# Chapter 18 Assignment 8: Dynamic Game

## 18.1 Simulate data

Suppose that there are $$m = 1, \cdots, M$$ markets and in each market there are $$i = 1, \cdots, N$$ firms and each firm makes decisions for $$t = 1, \cdots, \infty$$. In the following, I suppress the index of market, $$m$$. We solve the model under the infinite-horizon assumption, but generate data only for $$t = 1, \cdots, T$$. There are $$L = 3$$ states $$\{1, 2, 3\}$$ for each firm. Each firm can choose $$K + 1 = 2$$ actions $$\{0, 1\}$$. Thus, $$m_a := (K + 1)^N$$ and $$m_s := L^N$$. Let $$a_i$$ and $$s_i$$ be firm $$i$$’s action and state, and $$a$$ and $$s$$ are vectors of individual actions and states.

The mean period payoff to firm $$i$$ is: $\pi_i(a, s) := \tilde{\pi}(a_i, s_i, \overline{s}_{-i}) := \alpha \ln s_i - \eta \ln s_i \sum_{j \neq i} \ln s_j - \beta a_i,$ where $$\alpha, \beta, \eta> 0$$, and $$\alpha > \eta$$. The term $$\eta$$ means that the returns to investment decreases as rival’s average state profile improves. The period payoff is: $\tilde{\pi}(a_i, s_i, \overline{s}_{-i})+ \epsilon_i(a_i),$ and $$\epsilon_i(a_i)$$ is an i.i.d. type-I extreme random variable that is independent of all the other variables.

At the beginning of each period, state $$s$$ is realized and publicly observed. Then, choice-specific shocks $$\epsilon_i(a_i), a_i = 0, 1$$ are realized and privately observed by firm $$i = 1, \cdots, N$$. Then each firm simultaneously chooses her action. Then, the game moves to next period.

State transition is independent across firms conditional on individual state and action.

Suppose that $$s_i > 1$$ and $$s_i < L$$. If $$a_i = 0$$, the state stays at the same state with probability $$1 - \kappa$$ and moves down by 1 with probability $$\kappa$$. If $$a = 1$$, the state moves up by 1 with probability $$\gamma$$, moves down by 1 with probability $$\kappa$$, and stays at the same with probability $$1 - \kappa - \gamma$$.

Suppose that $$s_i = 1$$. If $$a_i = 0$$, the state stays at the same state with probability 1. If $$a_i = 1$$, the state moves up by 1 with probability $$\gamma$$ and stays at the same with probability $$1 - \gamma$$.

Suppose that $$s_i = L$$. If $$a_i = 0$$, the state stays at the same state with probability $$1 - \kappa$$ and moves down by 1 with probability $$\kappa$$. If $$a = 1$$, the state moves down by 1 with probability $$\kappa$$, and stays at the same with probability $$1 - \kappa$$.

The mean period profit is summarized in $$\Pi$$ as:

$\Pi := \begin{pmatrix} \pi(1, 1)\\ \vdots\\ \pi(m_a, 1)\\ \vdots \\ \pi(1, m_s)\\ \vdots\\ \pi(m_a, m_s)\\ \end{pmatrix}$

The transition law is summarized in $$G$$ as:

$g(a, s, s') := \mathbb{P}\{s_{t + 1} = s'|s_t = s, a_t = a\},$

$G := \begin{pmatrix} g(1, 1, 1) & \cdots & g(1, 1, m_s)\\ \vdots & & \vdots \\ g(m_a, 1, 1) & \cdots & g(m_a, 1, m_s)\\ & \vdots & \\ g(1, m_s, 1) & \cdots & g(1, m_s, m_s)\\ \vdots & & \vdots \\ g(m_a, m_s, 1) & \cdots & g(m_a, m_s, m_s)\\ \end{pmatrix}.$ The discount factor is denoted by $$\delta$$. We simulate data for $$M$$ markets with $$N$$ firms for $$T$$ periods.

