Choice Probabilities

Role of Choice Probabilities

• Essential in modeling discrete choices.
• Cannot predict choices definitively from representative utility alone because of the unobserved component.
• Choice probabilities crucial in discrete choice models.

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General Formula for Choice Probabilities

Calculation

• Probability of choosing alternative $i$: $P_{ni}$.
• Considers the cumulative distribution of unobserved utility components.

Formula

• The probability that decision maker $n$ chooses alternative $i$ is:

$$\begin{align*} P_{ni} &= Pr(U_{ni} > U_{nj}, \forall j \neq i)\\ &=Pr(V_{ni} + \epsilon_{ni} > V_{nj} + \epsilon_{nj} , \forall j \neq i)\\ &=Pr(\epsilon_{nj} - \epsilon_{ni} < V_{ni} - V_{nj}, \forall j \neq i)\\ &=\int_{\epsilon} \mathbb{1}(\epsilon_{nj} - \epsilon_{ni} < V_{ni} - V_{nj} \quad \forall j \neq i) f(\epsilon_n) d\epsilon_n \end{align*}$$

Note: Above integral is multi-dimensional integral and $f(\epsilon_n)$ is joint density function of $\epsilon_{n1},\epsilon_{n2},...,\epsilon_{nj}$. We have to make assumptions about this joint density function.

Example from statistics

Let,

$$\begin{align*} f_{X,Y}(x,y) &= \begin{cases} 6e^{-(2x+3y)}, & x, y \geq 0 \\ 0, & \text{otherwise} \end{cases}\\\\ \text{then, } P(X > Y) &= \int_{0}^{\infty} \int_{y}^{\infty} 6e^{-(2x+3y)} \, dx \, dy \\ &= \int_{0}^{\infty} 3e^{-3y} \, dy \\ &= \frac{3}{5}. \end{align*}$$

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Choice Probabilities Example

Scenario:

• A person decides whether to take a car $(c)$ or a bus $(b)$ to work.
• Observable attributes: time $(T)$ and cost $(M)$ for each mode of transport.

Representative Utility Specification:

• Car: $V_{nc} = \beta_{0c} + \beta_1T_{nc} + \beta_2M_{nc}$.
• Bus: $V_{nb} = \beta_{0b} + \beta_1T_{nb} + \beta_2M_{nb}$.

Assuming Known Coefficients:

• If $\beta$ coefficients are known, $V_{nc}$ and $V_{nb}$ can be calculated.
• However, unobserved factors $(\epsilon_{nc}$ and $\epsilon_{nb})$ also affect the decision.

Choice Probability Calculation:

• For driving:
  • $P_{nc} = Pr(\epsilon_{nb} - \epsilon_{nc} < V_{nc} - V_{nb})$.
  • Expanded: $P_{nc} = Pr(\epsilon_{nb} - \epsilon_{nc} < (\beta_{0c} - \beta_{0b}) + \beta_1(T_{nc} - T_{nb}) + \beta_2(M_{nc} - M_{nb}))$.

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Using Choice Probabilities To Estimate Parameters

Objective:

• Estimating structural parameters of the model that describe the decision maker’s preferences and behavior.

Role of Choice Probabilities:

• Choice probabilities assist in fitting the model to observed data.
• A good fit is indicated when the choice probability for the selected alternative is close to 1, and for all other alternatives, it is close to 0.

Fitting the Model:

• Finding structural parameters that align the model's choice probabilities with observed choices.

Methodology:

• The approach depends on the assumptions made about the distribution of the unobserved utility component $(\epsilon)$.