Estimating discrete-choice games of incomplete information: Simple static examples

Abstract

We investigate the computational aspect of estimating discrete-choice games under incomplete information. In these games, multiple equilibria can exist. Also, different values of structural parameters can result in different numbers of equilibria. Consequently, under maximum-likelihood estimation, the likelihood function is a discontinuous function of the structural parameters. We reformulate the maximum-likelihood estimation problem as a constrained optimization problem in the joint space of structural parameters and economic endogenous variables. Under this formulation, the objective function and structural equations are smooth functions. The constrained optimization approach does not require repeatedly solving the game or finding all the equilibria. We use two static-game models to demonstrate this approach, conducting Monte Carlo experiments to evaluate the finite-sample performance of the maximum-likelihood estimator, two-step estimators, and the nested pseudo-likelihood estimator.

Appendices

Appendix A: The constrained optimization formulation for the ML estimator under general equilibrium selection mechanism

In this appendix, we derive the constrained optimization formulation of the maximum likelihood estimation problem under a general equilibrium selection mechanism. To simplify the notation, we only consider the case of one market and drop the superscript m for market index in the notation used in Section 3.

Given the parameter vector 𝜃 and firms’ observed types x, let ${\mathcal E}(\boldsymbol {\theta }, \boldsymbol {x}) = \{ \bar {\boldsymbol {p}}_{k}(\boldsymbol {\theta }) \}_{k}$ denote the set of BN equilibria, where $\bar {\boldsymbol {p}}_{k}(\boldsymbol {\theta }) = (\bar {p}_{ka}(\boldsymbol {\theta }), \bar {p}_{kb}(\boldsymbol {\theta }))$ is the k-th equilibrium that solves the BN equilibrium (4). For the constrained optimization approach, given 𝜃, an equilibrium selection mechanism $\boldsymbol {\lambda } = \{\lambda _{k}\}_{k=1}^{|{\mathcal E}(\boldsymbol {\theta }, \boldsymbol {x})|}$ and any given set of probabilities $\{\boldsymbol {p}_{k} = (p_{ka}, p_{kb})\}_{k=1}^{|{\mathcal E}(\boldsymbol {\theta }, \boldsymbol {x})|}$ (which do not need to be the equilibrium probabilities that solve the BN equilibrium equation), we define the augmented logarithm of the likelihood function of observing the decisions $\boldsymbol {y} = ({y_{a}^{t}}, {y_{b}^{t}})_{t=1}^{T}$ as

$$\mathfrak{L} \left[ \boldsymbol{\theta}, \boldsymbol{\lambda}, \{\boldsymbol{p}_{k}\}_{k} \right] \, =\, \displaystyle \sum\limits_{t=1}^{T}\! \log \!\left\{\! \sum\limits_{k=1}^{|{\mathcal E}(\boldsymbol{\theta}, \boldsymbol{x})|} \!\lambda_{k} \left[\! (p_{ka})^{{y_{a}^{t}}} \!\right] \!\left[\! \left(1 \, -\, p_{ka} \right)^{1\, -\, {y_{a}^{t}}} \!\right]\! \left[\! (p_{kb})^{{y_{b}^{t}}} \!\right]\! \left[\! \left(1 \, -\, p_{kb}\right)^{1\, -\, {y_{b}^{t}}} \!\right] \!\right\}\!.$$

(26)

The constrained optimization formulation of the ML estimator is then defined as:

$$\begin{array}{ll} \displaystyle \mathop{\text{maximize}}_{\left\{ \boldsymbol{\theta}, \boldsymbol{\lambda}, \{\boldsymbol{p}_{k}\}_{k} \right\} } & \mathfrak{L} \left[ \boldsymbol{\theta}, \boldsymbol{\lambda}, \{\boldsymbol{p}_{k}\}_{k} \right] \\ \mbox{s.t.} & \displaystyle \sum\limits_{k=1}^{|{\mathcal E}(\boldsymbol{\theta}, \boldsymbol{x})| } \lambda_{k} = 1, \\ & \boldsymbol{p}_{k} = \boldsymbol{\Psi}( \boldsymbol{p}_{k}, \boldsymbol{x}; \boldsymbol{\theta}), \quad k = 1, \ldots, |{\mathcal E}(\boldsymbol{\theta}, \boldsymbol{x})|. \end{array}$$

(27)

Notice the constrained optimization problem formulated above is not computable because the number of constraints, $|{\mathcal E}(\boldsymbol {\theta }, \boldsymbol {x})|$, in Eq. 27 depends on the values of decision variables 𝜃. Nonetheless, we can characterize the equivalence in objective value of the two problems (10) and (27) in the following proposition.

