Example: One-shot games

A one-shot game (normal form game) is a stochastic game with only one state (\(\boldsymbol{\phi}=\boldsymbol{1}\)) and full discounting (\(\boldsymbol{\delta}=\boldsymbol{0}\)). Any one-shot game is fully specified by its payoff matrix \(\boldsymbol{u}\).

A famous example is the Prisoner’s Dilemma:

		player1
		cooperate	defect
player0	cooperate	2, 2	0, 3
player0	defect	3, 0	1, 1

The game is fully defined by the following game table.

state	a_player0	a_player1	u_player0	u_player1	phi_state0
delta			0	0
state0	cooperate	cooperate	2	2	0
state0	cooperate	defect	0	3	0
state0	defect	cooperate	3	0	0
state0	defect	defect	1	1	0

The game table specifies for each state and action profile the corresponding payoffs and state transitions. Additionally, the first row specifies the discount factors for each player.

To import the game table, use the SGame.from_table() method.

import sgamesolver
game = sgamesolver.SGame.from_table('path/to/table.xlsx')

One-shot games can be conveniently defined using the SGame.one_shot_game() method.

import sgamesolver
import numpy as np

payoff_matrix = np.array([[[2, 0],
                           [3, 1]],
                          [[2, 3],
                           [0, 1]]])

game = sgamesolver.SGame.one_shot_game(payoff_matrix=payoff_matrix)

# optional: overwrite action labels
game.action_labels = ['cooperate', 'defect']

The payoff_matrix should be an array of dimension num_players + 1, the first dimension indexing the player and the remaining dimensions indexing the strategy profile. Here, payoff_matrix has shape (2, 2, 2) with indices [player, action_0, action_1].