WebR — Sustentação de Colusão com Trigger Strategy

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# ============================================
# Colusao com trigger strategy
# Exercicio Resolvido 16.3
# ============================================

a <- 200; c <- 20  # Demanda: Q = a - p, CMg = c
n <- 3              # numero de firmas

cat("====== COLUSAO COM TRIGGER STRATEGY ======\n")
cat("Demanda: Q =", a, "- p\n")
cat("CMg =", c, "\n")
cat("Firmas:", n, "\n\n")

# --- Preco de monopolio ---
pm <- (a + c) / 2
Qm <- a - pm
pi_m <- (pm - c) * Qm
pi_coop <- pi_m / n

cat("--- Preco de monopolio ---\n")
cat(sprintf("p^m = %.0f  Q^m = %.0f  pi^m = %.0f\n", pm, Qm, pi_m))
cat(sprintf("Lucro por firma (cooperando) = %.0f / %d = %.0f\n\n",
    pi_m, n, pi_coop))

# --- Desvio ---
pi_desvio <- pi_m  # captura toda a demanda com p = pm - eps
pi_punição <- 0    # Bertrand: p = c => lucro = 0

cat("--- Desvio ---\n")
cat(sprintf("Lucro ao desviar (1 periodo) = %.0f\n", pi_desvio))
cat(sprintf("Lucro apos punicao (para sempre) = %.0f\n\n", pi_punição))

# --- Delta minimo ---
# Cooperar: pi_coop / (1 - delta) >= pi_desvio + 0
# (1/n) / (1 - delta) >= 1
# 1/n >= 1 - delta => delta >= 1 - 1/n
delta_min <- 1 - 1/n

cat("--- Delta minimo ---\n")
cat(sprintf("delta >= 1 - 1/n = 1 - 1/%d = %.4f\n\n", n, delta_min))

# --- Analise para diferentes n ---
cat("--- Delta minimo vs. numero de firmas ---\n")
cat(sprintf("%-8s %-12s %-15s\n", "n", "delta_min", "pi_coop"))
cat(strrep("-", 38), "\n")
for (ni in 2:10) {
  di <- 1 - 1/ni
  pi_i <- pi_m / ni
  cat(sprintf("%-8d %-12.4f %-15.0f\n", ni, di, pi_i))
}

cat("\nQuanto mais firmas, mais paciencia necessaria para\n")
cat("sustentar a colusao e menor o lucro de cada participante.\n")

# --- Grafico ---
par(mfrow = c(1, 2), mar = c(4.5, 4.5, 3, 1), bg = "#f8f9fa")

# Painel 1: Delta minimo vs n
n_seq <- 2:20
delta_seq <- 1 - 1/n_seq

plot(n_seq, delta_seq, type = "b", lwd = 3, col = "#0d6efd",
     pch = 19, cex = 1.2,
     xlab = "Numero de firmas (n)",
     ylab = expression(delta["min"]),
     main = "Dificuldade de colusao vs. n",
     ylim = c(0.4, 1))
abline(h = delta_min, col = "#dc3545", lty = 2, lwd = 1.5)
text(15, delta_min - 0.03,
     paste0("n = ", n, ": delta = ", round(delta_min, 3)),
     col = "#dc3545", cex = 0.75, font = 2)
abline(h = 1, col = "#adb5bd", lty = 3)

# Painel 2: Fluxo de lucros — cooperar vs desviar
T <- 15
delta_ex <- 0.7  # exemplo com delta > delta_min

lucro_coop <- rep(pi_coop, T)
lucro_desv <- c(pi_desvio, rep(0, T - 1))
VP_coop <- cumsum(pi_coop * delta_ex^(0:(T-1)))
VP_desv <- cumsum(c(pi_desvio, rep(0, T-1)) * delta_ex^(0:(T-1)))

plot(1:T, VP_coop, type = "l", lwd = 3, col = "#198754",
     xlab = "Periodo", ylab = "VP acumulado",
     main = paste0("VP acumulado (delta = ", delta_ex, ")"),
     ylim = c(0, max(VP_coop) * 1.1))
lines(1:T, VP_desv, lwd = 3, col = "#dc3545", lty = 2)
abline(v = 1, col = "#adb5bd", lty = 3)
text(T * 0.7, VP_coop[T] * 0.95, "Cooperar", col = "#198754",
     cex = 0.8, font = 2)
text(T * 0.7, VP_desv[T] * 1.1, "Desviar", col = "#dc3545",
     cex = 0.8, font = 2)

legend("bottomright",
       legend = c(paste0("Cooperar (VP = ", round(VP_coop[T]), ")"),
                  paste0("Desviar  (VP = ", round(VP_desv[T]), ")")),
       col = c("#198754", "#dc3545"), lwd = 3, lty = c(1, 2),
       cex = 0.65, bg = "white")

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