Movement Mechanic Needs Work

Let each bull be either red, green, or blue.
Each turn, roll one die of each color to determine how the bulls of that color will move. For example:

1-2: Trot (move forward 3 hexes)
3-4: Charge (move forward 6 hexes)
5-6: Rush (move 1/2 the distance to the nearest player, rounded up)

(Don't take my distances literally.)

For extra flavor, make custom dice and use symbols instead of number ranges to identify movement rules.

While at first this appears to make movement too random, thereby eliminating strategy, you can use it to bias groups toward favored behaviors. For example, red bulls might tend to trot, but occasionally charge, making them "usually" safe, but very dangerous otherwise.

Not quite. The "round up" condition allows the bulls to reach their target.

A swarming algorithm is a cool idea...

M.

In this case, the bull moves 1 hex because that's half the remaining distance. If you move 0 again next turn, the bull will move 1/2 a hex, which rounds to 1, which means it catches you. This is what I meant by the rounding making it different from Zeno's paradox.

You're right that, in this case, the bull never catches the player, but this is also not Zeno's paradox. In Zeno's paradox, the period of time over which the moves occur shrinks in each successive move, but in the above example, each turn represents the same amount of time. I admit, though, that I didn't think of this case when I wrote my original reply. Good thinking on your part.

Mark

And since .5 rounds up to 1, we're both 100% right!

True, but we don't induce critical thinking serving answers on silver platters.

The apparent problem with this mechanic is that the bulls will never catch or get ahead of the player. I say "apparent" because we don't actually know if the bulls are intended to catch the players although it is sort of implied, otherwise it's just a footrace between players.

I suggested a swarm mech earlier, but after re-reading Juggler5's comment...

...I think it would be too much since there would be considerable computation for each bull. One other thing we don't know is the shape of the play area. Is it a ring with players in the center trying to reach the perimeter? Is it the neighborhood streets of Pamplona? Is it a race track?
My point being that the intended path of the players will dictate the behavior of the bulls; e.g. if the play takes place in a ring and all players move in different directions, which player(s) do the bulls chase?

As for herd movement, the suggestion you (MarkK) made is probably the most convenient. It could even be further narrowed down into a quasi-swarm mech in which one alpha bull replaces the three color coded bull types. What the one leader bull does is what all bulls do, either trot, charge, or rush. Each turn, roll for the behavior of the lead bull, move ALL bulls accordingly.
The possible downside to this is that the bulls will always maintain a static shape of the herd. Not necessarily a bad thing per se, but could create problems close to path edges or obstacles. It could occur that the movement of the alpha bull forces bulls on the edges/flank of the herd off the map or into obstacles. Solving this could be as easy as taking them out of play or moving them to another position in the herd.

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Unrelated thoughts include considerations of the name of the game. I was under the impression that a "bull rush" is a type of reed plant found in marshes. Can't call the game "Bull Run" though either lest one risk the ire of Civil War gamers feeling misled.

What is a player's incentive for doing so? If the movement is random, spending a point and drawing a card could lead to the player getting himself trampled. Why would a player take that risk?
By extension, which bulls are controlled by players? If players have a choice, why is the movement random? If the choice is random, why isn't the movement predetermined?

Also, how does this prevent stragglers? If the herd is moving all at once and at the same speed, no bulls are falling behind. If you meant player stragglers, how is that possible without those players getting trampled by bulls?