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Consider an $n$-player infinitely repeated game.

First stage nature chooses for each player, $i$, a radius $r_{i}$.

For each later stage $t$ each player $i$:

  1. The payer chooses a "target" $p_{i, t}$ from the lattice $\mathbb{Z}^{n}$. It must of length at most $t$ from the origin using the taxicab metric.
  2. Nature chooses a "shot" $s_{i, t}$ from the uniform random variable over a ball of radius $r_{i}$ centred at the target, $B(p_{i, t}, r_{i})$.
  3. The player chooses whether to commit to this shot or commit to any shot they've made before.

At the end the "winning target" is the target closest to the centre of all the players committed shots.

The scores are then given by the components of the winning target: If player $i$ won round $t$ then player $j$ has total score $p_{i, t}[j]$ at that point in time.

Each player has knowledge of only the other players previous targets and committed shots.

  • Is this isomorphic to any existing game? If not, can one derive optimal strategies?
  • Is this related to the multiplayer war of attrition?
  • I can imagine the player with the smallest $r_{i}$ have a strategy to stay ahead indefinitely? What do the normalized scores tend to (as a function of the $r_{i}$'s) if every player plays optimally?
  • Do the dynamics change considerably if the underlying space is $\mathbb{R}^{n}$ with Euclidean distance?
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  • $\begingroup$ Wait, players don't get points for choosing the winning target, only for having the winning target have large component in their direction, right? So then how does having smaller radius (i.e. better aim) help the players? $\endgroup$ – Vincent Dec 12 '18 at 12:59
  • $\begingroup$ With better aim you can have more precise control over where your shot lands, where your shot lands controls where the centre is. $\endgroup$ – Harry Barber Dec 12 '18 at 14:30
  • $\begingroup$ For example, if everyone had an incredibly large radius they have a higher chance of pushing the centre further into an unfavorable direction. Whereas, if a player had radius 0 (limiting case) then, if he was able to pre-empt other players centre, he could place his shot at: predicted centre + vector pointing in his direction. $\endgroup$ – Harry Barber Dec 12 '18 at 14:35

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