Proposition 2

Let $(\bar {\boldsymbol {\theta }}, \bar {\boldsymbol {\lambda }}(\bar {\boldsymbol {\theta }}))$ be a solution of the ML estimation problem defined in Eq. 10 with corresponding equilibrium probabilities $\{\bar {\boldsymbol {p}}_{k}(\bar {\boldsymbol {\theta }})\}_{k=1}^{|{\mathcal E}(\bar {\boldsymbol {\theta }}, \boldsymbol {x})|}$. Let $(\boldsymbol {\theta }^{*}, \boldsymbol {\lambda }^{*}, \{\boldsymbol {p}^{*}_{k}\}_{k=1}^{|{\mathcal E}(\theta ^{*}, \boldsymbol {x})|})$ be a solution of the constrained optimization problem (27). Then $\mathbb {L}[ \bar {\boldsymbol {\theta }}, \bar {\boldsymbol {\lambda }}(\bar {\boldsymbol {\theta }}), \{ \bar {\boldsymbol {p}}_{k}(\bar {\boldsymbol {\theta }}) \}_{k}]=\mathfrak {L} \left [ \boldsymbol {\theta }^{*}, \boldsymbol {\lambda }^{*}, \{\boldsymbol {p}^{*}_{k}\}_{k} \right ]$. If the model is identified, then $\bar {\boldsymbol {\theta }} = \boldsymbol {\theta }^{*}$.

Proof

By definition, $\mathbb {L}[ \bar {\boldsymbol {\theta }}, \bar {\boldsymbol {\lambda }}(\bar {\boldsymbol {\theta }}), \{ \bar {\boldsymbol {p}}_{k}(\bar {\boldsymbol {\theta }}) \}_{k}] \geq \mathbb {L}[ \boldsymbol {\theta }, \boldsymbol {\lambda }(\boldsymbol {\theta }), \{ \bar {\boldsymbol {p}}_{k}(\boldsymbol {\theta }) \}_{k}]$ for any given 𝜃, λ(𝜃) and $\{ \bar {\boldsymbol {p}}_{k}(\boldsymbol {\theta })\}_{k}$ that satisfy the BN equilibrium equation p = Ψ(p, x, 𝜃). Since the pair $(\boldsymbol {\theta }^{*}, \{\boldsymbol {p}^{*}_{k}\}_{k=1}^{|{\mathcal E}(\theta ^{*}, \boldsymbol {x})|})$ satisfies the BN equation as the constraints in Eq. 27, it follows that

$$\mathbb{L}[ \bar{\boldsymbol{\theta}}, \bar{\boldsymbol{\lambda}}(\bar{\boldsymbol{\theta}}), \{ \bar{\boldsymbol{p}}_{k}(\bar{\boldsymbol{\theta}}) \}_{k}]] \geq \mathbb{L} \left[ \boldsymbol{\theta}^{*}, \boldsymbol{\lambda}^{*}, \{\boldsymbol{p}^{*}_{k}\}_{k} \right] = \mathfrak{L} \left[ \boldsymbol{\theta}^{*}, \boldsymbol{\lambda}^{*}, \{\boldsymbol{p}^{*}_{k}\}_{k} \right].$$

Conversely, since the tuple $(\bar {\boldsymbol {\theta }}, \bar {\boldsymbol {\lambda }}(\bar {\boldsymbol {\theta }}), \{\bar {\boldsymbol {p}}_{k}(\bar {\boldsymbol {\theta }})\}_{k=1}^{|{\mathcal E}(\bar {\boldsymbol {\theta }}, \boldsymbol {x})|})$ satisfies the constraints in Eq. 27, we have

$$\mathfrak{L} \left[ \boldsymbol{\theta}^{*}, \boldsymbol{\lambda}^{*}, \{\boldsymbol{p}^{*}_{k}\}_{k} \right] \geq \mathfrak{L}[ \bar{\boldsymbol{\theta}}, \bar{\boldsymbol{\lambda}}(\bar{\boldsymbol{\theta}}), \{ \bar{\boldsymbol{p}}_{k}(\bar{\boldsymbol{\theta}}) \}_{k}] = \mathbb{L}[ \bar{\boldsymbol{\theta}}, \bar{\boldsymbol{\lambda}}(\bar{\boldsymbol{\theta}}), \{ \bar{\boldsymbol{p}}_{k}(\bar{\boldsymbol{\theta}}) \}_{k}].$$

If the model is identified, the solution is unique, so $\bar {\boldsymbol {\theta }} = \boldsymbol {\theta }^{*}$.