1. Set constants and parameters as follows:
# set seed
set.seed(1)
# set constants
L <- 5
K <- 1
T <- 100
N <- 3
M <- 1000
lambda <- 1e-10
# set parameters
alpha <- 1
eta <- 0.3
beta <- 2
kappa <- 0.1
gamma <- 0.6
delta <- 0.95
1. Write a function compute_action_state_space(K, L, N) that returns a data frame for action and state space. Returned objects are list of data frame A and S. In A, column k is the index of an action profile, i is the index of a firm, and a is the action of the firm. In S, column l is the index of an state profile, i is the index of a firm, and s is the state of the firm.
output <- compute_action_state_space(L, K, N)
A <- output$A head(A) ## # A tibble: 6 x 3 ## k i a ## <int> <int> <int> ## 1 1 1 0 ## 2 1 2 0 ## 3 1 3 0 ## 4 2 1 1 ## 5 2 2 0 ## 6 2 3 0 tail(A) ## # A tibble: 6 x 3 ## k i a ## <int> <int> <int> ## 1 7 1 0 ## 2 7 2 1 ## 3 7 3 1 ## 4 8 1 1 ## 5 8 2 1 ## 6 8 3 1 S <- output$S
head(S)
## # A tibble: 6 x 3
##       l     i     s
##   <int> <int> <int>
## 1     1     1     1
## 2     1     2     1
## 3     1     3     1
## 4     2     1     2
## 5     2     2     1
## 6     2     3     1
tail(S)
## # A tibble: 6 x 3
##       l     i     s
##   <int> <int> <int>
## 1   124     1     4
## 2   124     2     5
## 3   124     3     5
## 4   125     1     5
## 5   125     2     5
## 6   125     3     5
# dimension
m_a <- max(A$k); m_a ##  8 m_s <- max(S$l); m_s
##  125
1. Write function compute_PI_game(alpha, beta, eta, L, K, N) that returns a list of $$\Pi_i$$.
PI <- compute_PI_game(alpha, beta, eta, A, S)
head(PI[[N]])
##      [,1]
## [1,]    0
## [2,]    0
## [3,]    0
## [4,]    0
## [5,]   -2
## [6,]   -2
dim(PI[[N]]) == m_s * m_a
##  TRUE
1. Write function compute_G_game(g, A, S) that converts an individual transition probability matrix into a joint transition probability matrix $$G$$.
G_marginal <- compute_G(kappa, gamma, L, K)
G <- compute_G_game(G_marginal, A, S)
head(G)
##         1    2 3 4 5    6    7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
## [1,] 1.00 0.00 0 0 0 0.00 0.00 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [2,] 0.40 0.60 0 0 0 0.00 0.00 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [3,] 0.40 0.00 0 0 0 0.60 0.00 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [4,] 0.16 0.24 0 0 0 0.24 0.36 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [5,] 0.40 0.00 0 0 0 0.00 0.00 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [6,] 0.16 0.24 0 0 0 0.00 0.00 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0
##      23 24 25   26   27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
## [1,]  0  0  0 0.00 0.00  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [2,]  0  0  0 0.00 0.00  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [3,]  0  0  0 0.00 0.00  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [4,]  0  0  0 0.00 0.00  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [5,]  0  0  0 0.60 0.00  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [6,]  0  0  0 0.24 0.36  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##      45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
## [1,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [2,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [3,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [4,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [5,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [6,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##      68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
## [1,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [2,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [3,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [4,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [5,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## [6,]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##      91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109
## [1,]  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0   0   0   0   0
## [2,]  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0   0   0   0   0
## [3,]  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0   0   0   0   0
## [4,]  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0   0   0   0   0
## [5,]  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0   0   0   0   0
## [6,]  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0   0   0   0   0
##      110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125
## [1,]   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
## [2,]   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
## [3,]   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
## [4,]   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
## [5,]   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
## [6,]   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
dim(G) == m_s * m_a
##  TRUE
dim(G) == m_s
##  TRUE

The ex-ante-value function for a firm is written as a function of a conditional choice probability as follows: $\varphi_i^{(\theta_1, \theta_2)}(p) := [I - \delta \Sigma(p) G]^{-1}[\Sigma(p)\Pi_i + D_i(p)],$ where $$\theta_1 = (\alpha, \beta, \eta)$$ and $$\theta_2 = (\kappa, \gamma)$$, $$p_i(a_i|s)$$ is the probability that firm $$i$$ choose action $$a_i$$ when the state profile is $$s$$, and: $p(a|s) = \prod_{i = 1}^N p_i(a_i|s),$

$p(s) = \begin{pmatrix} p(1|s) \\ \vdots \\ p(m_a|s) \end{pmatrix},$

$p = \begin{pmatrix} p(1)\\ \vdots\\ p(m_s) \end{pmatrix},$

$\Sigma(p) = \begin{pmatrix} p(1)' & & \\ & \ddots & \\ & & p(L)' \end{pmatrix}$

and:

