Wikipedia:Reference desk/Archives/Mathematics/2020 October 27

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October 27

Team lottery probabilities

I play Calvinball in groups that might total 4, 5, or 6 people, with two-person teams. Teams are chosen by drawing lots, with people drawing numbers equal to the number of people present. There is always one, and sometimes two, bad Calvinball players who I do not want to be on a team with. Player 1 is on a team with player 2, same with groups of person 3 & 4 and 5 & 6. (When there are 5 players, one player sits out every round, depending on the order of their player numbers. Player 1 would still be on a team with player 2, except on rounds when one of them sits out.) Say I can manipulate who chooses numbers when, but it's otherwise fair and random. Each player announces their number as soon as they draw it. When should I draw my number? My instincts say that I should draw immediately after a bad player draws, whether there are one or two bad players. But if two good players draw first and happen to be on the same team, I think I should draw right after them. Any help? Temerarius (talk) 18:16, 27 October 2020 (UTC)

In general, in problems like this, a strategy assigns an action to each state in which you can act. In this case the actions are either Draw or Pass – except of course if you have already drawn, or only one lot is left. A state is here a (total of partial) assignments of the players to the teams. In such a strategy, and therefore also in the optimal strategy, the action is determined by the current state, and not the last change of the state. It may be that in a specific problem knowing the last state change is sufficient, but in general that is not true. Is the next player to draw a lot (other than you) selected by chance with equal probability? If not, they may also be aiming for an optimal strategy, which could result in an after you – after you livelock.  --Lambiam 23:01, 27 October 2020 (UTC)

The other players do not have an option to pass. Temerarius (talk) 03:16, 28 October 2020 (UTC)

But what determines whose turn it is to draw the next lot? The order in which the other players draw may influence the expected outcome values of the possible strategies.  --Lambiam 08:23, 28 October 2020 (UTC)

The order to draw is random, except the player "me" can choose to draw at any time. Temerarius (talk) 17:27, 28 October 2020 (UTC)

In all cases that I have examined (quite a few), in any situation in which you have a choice, your chances are independent of whether you opt to draw or pass. So I think this is true in general, and I guess the proof, although perhaps not trivial, is not deep.  --Lambiam 21:30, 28 October 2020 (UTC)

Let $s$ and $t$ be two states, where $t$ differs from $s$ in that two more players have been assigned, one of which is you. There are two ways of reaching $t$ from $s$. The first is that you opt to draw, and then another player draws. Or you pass for now, letting another player draw, and then you draw. These two paths of going from $s$ to $t$ have equal probability of being traversed. It follows that it makes no difference whether at any instance you draw, or pass now and draw on the next occasion. But if that is the case for postponing drawing for one round, it is true for postponing for any number of rounds, until you are forced to draw because you are the only player left. So all strategies are just as good; none of them is better than any other.  --Lambiam 23:45, 28 October 2020 (UTC)

Okay, say there's a thousand players. At the beginning, odds look good that I won't get paired with the bad player. But if the bottle gets toward the end, and neither I nor the bad player have drawn yet, I'm starting to sweat, right? Temerarius (talk) 07:25, 29 October 2020 (UTC)

I guess so – unless all teams already have at least one member, in which case you are safe – but sweating won't improve your chances either. Suppose an impartial judge produces each time a randomly selected pair of balls from a pair of machines dispensing lottery balls, one containing balls marked with the names of the players, the other with team numbers. If, towards the end, you and the bad player still have not been selected and some teams have no members yet, you might also get jittery, while knowing there is nothing you can do to improve your luck. If the bad player has already been drawn but has no team mate yet, it is also an occasion to start feeling uncomfortable.  --Lambiam 08:51, 29 October 2020 (UTC)