Assuming only one equilibrium is played in the data, only one component in λ is 1 and all the others are 0. Then the constrained optimization problem (27) is equivalent to

$$\begin{array}{ll} \displaystyle \mathop{\text{maximize}}_{\left\{ \boldsymbol{\theta}, \{\boldsymbol{p}_{k}\}_{k} \right\} } & \displaystyle \max_{k = 1, \ldots, |{\mathcal E}(\boldsymbol{\theta}, \boldsymbol{x})|} \left[ \sum\limits_{t=1}^{T} \log \left\{ (p_{ka})^{{y_{a}^{t}}} \left(1 - p_{ka} \right)^{1- {y_{a}^{t}}} (p_{kb})^{{y_{b}^{t}}} \left(1 - p_{kb}\right)^{1- {y_{b}^{t}}} \right\} \right] \\ \mbox{s.t.} & \boldsymbol{p}_{k} = \boldsymbol{\Psi}( \boldsymbol{p}_{k}, \boldsymbol{x}; \boldsymbol{\theta}), \quad k = 1, \ldots, |{\mathcal E}(\boldsymbol{\theta}, \boldsymbol{x})|. \end{array}$$

(28)

Notice that the constrained optimization problem formulated in Eq. 28 is also not computable because its size (the number of constraints, $|{\mathcal E}(\boldsymbol {\theta }, \boldsymbol {x})|$ and decision variables {p _k } _k ) depend on the values of decision variables 𝜃; it also involves finding all the equilibrium solutions $\{ {\boldsymbol {p}_{k}} \}_{k =1}^{|{\mathcal E}(\boldsymbol {\theta }, \boldsymbol {x})|}$ at any given 𝜃. We present a computationally tractable reformulation of the problem (28) as follows:

$$\begin{array}{ll} \displaystyle \mathop{\text{maximize}}_{\left\{ \boldsymbol{\theta}, \boldsymbol{p} \right\} } & \displaystyle \sum\limits_{t=1}^{T} \log \left\{ (p_{a})^{{y_{a}^{t}}} \left(1 - p_{a} \right)^{1- {y_{a}^{t}}} (p_{b})^{{y_{b}^{t}}} \left(1 - p_{b}\right)^{1- {y_{b}^{t}}} \right\} \\ \mbox{s.t.} & \boldsymbol{p} = \boldsymbol{\Psi}( \boldsymbol{p}, \boldsymbol{x}; \boldsymbol{\theta}). \end{array}$$

(29)

Notice that the size (number of decision variables and constraints) of the problem (29) is fixed and does not depend on the value of 𝜃. Moreover, we do not need to find all equilibria at any given 𝜃; only one equilibrium p is needed since only one equilibrium constraint is imposed. The equivalence in the objective value and solutions between the problems (28) and (29) is stated below. We omit the proof since it follows similar arguments used in proving Proposition 2.

Proposition 3

Let $\left ( \boldsymbol {\theta }^{*}, \{\boldsymbol {p}^{*}_{k}\}_{k=1}^{|{\mathcal E}(\boldsymbol {\theta }^{*}, \boldsymbol {x})|} \right )$ denote the solution to the optimization problem (28) with $\boldsymbol {p}^{*}_{\bar {k}} \in \{\boldsymbol {p}^{*}_{k}\}_{k}$ being the equilibrium that maximizes the objective function, i.e.,

$$\bar{k} = \displaystyle \mathop{\text{argmax}}_{k = 1, \ldots, |{\mathcal E}(\boldsymbol{\theta}, \boldsymbol{x})|} \left[ \sum\limits_{t=1}^{T} \log \left\{ (p^{*}_{ka})^{{y_{a}^{t}}} \left(1 - p^{*}_{ka} \right)^{1- {y_{a}^{t}}} (p^{*}_{kb})^{{y_{b}^{t}}} \left(1 - p^{*}_{kb} \right)^{1- {y_{b}^{t}}} \right\} \right].$$

Then, the vector $\left ( \boldsymbol {\theta }^{*}, \boldsymbol {p}^{*}_{\bar {k}} \right )$ is also an optimal solution to the optimization problem (29). Conversely, let $\left ( \boldsymbol {\hat {\theta }}, \boldsymbol {\hat {p}} \right )$ denote a solution of the problem (29). Then the pair $\left ( \boldsymbol {\hat {\theta }}, \boldsymbol {\hat {p}} \right )$ together with all the other equilibrium probabilities $\{\boldsymbol {p}_{k}\}_{k} \setminus \boldsymbol {\hat {p}}$ at $\boldsymbol {\hat {\theta }}$ is an solution to the problem (28).

