Roll the dice, don't think twice

I'm a little behind on everything these days, and Scat strategy is one of them. Go back a month to review the rules.

When it is your turn, you have two strategies available:

1. Attempt to roll five 1's, thus winning the game.
2. Obtain a hand which is not likely to be the low hand for the round, thus avoiding elimination.

After your first roll, you have two opportunities to decide which dice to keep, and which dice to re-roll. This decision is a choice among three options:

1. Keep only the 1's, and re-roll all of the other dice.
2. Keep the 1's and your highest-ranking die or dice.
3. Keep the best hand you can make with the current dice.

For example, if your roll is 1-3-3-4-6, you can keep only the 1 (option 1), the 1 and the 6 (option 2), or the 1 and both 3's (option 3). There is no reason, ever, to keep the 4, or to not keep the 1.

In many cases, the decision is not even as complicated as the previous example; if the roll had been 1-3-4-6-6, or 1-3-4-5-6, options 2 and 3 would be the same. In fact, a little over 40% of the time (3126/7776, to be exact), there is only no decision to make - either you roll no 1's (3125 times), and must re-roll all five dice, or you roll five 1's (1 time), and win the game.

For those times when there is a decision to make, several additional factors should be considered:

• The current low hand. The low hand in each round is eliminated, so you will often (not always) choose the option with the best chance to beat the current low. More importantly, you will want to avoid options that have a very low probability of beating the low hand (keeping three 2's when the current low hand is four 4's, for example, is usually a bad plan).
• The number of players still to roll. Remember, you only need to beat one player each round, so if there are several players still to roll, you can choose a riskier option, based on the chance that not all of those players will beat you.
• The number of players still in the game. Without taking strategy into account, if there are n players at the beginning of the next round, your chance of winning by avoiding elimination this round is roughly 1/n.

I'll start getting into the probabilities of the game next time (and I hope that next time comes more quickly than this time did). However, I expect we'll find that, early in the game, the expectation gained by winning the game immediately will outweigh the risks of being eliminated early. In other words, rolling five 1's is the primary strategy (in my limited experience, most players make avoiding the low hand primary, and rarely go for the outright win unless they have already beaten the current low). This means that option 1 will frequently be the correct decision, because the other two options necessarily abandon the primary strategy. This will be particularly true after the first roll, because committing to any die other than a 1 restricts the ways for a hand to improve. I also expect that avoiding the low hand will become the primary strategy later in the game, when simply staying alive represents a greater increase in the probability of winning—my guess is that this happens when there are three players remaining, but we'll see.

Finally, and perhaps unfortunately, I expect we'll find that, while skillful play does increase one's expectation of winning, the overall effect won't be that great against people who do not play as well. In other words, you'll be able to win an n-player game against "normal" players more often than once every n games, but you'll likely need to play a few hundred games to see that result. Also, since each player's hand is independent of the other hands, we may find that there is a single pure strategy that maximizes one's probability of winning. In this case, the result of all players adopting that strategy makes the probability of winning an n-player game exactly 1/n. This is all conjecture for another time, though. Stay tuned.