Enriching interactions: Incorporating outcome data into static discrete games

Abstract

When modeling the behavior of firms, marketers and micro-economists routinely confront complex problems of strategic interaction. In competitive environments, firms make strategic decisions that not only depend on the features of the market, but also on their beliefs regarding the reactions of their rivals. Structurally modeling these interactions requires formulating and estimating a discrete game, a task which, until recently, was considered intractable. Fortunately, two-step estimation methods have cracked the problem, fueling a growing literature in both marketing and economics that tackles a host of issues from the optimal design of ATM networks to the choice of pricing strategy. However, most existing methods have focused on only the discrete choice of actions, ignoring a wealth of information contained in post-choice outcome data and severely limiting the scope for performing informative counterfactuals or identifying the deep structural parameters that drive strategic decisions. The goal of this paper is to provide a method for incorporating post-choice outcome data into static discrete games of incomplete information. In particular, our estimation approach adds a selection correction to the two-step games approach, allowing the researcher to use revenue data, for example, to recover the costs associated with alternative actions. Alternatively, a researcher might use R&D expenses to back out the returns to innovation.

Appendix

The probability of firm i choosing action k can be described as a function of state variables as follows (see Eq. 5 in the text):

$$ P_{i}\left( a_{i}=k|\mathbf{s}\right) =\int \int \mathbf{1}\left\{ d_{i}\left( s,\epsilon _{i}^{R},\epsilon _{i}^{C}\right) =k\right\} f\left( \epsilon _{i}^{R},\epsilon _{i}^{C}\right) d\epsilon _{i}^{R}d\epsilon _{i}^{C}. $$

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Recall as well that the revenue and cost equations are approximated as

$$ R_{i}^{k}\approx R\left( \mathbf{s},a_{i}=k,a_{-i};\theta _{R}^{k}\right) \mathbf{+}\epsilon _{i}^{R}\left( k\right) +\eta _{i}^{R}\left( k\right) , $$

$$ C_{i}^{k}\approx C\left( \mathbf{s},a_{i}=k,a_{-i};\theta _{C}^{k}\right) \mathbf{+}\epsilon _{i}^{C}\left( k\right) +\eta _{i}^{C}\left( k\right) . $$

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Define,

$$ \pi _{i}^{k} = \left[R_{i}^{k}\mathbf{+}\epsilon _{i}^{R}\left( k\right) +\eta _{i}^{R}\left( k\right) \right]-\left[C_{i}^{k}\mathbf{+}\epsilon _{i}^{C}\left( k\right) +\eta _{i}^{C}\left( k\right) \right] $$

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$$ E\left( \pi _{i}^{k}\right) = \bar{\pi}_{i}^{k}+\left[ \epsilon _{i}^{R}\left( k\right) -\epsilon _{i}^{C}\left( k\right) \right] $$

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$$ \overline{\pi }_{i}^{k} = R_{i}^{k}-C_{i}^{k} $$

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If strategy k was chosen by firm i (we ignore the market subscript in what follows) we know that

$$ E\left[ \pi _{i}^{k}\right] \geq E\left[ \pi _{i}^{k^{\prime }}\right] \text{ \ }\forall k\neq k^{\prime }. $$

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In other words

$$ \overline{\pi }_{i}^{k}+\epsilon _{i}^{R}\left( k\right) -\epsilon _{i}^{C}\left( k\right) \geq \max\limits_{k^{\prime }\neq k}\left\{ \overline{\pi } _{i}^{k^{\prime }}+\epsilon _{i}^{R}\left( k^{\prime }\right) -\epsilon _{i}^{C}\left( k^{\prime }\right) \right\} $$

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or

$$\begin{array}{rll} \epsilon _{i}^{R}\left( k\right) &\geq &\max_{k^{\prime }\neq k}\left\{ \overline{\pi }_{i}^{k^{\prime }}+\epsilon _{i}^{R}\left( k^{\prime }\right) -\epsilon _{i}^{C}\left( k^{\prime }\right) \right\} -\overline{\pi } _{i}^{k}+\epsilon _{i}^{C}\left( k\right) \\ \epsilon _{i}^{R}\left( k\right) &\geq &\Delta \tilde{\pi}_{i}^{k}+\epsilon _{i}^{C}\left( k\right) \end{array}$$

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where

$$ \Delta \tilde{\pi}_{i}^{k}=\max\limits_{k^{\prime }\neq k}\left\{ \overline{\pi } _{i}^{k^{\prime }}+\epsilon _{i}^{R}\left( k^{\prime }\right) -\epsilon _{i}^{C}\left( k^{\prime }\right) \right\} -\overline{\pi }_{i}^{k} $$