$D_i(p) = \begin{pmatrix} \sum_{k = 0}^K \mathbb{E}\{\epsilon_i^k|a_i = k, 1\}p_i(a_i = k|1)\\ \vdots\\ \sum_{k = 0}^K \mathbb{E}\{\epsilon_i^k|a_i = k, m_s\}p_i(a_i = k|m_s) \end{pmatrix}.$

1. Write a function initialize_p_marginal(A, S) that defines an initial marginal condition choice probability. In the output p_marginal, p is the probability for firm i to take action a conditional on the state profile being l. Next, write a function compute_p_joint(p_marginal, A, S) that computes a corresponding joint conditional choice probability from a marginal conditional choice probability. In the output p_joint, p is the joint probability that firms take action profile k condition on the state profile being l. Finally, write a function compute_p_marginal(p_joint, A, S) that compute a corresponding marginal conditional choice probability from a joint conditional choice probability.
# define a conditional choice probability for each firm
p_marginal <- initialize_p_marginal(A, S)
p_marginal
## # A tibble: 750 x 4
##        i     l     a     p
##    <int> <int> <int> <dbl>
##  1     1     1     0   0.5
##  2     1     1     1   0.5
##  3     1     2     0   0.5
##  4     1     2     1   0.5
##  5     1     3     0   0.5
##  6     1     3     1   0.5
##  7     1     4     0   0.5
##  8     1     4     1   0.5
##  9     1     5     0   0.5
## 10     1     5     1   0.5
## # ... with 740 more rows
dim(p_marginal) == N * m_s * (K + 1)
##  TRUE
# compute joint conitional choice probability from marginal probability
p_joint <- compute_p_joint(p_marginal, A, S)
p_joint
## # A tibble: 1,000 x 3
##        l     k     p
##    <int> <int> <dbl>
##  1     1     1 0.125
##  2     1     2 0.125
##  3     1     3 0.125
##  4     1     4 0.125
##  5     1     5 0.125
##  6     1     6 0.125
##  7     1     7 0.125
##  8     1     8 0.125
##  9     2     1 0.125
## 10     2     2 0.125
## # ... with 990 more rows
dim(p_joint) == m_s * m_a
##  TRUE
# compute marginal conditional chocie probability from joint probability
p_marginal_2 <- compute_p_marginal(p_joint, A, S)
max(abs(p_marginal - p_marginal_2))
##  0
1. Write a function compute_Sigma(p_marginal, A, S) that computes $$\Sigma(p)$$ given a joint conditional choice probability. Then, write a function compute_D(p_marginal) that returns a list of $$D_i(p)$$.
# compute Sigma for ex-ante value function calculation
Sigma <- compute_Sigma(p_marginal, A, S)
head(Sigma)
##  0.125 0.000 0.000 0.000 0.000 0.000
dim(Sigma) == m_s
##  TRUE
dim(Sigma) == m_s * m_a
##  TRUE
# compute D for ex-ante value function calculation
D <- compute_D(p_marginal)
head(D[[N]])
##          [,1]
## [1,] 1.270363
## [2,] 1.270363
## [3,] 1.270363
## [4,] 1.270363
## [5,] 1.270363
## [6,] 1.270363
dim(D[[N]]) == m_s
##  TRUE
1. Write a function compute_exante_value_game(p_marginal, A, S, PI, G, delta) that returns a list of matrices whose $$i$$-th element represents the ex-ante value function given a conditional choice probability for firm $$i$$.
# compute ex-ante value funciton for each firm
V <- compute_exante_value_game(p_marginal, A, S, PI, G, delta)
head(V[[N]])
##  10.786330 10.175982  9.606812  9.255459  9.115332 10.175982
dim(V[[N]]) == m_s
##  TRUE

The optimal conditional choice probability is written as a function of an ex-ante value function and a conditional choice probability of others as follows: $\Lambda_i^{(\theta_1, \theta_2)}(V_i, p_{-i})(a_i, s) := \frac{\exp\{\sum_{a_{-i}}p_{-i}(a_{-i}|s)[\pi_i(a_i, a_{-i}, s) + \delta \sum_{s'}V_i(s')g(a_i, a_{-i}, s, s')]\}}{\sum_{a_i'}\exp\{\sum_{a_{-i}}p_{-i}(a_{-i}|s)[\pi_i(a_i', a_{-i}, s) + \delta \sum_{s'}V_i(s')g(a_i', a_{-i}, s, s')]\}},$ where $$V$$ is an ex-ante value function.

