### When is it optimal to turn in cards?

Posted:

**Sun Mar 18, 2007 5:02 am**Waiting for Cards

One common decision one is presented with is whether to turn in an inferior set (i.e., not a mixed set but an all of a kind set worth 4, 6 or 8, instead of the 12 a mixed set is worth) but you have less than five cards. Should you wait for a mixed set? This all has to do with the probabilities of getting future sets (tangible) and the ?interest rate? you can get for your armies this turn (harder to quantify). Here, the interest rate on your armies is computed that would make it worth it to wait for a better set. If your armies make more than this rate, you should cash now. If they make less, you should wait for a better set. I?ll write those in a moment.

The computations going into these calculations are fairly lengthy. However, they take into account 1) forced cashes on your fourth turn in those instances you know you won?t have a mixed set, and 2) the fact that if you ?win the gamble? (i.e., when you had an inferior set, you decided to wait, and then got a mixed set after getting two more) the expected value of your cards in future turns goes down relative to if you had just cashed (I only computed expected values out five turns into the future from a situation where you?re holding three cards). For example, imagine you had three infantry cards. You wait for two more cards and (luckily) end up with a mixed set. After you cash it (on ?turn 5?), you?ll still hold two infantry cards. The probability you will get a mixed set in the next turn (?turn 6?) is now 0%. The probability on the next turn (?turn 7?) is now only 2/9ths. The probability on turn 8 is now only about 44%, versus about 62% under normal conditions.

Okay, back to some basics. If you have three cards of the same (say, three infantry), there is a 2/9ths chance that you?ll get a mixed set if you wait two turns. So the expected values of your cards under the strategy of turning them in immediately is the following. Turn 3: 4 armies. Turn 4: 0 armies. Turn 5: 0 armies. Say you wait for a mixed set. The expected values of your cards are the following. Turn 4: 0 armies. Turn 5: 1.33 (1/3rd of the time, you?ll draw another infantry, and thus cash in right then anyway). Turn 6: 4.44 (1/3rd of the time you?ve previously cashed (value of 0), 4/9ths of the time you?re stuck with your set of 4, and 2/9ths of the time you get a mixed set). These expected values in different turns are then multiplied by X, X-squared, X-cubed, etc. depending on how many turns of interest the armies will make, all out to 8 turns. These equations are then solved for a certain interest rate that will set them equal to each other. Above this amount, and you should turn in right now. Below this amount, and you?re better off waiting for another card. Here are the interest rates (as percents).

You have three cards, and a set worth 4: 18

You have three cards, and a set worth 6: 11

You have three cards, and a set worth 8: 8

You have four cards, and a set worth 4: with the fourth card giving you the chance of mixed set on your next turn: 54

You have four cards, and a set worth 6: with the fourth card giving you the chance of mixed set on your next turn: 34

You have four cards, and a set worth 8, with the fourth card giving you the chance of mixed set on your next turn: 24

What your expected interest rate on armies is varies with the situation, obviously. Armies can be used to 1) deny opponents income, 2) reduce opponent armies, 3) decrease the probability of attack and thus save armies or future income, and 4) increase income (occupying territories or taking new territories) either by boosting territory income, taking continents, or getting a card.

Your marginal interest rate on additional armies is likely to be lowest when you are building for expansion, but the time to expand has not come yet, and you perceive you are not likely to be attacked. It is probably rare for interest rates to be higher than 30%, so you should definitely wait if you have four cards and a set of infantry or cavalry. The other scenarios are harder to call, but I?m working on that.

One common decision one is presented with is whether to turn in an inferior set (i.e., not a mixed set but an all of a kind set worth 4, 6 or 8, instead of the 12 a mixed set is worth) but you have less than five cards. Should you wait for a mixed set? This all has to do with the probabilities of getting future sets (tangible) and the ?interest rate? you can get for your armies this turn (harder to quantify). Here, the interest rate on your armies is computed that would make it worth it to wait for a better set. If your armies make more than this rate, you should cash now. If they make less, you should wait for a better set. I?ll write those in a moment.

The computations going into these calculations are fairly lengthy. However, they take into account 1) forced cashes on your fourth turn in those instances you know you won?t have a mixed set, and 2) the fact that if you ?win the gamble? (i.e., when you had an inferior set, you decided to wait, and then got a mixed set after getting two more) the expected value of your cards in future turns goes down relative to if you had just cashed (I only computed expected values out five turns into the future from a situation where you?re holding three cards). For example, imagine you had three infantry cards. You wait for two more cards and (luckily) end up with a mixed set. After you cash it (on ?turn 5?), you?ll still hold two infantry cards. The probability you will get a mixed set in the next turn (?turn 6?) is now 0%. The probability on the next turn (?turn 7?) is now only 2/9ths. The probability on turn 8 is now only about 44%, versus about 62% under normal conditions.

Okay, back to some basics. If you have three cards of the same (say, three infantry), there is a 2/9ths chance that you?ll get a mixed set if you wait two turns. So the expected values of your cards under the strategy of turning them in immediately is the following. Turn 3: 4 armies. Turn 4: 0 armies. Turn 5: 0 armies. Say you wait for a mixed set. The expected values of your cards are the following. Turn 4: 0 armies. Turn 5: 1.33 (1/3rd of the time, you?ll draw another infantry, and thus cash in right then anyway). Turn 6: 4.44 (1/3rd of the time you?ve previously cashed (value of 0), 4/9ths of the time you?re stuck with your set of 4, and 2/9ths of the time you get a mixed set). These expected values in different turns are then multiplied by X, X-squared, X-cubed, etc. depending on how many turns of interest the armies will make, all out to 8 turns. These equations are then solved for a certain interest rate that will set them equal to each other. Above this amount, and you should turn in right now. Below this amount, and you?re better off waiting for another card. Here are the interest rates (as percents).

You have three cards, and a set worth 4: 18

You have three cards, and a set worth 6: 11

You have three cards, and a set worth 8: 8

You have four cards, and a set worth 4: with the fourth card giving you the chance of mixed set on your next turn: 54

You have four cards, and a set worth 6: with the fourth card giving you the chance of mixed set on your next turn: 34

You have four cards, and a set worth 8, with the fourth card giving you the chance of mixed set on your next turn: 24

What your expected interest rate on armies is varies with the situation, obviously. Armies can be used to 1) deny opponents income, 2) reduce opponent armies, 3) decrease the probability of attack and thus save armies or future income, and 4) increase income (occupying territories or taking new territories) either by boosting territory income, taking continents, or getting a card.

Your marginal interest rate on additional armies is likely to be lowest when you are building for expansion, but the time to expand has not come yet, and you perceive you are not likely to be attacked. It is probably rare for interest rates to be higher than 30%, so you should definitely wait if you have four cards and a set of infantry or cavalry. The other scenarios are harder to call, but I?m working on that.