## Thursday, January 26, 2012

### Basic Strategy for Stone Age: Probability Analysis and a Downloadable Strategy Guide

Jenny (my wife) and I recently started playing Stone Age (links: BoardGameGeek, Amazon) by Rio Grande Games.  We have really enjoyed several of Rio Grande's games, including our all-time favorite, Dominion.  Stone Age is also very fun.  It revolves around strategically allocating workers ("people") to do one of several tasks such as acquiring resources, creating tools or generating more people.  The resource acquisition game mechanic is of some interest because I found I didn't quite have an intuitive feel for the underlying probability distribution.

 Source: Wikipedia
Workers can acquire one of several resources, each having a particular cost: food (2), wood (3), brick (4), stone (5), and gold (6).  The number of resources earned is determine by rolling dice, calculating the total, dividing by the cost and rounding down.  The number of dice to be rolled is equal to the number of workers used to acquire the resource.
Example: If a player decided to acquire wood with four workers, four dice would be rolled.  If the dice display 6, 5, 5, and 3 then the sum is 19.  Dividing by the cost (3) and rounding down yields 6.  Therefore, the player would acquire six wood.
The maximum number of workers that can be allocated to any given resource acquisition project is seven, so a maximum of seven dice can be rolled.  I found I did not have a strong intuitive feel for how the probability distribution of resources earned changed as price and number of workers were varied.  Obviously, using more workers or trying to acquire cheaper resources both will increase the amount of resources obtained, but by how much?  Workers are scarce, so is it worth it to commit a seventh worker to the project or will six (probably) do just fine?  How many workers do I need to commit to my wood acquisition to be at least 70% confident that I will obtain at least four wood?

As it turns out, these questions are easily resolved by performing some basic probability calculations.  Since humans are not good at rapidly estimating those parameters, I decided to generate some probability tables.  The first table shows several things for each type of resource given that 1 - 7 dice are being rolled:
• Expected values - the number of resources you would expect to gain on average if you rolled the dice many times
• The typical minimum and maximum number of resources you should expect to obtain (90% of the time)
• Efficiency - The number of resources expected per worker
• Synergy - The gain in efficiency obtained by using more workers (two workers generate more resources than either would working alone)
There are a few basic trends to take away from analysis of these tables:
1. Every resource exhibits synergy.  Assigning more workers to a given acquisition increases not only the expected number of resources obtained, but also the efficiency (expected resources per worker).
2. Higher priced resources (e.g. gold) exhibit much more synergy than lower-priced resources (e.g. food).  The synergy score for seven workers obtaining gold is 3.143 which means that those workers are 214.3% more efficient at obtaining gold than a single worker working alone.  By contrast the maximal synergy score for food ("the hunt") is only 1.143 which means that seven workers working together are only 14.3% more efficient than a single worker alone.
3. 90% of the maximum synergy is always obtained by using at least four workers.
4. The expected value of wood is approximately equal to the number of workers assigned to obtain it.  Wood is the lowest priced-resource that can be used to buy civilization cards (which provide points towards victory, among other things), making wood an extremely useful resource.  An intuitive understanding how much wood can be expected as a function of workers assigned to the project is really useful.
The second table provides probabilities of obtaining at least N resources given D dice are rolled.  It is designed to answer questions of the type "How many workers do I need to commit to be 75% sure that I'll get at least 4 bricks?"
In this example, we look at the brick table in the row for "At least 4...".  This reveals that rolling five dice yields at 69% probability of obtaining at least 4 bricks and rolling six dice yields a 90% probability.  Thus, if we need to be more than 75% sure that we'll obtain 4 bricks, six workers will need to be committed to the acquisition.
This table provides some insights of strategic importance and it is practically useful during gameplay.
For example, during the endgame, gold is a strategically important resource.  However, the advantage of using seven workers instead of using six workers is not that significant.  In both cases, obtaining at least 2 gold is virtually assured and obtaining the 3rd gold occurs at high probability (94% vs 79%).  The primary tradeoff is the potential to obtain a fourth gold (59% vs 28%).  Thus, the advantage of using the seventh worker is to increase the probability of obtaining a fourth gold from low-probability to moderate-probability.

The in-game resource acquisition mechanic is slightly more complicated than I have described here (the use of "tools" slightly alters the probability distributions), but I think the tables provided are enough to give players an intuitive feel for how the resource acquisition probability distributions behave.

If you have a willing opponent, I strongly encourage you to play a few rounds with the tables in front of you.  I found that I used them a fair bit, but then started to get more intuition about how many resources I could expect to obtain with a given number of workers assigned.  Otherwise, I hope that at least some of the insights I've pointed out will help improve your strategy.  Obviously, there are many other strategic elements of the game, but having a more intuitive understanding of the value of your workers in their role as resource producers can help you make better tactical choices.

Standard disclaimers and statements
I am not affiliated Rio Grande Games, Hans-im-Gluck Verlags-GmbH or any of the retailers mentioned above. I have not accepted compensation in any form from any organization in support of this project. Use this information at your own risk. I am not responsible for any adverse effects your use of this information causes.

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