1. Write a function compute_profile_value_game(V, PI, G, delta, S, A) that returns a data frame that contains information on value function at a state and action profile for each firm. In the output value, i is the index of a firm, l is the index of a state profile, k is the index of an action profile, and value is the value for the firm at the state and action profile.
# compute state-action-profile value function
value <- compute_profile_value_game(V, PI, G, delta, S, A)
value
## # A tibble: 3,000 x 4
##        i     l     k value
##    <int> <int> <int> <dbl>
##  1     1     1     1 10.2
##  2     1     1     2  9.63
##  3     1     1     3  9.90
##  4     1     1     4  9.13
##  5     1     1     5  9.90
##  6     1     1     6  9.13
##  7     1     1     7  9.55
##  8     1     1     8  8.64
##  9     1     2     1 13.0
## 10     1     2     2 12.1
## # ... with 2,990 more rows
dim(value) == N * m_s * m_a
##  TRUE
1. Write a function compute_choice_value_game(p_marginal, V, PI, G, delta, A, S) that computes a data frame that contains information on a choice-specific value function given an ex-ante value function and a conditional choice probability of others.
# compute choice-specific value function
value <- compute_choice_value_game(p_marginal, V, PI, G, delta, A, S)
value
## # A tibble: 750 x 4
##        i     l     a value
##    <int> <int> <int> <dbl>
##  1     1     1     0  9.90
##  2     1     1     1  9.13
##  3     1     2     0 12.4
##  4     1     2     1 11.4
##  5     1     3     0 14.5
##  6     1     3     1 13.2
##  7     1     4     0 16.0
##  8     1     4     1 14.3
##  9     1     5     0 16.8
## 10     1     5     1 14.8
## # ... with 740 more rows
1. Write a function compute_ccp_game(p_marginal, V, PI, G, delta, A, S) that computes a data frame that contains information on a conditional choice probability given an ex-ante value function and a conditional choice probability of others.
# compute conditional choice probability
p_marginal <- compute_ccp_game(p_marginal, V, PI, G, delta, A, S)
p_marginal
## # A tibble: 750 x 4
##        i     l     a     p
##    <int> <int> <int> <dbl>
##  1     1     1     0 0.683
##  2     1     1     1 0.317
##  3     1     2     0 0.734
##  4     1     2     1 0.266
##  5     1     3     0 0.794
##  6     1     3     1 0.206
##  7     1     4     0 0.840
##  8     1     4     1 0.160
##  9     1     5     0 0.881
## 10     1     5     1 0.119
## # ... with 740 more rows
1. Write a function solve_dynamic_game(PI, G, L, K, delta, lambda, A, S) that find the equilibrium conditional choice probability and ex-ante value function by iterating the update of an ex-ante value function and a best-response conditional choice probability. The iteration should stop when $$\max_s|V^{(r + 1)}(s) - V^{(r)}(s)| < \lambda$$ with $$\lambda = 10^{-10}$$. There is no theoretical guarantee for the convergence.
# solve the dynamic game model
output <-
solve_dynamic_game(PI, G, L, K, delta, lambda, A, S)
save(output, file = "data/A8_equilibrium.RData")
load(file = "data/A8_equilibrium.RData")
p_marginal <- output$p_marginal; head(p_marginal) ## # A tibble: 6 x 4 ## i l a p ## <int> <int> <int> <dbl> ## 1 1 1 0 0.650 ## 2 1 1 1 0.350 ## 3 1 2 0 0.712 ## 4 1 2 1 0.288 ## 5 1 3 0 0.785 ## 6 1 3 1 0.215 V <- output$V[[N]]; head(V)
##  13.25670 12.39394 11.47346 10.82808 10.53018 12.39394
# compute joint conitional choice probability
p_joint <- compute_p_joint(p_marginal, A, S); head(p_joint)
## # A tibble: 6 x 3
##       l     k      p
##   <int> <int>  <dbl>
## 1     1     1 0.275
## 2     1     2 0.148
## 3     1     3 0.148
## 4     1     4 0.0795
## 5     1     5 0.148
## 6     1     6 0.0795
1. Write a function simulate_dynamic_game(p_joint, l, G, N, T, S, A, seed) that simulate the data for a market starting from an initial state for $$T$$ periods. The function should accept a value of seed and set the seed at the beginning of the procedure inside the function, because the process is stochastic.
