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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.

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Notes

  1. Multiplicity of equilibria can make it difficult to construct a likelihood since, in the absence of a clear selection rule, the model is effectively incomplete. If the researcher is willing to assume that only one equilibrium is played in the data, two-step methods restore completeness by allowing the (pseudo) likelihood function to condition on the equilibrium that was in fact selected. By estimating the first stage market by market, this equilibrium restriction can be weakened to requiring only that a single equilibrium be played in each market.

  2. There is a wide and growing literature on discrete games in both economics and marketing. Notable examples include Aguirregabiria and Mira (2007), Bajari et al. (2007), Berry (1992), Pakes et al. (2007), Draganska et al. (2009), Hartmann (2010), Ho (2009), Orhun (2006), Pesendorfer and Schmidt-Dengler (2007), Sweeting (2009), Vitorino (2007), and Zhu and Singh (2009). Ellickson and Misra (2011) provide an overview of this rapidly expanding field.

  3. For example, if the researcher had access to detailed price and quantity data and chose to specify a discrete choice demand system with a “Nash in prices” supply side (e.g. Berry et al. 1995), the relevant structural errors would most naturally enter in a highly non-linear manner that could not easily be accommodated here.

  4. Note that, because we have assumed that the expectational errors do not impact firm behavior, they drop out of the relevant decision rules.

  5. The extension of selectivity-correction techniques to settings with non-separable errors is beyond the scope of this paper.

  6. Note that the observed revenues of a given store are a function of the realized actions of competing stores, not their expected actions. The expected actions of a store’s rivals impact its expected revenues and, through those, its own action.

  7. Note that this does not preclude the econometrician from employing specific distributions (e.g. extreme value errors for ϵ) since these assumptions can be imposed in the final step of our algorithm. Remaining agnostic about these errors at this point simply retains flexibility while also being fairly simple to implement.

  8. Note that in cases where the cardinality of an individual firm’s choice set is large (i.e. there are many potential discrete actions), the researcher will likely face a dimensionality issue in modeling \(\Lambda _{k}\left( \widehat{\mathbf{P}}_{i}\right) \) that is analogous to the curse of dimensionality associated with many multinomial choice problems. For example, constructing a control function via a second order polynomial approximation with J alternatives would require estimating \(J+\Sigma _{i=1}^{J}i\) terms (so with 5 alternatives, one would have to estimate 20 parameters for the selectivity correction component alone). One possible solution is to assume that Dahl’s (2002) index sufficiency assumption holds, and rely only on \(\Lambda _{k}\left( \widehat{P_{iq}^{k}} :q\subset K\right) .\) Unfortunately, this reduction in dimension is somewhat ad-hoc in that it is not based on any utility theoretic primitive and is inconsistent with many canonical examples (e.g. multinomial probit).

  9. In other words, simply plugging in revenue data for the observed choice would lead to misspecification since these data include realizations of the private information components and measurement error shocks. The latter is explicitly not in the firm’s information set when the discrete choice is being made.

  10. We note here that there is no collinearity problem per se since the beliefs are typically nonlinear transformations of expected payoffs. However, in the absence of exclusion restrictions, the identification of strategic effects is based purely on parametric assumptions on the error structure and the functional form restrictions on the payoffs.

  11. This strategy implicitly requires that the equilibria that is played be stable. Pesendorfer and Schmidt-Dengler (2010) provide an example in which the equilibria is not stable and such iteration yields inconsistent estimates.

  12. Further details are available from the authors upon request.

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Correspondence to Sanjog Misra.

Additional information

The authors would like to thank participants at the 2007 Choice Symposium at the University of Pennsylvania and the 2007 QME conference at the University of Chicago. The authors have benefitted from conversations with Patrick Bajari, Jeremy Fox, Han Hong, and Harikesh Nair and would also like to thank the editors and two anonymous referees for many helpful comments. All remaining errors are our own.

Appendix

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):

$$ P_{i}\left( a_{i}=k|\mathbf{s}\right) =\int \int \mathbf{1}\left\{ d_{i}\left( s,\epsilon _{i}^{R},\epsilon _{i}^{C}\right) =k\right\} f\left( \epsilon _{i}^{R},\epsilon _{i}^{C}\right) d\epsilon _{i}^{R}d\epsilon _{i}^{C}. $$
(14)

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

$$ R_{i}^{k}\approx R\left( \mathbf{s},a_{i}=k,a_{-i};\theta _{R}^{k}\right) \mathbf{+}\epsilon _{i}^{R}\left( k\right) +\eta _{i}^{R}\left( k\right) , $$
$$ C_{i}^{k}\approx C\left( \mathbf{s},a_{i}=k,a_{-i};\theta _{C}^{k}\right) \mathbf{+}\epsilon _{i}^{C}\left( k\right) +\eta _{i}^{C}\left( k\right) . $$
(15)

