Error in "The Portrait of Development Cards as a Bad Idea"

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Error in "The Portrait of Development Cards as a Bad Idea"

Postby phantom042 » Tue May 10, 2011 11:50 pm

Finally, victory points are the best thing to get. However, this is just a 16% chance. That means you would need to spend 18 resources on average (3 per development card, 6 cards at a 16% chance each) to get one victory point. Obviously, this is a far worse choice than just building those roads and settlement, or building that city.


Unfortunately, you've calculated the average resource expenditure wrong. It's actually 12 resources (4 cards) not 18 resources (6 cards) if no development cards are purchased by anyone else.

First of all, you seem to have added 16% until you got near 100%, while to calculate the AVERAGE expenditure you should calculate to near 50%.

The correct formula is slightly complicated for two reasons:
1. After each card you buy that ISN'T a victory point, the chance that the next one IS a victory point is HIGHER. 4/25 = 16%; 4/24 = 16.7%
2. You only have to buy another card (to get one victory point) if the previous one was NOT a victory point. This means that after the first card you can't just use the chance the card is a victory point -- you need to calculate conditional probability. So the chance you get a victory point on the second card is (chance first card WASN'T a victory point) * (chance second card IS a victory point).

Put these together and you get:
Card 1:
(4 / 25)
Card 2 if Card 1 wasn't a victory point:
(4 / 24) * (1 - 4 / 25)
Card 3 if neither Card 1 nor Card 2 were victory points:
(4 / 23) * (1 - 4 / 25) * (1 - 4 / 24)
Card 4 if none of the previous cards were victory points:
(4 / 22) * (1 - 4 / 25) * (1 - 4 / 24) * (1 - 4 / 23)
Card 1 + Card 2 + Card 3 + Card 4 = 52.7% chance of getting a victory point

So on average, you will have received a victory card after only 12 resources have been spent, assuming you're starting from the full deck and no one else purchases cards.

EDIT:

Another (approximate) way to think of this is to imagine if you divide the deck into four equal parts. Each part will have 6.25 cards: 1 victory point, 3 soldiers, and 2.25 other cards. In this case, since there is only 1 victory point, buying half the cards gives you a 50% chance of getting a victory point: 3.125 cards; round this up to 4.

The table below shows three things:
1. The raw probability: the chance that a given draw is a victory point if none before were;
2. The conditional probability: the chance that you draw your first victory point on a given draw;
3. The cumulative probability: the chance you draw a victory point on or before a given draw.

You can see that the 4th card purchased yields a cumulative probability of over 50%.

Code: Select all
card   raw_prob  cond_prob  cum_prob
   1    16.00%    16.00%     16.00%
   2    16.67%    14.00%     30.00%
   3    17.39%    12.17%     42.17%
   4    18.18%    10.51%     52.69%
   5    19.05%     9.01%     61.70%
   6    20.00%     7.66%     69.36%
   7    21.05%     6.45%     75.81%
   8    22.22%     5.38%     81.19%
   9    23.53%     4.43%     85.61%
  10    25.00%     3.60%     89.21%
  11    26.67%     2.88%     92.09%
  12    28.57%     2.26%     94.35%
  13    30.77%     1.74%     96.09%
  14    33.33%     1.30%     97.39%
  15    36.36%     0.95%     98.34%
  16    40.00%     0.66%     99.00%
  17    44.44%     0.44%     99.45%
  18    50.00%     0.28%     99.72%
  19    57.14%     0.16%     99.88%
  20    66.67%     0.08%     99.96%
  21    80.00%     0.03%     99.99%
  22   100.00%     0.01%    100.00%


I'd say any of the other development cards are worth at least 2 resources (even if you're not going for largest army), so spending 12 resources, and getting 3 other cards means you're really only spending (on average, from an untouched deck) 6 resources to get the victory point --- much more reasonable than the 18 you suggest it would cost in your article. It's the lost additional resources that would have resulted from investing in settlements and cities, rather than their cost per victory point, that make this strategy a loser (unless you're also going for largest army).
phantom042
 
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Re: Error in "The Portrait of Development Cards as a Bad Idea"

Postby steve » Mon May 16, 2011 3:22 pm

Thanks for the extensive feedback and for calling me out on my incorrect math. I agree with your calculations, the 50% mark is more appropriate as a benchmark to compare the resources used to get a VP card. Also, I'm extremely grateful for pointing out that every time you don't get a VP card this increases the chances of getting a VP next time, I can't believe I didn't take that into account. 12 resources does seem right as the necessary amount for a decent chance at a VP. Factoring in the bonuses from the other cards (1 resource from a knight, X from monopoly, 4 from roads, 2 from year of plenty), this does make the wheat-ore-sheep strategy more viable, though I also agree that it's not enough to make it a better strategy than brick-lumber.

