## Aces Up

The rules of this game can be found here.

According to my solver, Aces Up is winnable 11.57% of the time. It can actually be won more often than that, but my solver refuses to play certain nonsensical moves such as filling an empty column using a card from a column consisting of only that one card. However, 11.57% is an unrealistically high rate, because some of these wins require making odd choices which happen to work out well given the remaining cards. To obtain a more realistic win rate, I wrote a simulator mode to play the game similarly as a human might.

In Aces Up, there is limited opportunity for strategy; the only decisions happen when choosing which card to use for filling an empty column. Some sources say to choose cards in such a way as to maximize further discards, which makes sense, because you will need to discard every non-Ace by the end of the game in order to win. Other sources say you should always try to place Aces at the top of a column when possible, which also makes sense, because you will need to end up with all four Aces at the top of the columns in order to win. But what to do if you have a choice between placing an Ace, or discarding a lot of cards (but not both)? I added options to the simulator to find out.

As a rule of thumb, it appears that you should be willing to sacrifice up to three discards (net) in order to place an Ace. Depending on the situation, there are variables that might change that (for example, it makes a difference how long the columns are, or how many cards remain to be dealt). But it really doesn't make too much difference how you prioritize placing Aces (relative to maximizing discards), your win rate will be about 1-in-35 regardless.

For a variation that is easier to win, declare victory if you can get each column to contain one Ace and no other cards (in other words, how a winning game looks at the end when using standard rules, as pictured above), even if there are additional cards remaining that could be dealt. Doing this increases the odds to 8.3%, or about 1-in-12.

Win Rates for Aces Up,
100 million deals each
StrategyGames
Won
Win
Rate %
Solver (superhuman)11,570,11311.57
Maximize discards (ignoring Aces)2,755,9422.756
Always favor discards2,820,9112.821
Always place Aces when possible2,840,1152.840
Sacrifice up to three discards
in order to place an Ace
2,844,3872.844
Declare victory early8,310,5098.31

See Michael Keller's discussion of Aces Up here.

Any comments are welcome, please send to Mark Masten at:

Last modified December 10, 2021

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