{"id":1442,"date":"2024-04-19T09:02:36","date_gmt":"2024-04-19T09:02:36","guid":{"rendered":"https:\/\/www.quantitativediscussions.com\/aloneinthemythos\/?p=1442"},"modified":"2024-04-19T09:02:39","modified_gmt":"2024-04-19T09:02:39","slug":"trading-through-a-test","status":"publish","type":"post","link":"https:\/\/www.quantitativediscussions.com\/aloneinthemythos\/2024\/04\/19\/trading-through-a-test\/","title":{"rendered":"Trading through a test"},"content":{"rendered":"\t\t
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Gambling in Arkham Horror LCG<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t
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In a previous post<\/a>, I introduced a model that allows to quantify the rate of a trade between different player resources<\/a>. In the reality of Arkham Horror, two types of trades occurs: trades with a certain outcome (such as playing Working a Hunch <\/em>to get a clue), and trades where the outcome has some randomness involved.\u00a0<\/p>

Because the resource trade rate model is sufficient to cover the first situation, this post will focus on trading for a random outcome (which can also be called gambling<\/strong>). In Arkham,\u00a0skill tests\u00a0<\/strong>are by far the most frequent gambling situations.\u00a0<\/p>

This post will build on the resource trade model to see if it can be extended to skill tests.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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When going high stakes…<\/figcaption><\/figure>
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… better be lucky!<\/figcaption><\/figure>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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The ugly mathematics<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Unfortunately, when dealing with random events, it is hard to escape probabilities. Fortunately, the concept needed to cover a skill test in Arkham are not that advanced.<\/p>

Let’s start with a player resource trade without randomness involved. In the previous post, we defined the Total Trade Rate (TTR)<\/strong>\u00a0of a trade as the sum of the resources gained minus the sum of the resources spent.<\/p>

TTR = sum(Resources Gained)-sum(Resources Spent)<\/p>

For a skill test, we introduce the Expected Total Trade Rate.\u00a0<\/strong>In probability theory, this relates to the expected value<\/strong> of a random event.<\/p>

The expected total trade rate is the sum of the resources gained pondered with the probability of gaining them, minus the resources spent, pondered with the probability of spending them.<\/p>

ETTR=sum(p_gain*resource_gained)-sum(p_spent*resource_spent).<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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Illustrative example<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Let’s immediately work through an example with one of the simplest tests: taking a basic investigate<\/b> with Intellect 3\u00a0<\/b>as the first action in the scenario\u00a0The Gathering<\/em>.\u00a0<\/strong>The starting location has Shroud 2.<\/strong><\/p>

Since it is a basic investigate, we are spending only an action, with a probability of 1 (the action is always spent). The test, if succeeded, will award us a clue. We are therefore gaining a clue with the probability P_success. The expected total trade rate for this test is:<\/p>

ETTR<\/b>=P_success*Cl-1*A<\/p>

To work out the numeric value of the trade, we need to know the probability P_success. For a 3v2 test with a standard difficulty bag in the beginning of The Gathering<\/b>, this probability is 0.625 (62.5%). We have therefore:<\/p>

ETTR<\/b>=0.625*3.5-1*2.5=-0.31R<\/b><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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Why is the expected total trade rate relevant to the model<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Now that we defined and can evaluate the expected total trade rate of a skill test, how can it be used in our Arkham model construction?<\/p>