Overview
The goal of rgamer is to help students learn Game Theory using R. The functions prepared by the package not only solve basic games such as two-person normal-form games but also provides the users with visual displays that highlight some aspects of the games — payoff matrix, best response correspondence, etc. In addition, it suggests some numerical solutions for games of which it is difficult — or even seems impossible — to derive a closed-form analytical solution.
Installation
You can install the development version from GitHub with:
# install.packages("remotes")remotes::install_github("yukiyanai/rgamer")or
# install.packages("devtools")devtools::install_github("yukiyanai/rgamer")Examples
library(rgamer)Example 1
An example of a normal-form game (prisoner’s dilemma).
- Player: Kamijo, Yanai
- Strategy: (Stays silent, Betrays), (Stays silent, Betrays)
- Payoff: (-1, 0, -3, -2), (-1, -3, 0, -2)
First, you define the game by normal_form():
game1 <- normal_form( players = c("Kamijo", "Yanai"), s1 = c("Stays silent", "Betrays"), s2 = c("Stays silent", "Betrays"), payoffs1 = c(-1, 0, -3, -2), payoffs2 = c(-1, -3, 0, -2))You can specify payoffs for each cell of the game matrix as follows.
game1b <- normal_form( players = c("Kamijo", "Yanai"), s1 = c("Stays silent", "Betrays"), s2 = c("Stays silent", "Betrays"), cells = list(c(-1, -1), c(-3, 0), c( 0, -3), c(-2, -2)), byrow = TRUE)Then, you can pass it to solve_nfg() function to get the table of the game and the Nash equilibrium.
s_game1 <- solve_nfg(game1, show_table = FALSE)#> Pure-strategy NE: [Betrays, Betrays]s_game1$tableYanai | |||
|---|---|---|---|
| strategy | Stays silent | Betrays | |
| Kamijo | Stays silent | -1, -1 | -3, 0^ |
| Betrays | 0^, -3 | -2^, -2^ | |
Example 2
An example of a coordination game.
- Player: Kamijo, Yanai
- Strategy: (Stag, Hare), (Stag, Hare)
- Payoff: (10, 8, 0, 7), (10, 0, 8, 7)
Define the game by normal_form():
game2 <- normal_form( players = c("Kamijo", "Yanai"), s1 = c("Stag", "Hare"), s2 = c("Stag", "Hare"), payoffs1 = c(10, 8, 0, 7), payoffs2 = c(10, 0, 8, 7))Then, you can pass it to solve_nfg() function to get NEs. Set mixed = TRUE to find mixed-strategy NEs well.
s_game2 <- solve_nfg(game2, mixed = TRUE, show_table = FALSE)#> Pure-strategy NE: [Stag, Stag], [Hare, Hare]#> Mixed-strategy NE: [(7/9, 2/9), (7/9, 2/9)]#> The obtained mixed-strategy NE might be only a part of the solutions.#> Please examine br_plot (best response plot) carefully.For a 2-by-2 game, you can plot the best response correspondences as well.
s_game2$br_plot
Example 3
An example of a normal-form game:
- Player:
- Strategy:
- Payoff:
You can define a game by specifying payoff functions as character vectors using normal_form():
game3 <- normal_form( players = c("A", "B"), payoffs1 = "-x^2 + (28 - y) * x", payoffs2 = "-y^2 + (28 - x) * y", par1_lim = c(0, 30), par2_lim = c(0, 30), pars = c("x", "y"))Then, you can pass it to solve_nfg(), which displays the best response correspondences by default.
s_game3 <- solve_nfg(game3)#> approximated NE: (9.3, 9.3)#> The obtained NE might be only a part of the solutions.#> Please examine br_plot (best response plot) carefully.
Example 4
An example of a normal-form game:
- Player:
- Strategy:
- Payoff:
You can define a normal-form game by specifying payoffs by R functions.
f_x <- function(x, y, a, b) { -x^a + (b - y) * x}f_y <- function(x, y, s, t) { -y^s + (t - x) * y}game4 <- normal_form( players = c("A", "B"), payoffs1 = f_x, payoffs2 = f_y, par1_lim = c(0, 30), par2_lim = c(0, 30), pars = c("x", "y"))Then, you can approximate a solution numerically by solve_nfg(). Note that you need to set the parameter values of the function that should be treated as constants by arguments cons1 and cons2, each of which accepts a named list. In addition, you can suppress the plot of best responses by plot = FALSE.
s_game4 <- solve_nfg( game = game4, cons1 = list(a = 2, b = 28), cons2 = list(s = 2, t = 28), plot = FALSE)#> approximated NE: (9.3, 9.3)#> The obtained NE might be only a part of the solutions.#> Please examine br_plot (best response plot) carefully.You can increase the precision of approximation by precision, which takes a natural number (default is precision = 1).
s_game4b <- solve_nfg( game = game4, cons1 = list(a = 2, b = 28), cons2 = list(s = 2, t = 28), precision = 3)#> approximated NE: (9.333, 9.333)#> The obtained NE might be only a part of the solutions.#> Please examine br_plot (best response plot) carefully.
You can extract the best response plot with NE marked as follows.
s_game4b$br_plot_NE
Example 5
You can define payoffs by R functions and evaluate them at some discretized values by setting discretize = TRUE. The following is a Bertrand competition example:
func_price1 <- function(p, q) { if (p < q) { profit <- p } else if (p == q) { profit <- 0.5 * p } else { profit <- 0 } profit}func_price2 <- function(p, q){ if (p > q) { profit <- q } else if (p == q) { profit <- 0.5 * q } else { profit <- 0 } profit}game5 <- normal_form( payoffs1 = func_price1, payoffs2 = func_price2, pars = c("p", "q"), par1_lim = c(0, 10), par2_lim = c(0, 10), discretize = TRUE)Then, you can examine the specified part of the game.
s_game5 <- solve_nfg(game5, mark_br = FALSE)Player 2 | |||||||
|---|---|---|---|---|---|---|---|
| strategy | 0 | 2 | 4 | 6 | 8 | 10 | |
| Player 1 | 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 |
| 2 | 0, 0 | 1, 1 | 2, 0 | 2, 0 | 2, 0 | 2, 0 | |
| 4 | 0, 0 | 0, 2 | 2, 2 | 4, 0 | 4, 0 | 4, 0 | |
| 6 | 0, 0 | 0, 2 | 0, 4 | 3, 3 | 6, 0 | 6, 0 | |
| 8 | 0, 0 | 0, 2 | 0, 4 | 0, 6 | 4, 4 | 8, 0 | |
| 10 | 0, 0 | 0, 2 | 0, 4 | 0, 6 | 0, 8 | 5, 5 | |
Example 6
You can draw a tree of an extensive form game.
game6 <- extensive_form( players = list("Yanai", rep("Kamijo", 2), rep(NA, 4)), actions = list(c("stat", "game"), c("stat", "game"), c("stat", "game")), payoffs = list(Yanai = c(2, 0, 0, 1), Kamijo = c(1, 0, 0, 2)), direction = "right")
And you can find the solution of the game by solve_efg().
s_game6 <- solve_efg(game6)#> backward induction: [(stat), (stat, game)]Then, you can see the path played under a solution by show_path().
show_path(s_game6)
