Estimating A Static Game of Traveling Doctors

Transcription

1 Estimating A Static Game of Traveling Doctors J. Jason Bell, Thomas Gruca, Sanghak Lee, and Roger Tracy January 26, 2015

2 Visiting Consultant Clinicians A Visiting Consultant Clinician, or a VCC is a traveling doctor. Usually, the doctor provides specialty services to a hospital in need of them. In return, the doctor gets a new pool of patients, some of which, hopefully, will decide to receive treatment at the doctor s home practice.

3 To Be or Not to Be a VCC? Most VCCs visit their non-home hospital twice a month, though some go as often as twice a week. The decisions of whether and how often to visit are likely influenced by: 1. The level of competition at home 2. The number of potential patients at the visited hospital 3. Travel costs 4. The number of other doctors of the same specialty expected to visit the same location.

4 The VCC Decision as a Game Our model sets up this decision in a game theoretic context. The game is a simultaneous move, imperfect information game between doctors. We model the decision using expected profit maximization, and we use a profit function of the form: log π li (a l, X li ) = log π li (a l, X li ) + ɛ li The first term not the right-hand side captures deterministic utility, i.e. utility which we do not model as a random variable. The second term is treated as a random variable, and represents information that the doctor knows, but we, the researchers, do not know. The letters l and i index locations and group-practices, respectively.

5 Time Constraints Doctors face a time constraint: ā i = l a li. They allocate this time to maximize profit. We solve this constrained maximization problem using the Lagrangian function: ( L = π i (a) + λ ā i a li ). l The decision variable for firm i is the vector (a 1i,..., a li,..., a Li,i).

6 Functional Forms We specify the deterministic portion of the profit function of firm i at location l as π li (a l, X li ) Γ li (a l, i, n l ) log(a li + 1) + δd li a li, where X li = (d li, n l ), d li is the distance between firm i s home and location l, and n l is the population at location l. The parameter Γ li partly controls the curvature, or marginal return, of the profit function, and is defined by Γ li (a l, i, n l ) = exp β 0 + β 1 n l + β 2 a lj. j i

7 Information Structure Note that firm i s profit is a function of the decisions of the other firms. In principle this makes sense, but the firms are playing a simultaneous move, imperfect information game, so they don t know what the other firms will do yet! All they know is a set of probabilities for various actions. Instead of placing the term j i a lj in the profit function for firm i, we instead place j i E [a lj]. This term is the expectation of firm i about how much time other firms will spend at location l.

8 Observation Woes Now we have a more appropriate model for firm behavior, but we have a new problem: we don t observe j i E [a lj]. This is an internal belief held by the decision makers. If we assume that firms are playing a single equilibrium, we can construct a non-parametric estimator for these beliefs. We model the game as static, as opposed to dynamic, since the VCCs in our dataset have been playing the game for many years. We think it is reasonable to believe they are in a long-run equilibrium, since fundamental conditions have remained the same for most of the time. We assume that firms are playing a single equilibrium, rather than jumping between equilibria. Given that decision makers are likely to avoid unnecessary complexity, we believe this assumption is defensible.

9 Equilibrium We define an equilibrium with respect to the expectations of the firms. Define the following matrix f (a11 ) f (a 12 )... f (a 1I ) f (a21 P ) f (a 22 )..... f (al1 )... f (a LI ) The matrix P is an L I matrix of pdfs, one pdf for each firm-location pair.

10 Equilibrium, continued Define ψ li (ali X, E[a l, i ]) as the choice rule of firm i conditional on X and expectations about other firms. This function returns a pdf for the random variable ali. Now define the matrix function Ψ(P X ) X, E[a1, I ]) ψ 21 (a21 X, E[a2, 1 ])..... ψ L1 (al1 X, E[a L, 1 ])... ψ LI (ali X, E[a L, I ]) ψ 11 (a11 X, E[a1, 1 ])... ψ 1I (a1i

11 An Equilibrium is a Fixed Point An equilibrium matrix P of conditional choice probability density functions is a fixed point of Ψ(P X ). In other words, P satisfies the equation P = Ψ( P X ) We use an asterisk to denote an optimal value for a choice variable, conditional on beliefs, or a best-response. We use a tilde to designate an equilibrium value for beliefs between firms.

12 Likelihood Function The likelihood function is analogous to the one in Kim et al. (2002). The probability that firm i enters k of the L locations is L i (Θ X li, P) = Pr {a li > 0, a mi = 0; l = 2,..., k and m = k + 1,..., L} = EgLi... Eg(k+1)i f ( Eg 2i,..., Eg ki, η (k+1)i,..., η Li ) J i dη (k+1)i... dη Li Where J is the Jacobian necessary for a transformation of variables. The rth row and cth column of J i is J i [r, c] = Eg (r+1)i a(c+1)i.

13 Likelihood, continued Firm i s contribution to the likelihood function above conditions on the expectations of firms P. The likelihood function for the entire equilibrium is therefore L(Θ X, P) = I L i (Θ X li, P). i If we assume that the variables ɛ li are distributed iid Gumbel (also known as Type II Extreme Value), the likelihood has a convenient and simple closed form, which is omitted for brevity. Proof due to Bhat (2005).

14 Estimation The likelihood function on the previous slide requires knowledge of equilibrium expectations between firms, which we do not have. We instead use something the literature calls a psuedo-likelihood function: L(Θ X, ˆP) = I L i (Θ X li, ˆP). The true equilibrium expectations, P are replaced by estimates of the equilibrium expectations, ˆP. We will obtain ˆP using non-parametric regression. We simply regress choices on all variables except for the competition variable, and use fitted values. Relying on the proof from Bajari et. al, we know this is a consistent estimator of equilibrium beliefs. i

15 Standard Errors Since our procedure is comprised of two stages, we have an extra layer of uncertainty, which should increase the standard errors of our parameter estimates. In order to obtain correct standard errors, we will use bootstrapping. The intended method is: 1. Sample the data with replacement until a dataset of the original size is created 2. Estimate the parameters using the dataset from step 1 3. Repeat steps 1 and 2 many times 4. Use the resulting distributions of parameter estimates to construct standard errors

16 Current Challenges 1. We ve recently discovered that some of the parameters are not separately identified for this model. Identification here refers to the model and the resulting likelihood, not the dataset. 2. Simulating data for this model in order to test the estimation routine turns out to be rather difficult (for me). I have code that simulates data, but the simulated data does not exhibit all of the necessary properties yet. 3. Because of 1 and 2, we have only successfully recovered 2 of 4 correct parameter estimates on simulated datasets.