# simulate a dynamic game
# set initial state profile
l <- 1
# draw simulation for a firm
seed <- 1
df <- simulate_dynamic_game(p_joint, l, G, N, T, S, A, seed)
df
## # A tibble: 300 x 6
##        t     i     l     k     s     a
##    <int> <int> <dbl> <dbl> <int> <int>
##  1     1     1     1     1     1     0
##  2     1     2     1     1     1     0
##  3     1     3     1     1     1     0
##  4     2     1     1     5     1     0
##  5     2     2     1     5     1     0
##  6     2     3     1     5     1     1
##  7     3     1    26     6     1     1
##  8     3     2    26     6     1     0
##  9     3     3    26     6     2     1
## 10     4     1    26     5     1     0
## # ... with 290 more rows
1. Write a function simulate_dynamic_decision_across_firms(p_joint, l, G, N, T, M, S, A, seed) that returns simulation data for $$N$$ firm. For firm $$i$$, set the seed at $$i$$
# simulate data across markets
df <- simulate_dynamic_decision_across_markets(p_joint, l, G, N, T, M, S, A)
save(df, file = "data/A8_df.RData")
load(file = "data/A8_df.RData")
df
## # A tibble: 300,000 x 7
##        m     t     i     l     k     s     a
##    <int> <int> <int> <dbl> <dbl> <int> <int>
##  1     1     1     1     1     1     1     0
##  2     1     1     2     1     1     1     0
##  3     1     1     3     1     1     1     0
##  4     1     2     1     1     5     1     0
##  5     1     2     2     1     5     1     0
##  6     1     2     3     1     5     1     1
##  7     1     3     1    26     6     1     1
##  8     1     3     2    26     6     1     0
##  9     1     3     3    26     6     2     1
## 10     1     4     1    26     5     1     0
## # ... with 299,990 more rows
summary(df)
##        m                t                i           l
##  Min.   :   1.0   Min.   :  1.00   Min.   :1   Min.   :  1.00
##  1st Qu.: 250.8   1st Qu.: 25.75   1st Qu.:1   1st Qu.: 28.00
##  Median : 500.5   Median : 50.50   Median :2   Median : 53.00
##  Mean   : 500.5   Mean   : 50.50   Mean   :2   Mean   : 55.08
##  3rd Qu.: 750.2   3rd Qu.: 75.25   3rd Qu.:3   3rd Qu.: 83.00
##  Max.   :1000.0   Max.   :100.00   Max.   :3   Max.   :125.00
##        k               s               a
##  Min.   :1.000   Min.   :1.000   Min.   :0.0000
##  1st Qu.:1.000   1st Qu.:2.000   1st Qu.:0.0000
##  Median :1.000   Median :3.000   Median :0.0000
##  Mean   :2.244   Mean   :2.753   Mean   :0.1776
##  3rd Qu.:3.000   3rd Qu.:4.000   3rd Qu.:0.0000
##  Max.   :8.000   Max.   :5.000   Max.   :1.0000
1. Write a function estimate_ccp_marginal_game(df) that returns a non-parametric estimate of the marginal conditional choice probability for each firm in the data. Compare the estimated conditional choice probability and the true conditional choice probability by a bar plot.
# non-parametrically estimate the conditional choice probability
p_marginal_est <- estimate_ccp_marginal_game(df)
check_ccp <- p_marginal_est %>%
dplyr::rename(estimate = p) %>%
dplyr::left_join(p_marginal, by = c("i", "l", "a")) %>%
dplyr::rename(true = p) %>%
dplyr::filter(a == 1)
ggplot(data = check_ccp, aes(x = true, y = estimate)) +
geom_point() +
labs(fill = "Value") + xlab("true") + ylab("estimate") 1. Write a function estimate_G_marginal(df) that returns a non-parametric estimate of the marginal transition probability matrix. Compare the estimated transition matrix and the true transition matrix by a bar plot.
# non-parametrically estimate individual transition probability
G_marginal_est <- estimate_G_marginal(df)
check_G <- data.frame(type = "true", reshape2::melt(G_marginal))