Define,

$$ \pi _{i}^{k} = \left[R_{i}^{k}\mathbf{+}\epsilon _{i}^{R}\left( k\right) +\eta _{i}^{R}\left( k\right) \right]-\left[C_{i}^{k}\mathbf{+}\epsilon _{i}^{C}\left( k\right) +\eta _{i}^{C}\left( k\right) \right] $$
(16)
$$ E\left( \pi _{i}^{k}\right) = \bar{\pi}_{i}^{k}+\left[ \epsilon _{i}^{R}\left( k\right) -\epsilon _{i}^{C}\left( k\right) \right] $$
(17)
$$ \overline{\pi }_{i}^{k} = R_{i}^{k}-C_{i}^{k} $$
(18)

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

$$ E\left[ \pi _{i}^{k}\right] \geq E\left[ \pi _{i}^{k^{\prime }}\right] \text{ \ }\forall k\neq k^{\prime }. $$
(19)

In other words

$$ \overline{\pi }_{i}^{k}+\epsilon _{i}^{R}\left( k\right) -\epsilon _{i}^{C}\left( k\right) \geq \max\limits_{k^{\prime }\neq k}\left\{ \overline{\pi } _{i}^{k^{\prime }}+\epsilon _{i}^{R}\left( k^{\prime }\right) -\epsilon _{i}^{C}\left( k^{\prime }\right) \right\} $$
(20)

or

$$\begin{array}{rll} \epsilon _{i}^{R}\left( k\right) &\geq &\max_{k^{\prime }\neq k}\left\{ \overline{\pi }_{i}^{k^{\prime }}+\epsilon _{i}^{R}\left( k^{\prime }\right) -\epsilon _{i}^{C}\left( k^{\prime }\right) \right\} -\overline{\pi } _{i}^{k}+\epsilon _{i}^{C}\left( k\right) \\ \epsilon _{i}^{R}\left( k\right) &\geq &\Delta \tilde{\pi}_{i}^{k}+\epsilon _{i}^{C}\left( k\right) \end{array}$$
(21)

where

$$ \Delta \tilde{\pi}_{i}^{k}=\max\limits_{k^{\prime }\neq k}\left\{ \overline{\pi } _{i}^{k^{\prime }}+\epsilon _{i}^{R}\left( k^{\prime }\right) -\epsilon _{i}^{C}\left( k^{\prime }\right) \right\} -\overline{\pi }_{i}^{k} $$
(22)

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

$$ E\left( \omega_{i}^{R}\left( k\right) |\epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi}_{i}^{k}+\epsilon _{i}^{C}\left( k\right) ,\overline{ \pi }_{i}\right) \neq 0 $$
(23)

Of course, given the independence of ϵ and η,

$$ E\left( \omega_{i}^{R}\left( k\right) |\epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi}_{i}^{k}\!+\!\epsilon _{i}^{C}\left( k\right) ,\overline{ \pi }_{i}\right) =E\left( \epsilon _{i}^{R}\left( k\right) |\epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi}_{i}^{k}+\epsilon _{i}^{C}\left( k\right) ,\overline{\pi }_{i}\right). $$
(24)

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,

$$ E\left( \omega _{i}^{R}\left( k\right) |\epsilon _{i}^{R}\left( k\right) \geq \Delta \tilde{\pi}_{i}^{k}+\Delta \epsilon _{i}^{k}+\epsilon _{i}^{C}\left( k\right) ,\overline{\pi }_{i}\right) =\Lambda _{k}\left( \overline{\pi }_{i}\right) $$
(25)

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

$$ R_{i}^{k}\left( \mathbf{s,a};\theta _{R}^{k}\right) =R\left( \mathbf{s,a} ;\theta _{R}^{k}\right) +\Lambda _{k}\left( \widehat{\mathbf{P}}_{i}\right) + \tilde{\omega}_{i}^{R}\left( k\right) $$
(26)

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.)

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Ellickson, P.B., Misra, S. Enriching interactions: Incorporating outcome data into static discrete games. Quant Mark Econ 10, 1–26 (2012). https://doi.org/10.1007/s11129-011-9112-5

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Keywords

  • Discrete games
  • Selection
  • Incomplete information
  • EDLP
  • Pricing strategy
  • Two step estimators

Keywords

  • C1
  • C7
  • M31
  • L81