Finally, the formulation in Eq. 29 can be generalized to the constrained optimization problem for the ML estimator defined in Eq. 18 for the case of observing data in multiple markets.

Appendix B: Creating examples with multiple equilibria

In this appendix, we derive the formula for calculating the spectral radius ρ[∇ _P Ψ(P, X;𝜃)] for the model in Section 5. We first derive the formula for calculating the spectral radius for one market and then generalize the formula to the case with multiple markets. Recall that in Section 5, for m = 1, …, M, we have

$$\begin{array}{rcl} {p_{W}^{m}} & = & \displaystyle \frac{ \text{exp}( \boldsymbol{\alpha}^{\prime} \boldsymbol{x}^{m} + \boldsymbol{\beta}_{W}^{\prime} \boldsymbol{x}^{m}_{W} - \delta {p_{K}^{m}} ) }{ 1 + \text{exp}( \boldsymbol{\alpha}^{\prime} \boldsymbol{x}^{m} + \boldsymbol{\beta}_{W}^{\prime} \boldsymbol{x}^{m}_{W} - \delta {p_{K}^{m}} ) } = \Psi_{W}( \boldsymbol{x}^{m}, \boldsymbol{x}_{W}^{m}, {p^{m}_{K}}; \boldsymbol{\theta}) \\ {p_{K}^{m}} & = &\displaystyle \frac{ \text{exp}( \boldsymbol{\alpha}^{\prime} \boldsymbol{x}^{m} + \boldsymbol{\beta}_{K}^{\prime} \boldsymbol{x}^{m}_{K} - \delta {p_{W}^{m}} ) }{ 1 + \text{exp}( \boldsymbol{\alpha}^{\prime} \boldsymbol{x}^{m} + \boldsymbol{\beta}_{K}^{\prime} \boldsymbol{x}^{m}_{K} - \delta {p_{W}^{m}} ) } = \Psi_{K}( \boldsymbol{x}^{m}, \boldsymbol{x}_{K}^{m}, {p^{m}_{W}}; \boldsymbol{\theta}), \end{array}$$

We represent the BN equilibrium equations above for market m as

$$\boldsymbol{p}^{m} = \boldsymbol{\Psi}( \boldsymbol{p}^{m}, \boldsymbol{X}^{m}; \boldsymbol{\theta}), $$

where $\boldsymbol {p}^{m} = ({p_{W}^{m}}, {p_{K}^{m}})$, Ψ = (Ψ _W , Ψ _K ), and $\boldsymbol {X}^{m} = (\boldsymbol {x}^{m}, \boldsymbol {x}_{W}^{m}, \boldsymbol {x}_{K}^{m} )$. The Jacobian of the mapping Ψ(p ^m, X ^m;𝜃) with respect to p ^m is

$$\nabla_{\boldsymbol{p}^{m}} \boldsymbol{\Psi}( \boldsymbol{p}^{m}, \boldsymbol{X}^{m}; \boldsymbol{\theta}) = \left[\begin{array}{cc} \frac{\partial \Psi_{W}}{\partial {p_{W}^{m}}} & \frac{\partial \Psi_{W}}{\partial {p_{K}^{m}}} \\ \frac{\partial \Psi_{K}}{\partial {p_{W}^{m}}} & \frac{\partial \Psi_{K}}{\partial {p_{K}^{m}}} \end{array}\right] = \left[\begin{array}{cc} 0 & {p_{W}^{m}}(1-{p_{W}^{m}}) \delta \\ {p_{K}^{m}}(1-{p_{K}^{m}}) \delta & 0 \end{array}\right].$$