check_G_est <- data.frame(type = "estimate", reshape2::melt(G_marginal_est))
check_G <- rbind(check_G, check_G_est)
check_G$variable = paste(check_G$Var1, check_G$Var2, sep = "_") ggplot(data = check_G, aes(x = variable, y = value, fill = type)) + geom_bar(stat = "identity", position = "dodge") + labs(fill = "Value") + xlab("action/state/state") + ylab("probability") + theme(axis.text.x = element_blank()) ## 18.2 Estimate parameters 1. Vectorize the parameters as follows: theta_1 <- c(alpha, beta, eta) theta_2 <- c(kappa, gamma) theta <- c(theta_1, theta_2) We estimate the parameters by a CCP approach. 1. Write a function estimate_theta_2_game(df) that returns the estimates of $$\kappa$$ and $$\gamma$$ directly from data by counting relevant events. # estimate theta_2 theta_2_est <- estimate_theta_2_game(df); theta_2_est ##  0.09995377 0.60136442 The objective function of the minimum distance estimator based on the conditional choice probability approach is: $\frac{1}{N K m_s} \sum_{i = 1}^N \sum_{l = 1}^{m_s} \sum_{k = 1}^{K}\{\hat{p}_i(a_k|s_l) - p_i^{(\theta_1, \theta_2)}(a_k|s_l)\}^2,$ where $$\hat{p}_i$$ is the non-parametric estimate of the marginal conditional choice probability and $$p_i^{(\theta_1, \theta_2)}$$ is the marginal conditional choice probability under parameters $$\theta_1$$ and $$\theta_2$$ given $$\hat{p}_i$$. $$a_k$$ is $$k$$-th action for a firm and $$s_l$$ is $$l$$-th state profile. 1. Write a function compute_CCP_objective_game(theta_1, theta_2, p_est, L, K, delta) that returns the objective function of the above minimum distance estimator given a non-parametric estimate of the conditional choice probability and $$\theta_1$$ and $$\theta_2$$. # compute the objective function of the minimum distance estimator based on the CCP approach objective <- compute_CCP_objective_game(theta_1, theta_2, p_marginal_est, A, S, delta, lambda) save(objective, file = "data/A8_objective.RData") load(file = "data/A8_objective.RData") objective ##  0.0002737567 1. Check the value of the objective function around the true parameter. # label label <- c("\\alpha", "\\beta", "\\eta") label <- paste("$", label, "$", sep = "") # compute the graph graph <- foreach (i = 1:length(theta_1)) %do% { theta_i <- theta_1[i] theta_i_list <- theta_i * seq(0.5, 2, by = 0.2) objective_i <- foreach (j = 1:length(theta_i_list), .combine = "rbind") %dopar% { theta_ij <- theta_i_list[j] theta_j <- theta_1 theta_j[i] <- theta_ij objective_ij <- compute_CCP_objective_game(theta_j, theta_2, p_marginal_est, A, S, delta, lambda) return(objective_ij) } df_graph <- data.frame(x = theta_i_list, y = objective_i) g <- ggplot(data = df_graph, aes(x = x, y = y)) + geom_point() + geom_vline(xintercept = theta_i, linetype = "dotted") + ylab("objective function") + xlab(TeX(label[i])) return(g) } save(graph, file = "data/A8_CCP_graph.RData") load(file = "data/A8_CCP_graph.RData") graph ## [] ## ## [] ## ## [] 1. Estimate the parameters by minimizing the objective function. To keep the model to be well-defined, impose an ad hoc lower and upper bounds such that $$\alpha \in [0, 1], \beta \in [0, 5], \delta \in [0, 1]$$. lower <- rep(0, length(theta_1)) upper <- c(1, 5, 0.3) CCP_result <- optim(par = theta_1, fn = compute_CCP_objective_game, method = "L-BFGS-B", lower = lower, upper = upper, theta_2 = theta_2_est, p_marginal_est = p_marginal_est, A = A, S = S, delta = delta, lambda = lambda) save(CCP_result, file = "data/A8_CCP_result.RData") load(file = "data/A8_CCP_result.RData") CCP_result ##$par
##  1.000000 2.011446 0.294446
##
## $value ##  0.0002702126 ## ##$counts
## $convergence ##  0 ## ##$message
##  "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
compare <-
); compare
##   true estimate
## 3  0.3 0.294446