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In the Plackett-Luce model, a 'worth' value, denoted as $$\alpha(y)$$, is assigned to each possible response $$y$$. This value is defined as the exponential of a reward score $$r(x, y)$$, which is associated with generating response $$y$$ given an input $$x$$. The formula is: $$\alpha(y) = \exp(r(x, y))$$

Worth Function in Plackett-Luce Model

In the Plackett-Luce model, the probability of selecting a specific response $$\mathbf{y}$$ from a set of possible responses $$Y$$ given an input $$\mathbf{x}$$, is calculated by normalizing its "worth" value, $$\alpha(\mathbf{y})$$. The selection probability is the worth of the selected response divided by the sum of the worths of all possible responses:
$$ \Pr(\mathbf{y}\text{ is selected}|\mathbf{x},Y) = \frac{\alpha(\mathbf{y})}{\sum_{\mathbf{y}' \in Y} \alpha(\mathbf{y}')} = \frac{\exp(r(\mathbf{x},\mathbf{y}))}{\sum_{\mathbf{y}' \in Y} \exp(r(\mathbf{x},\mathbf{y}'))} $$

Plackett-Luce Selection Probability Formula

Using the principle that a response's 'worth' is the exponential of its reward score, calculate the worth value for Response B. Round your answer to two decimal places.

Calculating Response Worth for an AI Assistant

In a system that assigns a 'worth' value to a response by taking the exponential of its reward score, doubling the reward score for a response will also double its assigned worth value.

A system assigns a 'worth' value to potential text completions, calculated as the exponential of a reward score. Initially, three completions (A, B, C) have reward scores of 2.0, 3.0, and 4.0, respectively. If the reward score for *each* completion is increased by a constant value of 1.0, how does this change affect the ratio of worth between any two completions (e.g., the ratio of worth(B) to worth(A))?

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