Anyways, thanks so much for the feedback, I'll be updating the site shortly to reflect your calculations.
steve
 
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Re: Error in "The Portrait of Development Cards as a Bad Idea"

Postby FinalVictory » Mon May 30, 2011 5:11 pm

It seems to me that knights are being less than appreciated. You list them as un undesirable draw when not going for largest army (and admittedly they possibly are) and say that they are worth one resource, whereas you invested three to buy it. Yes, directly you only gain the one resource drawn from an opponent when gaining, but moving the thief off of your settlements can gain you resources as well. At worst, a knight is worth only one resource, but it can be worth much more.

If the thief is sitting on a hex connected to one of your cities, and the number on that hex gets rolled, you lose two resources. Conversely, if you play a knight to remove the thief from that hex, and that number is rolled before the thief is placed back on that hex, the knight has just gained you two resources. If that number is rolled twice before the thief returns, then the knight has actually gained you more resources than it originally cost. Not only can it benefit you in this way, but the thief is also hurting your opponents.

Furthermore, drawing a knight gives you additional influence in the game. You get to determine where the thief sits, which can turn the tides of the game and help you take longest road or prevent an opponent from settling in the location you are fighting for. You can also sway the decision of others when they are moving the thief by rolling a seven or playing a knight of their own. You can tell them that if they put the thief on your hex that you will just play your knight and put it back on theirs. This way, you can keep the thief off your settlements without even playing the knight.

Knowing when to play a knight is also vital. Generally you should never play a knight unless the thief is on one of your settlements. The only time this does not necessarily apply is after you have drawn an additional knight. For example, you have one knight which you have not played, you buy another development card which is also a knight. In this case, you might play your first knight offensively to target the player in the lead or someone competing directly with you. If you have played one knight already, have one knight which you have not played, and draw a third knight, it is a good idea to play your second one on that turn so that you may take largest army on the next if others are competing with you for it. you should never play your second knight until you have drawn your third, unless the thief is on you or you really need to hurt an opponent immediately.

All this said, I do believe that the occasions on which you want to buy a development card are limited. Having too many cards and nothing to trade in would be one instance where you might buy a development card. Another would be if you are trying to settle rather than build a city and you have extra wheat and sheep. Obviously if you are going for largest army, then buying development cards is the only way to do it. If you are not going for largest army, then the two prior instances are probably the only real times to purchase a development card in my opinion.
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Re: Error in "The Portrait of Development Cards as a Bad Idea"

Postby steve » Tue May 31, 2011 7:10 pm

Good points, I agree with most of what you've said. There are definitely some bonus possible values of knights in preventing you from losing resources or blocking another player's resources. There are two reasons I don't give much weight to such benefits, though:

1) Moving the robber provides more random benefits than buying something which can never be taken away such as a road. You could have the robber on you for a long time in the game and never have it block any resources. You could move it away and have the next person roll a seven and put it right back. In general I prefer going with benefits of a known value instead of relying on chance. And I think if we bothered to calculate the expected utility of moving a robber off of your spot onto someone else's spot, it would be fairly low.

2) Moving the robber onto someone else's spot tends to make people angry, especially that person. The only time I think you can safely do this is when there is a clear leader (2 VP or more ahead). If you do this early in the game you might make someone not want to trade with you for the rest of the game. The utility of friendly relations with other players is rather hazy, but in my experience, very valuable. I've played enough games where upsetting a player early in the game can have nasty consequences later. I don't mind upsetting someone in order to get something valuable like an expansion slot, but it doesn't seem worth it for the minor benefit of a robber placement.

So, while I completely agree with you as to the extra benefits of the soldier card, I still think they aren't worth the price unless you need to dump resources or are going the ore-wheat route.
steve
 
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Re: Error in "The Portrait of Development Cards as a Bad Idea"

Postby FinalVictory » Wed Jun 01, 2011 1:31 pm

I suppose I'm not too afraid of angering people in the group I tend to play with. Generally they don't hold grudges and try to play the game in a way that helps them win rather than trying to just get revenge.
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Re: Error in "The Portrait of Development Cards as a Bad Idea"

Postby steve » Wed Jun 01, 2011 4:23 pm

I need to find more friends like yours. I've gotten to the point that some of them treat it as a victory as long as I lose, they don't care about winning themselves. It makes the game much more difficult and much less fun.
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