than the one hereinattempted is needed, allowing the solution whereby the $n$ players sit on the vertices of an $n$-gon (compatibl with the choices made for the neighbours): the main problem I have is how to define distances to allow for such case.

Previous clarification request: I just have some questions to ensure I understood this interesting setting well.

The main problem is, will the equilibrium position attained, starting from any configuration of players around a circle and any choice of equidistant neighbours made by any player?

Firstly, I believe there are some choices the players could make, which cannot yield an equilibrium configuration. For example in the case say $n = 5$, if the player $p_4$ wants to stay between players $p_3$ and $p_5$, and the player $p_3$ wants to stay between $p_4$ and $p_5$, no equilibrium configuration can be attained (if I understood your comment on shortest possible equidistance).

Let us assume for a moment that we can characterize all “compatible” choices, i.e. initial choices of equidistant neighbours made by each player such that an equilibrium configuration exists.

Then I thought, we could see if such configuration is attained by reversing the game, i.e. starting from the equilibrium configuration and going backwards in time: if we verify that any starting configuration is attainable (something akin “ergodicity”), some progress would be made.

At this point I notice I am unsure on what rules you implemented for your game and your (nice) plots.

At each time step, how is the motion of the $p_i$ player defined? Let me clarify with an example.

At the $k$-th time step, the configuration reads $ 1, 3, 4, 2, 5$ (clockwise sense). Let us assume the $p_5$ player has chosen $p_3$ and $p_2$ as desired neighbours.

In your simulation, where will $p_5$ go? He can move between $p_3$ and $p_4$, or between $p_4$ and $p_2$. Also, do you decide the movements of each $p_i$ player at the beginning of the $k$-th time step, given the configuration attained at the $k-1$-th time step, or do you move $p_1$, and then decide the moves of the other players based on the updated configuration (after $p_1$ has moved)?

I am far from confident I can contribute much to this problem, but it is fun to think about. Once the dynamics you use is clear, it would be interesting to check if the reverse dynamics can be somehow characterized, starting from the equilibrium position.