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Recalling that $\omega_{i}^{R}\left( k\right) =\epsilon _{i}^{R}\left( k\right) +\eta _{i}^{R}\left( k\right)$, it is clear that

$$ E\left( \omega_{i}^{R}\left( k\right) |\epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi}_{i}^{k}+\epsilon _{i}^{C}\left( k\right) ,\overline{ \pi }_{i}\right) \neq 0 $$

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Of course, given the independence of ϵ and η,

$$ E\left( \omega_{i}^{R}\left( k\right) |\epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi}_{i}^{k}\!+\!\epsilon _{i}^{C}\left( k\right) ,\overline{ \pi }_{i}\right) =E\left( \epsilon _{i}^{R}\left( k\right) |\epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi}_{i}^{k}+\epsilon _{i}^{C}\left( k\right) ,\overline{\pi }_{i}\right). $$

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Now, letting $g\left( \Delta \tilde{\pi}_{i}^{k}|\overline{\pi }_{i}\right)$ denote the density of $\Delta \tilde{\pi}_{i}^{k},$ this expectation can be written as

$$ \begin{array}{rll} &&\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }\int_{\Delta \tilde{\pi} _{i}^{k}+\epsilon _{i}^{C}\left( k\right) }^{\infty }\frac{\epsilon _{i}^{R}\left( k\right) f\left( \epsilon _{i}^{R}\left( k\right) ,\Delta \tilde{\pi}_{i}^{k},\epsilon _{i}^{C}\left( k\right) |\overline{\pi } _{i}\right) }{P\left( \epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi }_{i}^{k}+\epsilon _{i}^{C}\left( k\right) |\overline{\pi }_{i}\right) } d\epsilon _{i}^{R}\left( k\right) d\Delta \tilde{\pi}_{i}^{k}d\epsilon_{i}^{C}\left( k\right) \notag \\ && \;\; =\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }\int_{\Delta \tilde{\pi} _{i}^{k}+\epsilon _{i}^{C}\left( k\right) }^{\infty }\frac{\epsilon _{i}^{R}\left( k\right) f\left( \epsilon _{i}^{R}\left( k\right) ,\epsilon _{i}^{C}\left( k\right) |\overline{\pi }_{i}\right) g\left( \Delta \tilde{\pi }_{i}^{k}|\overline{\pi }_{i}\right) }{P\left( \epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi}_{i}^{k}+\epsilon _{i}^{C}\left( k\right) | \overline{\pi }_{i}\right)}\\ && \;\;\; \times\, d\epsilon _{i}^{R}\left( k\right) d\Delta \tilde{ \pi}_{i}^{k}d\epsilon _{i}^{C}\left( k\right) \end{array}$$

Since, by the i.i.d. assumption, $\epsilon _{i}^{C}\left( k\right)$ is independent of $\epsilon _{i}^{R}\left( k^{\prime }\right)$ and $\epsilon _{im}^{C}\left( k^{\prime }\right) ,$ it is easy to see that this expectation will only be a function of profit indices, $\overline{\pi } _{i}=\left\{ \overline{\pi }_{i}^{1},\overline{\pi }_{i}^{2},...,\overline{ \pi }_{i}^{K}\right\} .$

In other words,

$$ E\left( \omega _{i}^{R}\left( k\right) |\epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi}_{i}^{k}+\Delta \epsilon _{i}^{k}+\epsilon _{i}^{C}\left( k\right) ,\overline{\pi }_{i}\right) =\Lambda _{k}\left( \overline{\pi }_{i}\right) $$

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where Λ _k is some unknown function. Given the one to one correspondence between $\overline{\pi }_{i}$ and P _i this can equivalently be expressed as,

$$ E\left( \omega _{i}^{R}\left( k\right) |\epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi}_{i}^{k}+\Delta \epsilon _{i}^{k}+\epsilon _{i}^{C}\left( k\right) ,\overline{\pi }_{i}\right) =\Lambda _{k}\left( \mathbf{P}_{i}\right) $$

where we have abused notation slightly in using Λ(·) to represent both functions. The selectivity corrected regression can then be run as

$$ R_{i}^{k}\left( \mathbf{s,a};\theta _{R}^{k}\right) =R\left( \mathbf{s,a} ;\theta _{R}^{k}\right) +\Lambda _{k}\left( \widehat{\mathbf{P}}_{i}\right) + \tilde{\omega}_{i}^{R}\left( k\right) $$

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where $\Lambda _{k}\left( \mathbf{z}\right)$ is a function of the vector z, and $\widehat{\mathbf{P}}_{i}$ is a consistent estimator of P _i and $\tilde{\omega}_{i}^{R}\left( k\right)$ is a homoskedastic, mean zero error term. In practice the function Λ _k can be approximated by standard methods (polynomial series, splines, etc.)