The eigenvalues λ ^m of $\nabla _{\boldsymbol {p}^{m}} \boldsymbol {\Psi }( \boldsymbol {p}^{m}, \boldsymbol {X}^{m}; \boldsymbol {\theta })$ satisfy the following conditions:

$$\text{det} \left(\nabla_{\boldsymbol{p}^{m}} \boldsymbol{\Psi}( \boldsymbol{p}^{m}, \boldsymbol{X}^{m}; \boldsymbol{\theta}) - \lambda^{m} I \right) = (\lambda^{m})^{2} - {p_{W}^{m}}(1-{p_{W}^{m}}) {p_{K}^{m}} (1-{p_{K}^{m}}) \delta^{2} = 0, $$

where I is the identity matrix. Hence, we have

$$\lambda^{m} = \pm \delta \sqrt{{p_{W}^{m}}(1-{p_{W}^{m}}) {p_{K}^{m}} (1-{p_{K}^{m}})}.$$

The spectral radius of the Jacobian mapping $\nabla _{\boldsymbol {p}^{m}} \boldsymbol {\Psi }( \boldsymbol {p}^{m}, \boldsymbol {X}^{m}; \boldsymbol {\theta })$ for market m is then

$$\rho[\nabla_{\boldsymbol{p}^{m}} \boldsymbol{\Psi}( \boldsymbol{p}^{m}, \boldsymbol{X}^{m}; \boldsymbol{\theta})] = \max |\lambda^{m}| = | \delta | \sqrt{{p_{W}^{m}}(1-{p_{W}^{m}}) {p_{K}^{m}} (1-{p_{K}^{m}})}.$$

Next we generalize the derivation above to the case with multiple markets. Recall we represent the BN equilibrium equations for all markets m = 1, …, M as

$$\boldsymbol{P} = \boldsymbol{\Psi}( \boldsymbol{P}, \boldsymbol{X}; \boldsymbol{\theta}), $$

where $\boldsymbol {P} = ({p^{m}_{W}}, {p^{m}_{K}})_{m = 1}^{M}$, Ψ = (Ψ _W , Ψ _K ) and $\boldsymbol {X} = (\boldsymbol {x}^{m}, \boldsymbol {x}_{W}^{m}, \boldsymbol {x}_{K}^{m} )_{m=1}^{M}$. Observe that the Jacobian matrix ∇ _P Ψ(P, X;𝜃) has the block-diagonal structure

$$\nabla_{\boldsymbol{P}} \boldsymbol{\Psi}( \boldsymbol{P}, \boldsymbol{X}; \boldsymbol{\theta}) \, =\, \begin{bmatrix} \nabla_{\boldsymbol{p}^{1}} \boldsymbol{\Psi}( \boldsymbol{p}^{1}, \boldsymbol{X}^{1}; \boldsymbol{\theta}) & 0 & \cdots & 0 \\ 0 & \nabla_{\boldsymbol{p}^{2}} \boldsymbol{\Psi}( \boldsymbol{p}^{2}, \boldsymbol{X}^{2}; \boldsymbol{\theta}) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \nabla_{\boldsymbol{p}^{M}} \boldsymbol{\Psi}( \boldsymbol{p}^{M}, \boldsymbol{X}^{M}; \boldsymbol{\theta}) \end{bmatrix}.$$

Consequently, the spectral radius of the Jacobian mapping ∇ _P Ψ(P, X;𝜃) is given as

$$\rho[\nabla_{\boldsymbol{P}} \boldsymbol{\Psi}( \boldsymbol{P}, \boldsymbol{X}; \boldsymbol{\theta})] = \max_{m = 1, \ldots, M} | \lambda^{m} | = \max_{m = 1, \ldots, M} \left\{ | \delta | \sqrt{{p_{W}^{m}}(1-{p_{W}^{m}}) {p_{K}^{m}} (1-{p_{K}^{m}})} \right\}.$$

To create an example with multiple equilibria in some markets, we need the spectral radius ρ[∇ _P Ψ(P ^∗, X;𝜃 ⁰)] evaluated at structural parameters 𝜃 ⁰ and the corresponding equilibrium P to be greater than 1, which implies $| \delta ^{0} | \sqrt {{p_{W}^{m}}(1-{p_{W}^{m}}) {p_{K}^{m}} (1-{p_{K}^{m}})} > 1$ for some m. This, in turns, requires that

$$|\delta^{0}| > 4, $$

since $\sqrt {{p_{W}^{m}}(1-{p_{W}^{m}}) {p_{K}^{m}} (1-{p_{K}^{m}})} < 1/4$. As a conservative choice, we choose δ ⁰ = 6 in the second set of true parameter values in Eq. 25.